Math g4 m3 Full Module
Math g4 m3 Full Module
Math g4 m3 Full Module
4
GRADE
Mathematics Curriculum
GRADE 4 • MODULE 3
Table of Contents
GRADE 4 • MODULE 3
Multi-Digit Multiplication and Division
Module Overview ......................................................................................................... 2
Topic E: Division of Tens and Ones with Successive Remainders ............................. 202
Grade 4 • Module 3
Multi-Digit Multiplication and
Division
OVERVIEW
In this 43-day module, students use place value understanding and visual representations to solve
multiplication and division problems with multi-digit numbers. As a key area of focus for Grade 4, this
module moves slowly but comprehensively to develop students’ ability to reason about the methods and
models chosen to solve problems with multi-digit factors and dividends.
Students begin in Topic A by investigating the formulas for area and perimeter. They then solve multiplicative
comparison problems including the language of times as much as with a focus on problems using area and
perimeter as a context (e.g., “A field is 9 feet wide. It is 4 times as long as it is wide. What is the perimeter of
the field?”). Students create diagrams to represent these problems as well as write equations with symbols
for the unknown quantities (4.OA.1). This is foundational for understanding multiplication as scaling in Grade
5 and sets the stage for proportional reasoning in Grade 6. This Grade 4 module, beginning with area and
perimeter, allows for new and interesting word problems as students learn to calculate with larger numbers
and interpret more complex problems (4.OA.2, 4.OA.3, 4.MD.3).
In Topic B, students use place value disks to multiply single-digit numbers by multiples of 10, 100, and 1,000
and two-digit multiples of 10 by two-digit multiples of 10 (4.NBT.5). Reasoning between arrays and written
numerical work allows students to see the role of place value units in multiplication (as pictured below).
Students also practice the language of units to prepare them for multiplication of a single-digit factor by a
factor with up to four digits and multiplication of two two-digit factors.
In preparation for two-digit by two-digit multiplication, students practice the new complexity of multiplying
two two-digit multiples of 10. For example, students have multiplied 20 by 10 on the place value chart and
know that it shifts the value one place to the left, 10 × 20 = 200. To multiply 20 by 30, the associative
property allows for simply tripling the product, 3 × (10 × 20), or multiplying the units, 3 tens × 2 tens = 6
hundreds (alternatively, (3 × 10) × (2 × 10) = (3 × 2) × (10 × 10)). Introducing this early in the module allows
students to practice during fluency so that, by the time it is embedded within the two-digit by two-digit
multiplication in Topic H, understanding and skill are in place.
Building on their work in Topic B, students begin in Topic C decomposing numbers into base ten units in order
to find products of single-digit by multi-digit numbers. Students use the distributive property and multiply
using place value disks to model. Practice with place value disks is used for two-, three-, and four-digit by
one-digit multiplication problems with recordings as partial products. Students bridge partial products to the
recording of multiplication via the standard algorithm. 1 Finally, the partial products method, the standard
algorithm, and the area model are compared and connected by the distributive property (4.NBT.5).
1,423 x 3
Topic D gives students the opportunity to apply their new multiplication skills to solve multi-step word
problems (4.OA.3, 4.NBT.5) and multiplicative comparison problems (4.OA.2). Students write equations from
statements within the problems (4.OA.1) and use a combination of addition, subtraction, and multiplication
to solve.
In Topic E, students synthesize their Grade 3 knowledge of division types (group size unknown and number of
groups unknown) with their new, deeper understanding of place value.
1Students become fluent with the standard algorithm for multiplication in Grade 5 (5.NBT.5). Grade 4 students are introduced to the
standard algorithm in preparation for fluency and as a general method for solving multiplication problems based on place value
strategies, alongside place value disks, partial products, and the area model. Students are not assessed on the standard algorithm in
Grade 4.
Students focus on interpreting the remainder within division problems, both in word problems and long division
(4.OA.3). A remainder of 1, as exemplified below, represents a leftover flower in the first situation and a
remainder of 1 ten in the second situation. 2
While we have no reason to subdivide a remaining flower, there are good reasons to subdivide a remaining ten.
Students apply this simple idea to divide two-digit numbers unit by unit: dividing the tens units first, finding
the remainder (the number of tens unable to be divided), and decomposing remaining tens into ones to then
be divided. Students represent division with single-digit divisors using arrays and the area model before
practicing with place value disks. The standard division algorithm 3 is practiced using place value knowledge,
decomposing unit by unit. Finally, students use the area model to solve division problems, first with and then
without remainders (4.NBT.6).
In Topic F, armed with an understanding of remainders, students explore factors, multiples, and prime and
composite numbers within 100 (4.OA.4), gaining valuable insights into patterns of divisibility as they test for
primes and find factors and multiples. This prepares them for Topic G’s work with multi-digit dividends.
Topic G extends the practice of division with three- and four-digit dividends using place value understanding.
A connection to Topic B is made initially with dividing multiples of 10, 100, and 1,000 by single-digit numbers.
Place value disks support students visually as they decompose each unit before dividing. Students then
practice using the standard algorithm to record long division. They solve word problems and make
connections to the area model as was done with two-digit dividends (4.NBT.6, 4.OA.3).
2Note that care must be taken in the interpretation of remainders. Consider the fact that 7 ÷ 3 is not equal to 5 ÷ 2 because the
1 1
remainder of 1 is in reference to a different whole amount (2 3 is not equal to 2 2).
3Students become fluent with the standard division algorithm in Grade 6 (6.NS.2). For adequate practice in reaching fluency, students
are introduced to, but not assessed on, the division algorithm in Grade 4 as a general method for solving division problems.
4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number
answers using the four operations, including problems in which remainders must be
interpreted. Represent these problems using equations with a letter standing for the
unknown quantity. Assess the reasonableness of answers using mental computation and
estimation strategies including rounding.
Solve problems involving measurement and conversion of measurements from a larger unit
to a smaller unit. 5
4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical
problems. For example, find the width of a rectangular room given the area of the flooring
and the length, by viewing the area formula as a multiplication equation with an unknown
factor.
Foundational Standards
3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal
groups, arrays, and measurement quantities, e.g., by using drawings and equations with a
symbol for the unknown number to represent the problem. (See CCLS Glossary, Table 2.)
3.OA.4 Determine the unknown whole number in a multiplication or division equation relating three
whole numbers. For example, determine the unknown number that makes the equation true
in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?.
4
4.NBT.4 is addressed in Module 1 and is then reinforced throughout the year.
5
4.MD.1 is addressed in Modules 2 and 7; 4.MD.2 is addressed in Modules 2, 5, 6, and 7.
3.OA.5 Apply properties of operations as strategies to multiply and divide. (Students need not use
formal terms for these properties.) Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also
known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then
15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.)
Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 +
16 = 56. (Distributive property.)
3.OA.6 Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the
number that makes 32 when multiplied by 8.
3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between
multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties
of operations. By the end of Grade 3, know from memory all products of two one-digit
numbers.
3.OA.8 Solve two-step word problems using the four operations. Represent these problems using
equations with a letter standing for the unknown quantity. Assess the reasonableness of
answers using mental computation and estimation strategies including rounding. (This
standard is limited to problems posed with whole numbers and having whole-number
answers; students should know how to perform operations in the conventional order when
there are no parentheses to specify a particular order, i.e., Order of Operations.)
3.NBT.3 Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60)
using strategies based on place value and properties of operations.
3.MD.7 Relate area to the operations of multiplication and addition.
3.MD.8 Solve real world and mathematical problems involving perimeters of polygons, including
finding the perimeter given the side lengths, finding an unknown side length, and exhibiting
rectangles with the same perimeter and different areas or with the same area and different
perimeters.
Terminology
New or Recently Introduced Terms
Associative property (e.g., 96 = 3 × (4 × 8) = (3 × 4) × 8)
Composite number (positive integer having three or more whole number factors)
Distributive property (e.g., 64 × 27 = (60 × 20) + (60 × 7) + (4 × 20) + (4 × 7))
Divisible
Divisor (the number by which another number is divided)
Formula (a mathematical rule expressed as an equation with numbers and/or variables)
Long division (process of dividing a large dividend using several recorded steps)
6These are terms and symbols students have used or seen previously.
Tape diagram
Ten thousands place value chart (Lesson 7 Template)
Thousands place value chart (Lesson 4 Template)
Scaffolds7
The scaffolds integrated into A Story of Units give alternatives for how students access information as well as
express and demonstrate their learning. Strategically placed margin notes are provided within each lesson
elaborating on the use of specific scaffolds at applicable times. They address many needs presented by
English language learners, students with disabilities, students performing above grade level, and students
performing below grade level. Many of the suggestions are organized by Universal Design for Learning (UDL)
principles and are applicable to more than one population. To read more about the approach to
differentiated instruction in A Story of Units, please refer to “How to Implement A Story of Units.”
7
Students with disabilities may require Braille, large print, audio, or special digital files. Please visit the website
www.p12.nysed.gov/specialed/aim for specific information on how to obtain student materials that satisfy the National Instructional
Materials Accessibility Standard (NIMAS) format.
Assessment Summary
Type Administered Format Standards Addressed
Mid-Module After Topic D Constructed response with rubric 4.OA.1
Assessment Task 4.OA.2
4.OA.3
4.NBT.5
4.MD.3
End-of-Module After Topic H Constructed response with rubric 4.OA.1
Assessment Task 4.OA.2
4.OA.3
4.OA.4
4.NBT.5
4.NBT.6
4.MD.3
4
GRADE
Mathematics Curriculum
GRADE 4 • MODULE 3
Topic A
Multiplicative Comparison Word
Problems
4.OA.1, 4.OA.2, 4.MD.3, 4.OA.3
Multiplicative comparison is foundational for understanding multiplication as scaling in Grade 5 and sets the
stage for proportional reasoning in Grade 6. Students determine, using times as much as, the length of one
side of a rectangle as compared to its width. Beginning this Grade 4 module with area and perimeter allows
students to review their multiplication facts, apply them to new and interesting word problems, and develop
a deeper understanding of the area model as a method for calculating with larger numbers.
Objective 2: Solve multiplicative comparison word problems by applying the area and perimeter
formulas.
(Lesson 2)
Objective 3: Demonstrate understanding of area and perimeter formulas by solving multi-step real-
world problems.
(Lesson 3)
Lesson 1
Objective: Investigate and use the formulas for area and perimeter of
rectangles.
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
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Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
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Materials: (T) Grid paper (with ability to project or enlarge grid paper), chart paper (S) Grid paper, personal
white board
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
19
S: 12 units.
T: How does 12 relate to the length and width of the rectangle?
S: It’s the sum of the length and width.
T: How does the sum of the length and width relate to finding the perimeter of the rectangle?
S: It’s halfway around. I can double the length and double the width to find the perimeter instead of
adding all the sides (2l + 2w). I could also add the length and the width and double that sum,
2 × (l + w). Both of those work since the opposite sides are equal.
T: You have just mentioned many formulas, like counting along the sides of the rectangle or adding
sides or doubling, to find the perimeter. Let’s create a chart to keep track of the formulas for finding
the perimeter of a rectangle. Talk to your partner about the most efficient way to find the
perimeter.
S: If I draw the shape on grid paper, I can just count along the edge. I am
good at adding, so I will add all four sides. It is faster to double the sum
of the length and width. It’s only two steps.
T: We can write the formula as P = 2 × (l + w) on our chart, meaning we add
the length and width first and then multiply that sum by 2. What is the
length plus width of this rectangle?
S: 3 plus 9 equals 12. 12 units.
T: 12 units doubled, or 12 units times 2, equals …?
S: 24 units.
T: Now, draw a rectangle that is 2 units wide and 4 units long. Find the perimeter by using the formula
I just mentioned. Then, solve for the perimeter using a different formula to check your work.
S: 2 + 4 = 6 and 6 × 2 = 12. The perimeter is 12 units. Another way is to double 2, double 4, and
then add the doubles together. 4 plus 8 is 12 units. Both formulas give us the same answer.
Repeat with a rectangle that is 5 units wide and 6 units long.
Instruct students to sketch a rectangle with a width of 5 units and a perimeter
of 26 units on their personal white boards, not using graph paper.
T: Label the width as 5 units. Label the length as an unknown of x units.
How can we determine the length? Discuss your ideas with a partner.
S: If I know that the width is 5, I can label the opposite side as 5 units since they are the same. If the
perimeter is 26, I can take away the widths to find the sum of the other two sides. 26 – 10 = 16. If
the sum of the remaining two sides is 16, I know that each side must be 8 since I know that they are
equal and that 8 + 8 = 16, so x = 8 (shows sketch to demonstrate her thinking).
S: We could also find the length another way. I know that if I add the length and the width of the
rectangle together, I will get half of the perimeter. In this rectangle, because the perimeter is 26
units, the length plus the width equals 13 units. If the width is 5, that means that the length has to
be 8 units because 5 + 8 = 13. 26 ÷ 2 = 13, x + 5 = 13 or 13 – 5 = x, so x = 8.
Repeat for P = 28 cm, l = 8 cm.
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
20
Problem 3: Use the area formula (l × w) to solve for area and to solve for the unknown side length of a
rectangle.
T: Look back at the rectangle with the width of 3 units and the length of 9 units. How can we find the
area of the rectangle?
S: We can count all of the squares. We could also count the number of squares in one row and then
skip-count that number for all of the rows. That’s just multiplying the number of rows by the
number in each row. A quicker way is to multiply the length times the width. Nine rows of 3
units each is like an array. We can just multiply 9 × 3.
T: Talk to your partner about the most efficient way to find the area of a rectangle.
Discuss how to find the area for the 2 × 4 rectangle and the 5 × 6 rectangle drawn earlier in the lesson.
Encourage students to multiply length times width to solve. Ask students to tell how the area of each
rectangle needs to be labeled and why.
T: We discussed a formula for finding the perimeter of a rectangle. We just discovered a formula for
finding the area of a rectangle. If we use A for area, l for length, and w for width, how could we
write the formula?
S: A = l × w.
T: (Sketch a rectangle on the board, and label the area as 50 square
centimeters.) If we know that the area of a rectangle is 50 square
centimeters and that the length of the rectangle is 10 centimeters,
how can we determine the measurement of the width of the
rectangle?
S: I can use the area formula. 50 square centimeters is equal to 10 centimeters times the width. 10
times 5 equals 50, so the width is 5 centimeters. The area formula says 50 = 10 × . I can solve
that with division! So, 50 square centimeters divided by 10 centimeters is 5 centimeters.
Repeat for A = 32 square m, l = 8 m and for A = 63 square cm, w = 7 cm.
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
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Problem 4: Given the area of a rectangle, find all possible whole number combinations of the length and
width, and then calculate the perimeter.
T: If a rectangle has an area of 24 square units, what whole numbers could be the length and width of
the rectangle? Discuss with your partner.
S: The length is 3 units, and the width is 8 units. Yes, but the length could also be 4 units and the
width 6 units. Or, the other way around: length of 6 units and width of 4 units. There are many
combinations of length and width to make a rectangle with an area of 24 square units.
T: With your partner, draw and complete a table similar to mine until you have found all possible whole
number combinations for the length and width.
Circulate, checking to see that students are using the length times width formula to find the dimensions.
Complete the table with all combinations as a class.
T: Now, sketch each rectangle, and solve for the perimeter using the perimeter formula.
Circulate, checking to see that students draw rectangles to scale and solve for perimeter using the formula.
Check answers as a class.
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
22
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
23
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
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Name Date
a. A = _______________ A = _______________
b. P = _______________ P = _______________
5 cm P = ____________
8 cm P = ____________
A = ____________
A = ____________
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
25
99 m 75 cm
P = _______________ P = _______________
49 x cm
80 square
square x cm
cm
cm
x = ____________ x = ____________
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
26
20 cm xm
250 m
x cm
x = _______________ x = ______________
6. Each of the following rectangles has whole number side lengths. Given the area and perimeter, find the
length and width.
a. P = 20 cm b. P = 28 m
l = _________
24
24 w = _______ square
square m
cm w = _______
l = _________
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
27
Name Date
8 cm
2 cm
347 m
99 m
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
28
Name Date
a. A = _______________ A = _______________
b. P = _______________ P = _______________
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
29
76 m 45 cm
P = _______________
P = _______________
25
60 square xm
square x cm m
cm
x = ____________ x = ____________
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
30
a. P = 180 cm b. P = 1,000 m
40 cm xm
150 m
x cm
x = _______________ x = ______________
6. Each of the following rectangles has whole number side lengths. Given the area and perimeter, find the
length and width.
a. A = 32 square cm b. A = 36 square m
P = 24 cm P = 30 m
w = _______
l = _________
36
w = _______ square l = _________
32 square cm
m
Lesson 1: Investigate and use the formulas for area and perimeter of rectangles.
31
Lesson 2
Objective: Solve multiplicative comparison word problems by applying the
area and perimeter formulas.
Note: Multiplying a number by itself helps students quickly compute the areas of squares.
Repeat the process from Lesson 1, using more choral response.
Extension: Tommy’s dad is making a table 6 feet wide and 8 feet long. When both tables are placed
together, what will their combined area be?
Note: This Application Problem builds from 3.MD.5, 3.MD.6, and 3.MD.8 and bridges back to the Concept
Development of Lesson 1, during which students investigated and used the formulas for the area and
perimeter of rectangles.
Materials: (T) Chart of formulas for perimeter and area from Lesson 1 (S) Personal white board, square-inch
tiles
Problem 1: A rectangle is 1 inch wide. It is 3 times as long as it is wide. Use square tiles to find its length.
T: Place 3 square-inch tiles on
your personal white
board. Talk to your
partner about what the
width and length of this
rectangle are.
S: (Discuss.)
T: I heard Alyssa say that
the width is 1 inch and
the length is 3 inches.
Now, make it 2 times as
long. (Add 3 more square
tiles.) It’s now 6 inches
long. Three times as long
(add 3 more tiles) would
be 9 inches. Using the
original length of 3 inches,
tell your partner how to
determine the current
length that is three times as
many.
S: I multiply the original length times 3. Three times as long as 3 inches is the same as 3 times 3
inches.
Repeat using tiles to find a rectangle that is 3 inches wide and 3 times as long as it is wide.
Problem 2: A rectangle is 2 meters wide. It is 3 times as long as it is wide. Draw to find its length.
T: The rectangle is 2 meters wide. (Draw a vertical line and label it as 2 meters.)
T: It is 3 times as long as it is wide. That means the length can be thought of as three segments, or
short lines, each 2 meters long. (Draw the horizontal lines to create a square 2 meters by 2 meters.)
T: Here is the same length, 2 times as long, 3 times as long. (Extend the rectangle as shown.) What is
the length when there are 3 segments, each 2 meters long?
S: 6 meters.
Problem 3: Solve a multiplicative comparison word problem using the area and perimeter formulas.
Christine painted a mural with an area of 18 square meters and a length of 6 meters. What is the width of her
mural? Her next mural will be the same length as the first but 4 times as wide. What is the perimeter of her
next mural?
Display the first two statements of the problem.
T: With your partner, determine the width of the first
mural.
S: The area is 18 square meters. 18 square meters
divided by 6 meters is 3 meters. The width is
3 meters.
T: True. (Display the last two statements of the
problem.) Using those dimensions, draw and label
Christine’s next mural. Begin with the side length
you know, 6 meters. How many copies of Christine’s
first mural will we see in her next mural? Draw
them.
S: Four copies. (Draw.)
T: Tell me a multiplication sentence to find how wide her
next mural will be.
S: 3 meters times 4 equals 12 meters. NOTES ON
T: Finish labeling the diagram. MULTIPLE MEANS
OF ENGAGEMENT:
T: Find the perimeter of Christine’s next mural. For help,
English language learners may benefit
use the chart of formulas for perimeter that we
from frequent checks for
created during Lesson 1.
understanding as the word problem is
S: 12 meters plus 6 meters is 18 meters. 18 meters read aloud. Explain how the term
doubled is 36 meters. The perimeter is 36 meters. square meters denotes the garden’s
area. Instead of twice, say two times.
Problem 4: Observe the relationship of area and perimeter Use gestures and illustrations to clarify
while solving a multiplicative comparison word problem using the meaning. In addition, after
the area and perimeter formulas. students discover the relationship
between area and perimeter, challenge
Sherrie’s rectangular garden is 8 square meters. The longer side them to explore further. Ask, “If you
of the garden is 4 meters. Nancy’s garden is twice as long and draw another rectangle with a different
length, will a similar doubling of the
twice as wide as Sherrie’s rectangular garden.
perimeter and quadrupling of the area
Display the first two statements. result?”
T: (Display the next statement.) Help me draw Nancy’s garden. Twice as long as 4 meters is how many
meters?
S: 8 meters.
T: Twice as wide as 2 meters is how many meters?
S: 4 meters.
T: Draw Nancy’s garden and find
the perimeters of both
gardens.
S: (Draw and solve to find the
perimeters.)
T: Tell your partner the
relationship between the two
perimeters.
S: Sherrie’s garden has a
perimeter of 12 meters.
Nancy’s garden has a
perimeter of 24 meters.
The length doubled, and the
width doubled, so the
perimeter doubled! 12
meters times 2 is 24 meters.
T: If Sherrie’s neighbor had a garden 3 times as long and 3 times as wide as her garden, what would be
the relationship of the perimeter between those gardens?
S: The perimeter would triple!
T: Solve for the area of Nancy’s garden and the neighbor’s garden. What do you notice about the
relationship among the perimeters and areas of the three gardens?
S: Nancy’s garden has an area of 32 square meters. The neighbor’s garden has an area of 72 square
meters. The length and width of Nancy’s garden is double that of Sherrie’s garden, but the area
did not double. The length is doubled and the width is doubled. 2 times 2 is 4, so the area will be
4 times as large. Right, the area quadrupled! I can put the area of Sherrie’s garden inside
Nancy’s garden 4 times. The length and width of the neighbor’s garden tripled, and 3 times 3 is 9.
The area of the neighbor’s garden is 9 times that of Sherrie’s.
Create a table to show the relationship among the areas and perimeters of the three gardens.
Name Date
b. Charlie wants to draw a second rectangle that is the same length but is 3 times as wide. Draw and
label Charlie’s second rectangle.
4. The area of Betsy’s rectangular sandbox is 20 square feet. The longer side measures 5 feet. The sandbox
at the park is twice as long and twice as wide as Betsy’s.
a. Draw and label a diagram of Betsy’s b. Draw and label a diagram of the sandbox at
sandbox. What is its perimeter? the park. What is its perimeter?
e. The sandbox at the park has an area that is how many times that of Betsy’s sandbox?
f. Compare how the perimeter changed with how the area changed between the two sandboxes.
Explain what you notice using words, pictures, or numbers.
Name Date
Name Date
b. Elsa wants to draw a second rectangle that is the same length but is 3 times as wide. Draw and label
Elsa’s second rectangle.
4. The area of Nathan’s bedroom rug is 15 square feet. The longer side measures 5 feet. His living room rug
is twice as long and twice as wide as the bedroom rug.
a. Draw and label a diagram of Nathan’s b. Draw and label a diagram of Nathan’s living
bedroom rug. What is its perimeter? room rug. What is its perimeter?
d. Find the area of the living room rug using the formula A = l × w.
e. The living room rug has an area that is how many times that of the bedroom rug?
f. Compare how the perimeter changed with how the area changed between the two rugs. Explain
what you notice using words, pictures, or numbers.
Lesson 3
Objective: Demonstrate understanding of area and perimeter formulas by
solving multi-step real-world problems.
Note: This Sprint reviews skills that help students as they solve area problems.
Note: For this lesson, the Problem Set comprises word NOTES ON
problems from the Concept Development and should therefore MULTIPLE MEANS
be used during the lesson itself. OF ENGAGEMENT:
Students may work in pairs to solve Problems 1─4 below using To maximize productivity, choose to
make team goals for sustained effort,
the RDW approach to problem solving.
perseverance, and cooperation.
Motivate improvement by providing
1. Model the problem.
specific feedback after each problem.
Have two pairs of students who can be successful with modeling Resist feedback that is comparative or
the problem work at the board while the others work competitive. Showcase students who
independently or in pairs at their seats. Review the following incorporated feedback into their
questions before beginning the first problem. subsequent work.
Problem 1
The rectangular projection screen in the school auditorium is 5 times as long and 5 times as wide as the
rectangular screen in the library. The screen in the library is 4 feet long with a perimeter of 14 feet. What is
the perimeter of the screen in the auditorium?
The structure of this problem and what it demands of students is similar to that found within the first and
second lessons of this module. Elicit from students why both the length and the width were multiplied by 5 to
find the dimensions of the larger screen. Students use the dimensions to find the perimeter of the larger
screen. Look for students to use formulas for perimeter other than 2 × (l + w) for this problem, such as the
formula 2l + 2w.
Problem 2
The width of David’s rectangular tent is 5 feet. The length is twice the width. David’s rectangular air mattress
measures 3 feet by 6 feet. If David puts the air mattress in the tent, how many square feet of floor space will
be available for the rest of his things?
The new complexity here is that students are finding an area within an area and determining the difference
between the two. Have students draw and label the larger area first and then draw and label the area of the
air mattress inside as shown above. Elicit from students how the remaining area can be found using
subtraction.
Problem 3
Jackson’s rectangular bedroom has an area of 90 square feet. The area of his bedroom is 9 times that of his
rectangular closet. If the closet is 2 feet wide, what is its length?
This multi-step problem requires students to work backwards, taking the area of Jackson’s room and dividing
by 9 to find the area of his closet. Students use their learning from the first and second lessons of this module
to help solve this problem.
Problem 4
The length of a rectangular deck is 4 times its width. If the deck’s perimeter is 30 feet, what is the deck’s
area?
Students need to use what they know about multiplicative comparison and perimeter to find the dimensions
of the deck. Students find this rectangle has 10 equal-size lengths around its perimeter. Teachers can
support students who are struggling by using square tiles to model the rectangular deck. Emphasize finding
the number of units around the perimeter of the rectangle. Once the width is determined, students are able
to solve for the area of the deck. If students have solved using square tiles, encourage them to follow up by
drawing a picture of the square tile representation. This allows students to bridge the gap between the
concrete and pictorial stages.
Problem Set
Please note that the Problem Set for Lesson 3 comprises
this lesson’s problems, as stated in the introduction of the
lesson.
A
Number Correct: _______
1. 2×2= 23. 3× = 21
2. 2× =4 24. 3×3=
3. 3×3= 25. 4× = 20
4. 3× =9 26. 4× = 32
6. 5× = 25 28. 5× = 20
7. 1× =1 29. 5× = 40
9. 4× = 16 31. 6× = 18
15. 10 × 10 = 37. 8× = 24
19. 2× = 10 41. 9× = 63
22. 3× = 12 44. 10 × 10 =
B
Number Correct: _______
Improvement: _______
Squares and Unknown Factors
1. 5×5= 23. 3× = 24
2. 5× = 25 24. 3×3=
3. 2×2= 25. 4× = 12
4. 2× =4 26. 4× = 28
6. 3× =9 28. 5× = 10
7. 1×1= 29. 5× = 35
8. 1× =1 30. 5×5=
9. 4× = 16 31. 6× = 24
15. 10 × 10 = 37. 8× = 32
19. 2× =8 41. 9× = 72
22. 3× = 15 44. 10 × 10 =
Name Date
Solve the following problems. Use pictures, numbers, or words to show your work.
1. The rectangular projection screen in the school auditorium is 5 times as long and 5 times as wide as the
rectangular screen in the library. The screen in the library is 4 feet long with a perimeter of 14 feet. What
is the perimeter of the screen in the auditorium?
2. The width of David’s rectangular tent is 5 feet. The length is twice the width. David’s rectangular air
mattress measures 3 feet by 6 feet. If David puts the air mattress in the tent, how many square feet of
floor space will be available for the rest of his things?
3. Jackson’s rectangular bedroom has an area of 90 square feet. The area of his bedroom is 9 times that of
his rectangular closet. If the closet is 2 feet wide, what is its length?
4. The length of a rectangular deck is 4 times its width. If the deck’s perimeter is 30 feet, what is the deck’s
area?
Name Date
Solve the following problem. Use pictures, numbers, or words to show your work.
A rectangular poster is 3 times as long as it is wide. A rectangular banner is 5 times as long as it is wide. Both
the banner and the poster have perimeters of 24 inches. What are the lengths and widths of the poster and
the banner?
Name Date
Solve the following problems. Use pictures, numbers, or words to show your work.
1. Katie cut out a rectangular piece of wrapping paper that was 2 times as long and 3 times as wide as the
box that she was wrapping. The box was 5 inches long and 4 inches wide. What is the perimeter of the
wrapping paper that Katie cut?
2. Alexis has a rectangular piece of red paper that is 4 centimeters wide. Its length is twice its width. She
glues a rectangular piece of blue paper on top of the red piece measuring 3 centimeters by 7 centimeters.
How many square centimeters of red paper will be visible on top?
3. Brinn’s rectangular kitchen has an area of 81 square feet. The kitchen is 9 times as many square feet as
Brinn’s pantry. If the rectangular pantry is 3 feet wide, what is the length of the pantry?
4. The length of Marshall’s rectangular poster is 2 times its width. If the perimeter is 24 inches, what is the
area of the poster?
4
GRADE
Mathematics Curriculum
GRADE 4 • MODULE 3
Topic B
Multiplication by 10, 100, and 1,000
4.NBT.5, 4.OA.1, 4.OA.2, 4.NBT.1
Focus Standard: 4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and
multiply two two-digit numbers, using strategies based on place value and the
properties of operations. Illustrate and explain the calculation by using equations,
rectangular arrays, and/or area models.
Instructional Days: 3
Coherence -Links from: G3–M1 Properties of Multiplication and Division and Problem Solving with Units of 2–5 and 10
-Links to: G5–M1 Place Value and Decimal Fractions
In Topic B, students examine multiplication patterns when multiplying by 10, 100, and 1,000. Reasoning
between arrays and written numerical work allows students to see the role of place value units in
multiplication (as pictured below). Students also practice the language of units to prepare them for
multiplication of a single-digit factor by a factor with up to four digits. Teachers also continue using the
phrase “____ is ____ times as much as ____” (e.g., 120 is 3 times as much as 40). This carries forward
multiplicative comparison from Topic A, in the context of area, to Topic B, in the context of both calculations
and word problems.
Introducing this early in the module allows students to practice this multiplication during fluency activities so
that by the time it is embedded within the two-digit by two-digit multiplication in Topic H, both
understanding and procedural fluency have been developed.
In Lesson 4, students interpret and represent patterns when multiplying by 10, 100, and 1,000 in arrays and
numerically. Next, in Lesson 5, students draw disks to multiply single-digit numbers by multiples of 10, 100,
and 1,000. Finally, in Lesson 6, students use disks to multiply two-digit multiples of 10 by two-digit multiples
of 10 (4.NBT.5) with the area model.
Objective 2: Multiply multiples of 10, 100, and 1,000 by single digits, recognizing patterns.
(Lesson 5)
Objective 3: Multiply two-digit multiples of 10 by two-digit multiples of 10 with the area model.
(Lesson 6)
Lesson 4
Objective: Interpret and represent patterns when multiplying by 10, 100,
and 1,000 in arrays and numerically.
Note: Renaming units helps prepare students for the next fluency activity and for this lesson’s content.
Repeat the process from Lesson 2 using the following suggested sequence: 8 tens, 9 tens, 11 tens,
14 tens, 14 hundreds, 14 thousands, 18 tens, 28 tens, 28 hundreds, and 28 thousands.
Lesson 4: Interpret and represent patterns when multiplying by 10, 100, and
1,000 in arrays and numerically. 64
T: Continue.
S: 12 tens, 15 tens.
T: (Raise hand.) Say the number.
S: 150.
Repeat the process for 21 tens, 27 tens, and 30 tens.
Repeat the process, counting by 4 hundreds, stopping to convert at 12 hundreds, 20 hundreds, 32 hundreds,
and 40 hundreds.
Repeat the process, counting by 6 hundreds, stopping to convert at 18 hundreds, 30 hundreds, 48 hundreds,
and 60 hundreds.
Lesson 4: Interpret and represent patterns when multiplying by 10, 100, and
1,000 in arrays and numerically. 65
Lesson 4: Interpret and represent patterns when multiplying by 10, 100, and
1,000 in arrays and numerically. 66
Problem 2: Draw place value disks to represent products when multiplying by a two-digit number.
Display 15 × 10 on the board.
T: Draw place value disks to represent 15, and then show 15 × 10. Explain what you did.
S: I drew an arrow to the next column. I drew an arrow to
show times 10 for the 1 ten and also for the 5 ones.
T: Right. We need to show times 10 for each of our units.
T: What is 1 ten × 10?
S: 1 hundred.
T: What is 5 ones × 10?
S: 5 tens.
T: 15 × 10 equals?
S: 150.
Display 22 × 100 on the board.
T: With your partner, represent 22 × 100 using place value
disks. What did you draw?
S: I drew 2 tens and 2 ones and showed times 10. Then, I did
times 10 again. I drew 2 tens and 2 ones and showed
MP.4 times 100 by moving two place values to the left.
T: How can we express your solution strategies as
multiplication sentences?
S: 22 × 10 × 10. 22 × 100.
T: What is 22 × 100?
S: 2,200.
Lesson 4: Interpret and represent patterns when multiplying by 10, 100, and
1,000 in arrays and numerically. 67
Lesson 4: Interpret and represent patterns when multiplying by 10, 100, and
1,000 in arrays and numerically. 68
Any combination of the questions below may be used to lead the discussion.
What is the difference between saying 10 more
and 10 times as many?
What is another expression that has the same
value as 10 × 800 and 1,000 × 8?
Think about the problems we solved during the
lesson and the problems you solved in the
Problem Set. When does the number of zeros in
the factors not equal the number of zeros in the
product?
For Problem 4, 12 × 10 = 120, discuss with your
partner whether or not this equation is true:
12 × 10 = 3 × 40. (Problem 7 features 3 × 40.)
How did the Application Problem connect to
today’s lesson?
Lesson 4: Interpret and represent patterns when multiplying by 10, 100, and
1,000 in arrays and numerically. 69
Name Date
Example:
5 × 10 = _______
5 × 10 × 10 = __________
5 × 10 × 10 × 10 = __________
Lesson 4: Interpret and represent patterns when multiplying by 10, 100, and
1,000 in arrays and numerically. 70
4. 12 × 10 = __________
thousands hundreds tens ones
5. 18 × 100 = __________
thousands hundreds tens ones
18 × 10 × 10 = __________
________________
= _________ = _________
Lesson 4: Interpret and represent patterns when multiplying by 10, 100, and
1,000 in arrays and numerically. 71
Name Date
Lesson 4: Interpret and represent patterns when multiplying by 10, 100, and
1,000 in arrays and numerically. 72
Name Date
Example:
5 × 10 = _______
Draw place value disks and arrows as shown to represent each product.
1. 7 × 100 = __________
thousands hundreds tens ones
7 × 10 × 10 = __________
2. 7 × 1,000 = __________
thousands hundreds tens ones
7 × 10 × 10 × 10 = __________
___________________
Lesson 4: Interpret and represent patterns when multiplying by 10, 100, and
1,000 in arrays and numerically. 73
5. 17 × 100 = __________
thousands hundreds tens ones
17 × 10 × 10 = __________
= _________ = ________
Lesson 4: Interpret and represent patterns when multiplying by 10, 100, and
1,000 in arrays and numerically. 74
Lesson 4: Interpret and represent patterns when multiplying by 10, 100, and
1,000 in arrays and numerically. 75
Lesson 5
Objective: Multiply multiples of 10, 100, and 1,000 by single digits,
recognizing patterns.
Note: This fluency activity gives students practice reviewing content from Lesson 4.
T: (Write 3 × 2 = .) Say the multiplication sentence in unit form.
S: 3 ones × 2 = 6 ones.
T: On your personal white boards, write the answer in standard form.
S: (Write 6.)
T: (Write 30 × 2 = .) Say the multiplication sentence in unit form.
S: 3 tens × 2 = 6 tens.
T: Write the answer in standard form.
S: (Write 60.)
Repeat for the following possible sequence: 3 hundreds × 2, 3 thousands × 2, 5 ones × 3, 5 tens × 3,
5 thousands × 3, 5 thousands × 4, 5 tens × 4, 5 ones × 8, 5 hundreds × 8, and 9 tens × 7.
Lesson 5: Multiply multiples of 10, 100, and 1,000 by single digits, recognizing
patterns. 76
Lesson 5: Multiply multiples of 10, 100, and 1,000 by single digits, recognizing
patterns. 77
Lesson 5: Multiply multiples of 10, 100, and 1,000 by single digits, recognizing
patterns. 78
Lesson 5: Multiply multiples of 10, 100, and 1,000 by single digits, recognizing
patterns. 79
Lesson 5: Multiply multiples of 10, 100, and 1,000 by single digits, recognizing
patterns. 80
Name Date
Draw place value disks to represent the value of the following expressions.
1. 2 × 3 = ______
thousands hundreds tens ones
3
2 times _____ ones is _____ ones.
× 2
2. 2 × 30 = ______
thousands hundreds tens ones
3. 2 × 300 = ______
4. 2 × 3,000 = ______
2
×
Lesson 5: Multiply multiples of 10, 100, and 1,000 by single digits, recognizing
patterns. 81
a. 20 × 7 b. 3 × 60 c. 3 × 400 d. 2 × 800
e. 7 × 30 f. 60 × 6 g. 400 × 4 h. 4 × 8,000
i. 5 × 30 j. 5 × 60 k. 5 × 400 l. 8,000 × 5
6. Brianna buys 3 packs of balloons for a party. Each pack has 60 balloons. How many balloons does
Brianna have?
Lesson 5: Multiply multiples of 10, 100, and 1,000 by single digits, recognizing
patterns. 82
7. Jordan has twenty times as many baseball cards as his brother. His brother has 9 cards. How many cards
does Jordan have?
8. The aquarium has 30 times as many fish in one tank as Jacob has. The aquarium has 90 fish. How many
fish does Jacob have?
Lesson 5: Multiply multiples of 10, 100, and 1,000 by single digits, recognizing
patterns. 83
Name Date
Draw place value disks to represent the value of the following expressions.
1. 4 × 200 = ______
2. 4 × 2,000 = ______
a. 30 × 3 b. 8 × 20 c. 6 × 400 d. 2 × 900
e. 8 × 80 f. 30 × 4 g. 500 × 6 h. 8 × 5,000
4. Bonnie worked for 7 hours each day for 30 days. How many hours did she work altogether?
Lesson 5: Multiply multiples of 10, 100, and 1,000 by single digits, recognizing
patterns. 84
Name Date
Draw place value disks to represent the value of the following expressions.
1. 5 × 2 = ______
2
2. 5 × 20 = ______ × 5
3. 5 × 200 = ______
4. 5 × 2,000 = ______
Lesson 5: Multiply multiples of 10, 100, and 1,000 by single digits, recognizing
patterns. 85
a. 20 × 9 b. 6 × 70 c. 7 × 700 d. 3 × 900
e. 9 × 90 f. 40 × 7 g. 600 × 6 h. 8 × 6,000
i. 5 × 70 j. 5 × 80 k. 5 × 200 l. 6,000 × 5
6. At the school cafeteria, each student who orders lunch gets 6 chicken nuggets. The cafeteria staff
prepares enough for 300 kids. How many chicken nuggets does the cafeteria staff prepare altogether?
Lesson 5: Multiply multiples of 10, 100, and 1,000 by single digits, recognizing
patterns. 86
7. Jaelynn has 30 times as many stickers as her brother. Her brother has 8 stickers. How many stickers does
Jaelynn have?
8. The flower shop has 40 times as many flowers in one cooler as Julia has in her bouquet. The cooler has
120 flowers. How many flowers are in Julia’s bouquet?
Lesson 5: Multiply multiples of 10, 100, and 1,000 by single digits, recognizing
patterns. 87
Lesson 6
Objective: Multiply two-digit multiples of 10 by two-digit multiples of 10
with the area model.
There are 400 children at Park Elementary School. Park High School has 4 times as many students.
a. How many students in all attend both schools?
b. Lane High School has 5 times as many students as Park Elementary. How many more students attend
Lane High School than Park High School?
NOTES ON
MULTIPLE MEANS
OF ENGAGEMENT:
Differentiate the difficulty of the
Application Problem by adjusting the
numbers. Extend for students working
above grade level with these open-
ended questions:
How many students would you
predict attend the middle school?
Note: These problems are a review of work from Lesson 5. Explain your reasoning.
If these were estimates of the
number of students, what might be
the actual numbers?
Materials: (T) Thousands place value chart (Lesson 4 Template) (S) Personal white board, thousands place
value chart (Lesson 4 Template)
Problem 1: Use the place value chart to multiply a two-digit multiple of 10 by a two-digit multiple of 10.
Display 30 × 20 on the board.
T: Here we are multiplying a two-digit number by another
two-digit number. What are some other ways we NOTES ON
could express 30 × 20? MULTIPLE MEANS
S: 3 tens × 2 tens. OF REPRESENTATION:
10 × 20 × 3. For students developing oral language
10 × 30 × 2. skills, alternate between choral
response and written response.
2 × 30 × 10.
Encourage students to explain their
3 × 20 × 10. math thinking in the language of their
T: Let’s use 10 × 20 × 3 in a place value chart to help us choice. Allow added response time for
solve 30 × 20. (Project place value chart as shown to English language learners to gather
the right.) their thoughts.
Problem 2: Create an area model to represent the multiplication of a two-digit multiple of 10 by a two-digit
multiple of 10.
T: (Display 40 × 20.) Let’s model 40 × 20 as an area. Tell your partner what 40 × 20 is.
S: 4 tens times 20. That’s 80 tens, or 800.
T: (Record student statement.) What is 20 in unit form?
S: 2 tens.
T: So, then, what is 4 tens times 2 tens?
S: I know 4 times 2 is 8. I don’t know what to do with the units. I know 4
times 2 is 8. That leaves both tens. 10 tens. It’s like saying 4 times 2 times
10 tens!
T: Let’s prove how we can multiply the units. Draw a 40 by 20 rectangle on
your personal white board. Partition the horizontal side into 2 tens and the
vertical side into 4 tens. Label each side. What is the area of one square?
(Point to a 10 by 10 square.)
S: 10 × 10 = 100.
T: Say a multiplication sentence for how many of the squares there are.
S: 4 × 2 = 8.
T: Tell your partner how this rectangle shows 4 tens times 2 tens equals
8 hundreds.
S: Each square is 10 by 10. That makes 100. There are 8 hundreds.
Name Date
Represent the following problem by drawing disks in the place value chart.
(2 tens × 4) × 10 = ________
20 × (4 × 10) = ________
20 × 40 = _______
30 × 40 = ______
20 × 50 = _______
5. 20 × 20 = ________ 6. 60 × 20 = _______
7. 70 × 20 = _______ 8. 70 × 30 = _______
_____ tens × _____ tens = 14 _________ ____ _______ × ____ _______ = _____ hundreds
9. If there are 40 seats per row, how many seats are in 90 rows?
10. One ticket to the symphony costs $50. How much money is collected if 80 tickets are sold?
Name Date
Represent the following problem by drawing disks in the place value chart.
(2 tens × 3) × 10 = ________
20 × (3 × 10) = ________
20 × 30 = _______
3. Every night, Eloise reads 40 pages. How many total pages does she read at night during the 30 days of
November?
Name Date
Represent the following problem by drawing disks in the place value chart.
(3 tens × 6) × 10 = ________
30 × (6 × 10) = ________
30 × 60 = _______
20 × 20 = ______
40 × 60 = _______
5. 50 × 20 = ________ 6. 30 × 50 = ________
7. 60 × 20 = ________ 8. 40 × 70 = ________
_____ tens × _____ tens = 12 _________ ____ _______ × ____ _______ = _____ hundreds
9. There are 60 seconds in a minute and 60 minutes in an hour. How many seconds are in one hour?
10. To print a comic book, 50 pieces of paper are needed. How many pieces of paper are needed to print
40 comic books?
4
GRADE
Mathematics Curriculum
GRADE 4 • MODULE 3
Topic C
Multiplication of up to Four Digits by
Single-Digit Numbers
4.NBT.5, 4.OA.2, 4.NBT.1
Focus Standard: 4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and
multiply two two-digit numbers, using strategies based on place value and the
properties of operations. Illustrate and explain the calculation by using equations,
rectangular arrays, and/or area models.
Instructional Days: 5
Coherence -Links from: G3–M1 Properties of Multiplication and Division and Problem Solving with Units of 2–5 and 10
G3–M3 Multiplication and Division with Units of 0, 1, 6–9, and Multiples of 10
-Links to: G5–M2 Multi-Digit Whole Number and Decimal Fraction Operations
Building on their work in Topic B, students begin in Topic C decomposing numbers into base ten units in order
to find products of single-digit by multi-digit numbers. Students practice multiplying by using models before
being introduced to the standard algorithm. Throughout the topic, students practice multiplication in the
context of word problems, including multiplicative comparison problems.
In Lessons 7 and 8, students use place value disks to represent the multiplication of two-, three-, and
four-digit numbers by a one-digit whole number.
Lessons 9 and 10 move students to the abstract level as they multiply three- and four-digit numbers by
one-digit numbers using the standard algorithm.
Finally, in Lesson 11, partial products, the standard algorithm, and the area model are compared and
connected via the distributive property (4.NBT.5).
Objective 2: Extend the use of place value disks to represent three- and four-digit by one-digit
multiplication.
(Lesson 8)
Objective 3: Multiply three- and four-digit numbers by one-digit numbers applying the standard
algorithm.
(Lessons 9–10)
Objective 4: Connect the area model and the partial products method to the standard algorithm.
(Lesson 11)
Lesson 7
Objective: Use place value disks to represent two-digit by one-digit
multiplication.
Note: This Sprint reinforces concepts taught and reviewed in Lessons 1─6.
The basketball team is selling T-shirts for $9 each. On Monday, they sold 4 T-shirts. On Tuesday, they sold 5
times as many T-shirts as on Monday. How much money did the team earn altogether on Monday and
Tuesday?
NOTES ON
MULTIPLE MEANS
OF ENGAGEMENT:
Extend the Application Problem for
students above grade level with open-
ended questions, such as the following:
What might be an explanation for
the large difference in T-shirt sales
between Monday and Tuesday?
Based on your thoughts, what
might be a strategy for generating
the most money from T-shirt sales?
Given the increase in T-shirts sold,
should the team increase or
decrease the price of the shirt?
Note: This is a multi-step word problem reviewing multiplying Explain your reasoning.
by multiples of 10 from Lesson 5, including multiplicative
comparison.
Talk to your partner about which method you prefer. Do you prefer writing the partial products or
using a place value chart with disks? Is one of these methods easier for you to understand? Does
one of them help you solve the problem faster?
A
Number Correct: _______
2. 30 × 2 = 24. 700 × 5 =
4. 3,000 × 2 = 26. 80 × 3 =
7. 2 × 40 = 29. 7×6=
20. 70 × 2 = 42. 5 × 60 =
B
Number Correct: _______
Improvement: _______
Multiply Multiples of 10, 100, and 1,000
2. 40 × 2 = 24. 900 × 5 =
4. 4,000 × 2 = 26. 80 × 4 =
7. 3 × 30 = 29. 6×7=
16. 30 × 4 = 38. 6 × 60 =
Name Date
1. Represent the following expressions with disks, regrouping as necessary, writing a matching expression,
and recording the partial products vertically as shown below.
a. 1 × 43
4 3
tens ones
× 1
3 1 × 3 ones
+ 4 0 1 × 4 tens
4 3
b. 2 × 43
tens ones
c. 3 × 43
hundreds tens ones
d. 4 × 43
2. Represent the following expressions with disks, regrouping as necessary. To the right, record the partial
products vertically.
a. 2 × 36
b. 3 × 61
c. 4 × 84
Name Date
Represent the following expressions with disks, regrouping as necessary. To the right, record the partial
products vertically.
1. 6 × 41
2. 7 × 31
Name Date
1. Represent the following expressions with disks, regrouping as necessary, writing a matching expression,
and recording the partial products vertically.
a. 3 × 24
tens ones
b. 3 × 42
c. 4 × 34
2. Represent the following expressions with disks, regrouping as necessary. To the right, record the partial
products vertically.
a. 4 × 27
b. 5 × 42
3. Cindy says she found a shortcut for doing multiplication problems. When she multiplies 3 × 24, she says,
“3 × 4 is 12 ones, or 1 ten and 2 ones. Then, there’s just 2 tens left in 24, so add it up, and you get 3 tens
and 2 ones.” Do you think Cindy’s shortcut works? Explain your thinking in words, and justify your
response using a model or partial products.
Lesson 8
Objective: Extend the use of place value disks to represent three- and four-
digit by one-digit multiplication.
Note: Reviewing standard form versus expanded form prepares students to decompose multi-digit
multiplication sentences into a series of multiplication sentences.
T: (Write 200 + 30 + 4.) Say the addition sentence with the answer in standard form.
S: 200 + 30 + 4 = 234.
Repeat the process for the following possible sequence: 3,000 + 500 + 60 + 8 and 400 + 7 + 90.
T: (Write 572.) Say the number.
S: 572.
T: On your personal white board, write 572 in expanded form.
S: (Write 572 = 500 + 70 + 2.)
Repeat the process using the following possible sequence: 8,463 and 9,075.
Lesson 8: Extend the use of place value disks to represent three- and four-digit
by one-digit multiplication. 116
Andre buys a stamp to mail a letter. The stamp costs 46 cents. Andre also mails a package. The postage to
mail the package costs 5 times as much as the cost of the stamp. How much does it cost to mail the package
and letter?
Note: This problem is a review of Lesson 7 and incorporates multiplicative comparison. Students who
examine the tape diagram find a more rapid solution is to multiply to find 6 units of 46 cents.
Lesson 8: Extend the use of place value disks to represent three- and four-digit
by one-digit multiplication. 117
Materials: (T) Ten thousands place value chart (Lesson 7 Template) (S) Personal white board, ten thousands
place value chart (Lesson 7 Template)
Note: Today’s lesson is an extension of Lesson 7. Students solve three- and four-digit by one-digit
multiplication using the same method as they used in Lesson 7. Students should be given more autonomy to
work on the problems in partnerships or individually. A connection regarding the process should be made so
that students understand that although the numbers are larger, the process is the same.
Problem 1: Represent 2 × 324 with disks. Write a matching equation, and record the partial products
vertically.
T: Use your place value chart to
represent 2 times 324.
T: What is the value in the
ones?
S: 2 times 4 ones is 8 ones or 8.
T: The tens?
S: 2 times 2 tens is 4 tens or 40.
T: The hundreds?
S: 2 times 3 hundreds is
6 hundreds or 600.
T: Beneath your place value chart, as we did in
yesterday’s lesson, write an expression that shows the
total value expressed in the chart.
NOTES ON
S: (Write 2 × 3 hundreds + 2 × 2 tens + 2 × 4 ones.)
MULTIPLE MEANS
T: Write 2 times 324 vertically on your personal white OF REPRESENTATION:
board. Record the partial products for the ones, tens,
Clarify math language such as
and hundreds.
expression, value, vertically, partial
T: What is the value of the disks represented on the products, equation, and sum for English
chart? language learners. Offer explanations
S: 648. in students’ first language if possible.
Link vocabulary to words they may be
T: Add the values that you wrote in the problem. What is more familiar with, for example, sum
their sum? has a similar meaning to total. Make
S: 648. It’s another way to represent the answer! sure to distinguish some from sum.
T: Work with a partner to solve 3 × 231.
Monitor and provide assistance as students work in pairs to solve.
Lesson 8: Extend the use of place value disks to represent three- and four-digit
by one-digit multiplication. 118
Problem 3: Solve 3 × 851 using a partial products drawing on the place value chart.
T: Write the problem 3 × 851 vertically. This time, rather than recording 3 groups of 851 to begin, let’s
record the partial products as we multiply each unit.
T: 3 times 1 one is…?
S: 3 ones.
T: Record that in your place value chart at the top of the ones place.
T: 3 times 5 tens?
S: 15 tens.
Lesson 8: Extend the use of place value disks to represent three- and four-digit
by one-digit multiplication. 119
T: Record that in your place value chart as 1 hundred 5 tens a bit lower than the ones so you can see
the separate partial product.
T: 3 times 8 hundreds?
S: 24 hundreds.
T: Record that in your place value chart as…?
S: 2 thousands 4 hundreds.
T: Where?
S: A bit lower than the 1 hundred 5 tens.
T: Just as we record the partial products
numerically, we draw them. This does not
show the connection to addition well, but
it does show the partial products well.
Can you see the three partial products?
S: Yes.
T: Just looking at the place value chart for now, what are the products from least to greatest in unit
form?
S: 3 ones, 1 hundred 5 tens, and 2 thousands 4 hundreds.
T: What is the total product recorded both in your vertical problem and in your place value chart?
S: 2,553.
Repeat with 3 × 763.
Problem 4: Solve 4 × 6,379 using a partial products drawing on the place value chart.
T: Write the equation 4 × 6,379. Let’s record the partial products as we multiply each unit.
T: 4 times 9 ones is…?
S: 36 ones or 3 tens 6 ones.
T: Record that in your place value chart
at the top.
T: 4 times 7 tens?
S: 28 tens.
T: Record that in your place value chart
as 2 hundreds 8 tens a bit lower than
the 3 tens 6 ones so you can see the
separate partial product.
T: 4 times 3 hundreds?
S: 12 hundreds.
T: Record that in your place value chart as…?
S: 1 thousand 2 hundreds.
T: Where?
S: A bit lower than the 2 hundreds 8 tens.
Lesson 8: Extend the use of place value disks to represent three- and four-digit
by one-digit multiplication. 120
T: 4 times 6 thousands?
S: 24 thousands. 2 ten thousands 4 thousands.
T: Where?
S: A bit lower than the 1 thousand 2 hundreds.
T: Can you see the four partial products?
S: Yes.
T: Find the total of the partial products both in your problem and in your place value chart. Notice that
you will need to regroup when you find the total of the partial products. What is the total?
S: 25,516.
T: Work with a partner to solve 3 × 2,567.
Give students time to work through the problem and provide guidance as needed.
Lesson 8: Extend the use of place value disks to represent three- and four-digit
by one-digit multiplication. 121
Lesson 8: Extend the use of place value disks to represent three- and four-digit
by one-digit multiplication. 122
Name Date
1. Represent the following expressions with disks, regrouping as necessary, writing a matching expression,
and recording the partial products vertically as shown below.
a. 1 × 213
b. 2 × 213
c. 3 × 214
Lesson 8: Extend the use of place value disks to represent three- and four-digit
by one-digit multiplication. 123
2. Represent the following expressions with disks, using either method shown during class, regrouping as
necessary. To the right, record the partial products vertically.
a. 3 × 212
b. 2 × 4,036
Lesson 8: Extend the use of place value disks to represent three- and four-digit
by one-digit multiplication. 124
c. 3 × 2,546
d. 3 × 1,407
3. Every day at the bagel factory, Cyndi makes 5 different kinds of bagels. If she makes 144 of each kind,
what is the total number of bagels that she makes?
Lesson 8: Extend the use of place value disks to represent three- and four-digit
by one-digit multiplication. 125
Name Date
Represent the following expressions with disks, regrouping as necessary. To the right, record the partial
products vertically.
1. 4 × 513
2. 3 × 1,054
Lesson 8: Extend the use of place value disks to represent three- and four-digit
by one-digit multiplication. 126
Name Date
1. Represent the following expressions with disks, regrouping as necessary, writing a matching expression,
and recording the partial products vertically as shown below.
a. 2 × 424
b. 3 × 424
c. 4 × 1,424
Lesson 8: Extend the use of place value disks to represent three- and four-digit
by one-digit multiplication. 127
2. Represent the following expressions with disks, using either method shown in class, regrouping as
necessary. To the right, record the partial products vertically.
a. 2 × 617
b. 5 × 642
c. 3 × 3,034
Lesson 8: Extend the use of place value disks to represent three- and four-digit
by one-digit multiplication. 128
3. Every day, Penelope jogs three laps around the playground to keep in shape. The playground is
rectangular with a width of 163 m and a length of 320 m.
a. Find the total amount of meters in one lap.
Lesson 8: Extend the use of place value disks to represent three- and four-digit
by one-digit multiplication. 129
Lesson 9
Objective: Multiply three- and four-digit numbers by one-digit numbers
applying the standard algorithm.
Note: Reviewing standard form versus expanded form prepares students to decompose multi-digit
multiplication sentences into a series of multiplication sentences.
Repeat the process from Lesson 8 for the following possible sequence: 300 + 40 + 3; 4,000 + 600 + 70 + 9;
500 + 8 + 20; 275; 4,638; and 9,705.
Note: This fluency activity reviews Lesson 8’s Concept Development. Repeat the process from Lesson 8,
expanding to three- and four-digit numbers, for the following possible sequence: 1 × 312, 2 × 312, 3 × 312,
2 × 2,154, 4 × 212, and 3 × 1,504.
Problem 1: Represent and solve 6 × 162 in the place value chart. Relate the process to solving using the
standard algorithm.
T: Represent 6 × 162 on your place value chart using
the repeated addition way. Work with a partner
to solve. Was it necessary to regroup?
S: Yes. On my place value chart, I had 6 hundreds,
36 tens, and 12 ones. I regrouped 10 ones for
1 ten and 30 tens for 3 hundreds. My answer is
9 hundreds, 7 tens, and 2 ones.
T: Write the expression 6 × 162 again vertically on
your personal white boards. Let’s find a faster
way to express your answer. Use the place value
chart to help.
T: Tell me what happened in the ones column of your place value chart.
S: I multiplied 6 times 2 ones to get 12 ones. We regrouped 10 ones for 1 ten and were left with
2 ones.
T: Record the number of regrouped tens on the line under the tens column. Record the number of
ones in the ones place.
T: Tell me what happened in the tens column of your place value chart.
S: I multiplied 6 times 6 tens and got 36 tens. We exchanged 30 tens for 3 hundreds and were left with
6 tens. But, we have the 1 ten regrouped from the ones, so 36 tens plus the 1 ten makes 37 tens.
So, we have 3 hundreds and 7 tens after we regroup.
T: Record the number of hundreds on the line in the hundreds column. Record the number of tens in
the tens place. What about the 1 that was written on the line in the tens place, do I need it
anymore?
S: No, we counted it already.
T: Right, so if we’re done with it, let’s get rid of it. Cross it out.
T: Now, let’s look at the hundreds. What was the value of the hundreds?
S: We had 6 times 1 hundred equals 6 hundreds. 6 hundreds plus the 3 hundreds we regrouped equals
9 hundreds.
T: Since there’s no need to regroup, write the number of hundreds in the hundreds place. Have we
already counted the 3 hundreds we regrouped?
S: Yes!
T: Cross it out. What’s the product?
S: 972. That’s the same number we got with the place value chart!
Problem 2: Solve 5 × 237 using the partial products algorithm. Then solve using the standard algorithm, and
relate the two methods to each other.
T: Write the expression 5 × 237 vertically on your board. Draw and solve
using partial products.
Students work individually or in pairs to draw and solve using partial products.
T: Now, let’s solve using the standard algorithm. Starting in the ones
column, what do we do?
S: We multiply 5 times 7 ones and get 35 ones.
T: Tell your partner how you record 35 ones as partial products.
S: 35 ones is 3 tens 5 ones, so we record 3 tens in the tens column and 5 ones in the ones column on
MP.2 the same line.
T: Let’s record 3 tens 5 ones using the standard algorithm. (Record 3 tens on the line and 5 ones in the
ones column.) Tell your partner what you notice about this recording.
S: The 3 tens is on the line in the tens like in addition and the 5 ones is in the ones place, so it still
shows 35 ones. We add partial products together, so the 3 tens on the line means it will get
added to the product.
T: Working in the tens column, what do we do next?
S: We multiply 5 times 3 tens and get 15 tens.
T: 15 tens was recorded on the second line in the partial products method. For the standard algorithm,
add 3 tens to 15 tens.
S: 18 tens.
T: Say 18 tens as hundreds and tens.
S: 1 hundred 8 tens.
T: Record 1 hundred on the line in the hundreds column and 8 tens in the tens column. Cross out 3
tens because it was added.
T: What do we do next?
S: Next, we multiply 5 times 2 hundreds and get 10 hundreds. But, in the standard algorithm we
have to add the 1 hundred that is on the line to make 11 hundreds.
T: Remember to cross off the 1 since we have already included it in our answer. Because there are no
more numbers to multiply, we can just record 11 hundreds directly in the product, which is…?
S: 1,185.
T: Look back at the work that you did when you solved using partial products. What was the product?
S: 1,185. It’s the same thing. We came up with the same
product even though our methods were different.
T: What are the advantages to the standard algorithm? NOTES ON
S: We record our answer on one line. We are doing all MULTIPLE MEANS
of the calculations in a few steps. OF ENGAGEMENT:
Have students use and compare the
Repeat using 6 × 716.
two methods: partial products and the
standard algorithm. Encourage
Problem 3: Multiply three- and four-digit numbers by one-digit learners to analyze their proficiency
numbers applying the standard algorithm. and efficiency using each method.
Write or project the following: Guide students to ask, “What mistakes
do I make? When? Which method is
Shane measured 457 mL of water in a beaker. Olga measured easier for me? When? Why?”
3 times as much water. How much water did they measure
altogether?
T: Draw a tape diagram and discuss with a partner how you
would solve this problem.
S: We would multiply 4 times 457 to solve this problem. If Olga
measured 3 times as much water as Shane, we multiply by
four to find the total.
T: Solve using the standard algorithm. What do we multiply
first?
S: Four times 7 ones equals 28 ones.
T: Show me how to record 28 ones.
S: I write the 2 on the line under the tens place. I write the 8 in
the ones.
T: What do we multiply next?
S: Four times 5 tens is 20 tens plus 2 tens that were changed from the ones. I have 22 tens. I cross off
the 2 because I just included it in the total for the tens. I write 22 tens in my answer. The 2 is
written in the hundreds place on the line to show that we regrouped to the hundreds, and the 2 is
written in the tens.
T: What do we multiply next?
S: Four times 4 hundreds equals 16 hundreds. Then, I add 2 hundreds to get 18 hundreds. I cross off
the 2 because I included it in the total hundreds. Because there are no more numbers to multiply, I
record 18 hundreds directly in my product. The product is 1,828.
Name Date
a. b. c.
2 5 1 1 3 5 3 0 4
× 3 × 6 × 9
d. e. f.
4 0 5 3 1 6 3 9 2
× 4 × 5 × 6
6. One game system costs $238. How much will 4 game systems cost?
7. A small bag of chips weighs 48 grams. A large bag of chips weighs three times as
much as the small bag. How much will 7 large bags of chips weigh?
Name Date
a. b.
6 0 8 5 7 4
× 9 × 7
2. Morgan is 23 years old. Her grandfather is 4 times as old. How old is her grandfather?
Name Date
_× 2 × 2 × 4 × 4
a. b. c.
2 3 2 1 4 2 3 1 4
× 4 × 6 × 7
d. e. f.
4 4 0 5 0 7 3 8 4
× 3 × 8 × 9
4. Isabel earned 350 points while she was playing Blasting Robot. Isabel’s mom earned 3 times as many
points as Isabel. How many points did Isabel’s mom earn?
5. To get enough money to go on a field trip, every student in a club has to raise $53 by selling chocolate
bars. There are 9 students in the club. How much money does the club need to raise to go on the field
trip?
6. Mr. Meyers wants to order 4 tablets for his classroom. Each tablet costs $329. How much will all four
tablets cost?
7. Amaya read 64 pages last week. Amaya’s older brother, Rogelio, read twice as many pages in the same
amount of time. Their big sister, Elianna, is in high school and read 4 times as many pages as Rogelio did.
How many pages did Elianna read last week?
Lesson 10
Objective: Multiply three- and four-digit numbers by one-digit numbers
applying the standard algorithm.
Note: This activity serves as a review of the Concept Development in Lessons 7 and 8.
T: (Write 322 × 7.) Say the multiplication expression.
S: 322 × 7.
T: Say it as a three-product addition expression in unit form.
S: (3 hundreds × 7) + (2 tens x 7) + (2 ones × 7).
T: Write 322 × 7 vertically, and solve using the partial product strategy.
Repeat the process for the following possible sequence: 7 thousands 1 hundred 3 tens 5 ones × 5 and
3 × 7,413.
The principal wants to buy 8 pencils for every student at her school. If there are 859 students, how many
pencils does the principal need to buy?
Note: This problem is a review of Lesson 9. Students may solve using the algorithm or partial products. Both
are place value strategies.
Problem 1: Solve 5 × 2,374 using partial products, and then connect to the algorithm.
Display 5 × 2,374 vertically on the board.
T: With your partner, solve 5 × 2,374 using the partial products method.
Allow two minutes to solve.
T: Now, let’s solve using the algorithm. Say a multiplication sentence for the ones column.
Problem 3: Solve a word problem that requires four-digit by one-digit multiplication using the algorithm.
There are 5,280 feet in a mile. If Bryan ran 4 miles, how many feet did he run?
T: Discuss with your partner how you would solve this problem.
T: On your own, use the algorithm to solve for how many feet Bryan ran.
S: 5,280 × 4 is 21,120. Bryan ran 21,120 feet.
Name Date
c. 6 × 431 d. 3 × 431
e. 3 × 6,212 f. 3 × 3,106
g. 4 × 4,309 h. 4 × 8,618
2. There are 365 days in a common year. How many days are in 3 common years?
3. The length of one side of a square city block is 462 meters. What is the perimeter of the block?
4. Jake ran 2 miles. Jesse ran 4 times as far. There are 5,280 feet in a mile. How many feet did Jesse run?
Name Date
2. A farmer planted 4 rows of sunflowers. There were 1,205 plants in each row. How many sunflowers did
he plant?
Name Date
a. 3 × 41 b. 9 × 41
c. 7 × 143 d. 7 × 286
e. 4 × 2,048 f. 4 × 4,096
g. 8 × 4,096 h. 4 × 8,192
2. Robert’s family brings six gallons of water for the players on the football team. If one gallon of water
contains 128 fluid ounces, how many fluid ounces are in six gallons?
3. It takes 687 Earth days for the planet Mars to revolve around the sun once. How many Earth days does it
take Mars to revolve around the sun four times?
4. Tammy buys a 4-gigabyte memory card for her camera. Dijonea buys a memory card with twice as much
storage as Tammy’s. One gigabyte is 1,024 megabytes. How many megabytes of storage does Dijonea
have on her memory card?
Lesson 11
Objective: Connect the area model and the partial products method to the
standard algorithm.
Note: This fluency activity reviews the Concept Development in Lessons 7─10.
Lesson 11: Connect the area model and the partial products method to the
standard algorithm. 153
Write an equation for the area of each rectangle. Then, find the sum of the two areas.
Extension: Find a faster method for finding the area of the combined rectangles.
NOTES ON
MULTIPLE MEANS
OF ACTION AND
EXPRESSION:
Note: This problem is designed to bridge learning from Topic A, Scaffold student use of the area model
in which students solved for the area, to this lesson, where they to solve with the following options:
learn to model multiplication problems using the area model. Provide a blank area model
The placement of the small rectangle to the right of the larger template for students to slip into
rectangle is intentional for showing the tens and ones of the their personal white boards.
area model. It is recommended that this problem be presented Review expanded form with place
value cards or place value disks.
immediately prior to the Concept Development.
Simplify the multiplication. For
example, use 4 as a factor rather
Concept Development (31 minutes) than 8.
Problem 1: Multiply a three-digit number by a one-digit number using the area model.
T: Draw a rectangle with a width of 8 and a length of 200.
S: (Draw.)
T: Tell your neighbor how to find the area.
S: Multiply 8 times 200. That equals 1,600.
T: Write the area inside your rectangle.
T: Think back to the Application Problem (above). We had
two rectangles also with the width of 8. Let’s combine all
three rectangles: this one and the two from the
Application Problem. (Draw them.) With
your partner, discuss how to find the area of all three
rectangles put together.
Lesson 11: Connect the area model and the partial products method to the
standard algorithm. 154
S: In the Application Problem, I multiplied 8 times 4 and 8 times 30. So, then I can also multiply 8 times
200 and add all the sums together.
T: Record that as one continuous addition problem with your partner.
Guide students to record (8 × 200) + (8 × 30) + (8 × 4).
T: You are saying to multiply each section of the lengths by 8? (Record 8 (200 + 30 + 4) on the board.)
S: Yes.
T: Solve to find the area of the entire rectangle. Let’s begin with the largest rectangle.
T: 8 times 200?
S: 1,600. (Record 1,600 as a partial product in the area model and in the
written method.)
T: 8 times 30?
S: 240. (Record 240 as a partial product in the area model and in the written
method.)
T: Show your partner where to record 8 times 4. Tell your partner the
multiplication sentence represented by the area model.
S: 8 times 234 equals 1,872.
T: Compare the partial products to the rectangular area model.
S: The area inside each smaller rectangle is the same as
each of the partial products.
T: We recorded the partial products starting with the NOTES ON
largest unit, the hundreds. Does the order of partial MULTIPLE MEANS
products change the final product? Work with your OF ACTION AND
partner to solve 8 times 234 using partial products, EXPRESSION:
beginning with the smallest unit, the ones. One advantage of the area model is its
S: The answer is the same. I can multiply in any order flexibility for learners. Students can
using partial products. The order of addends does represent their partial products as
not matter. That’s the commutative property of arrays of place value disks, in unit
addition. I can record partial products using the form, or standard form. Though not as
smallest or largest unit first. efficient as the standard algorithm, it
may be an effective scaffold for
T: Yes, the rectangle, or area model, is another way to students working below grade level.
represent the partial products in multiplication.
Problem 2: Multiply a three-digit number by a one-digit number, connecting the area model to the standard
algorithm.
Display 316 × 4.
T: How many hundreds, tens, and ones are in 316?
S: 3 hundreds 1 ten 6 ones.
T: Draw an area model with a length of 3 hundreds 1 ten 6 ones and a width of 4.
T: Tell your partner how to solve using the area model.
Lesson 11: Connect the area model and the partial products method to the
standard algorithm. 155
Problem 3: Solve a word problem using the standard algorithm, area model, or partial products strategy.
A cafeteria makes 4,408 lunches each day.
How many lunches are made Monday
through Friday?
T: Discuss with your partner how to
solve this problem.
T: What are some methods you
could use to solve this?
S: An area model could help. I
like using the partial products
method. I think I can just use
the algorithm.
T: You could also use the distributive property to help break apart and solve. Choose your method and
solve.
Lesson 11: Connect the area model and the partial products method to the
standard algorithm. 156
S: 4,408 × 5 is 22,040. The cafeteria makes 22,040 lunches Monday through Friday.
When debriefing the solution, make note of how to draw an area model without a digit in the tens column.
Lesson 11: Connect the area model and the partial products method to the
standard algorithm. 157
Lesson 11: Connect the area model and the partial products method to the
standard algorithm. 158
Name Date
1. Solve the following expressions using the standard algorithm, the partial products method, and the area
model.
a. 4 2 5 × 4
4 (400 + 20 + 5)
b. 5 3 4 × 7
c. 2 0 9 × 8
__ ( ____ + ____ )
( __ × _____ ) + ( __ × _____ )
Lesson 11: Connect the area model and the partial products method to the
standard algorithm. 159
Cayla’s school has 258 students. Janet’s school has 3 times as many students as Cayla’s. How many
students are in Janet’s school?
Solve using the standard algorithm, the area model, the distributive property, or the partial products method.
4. 5,131 × 7
Lesson 11: Connect the area model and the partial products method to the
standard algorithm. 160
6. A restaurant sells 1,725 pounds of spaghetti and 925 pounds of linguini every month. After 9 months,
how many pounds of pasta does the restaurant sell?
Lesson 11: Connect the area model and the partial products method to the
standard algorithm. 161
Name Date
1. Solve using the standard algorithm, the area model, the distributive property, or the partial products
method.
2,809 × 4
2. The monthly school newspaper is 9 pages long. Mrs. Smith needs to print 675 copies. What will be the
total number of pages printed?
Lesson 11: Connect the area model and the partial products method to the
standard algorithm. 162
Name Date
1. Solve the following expressions using the standard algorithm, the partial products method, and the area
model.
a. 3 0 2 × 8
8 (300 + 2)
(8 × _____ ) + (8 × _____ )
b. 2 1 6 × 5
c. 5 9 3 × 9
Lesson 11: Connect the area model and the partial products method to the
standard algorithm. 163
On Monday, 475 people visited the museum. On Saturday, there were 4 times as many visitors as there
were on Monday. How many people visited the museum on Saturday?
Solve using the standard algorithm, the area model, the distributive property, or the partial products method.
4. 6,253 × 3
Lesson 11: Connect the area model and the partial products method to the
standard algorithm. 164
6. A cafeteria makes 2,516 pounds of white rice and 608 pounds of brown rice every month. After
6 months, how many pounds of rice does the cafeteria make?
Lesson 11: Connect the area model and the partial products method to the
standard algorithm. 165
4
GRADE
Mathematics Curriculum
GRADE 4 • MODULE 3
Topic D
Multiplication Word Problems
4.OA.1, 4.OA.2, 4.OA.3, 4.NBT.5
Topic D gives students the opportunity to apply their new multiplication skills (4.NBT.5). In Lesson 12,
students extend their work with multiplicative comparison from Topic A to solve real-world problems
(4.OA.2). As shown on the next page, students use a combination of addition, subtraction, and multiplication
to solve multi-step problems in Lesson 13 (4.OA.3).
Lesson 12
Objective: Solve two-step word problems, including multiplicative
comparison.
Note: For this lesson, the Problem Set comprises word NOTES ON
problems from the Concept Development and is therefore to be MULTIPLE MEANS
used during the lesson itself. OF ENGAGEMENT:
Students may work in pairs to solve Problems 1─4 below using Give everyone a fair chance to be
the RDW approach to problem solving. successful by providing appropriate
scaffolds. Demonstrating students may
1. Model the problem. use translators, interpreters, or
sentence frames to present and
Have two pairs of students who can be successful with modeling respond to feedback. Models shared
the problem work at the board while the others work may include concrete manipulatives.
independently or in pairs at their seats. Review the following If the pace of the lesson is a
questions before beginning the first problem. consideration, prepare presenters
beforehand. The first problem may be
Can you draw something? most approachable for students
What can you draw? working below grade level.
What conclusions can you make from your drawing?
As students work, circulate. Reiterate the questions above.
After two minutes, have the two pairs of students share only their labeled diagrams.
For about one minute, have the demonstrating students receive and respond to feedback and questions from
their peers.
Problem 1
The table shows the cost of party favors. Each party guest receives a bag with 1 balloon, 1 lollipop, and
1 bracelet. What is the total cost for 9 guests?
Item Cost
1 balloon 26¢
1 lollipop 14¢
1 bracelet 33¢
This two-step problem requires students to determine the cost of party favors for one guest and then use
that information to determine the total cost of party favors for 9 guests. Although RDW is reviewed prior to
beginning work on this problem, because of its simplicity, many students might elect to begin solving
immediately. Some students may choose to multiply each item by 9 before adding those amounts. Based on
their prior experience with money, some students may represent the total amount of 657 cents as $6.57, but
they are not required to do so.
Problem 2
The Turner family uses 548 liters of water per day. The Hill family uses 3 times as much water per day. How
much water does the Hill family use per week?
In solving this problem, students use information from the problem and their knowledge of language
denoting multiplicative comparison to determine their answer. They must also remember that there are 7
days in a week in order to complete the computation necessary to finish the problem. Models chosen for this
problem may include tape diagrams as shown.
Problem 3
Jayden has 347 marbles. Elvis has 4 times as many as Jayden. Presley has 799 fewer than Elvis. How many
marbles does Presley have?
This two-step problem affords students another opportunity to model with tape diagrams. They are required
to apply what they have learned about multiplying multi-digit numbers by single digits, as well as practice
their subtraction with multiple regrouping skills from Module 1. Encourage students to also practice mental
math, such as when subtracting 799 from 1,388. As illustrated above, note that the diagram may or may not
accurately show the relationship between 799 and the unit size, 347. Nevertheless, discuss how one might
use mental math to estimate how long Presley’s bar should be.
Problem 4
a. Write an equation that would allow someone to find NOTES ON
the value of R. MULTIPLE MEANS
OF ACTION AND
EXPRESSION:
1,167 1,167 1,167 Support English language learners as
they write their own word problems.
239 Provide sentence starters and a word
bank.
Sentence starters may include:
R
“I had (units),”
“How many in all?”
Possible words for the word bank may
include:
Student equations may include one or both of the equations times as many fewer than
above. They must include the use of the R for the unknown more than total difference
quantity and show that R is equal to 239 less than three times
1,167.
b. Write your own word problem to correspond to the tape diagram, and then solve.
Responses will vary. Guide students with a context for creating a problem, such as the number of students
who attend two schools or the weights of objects.
Problem Set
Please note that the Problem Set for Lesson 12 comprises
the problems from the Concept Development, as stated in
the introduction of the lesson.
Any combination of the questions below may be used to lead the discussion.
How was Problem 1 similar to the other problems
we did today? How was it different?
How was setting up Problem 2 similar to setting
up Problem 3? At what point did the two
problems become quite different?
What piece of information did you need to know
to solve Problem 2 that was not given to you in
the problem?
Share the word problem you created for Problem
4(b) with your partner. Solve your partner’s
problem. Explain the strategy you used to solve
it.
Name Date
2. The Turner family uses 548 liters of water per day. The Hill family uses 3 times as much water per day.
How much water does the Hill family use per week?
3. Jayden has 347 marbles. Elvis has 4 times as many as Jayden. Presley has 799 fewer than Elvis. How
many marbles does Presley have?
239
b. Write your own word problem to correspond to the tape diagram, and then solve.
Name Date
Jennifer has 256 beads. Stella has 3 times as many beads as Jennifer. Tiah has 104 more beads than Stella.
How many beads does Tiah have?
Name Date
2. The small copier makes 437 copies each day. The large copier makes 4 times as many copies each day.
How many copies does the large copier make each week?
3. Jared sold 194 Boy Scout chocolate bars. Matthew sold three times as many as Jared. Gary sold 297
fewer than Matthew. How many bars did Gary sell?
973 meters
M
b. Write your own word problem to correspond to the tape diagram, and then solve.
Lesson 13
Objective: Use multiplication, addition, or subtraction to solve multi-step
word problems.
Note: This fluency activity reviews the Concept Development from Lessons 10 and 11.
T: (Write 773 × 2.) On your personal white board, solve the expression using the standard algorithm.
Repeat the process for the following possible sequence: 147 × 3, 1,605 × 3, and 5,741 × 5.
This multi-step problem requires students to apply their knowledge of multiplication of a multi-digit number by
a single-digit number. While most students may apply the multiplication algorithm, they should be encouraged
to use whichever strategy they are most comfortable with to complete the multiplication. The sum of $375 and
$137 may be found before subtracting it from Kate’s total salary, or the two amounts may be subtracted
separately.
Problem 2
Sylvia weighed 8 pounds when she was born. By her first
birthday, her weight had tripled. By her second birthday, she NOTES ON
had gained 12 more pounds. At that time, Sylvia’s father MULTIPLE MEANS
weighed 5 times as much as she did. What was Sylvia and her OF ACTION AND
dad’s combined weight? EXPRESSION:
Learners differ in their solution
strategies, and classroom discussion is
enriched with the sharing of diverse,
innovative, efficient, and thoughtful
solutions.
Students may choose to omit the
modeling part of a multi-step problem.
For example, the work to the left does
not show the tripling of 8. Therefore,
the sharing of student work when
solving multi-step problems can be
even more interesting.
It is best to be prepared to model each
In this problem, students need to compute Sylvia’s weight at step of the problem since students may
two separate points in time. Some students may gravitate to be overwhelmed by the simplest words
calculations. Others may use tape diagrams or other models to when they are embedded within a
represent the problem. Either is acceptable. Then, they may multi-step problem.
multiply Sylvia’s current weight by 6 to find her and her father’s
combined weight.
Problem 3
Three boxes weighing 128 pounds each and one box weighing 254 pounds were loaded onto the back of an
empty truck. A crate of apples was then loaded onto the same truck. If the total weight loaded onto the
truck was 2,000 pounds, how much did the crate of apples weigh?
This multi-step problem may be modeled or simply solved using algorithms. Students need to recognize that
128 must be tripled before that total is added to 254. To arrive at the answer to the problem, this new sum
must be subtracted from 2,000, requiring students to use a simplifying strategy or to regroup across multiple
zeros (a skill they mastered in Module 1).
Problem 4
In one month, Charlie read 814 pages. In the same month, his mom read 4 times as many pages as Charlie,
and that was 143 pages more than Charlie’s dad read. What was the total number of pages read by Charlie
and his parents?
Solution A:
Solution B:
In this multi-step problem, students may find that each calculation is dependent upon the following
calculation. Encourage students to use simplifying strategies when solving, such as seeing in the model that
there are 9 equal-size rectangles worth 814 pages, minus 143 pages.
Problem Set
Please note that the Problem Set for Lesson 13 comprises
the problems from the Concept Development, as stated in
the introduction.
Any combination of the questions below may be used to lead the discussion.
Explain to your partner how you solved Problem
1. If you used different strategies, discuss how
you arrived at the same answer.
Let’s look at how two different students modeled
Problem 2. How are they similar? How are they
different?
Student A, in Problem 4, why did you multiply
814 by 9 and subtract 143? From the model,
I only see 5 units of 814. (Also, draw out the
alternate strategies from Problem 3.)
Student B, would you present your solution?
(Student presents.) Does anyone have comments
or questions for Student B?
How did you know what to do when you saw the
word tripled in Problem 2?
When might it be better to use multiplication
rather than addition?
What are the advantages of knowing several
methods for solving a multiplication problem?
A
Number Correct: _______
Mental Multiplication
1. 1×4= 23. 21 × 3 =
2. 10 × 4 = 24. 121 × 3 =
3. 11 × 4 = 25. 42 × 2 =
5. 20 × 2 = 27. 242 × 2 =
6. 21 × 2 = 28. 342 × 2 =
8. 30 × 3 = 30. 3×3=
9. 32 × 3 = 31. 13 × 3 =
15. 43 × 3 = 37. 24 × 4 =
17. 70 × 2 = 39. 54 × 3 =
18. 74 × 2 = 40. 63 × 6 =
B
Number Correct: _______
Improvement: _______
Mental Multiplication
1. 1×6= 23. 21 × 4 =
2. 10 × 6 = 24. 121 × 4 =
3. 11 × 6 = 25. 24 × 2 =
5. 30 × 2 = 27. 224 × 2 =
6. 31 × 2 = 28. 324 × 2 =
8. 20 × 3 = 30. 3×2=
9. 23 × 3 = 31. 13 × 2 =
15. 34 × 4 = 37. 23 × 4 =
17. 90 × 2 = 39. 45 × 3 =
18. 94 × 2 = 40. 36 × 6 =
Name Date
1. Over the summer, Kate earned $180 each week for 7 weeks. Of that money, she spent $375 on a new
computer and $137 on new clothes. How much money did she have left?
2. Sylvia weighed 8 pounds when she was born. By her first birthday, her weight had tripled. By her second
birthday, she had gained 12 more pounds. At that time, Sylvia’s father weighed 5 times as much as she
did. What was Sylvia and her dad’s combined weight?
3. Three boxes weighing 128 pounds each and one box weighing 254 pounds were loaded onto the back of
an empty truck. A crate of apples was then loaded onto the same truck. If the total weight loaded onto
the truck was 2,000 pounds, how much did the crate of apples weigh?
4. In one month, Charlie read 814 pages. In the same month, his mom read 4 times as many pages as
Charlie, and that was 143 pages more than Charlie’s dad read. What was the total number of pages read
by Charlie and his parents?
Name Date
1. Michael earns $9 per hour. He works 28 hours each week. How much does he earn in 6 weeks?
2. David earns $8 per hour. He works 40 hours each week. How much does he earn in 6 weeks?
3. After 6 weeks, who earned more money? How much more money?
Name Date
1. A pair of jeans costs $89. A jean jacket costs twice as much. What is the total cost of a jean jacket and 4
pairs of jeans?
2. Sarah bought a shirt on sale for $35. The original price of the shirt was 3 times that amount. Sarah also
bought a pair of shoes on sale for $28. The original price of the shoes was 5 times that amount.
Together, how much money did the shirt and shoes cost before they went on sale?
3. All 3,000 seats in a theater are being replaced. So far, 5 sections of 136 seats and a sixth section
containing 348 seats have been replaced. How many more seats do they still need to replace?
4. Computer Depot sold 762 reams of paper. Paper Palace sold 3 times as much paper as Computer Depot
and 143 reams more than Office Supply Central. How many reams of paper were sold by all three stores
combined?
Name Date
1. Draw an area model to solve the following. Find the value of the following expressions.
a. 30 × 60 b. 3 × 269
a. 3 × 68 b. 4 × 371
c. 7 × 1,305 d. 6,034 × 5
Solve using a model or equation. Show your work and write your answer as a statement.
3. A movie theater has two rooms. Room A has 9 rows of seats with 18 seats in each row. Room B has
three times as many seats as Room A. How many seats are there in both rooms?
4. The high school art teacher has 9 cases of crayons with 52 boxes in each case. The elementary school art
teacher has 6 cases of crayons with 104 boxes in each case. How many total boxes of crayons do both
teachers have? Is your answer reasonable? Explain.
5. Last year, Mr. Petersen’s rectangular garden had a width of 5 meters and an area of 20 square meters.
This year, he wants to make the garden three times as long and two times as wide.
a. Solve for the length of last year’s garden using the area formula. Then, draw and label the
measurements of this year’s garden.
20
5m square
meters
_____ m
b. How much area for planting will Mr. Petersen have in the new garden?
c. Last year, Mr. Petersen had a fence all the way around his garden. He can reuse all of the fence he
had around the garden last year, but he needs to buy more fencing to go around this year’s garden.
How many more meters of fencing is needed for this year’s garden than last year’s?
d. Last year, Mr. Petersen was able to plant 4 rows of carrots with 13 plants in each row. This year, he
plans to plant twice as many rows with twice as many carrot plants in each. How many carrot plants
will he plant this year? Write a multiplication equation to solve. Assess the reasonableness of your
answer.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply
two two-digit numbers, using strategies based on place value and the properties of
operations. Illustrate and explain the calculation by using equations, rectangular arrays,
and/or area models.
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller
unit.
4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical
problems. For example, find the width of a rectangular room given the area of the flooring
and the length, by viewing the area formula as a multiplication equation with an unknown
factor.
2 The student is unable The student correctly The student correctly The student correctly
to solve more than one solves at least two of solves at least three of answers all parts,
4.NBT.5 problem correctly. the four problems with the problems showing showing all work using
evidence of some place reasoning through a area models, partial
value knowledge. place value strategy, or products, or the
the student correctly general method:
answers four problems, a. 204
only showing solid
b. 1,484
reasoning for three
problems. c. 9,135
d. 30,170
3 The student answers The student attempts The student solves the The student correctly
incorrectly with little to use an equation or problem using an answers 648 seats in an
4.OA.1 attempt at solving the model, resulting in an equation or model but answer statement and
4.OA.2 problem. incorrect answer. with an incorrect uses an equation or
4.OA.3 answer, or the student model correctly to
4.NBT.5 answers correctly solve.
showing only some
reasoning.
4 The student answers The student answers The student correctly The student correctly
incorrectly and incorrectly but shows answers 1,092 boxes answers 1,092 boxes in
4.NBT.5 provides little or no some evidence in using a model or an answer statement,
4.OA.1 evidence of reasoning reasoning through equation accurately uses an area model or
4.OA.3 through estimation. estimation. but is unable to clearly equation to solve, and
reason using validates the
estimation, or the reasonableness of his
student provides clear answer through
reasoning and an estimation.
attempt at solving but
provides an incorrect
answer.
5 The student shows The student correctly The student answers The student correctly
little to no reasoning answers two of four three of the four parts answers:
4.NBT.5 and answers more than parts, showing little correctly, or the a. 5 m × 4 m = 20
4.OA.1 two parts incorrectly. reasoning in Part (d) student answers all square meters and
4.OA.2 and little evidence of four parts correctly expresses length
4.OA.3 place value with unclear reasoning as 4 m; draws a
understanding. in Part (d), or the
4.MD.3 rectangle; labels
student does not show the width as 10
solid evidence of place meters and length
value understanding in as 12 meters.
all solutions.
b. 120 square
meters.
c. 26 meters.
d. 208 plants; shows
a written equation
and reasons
correctly through
estimation.
Name Date
1. Draw an area model to solve the following. Find the value of the following expressions.
a. 30 × 60 b. 3 × 269
Solve using a model or equation. Show your work and write your answer as a statement.
3. A movie theater has two rooms. Room A has 9 rows of seats with 18 seats in each row. Room B has
three times as many seats as Room A. How many seats are there in both rooms?
4. The high school art teacher has 9 cases of crayons with 52 boxes in each case. The elementary school art
teacher has 6 cases of crayons with 104 boxes in each case. How many total boxes of crayons do both
teachers have? Is your answer reasonable? Explain.
5. Last year, Mr. Petersen’s rectangular garden had a width of 5 meters and an area of 20 square meters.
This year, he wants to make the garden three times as long and two times as wide.
a. Solve for the length of last year’s garden using the area formula. Then, draw and label the
measurements of this year’s garden.
b. How much area for planting will Mr. Petersen have in the new garden?
c. Last year, Mr. Petersen had a fence all the way around his garden. He can reuse all of the fence he
had around the garden last year, but he needs to buy more fencing to go around this year’s garden.
How many more meters of fencing is needed for this year’s garden than last year’s?
4
GRADE
Mathematics Curriculum
GRADE 4 • MODULE 3
Topic E
Division of Tens and Ones with
Successive Remainders
4.NBT.6, 4.OA.3
Focus Standard: 4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-
digit divisors, using strategies based on place value, the properties of operations,
and/or the relationship between multiplication and division. Illustrate and explain the
calculation by using equations, rectangular arrays, and/or area models.
Instructional Days: 8
Coherence -Links from: G3–M1 Properties of Multiplication and Division and Solving Problem with Units of 2–5 and 10
-Links to: G3–M3 Multiplication and Division with Units of 0, 1, 6–9, and Multiples of 10
G5–M2 Multi-Digit Whole Number and Decimal Fraction Operations
In Topic E, students synthesize their Grade 3 knowledge of division types (group size unknown and number of
groups unknown) with their new, deeper understanding of place value.
Students focus on interpreting the remainder within division problems both in word problems and long
division (4.OA.3). A remainder of 1, as exemplified below, represents a leftover flower in the first situation
and a remainder of 1 ten in the second situation. 1
1
Note that care must be taken in the interpretation of remainders. Consider the fact that 7 ÷ 3 is not equal to 5 ÷ 2 because the
1 1
remainder of 1 is in reference to a different whole amount (2 is not equal to 2 ).
3 2
While we have no reason to subdivide a remaining flower, there are good reasons to subdivide a remaining
ten. Students apply this simple idea to divide two-digit numbers unit by unit: dividing the tens units first,
finding the remainder (the number of tens unable to be divided), and decomposing remaining tens into ones
to then be divided.
Lesson 14 begins Topic E by having students solve division word problems involving remainders. In Lesson 15,
students deepen their understanding of division by solving problems with remainders using both arrays and
the area model. Students practice dividing two-digit dividends with a remainder in the ones place using place
value disks in Lesson 16 and continue that modeling in Lesson 17 where the remainder in the tens place is
decomposed into ones.
The long division algorithm 2 is introduced in Lesson 16 by directly relating the steps of the algorithm to the
steps involved when dividing using place value disks. Introducing the algorithm in this manner helps students
to understand how place value plays a role in the steps of the algorithm. The same process of relating the
standard algorithm to the concrete representation of division continues in Lesson 17.
Lesson 18 moves students to the abstract level by requiring them to solve division problems numerically
without drawing. In Lesson 19, students explain the successive remainders of the algorithm by using place
value understanding and place value disks. Finally, in Lessons 20 and 21, students use the area model to solve
division problems and then compare the standard algorithm to the area model (4.NBT.6). Lesson 20 focuses
on division problems without remainders, while Lesson 21 involves remainders.
Quotients and remainders are independent of each other but must both be included to give a complete
response. A quotient and a remainder cannot be recorded after an equal sign because the symbol R or the
words with a remainder of are invalid in an equation. Therefore, a quotient and a remainder can be written
as a statement such as seven divided by two is three with a remainder of one, or the quotient is three and the
remainder is one. It is mathematically correct to record the quotient and the remainder together at the top
of the long division algorithm.
2
Students become fluent with the standard division algorithm in Grade 6 (6.NS.2). For adequate practice in reaching fluency, students
are introduced to, but not assessed on, the division algorithm in Grade 4 as a general method for solving division problems.
A Teaching Sequence Toward Mastery of Division of Tens and Ones with Successive Remainders
Objective 1: Solve division word problems with remainders.
(Lesson 14)
Objective 2: Understand and solve division problems with a remainder using the array and area models.
(Lesson 15)
Objective 3: Understand and solve two-digit dividend division problems with a remainder in the ones
place by using place value disks.
(Lesson 16)
Objective 4: Represent and solve division problems requiring decomposing a remainder in the tens.
(Lesson 17)
Objective 7: Solve division problems without remainders using the area model.
(Lesson 20)
Objective 8: Solve division problems with remainders using the area model.
(Lesson 21)
Lesson 14
Objective: Solve division word problems with remainders.
Note: This fluency activity prepares students for Lesson 15’s Concept Development.
T: (Project a 3 × 4 array.) How many boxes do you see altogether?
S: 12.
T: Let’s count by threes to check. (Point at columns as students count.)
S: 3, 6, 9, 12.
T: Let’s count by fours to check. (Point at rows as students count.)
S: 4, 8, 12.
T: On your personal white board, write two multiplication sentences
to show how many boxes are in this array.
S: (Write 3 × 4 = 12 and 4 × 3 = 12.)
T: (Write 12 ÷ __ =__ . Write 12 ÷ __ =__.) Write two division sentences for this array.
S: (Write 12 ÷ 3 = 4 and 12 ÷ 4 = 3.)
Continue with the following possible sequence: 5 × 2 array and 7 × 3 array.
S: 12 ÷ 3.
T: Does the quotient tell us the size of the group or the number of groups?
S: The number of groups.
T: The same array can represent a situation with the group size unknown or the number of groups
unknown.
Problem 2: Divide a two-digit number by a one-digit number with a remainder modeled with an array.
13 ÷ 4
T: One more student joined the class described at the A NOTE ON
beginning of Problem 1. There are now 13 students to THE RECORDING
be divided into 4 equal teams. Draw an array to find OF QUOTIENTS AND
how many students are on each team. What did you REMAINDERS:
find? When writing 13 ÷ 4 = 3 R1, one may
conclude that since 7 ÷ 2 = 3 R1, the
S: I can represent 13 in four groups. Four groups of 3
following must be true: 7 ÷ 2 = 13 ÷ 4.
make 12, but I have 1 left over. One student won’t 1 1
However, this translates into 3 2 = 3 3,
be on a team.
which is a false number sentence. To
T: Tell me an expression to represent this problem. avoid this incorrect use of the equal
S: 13 ÷ 4. sign and the misconceptions it creates,
the remainder is stated separately from
T: When we divide a number into equal groups, the quotient, and the R notation
sometimes there is an amount leftover. We call the directly following the equal sign is not
number that we have left a remainder. used.
T: What is the quotient?
S: The quotient is 3.
T: What is the amount left over, the remainder?
S: 1.
T: We state our answer by saying the quotient and then the
remainder. The quotient is 3. The remainder is 1. We can also
say or write, “The quotient is 3 with a remainder of 1.”
T: Discuss with your partner how you can use multiplication to
check your work for this answer.
S: Four threes is 12. That doesn’t prove our answer is right.
We can add the remainder to the product. Four times 3 is 12.
Add 1 to get 13.
T: Let’s return again to a second story. There are 13 students in
PE class. Exactly 3 students are needed on each team. How
many teams can be made?
T: Tell me the new expression.
S: 13 ÷ 3.
T: State the quotient and remainder.
T: The quotient is 4, and the remainder is 1.
T: Talk to your partner. What do the quotient and the remainder mean in the second story?
S: Four teams can be made, and there is 1 extra person.
Draw the number bond as shown, and have students compare it with the quotient and the remainder. Notice
the part on the left represents the equal groups, and the part on the right is the remainder.
Problem 3: Divide a two-digit number by a one-digit number with a remainder modeled with a tape diagram.
Kristy bought 13 roses. If she puts 6 roses in each vase, how many vases will she use? Will there be any roses
left over?
T: Draw an array. Solve for 13 ÷ 6.
S: I can’t because 13 is an odd number, and 6 + 6 = 12. An even number plus an even number won’t
give you an odd number. You can divide by 6, but there will be 1 extra flower left over. I can
fill 2 vases and have 1 flower left over.
T: Tell your partner a statement that tells the quotient
and remainder for this problem.
S: The quotient is 2, and the remainder is 1.
T: Describe to your partner what that statement tells us.
S: We started with 13 and made groups of 6. We made 2
groups with 1 rose remaining. Kristy can fill 2 vases.
She will have 1 rose left over.
T: Again, let’s revise our story a bit. Now, Kristy bought
13 roses and wants to put them equally in 2 vases.
How many roses will be in each vase? Is this the same
array?
S: Yes.
T: Talk to your partner. How has our interpretation of
the array changed?
S: In the first story, we didn’t know the number of vases.
In the second story, we didn’t know the number in
each vase. We changed the story from finding the
number of groups to finding the size of the group.
T: How can we check our work for both situations?
S: We can draw a number bond to show 2 groups of 6, and then
1 more. Two times 6 is 12, and 12 plus 1 is 13.
T: Let’s turn our array into a tape diagram to show 13 in 2 groups
of 6 with a remainder of 1. (Demonstrate.)
T: Using the array, draw a rectangle around the flowers. Erase
the flowers, and label the diagram.
S: You should divide the bar into two parts. I know each part is
worth 6, but 6 plus 6 isn’t 13.
T: Our tape diagram must have a third part to represent the remainder. Let’s separate the bar into
two equal parts and make a very small third part. Shade to show the remaining flower.
(Demonstrate.)
T: With your partner, draw a tape diagram to show 13 roses divided equally into 4 vases.
Students draw a tape diagram, dividing it into four parts. Using their basic facts, they know 13 cannot be
divided into four equal parts. They shade a fifth part of the tape diagram to show the remainder.
S: The quotient is 3. The remainder is 1. We can
check our work by drawing a number bond and
adding the parts or multiplying 4 times 3 and
adding 1. Whatever method we use, we get back to
the original total when our quotient and remainder
are correct.
T: Look at your tape diagram. Is the model the same
when we don’t know the number of groups, when we
know that there are 3 flowers in each vase, but we
don’t know the number of vases?
S: Yes!
What complications are there in modeling a division problem with a remainder using a tape diagram?
What new math vocabulary did we use today to communicate precisely?
Name Date
1. There are 19 identical socks. How many pairs of socks are there? Will there be any socks without a
match? If so, how many?
2. If it takes 8 inches of ribbon to make a bow, how many bows can be made from 3 feet of ribbon
(1 foot = 12 inches)? Will any ribbon be left over? If so, how much?
3. The library has 27 chairs and 5 tables. If the same number of chairs is placed at each table, how many
chairs can be placed at each table? Will there be any extra chairs? If so, how many?
4. The baker has 42 kilograms of flour. She uses 8 kilograms each day. After how many days will she need
to buy more flour?
5. Caleb has 76 apples. He wants to bake as many pies as he can. If it takes 8 apples to make each pie, how
many apples will he use? How many apples will not be used?
6. Forty-five people are going to the beach. Seven people can ride in each van. How many vans will be
required to get everyone to the beach?
Name Date
Fifty-three students are going on a field trip. The students are divided into groups of 6 students. How many
groups of 6 students will there be? If the remaining students form a smaller group, and one chaperone is
assigned to every group, how many total chaperones are needed?
Name Date
1. Linda makes booklets using 2 sheets of paper. She has 17 sheets of paper. How many of these
booklets can she make? Will she have any extra paper? How many sheets?
2. Linda uses thread to sew the booklets together. She cuts 6 inches of thread for each booklet. How
many booklets can she stitch with 50 inches of thread? Will she have any unused thread after
stitching up the booklets? If so, how much?
3. Ms. Rochelle wants to put her 29 students into groups of 6. How many groups of 6 can she make?
If she puts any remaining students in a smaller group, how many students will be in that group?
4. A trainer gives his horse, Caballo, 7 gallons of water every day from a 57-gallon container. How many
days will Caballo receive his full portion of water from the container? On which number day will the
trainer need to refill the container of water?
5. Meliza has 43 toy soldiers. She lines them up in rows of 5 to fight imaginary zombies. How many of
these rows can she make? After making as many rows of 5 as she can, she puts the remaining soldiers
in the last row. How many soldiers are in that row?
6. Seventy-eight students are separated into groups of 8 for a field trip. How many groups are there?
The remaining students form a smaller group of how many students?
Lesson 15
Objective: Understand and solve division problems with a remainder using
the array and area models.
Note: This fluency activity prepares students for Lesson 16’s Concept Development.
T: (Project the place value chart with 2 tens disks and 4 ones disks.) On your personal white board,
write the number in standard form.
S: (Write 24.)
Repeat process for 5 tens and 3 ones, 4 tens and 1 one, 3 tens and 11 ones, and 3 tens and 17 ones.
T: (Write 32.) Say the number.
S: 32.
T: Show 32 using place value disks.
S: (Draw disks for 3 tens and 2 ones.)
Continue with the following possible sequence: 21 and 43.
Lesson 15: Understand and solve division problems with a remainder using the
array and area models. 218
Note: This fluency activity prepares students for this lesson’s Concept
Development.
T: (Project a 5 × 3 + 1 array.) How many boxes do you see altogether?
S: 16.
T: Let’s count by fives to check. (Point at columns as students count.)
S: 5, 10, 15.
T: Plus 1? (Point to the extra square outside of the rectangle.)
S: 16.
T: Count by threes to check.
S: 3, 6, 9, 12, 15.
T: Plus 1? (Point to the extra square outside of the rectangle.)
S: 16.
T: On your personal white board, write two multiplication number sentences to show how many boxes
are in this array.
S: (Write (5 × 3) + 1 = 16 and (3 × 5) + 1 = 16.)
T: Write two division sentences for this array.
S: (Write 16 ÷ 3 = 5 with a remainder of 1 and 16 ÷ 5 = 3 with a remainder of 1.)
Repeat using the following possible sequence: (3 × 6) + 1 and (3 × 4) + 2.
Lesson 15: Understand and solve division problems with a remainder using the
array and area models. 219
Problem 1: Solve a division problem with and without a remainder using the area model.
Display 10 ÷ 2.
T: Draw an array to represent 10 ÷ 2. Explain to your partner how
you solved.
S: (Draw.) I drew 2 circles and placed 10 dots evenly among the
circles. I drew 10 dots as 2 rows of 5 dots.
T: Let’s use grid paper to draw a rectangle with an area of 10 square
centimeters and one side length of 2 centimeters. Tell your
partner how we can find the unknown side length.
S: The area is 10, so we know it is 5. If the width is 2 centimeters,
that means the length is 5 centimeters, and 2 centimeters times
5 centimeters gives an area of 10 square centimeters. We can
count and mark off by twos until we get to 10.
T: Discuss with your partner how the length of 5 centimeters is represented in the area model.
S: The length is 5, and the quotient is 5. The length of the area model represents the quotient of
this division problem.
Display 11 ÷ 2.
T: With your partner, discuss how you would draw an area model
for 11 ÷ 2.
S: Two can be the length or the width. I can’t just draw 2 rows of
square units because of the remainder. If I mark off 2 squares
at a time, I count 2, 4, 6, 8, 10. I can’t do another group of 2
because it would be 12. There aren’t enough.
T: Eleven square centimeters is the total area. Let’s draw a rectangle
starting with a width of 2 centimeters. We’ll continue lengthening
it until we get as close to 11 square centimeters as we can.
S: A length of 5 centimeters and width of 2 centimeters is as close as we can get to 11 square
centimeters. We can’t do 2 × 6 because that’s 12 square centimeters, and the total area is 11
square centimeters.
T: We can show a total area of 11 square centimeters by modeling 1 more square centimeter. The
remainder of 1 represents 1 more square centimeter.
Repeat for 16 ÷ 3 and 23 ÷ 4.
Lesson 15: Understand and solve division problems with a remainder using the
array and area models. 220
Problem 2: Solve a division problem using an array and the area model.
Display 38 ÷ 4.
T: In the Application Problem, you drew an array (pictured to
the right) to solve. Represent the same problem using the
area model on grid paper. (Allow two minutes to work.)
T: What do you notice about the array compared to the
area model on graph paper?
S: The area model is faster to draw. Thirty-eight dots is a
lot to draw. There are the same number of dots
and squares when I used graph paper. Both get us
the same answer of a quotient 9 with a remainder of 2.
MP.4 T: Let’s represent 38 ÷ 4 even more efficiently without
grid paper since it’s hard to come by grid paper every
time you want to solve a problem.
T: (Give students one minute to draw.) Talk to your
partner about how the array model and grid paper
model supported you in drawing the rectangle with a
given structure.
S: I knew the length was a little more than twice the
width. I knew that the remainder was half a
column. I knew that there was a remainder. It was
really obvious with the array and grid paper.
Lesson 15: Understand and solve division problems with a remainder using the
array and area models. 221
Lesson 15: Understand and solve division problems with a remainder using the
array and area models. 222
Name Date
1. 18 ÷ 6
Quotient = _________
2. 19 ÷ 6
Quotient = _________
Lesson 15: Understand and solve division problems with a remainder using the
array and area models. 223
Solve using an array and an area model. The first one is done for you.
Example: 25 ÷ 2
12
a. b.
2
Quotient = 12 Remainder = 1
3. 29 ÷ 3
a. b.
4. 22 ÷ 5
a. b.
5. 43 ÷ 4
a. b.
6. 59 ÷ 7
a. b.
Lesson 15: Understand and solve division problems with a remainder using the
array and area models. 224
Name Date
1. 27 ÷ 5
a. b.
2. 32 ÷ 6
a. b.
Lesson 15: Understand and solve division problems with a remainder using the
array and area models. 225
Name Date
1. 24 ÷ 4
Quotient = _________
Can you show 24 ÷ 4 with one rectangle? ______
Remainder = _______
2. 25 ÷ 4
Quotient = _________
Can you show 25 ÷ 4 with one rectangle? ______
Remainder = _______ Explain how you showed the remainder:
Lesson 15: Understand and solve division problems with a remainder using the
array and area models. 226
Solve using an array and area model. The first one is done for you.
Example: 25 ÷ 3
8
a. b.
3
Quotient = 8 Remainder = 1
3. 44 ÷ 7
a. b.
4. 34 ÷ 6
a. b.
5. 37 ÷ 6
a. b.
6. 46 ÷ 8
a. b.
Lesson 15: Understand and solve division problems with a remainder using the
array and area models. 227
Lesson 16
Objective: Understand and solve two-digit dividend division problems with
a remainder in the ones place by using place value disks.
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 228
Materials: (T) Tens place value chart (Template) (S) Personal white board, tens place value chart (Template)
Problem 1
6 ones ÷ 3
3 tens 6 ones ÷ 3
Display 6 ÷ 3 on the board.
T: 6 ones represents what?
S: The whole. The total. What you are dividing.
T: Show 6 using place value disks. What is the number we are dividing by?
S: 3.
T: Let’s assume it’s telling us how many groups to make. Draw 3 groups
below. Can we distribute 6 ones into 3 groups? Think of it like
dealing cards evenly among 3 players. (Model as students follow
along.) First, put one in each group. Cross off the ones one at a time
as you distribute them evenly. Next, put another one in each group
if you are able. Continue this until all of the ones are distributed.
S: We can put 2 ones in each group.
T: Are there any ones left over?
S: No.
T: How many ones are in each of our 3 groups?
S: 2 ones.
T: What is 6 ones ÷ 3? Give me the number sentence.
S: 6 ones ÷ 3 equals 2 ones.
T: Let’s represent 6 ÷ 3 in a new way. Let’s record the whole and the divisor. (Record with long
division symbol as shown above.) Look back to your model. 6 ones divided by 3 is…?
S: 2 ones.
T: (Record 2 ones.)
T: (Point to the place value chart.) You distributed 2 ones 3 times. 2 ones times 3 is…?
S: 6 ones.
T: (Refer to the numbers carefully, pointing to 2 ones and the divisor, and recording 6 ones.)
T: (Point to the place value chart.) We divided 6 ones and have no ones remaining. 6 ones minus
6 ones equals 0 ones. (Write the subtraction line.) What does this zero mean?
S: There is no remainder. All the ones were divided with none left over. We subtracted the total
number distributed from the total number of ones.
T: We can see the 3 groups of 2 both in our model and in our numbers and know our answer is correct
since 3 times 2 equals 6.
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 229
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 230
Problem 2
5 ones ÷ 4
4 tens 5 ones ÷ 4
Display 5 ÷ 4 on the board.
T: With your partner, represent the whole and the divisor, 4, on the
place value chart, and record the written problem.
S: (Draw 5 ones and 4 groups below in the place value chart, and
record the written problem.)
T: 5 ones divided by 4 equals?
S: It doesn’t divide evenly. I can place 1 one in each group,
but I will have 1 one left over.
T: Distribute as many ones as you can, crossing off the ones you use.
What is the quotient for 5 ones divided by 4?
S: 1 one.
T: Record your quotient numerically. Say a multiplication sentence
for how many ones were distributed.
S: 1 one times 4 equals 4 ones.
T: Record 4 ones numerically and subtract.
S: 5 ones minus 4 ones is 1 one.
T: Record 1 one numerically. How many ones are remaining in the place value chart?
S: 1 one.
T: Circle 1 one. Tell your partner why 1 one is a remainder.
S: It is what is left over after we made our groups. Our groups must be equal. If we put this 1 one
into a group, the groups will not be equal.
T: Watch as I record the remainder numerically using R1.
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 231
Problem 3
8 ones ÷ 3
6 tens 8 ones ÷ 3
Display 8 ÷ 3 on the board.
T: Solve for 8 ÷ 3 using place value disks. Represent the problem
using long division with your partner.
Circulate. Listen for students using place value as they divide, multiply,
and subtract.
S: The quotient is 2 and the remainder is 2.
T: How do we use multiplication and addition to check our quotient and remainder in division?
S: Two times 3 is 6. Six plus 2 is 8. We multiply the quotient times the divisor and add the
remainder. We multiply the number in each group by the number of groups and then add the
remainder.
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 232
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 233
Any combination of the questions below may be used to lead the discussion.
How did solving Problem 1 prepare you for
solving Problem 2?
Explain to your partner why only 6 ones could be
distributed in Problem 3. What happens to the
remaining ones?
Solve 12 divided by 3. Solve 12 divided by 4. As a
divisor gets larger, what will happen to the
quotient if the whole stays the same?
Was the remainder ever larger than the divisor?
Why not?
In the Problem Set, we only had remainders of 1
and 2. Give me an example of a problem that
might have a larger remainder.
Explain the connection between using place value
disks and long division. Why do you think it is
called long division?
What new math vocabulary did we use today to
communicate precisely?
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 234
Name Date
Show the division using disks. Relate your work on the place value chart to long division. Check your quotient
and remainder by using multiplication and addition.
1. 7 ÷ 2
Check Your Work
2 7
3
× 2
quotient = __________
remainder = __________
2. 27 ÷ 2
2 27
quotient = __________
remainder = __________
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 235
3. 8 ÷ 3
Check Your Work
3 8
quotient = __________
remainder = __________
4. 38 ÷ 3
3 38
quotient = __________
remainder = __________
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 236
4 6
quotient = __________
remainder = __________
6. 86 ÷ 4
4 86
quotient = __________
remainder = __________
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 237
Name Date
Show the division using disks. Relate your work on the place value chart to long division. Check your quotient
and remainder by using multiplication and addition.
1. 5 ÷ 3
Check Your Work
3 5
quotient = __________
remainder = __________
2. 65 ÷ 3
3 65
quotient = __________
remainder = __________
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 238
Name Date
Show the division using disks. Relate your work on the place value chart to long division. Check your quotient
and remainder by using multiplication and addition.
quotient = __________ × 3
remainder = __________
2. 67 ÷ 3
3 67
quotient = _________
remainder = __________
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 239
2 5
quotient = __________
remainder = __________
4. 85 ÷ 2
2 85
quotient = __________
remainder = ________
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 240
5. 5 ÷ 4
Check Your Work
4 5
quotient = __________
remainder = __________
6. 85 ÷ 4
4 85
quotient = ________
remainder = __________
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 241
Lesson 16: Understand and solve two-digit dividend division problems with a
remainder in the ones place by using place value disks. 242
Lesson 17
Objective: Represent and solve division problems requiring decomposing a
remainder in the tens.
Audrey and her sister found 9 dimes and 8 pennies. If they share the money equally, how much money will
each sister get?
Note: This Application Problem reviews division of ones. Sharing 9 dimes connects to Problems 1 and 2 of
today’s Concept Development, asking students to decompose 1 ten for 10 ones.
Materials: (T) Tens place value chart (Lesson 16 Template) (S) Personal white board, tens place value chart
(Lesson 16 Template)
Problem 1: Divide two-digit numbers by one-digit numbers using place value disks, regrouping in the tens.
3 ones ÷ 2
3 tens ÷ 2
T: (Record 1 ten. Point to the place value chart.) You recorded 1 ten, twice. Say a multiplication
equation that tells that.
S: 1 ten times 2 equals 2 tens.
As students say the multiplication equation, refer to the problem, pointing to 1 ten and the divisor,
and record 2 tens.
T: (Point to the place value chart.) We started with 3 tens, distributed 2 tens, and have 1 ten
remaining. Tell me a subtraction equation for that.
S: 3 tens minus 2 tens equals 1 ten.
As students say the subtraction equation, refer to the problem, pointing to the tens column, drawing a
subtraction line, and recording 1 ten.
T: (Point to the place value chart.) How many ones remain to be divided?
S: 10 ones.
T: Yes. We changed 1 ten for 10 ones. Say a division equation for how you distributed 1 ten
or 10 ones.
S: 10 ones divided by 2 equals 5 ones.
As students say the division equation, refer to the problem, pointing to the 10 ones and the divisor, and
record 5 ones.
T: (Point to the place value chart.) You recorded 5 ones twice. Say a multiplication equation that tells
that.
S: 5 ones times 2 equals 10 ones.
As students say the multiplication equation, refer to the problem, pointing to 5 ones and the divisor, and
record 10 ones.
T: (Point to the place value chart.) We renamed 10 ones, distributed 10 ones, and have no ones
remaining. Say a subtraction equation for that.
S: 10 ones minus 10 ones equals 0 ones.
As students say the subtraction equation, refer to the problem, drawing a subtraction line, and record 0 ones.
Have students share with a partner how the model matches the steps of the algorithm. Note that both show
equal groups and how both can be used to check their work using multiplication.
Problem 2
4 ones ÷ 3
4 tens 2 ones ÷ 3
Problem 3
8 tens 4 ones ÷ 3
Lesson Objective: Represent and solve division problems Cross out to track the number
requiring decomposing a remainder in the tens. distributed.
Draw dots in arrays. The hands way
The Student Debrief is intended to invite reflection and active array may be helpful.
processing of the total lesson experience.
Circle the remainder.
Invite students to review their solutions for the Problem Set. • Try disks, dots, numbers, etc. Use
They should check work by comparing answers with a partner what is most efficient for you.
before going over answers as a class. Look for misconceptions
or misunderstandings that can be addressed in the Debrief.
Guide students in a conversation to debrief the Problem
Set and process the lesson.
Any combination of the questions below may be used to
lead the discussion.
How did Problem 2 allow you to see only the
remaining 1 ten in the ones column?
Explain why 1 ten remains in Problem 4.
How is the long division recording different in
today’s lesson compared to yesterday’s lesson?
What different words are we using to describe
what we do when we have a remaining ten or
tens? (Break apart, unbundle, change, rename,
decompose, regroup) Which of these words are
you most comfortable using yourself?
What other operation involves changing 1 ten for
10 ones at times? (Subtraction.) What
operations involve the opposite, changing 10
ones for 1 ten at times?
What would happen if we divided the ones before
the tens?
Name Date
Show the division using disks. Relate your model to long division. Check your quotient and remainder by
using multiplication and addition.
1. 5 ÷ 2
× 2
quotient = __________
remainder = __________
2. 50 ÷ 2
2 5 0
quotient = __________
remainder = __________
3. 7 ÷ 3
Check Your Work
3 7
quotient = __________
remainder = __________
4. 75 ÷ 3
3 7 5
quotient = _________
remainder = ________
5. 9 ÷ 4
Check Your Work
4 9
quotient = __________
remainder = __________
6. 92 ÷ 4
4 9 2
quotient = ______
remainder = _____
Name Date
Show the division using disks. Relate your model to long division. Check your quotient by using multiplication
and addition.
1. 5 ÷ 4
Check Your Work
4 5
quotient = __________
remainder = __________
2. 56 ÷ 4
4 5 6
quotient = ________
remainder = _______
Name Date
Show the division using disks. Relate your model to long division. Check your quotient and remainder by
using multiplication and addition.
1. 7 ÷ 2
Check Your Work
2 7
quotient = __________
remainder = __________
2. 73 ÷ 2
2 7 3
quotient = ________
remainder = _______
3. 6 ÷ 4
Check Your Work
4 6
quotient = __________
remainder = __________
4. 62 ÷ 4
4 6 2
quotient = _______
remainder = ______
5. 8 ÷ 3
Check Your Work
3 8
quotient = __________
remainder = __________
6. 84 ÷ 3
3 8 4
quotient = _______
remainder = ______
Lesson 18
Objective: Find whole number quotients and remainders.
Materials: (T) Tens place value chart (Lesson 16 Template) (S) Personal white board, tens place value chart
(Lesson 16 Template)
Problem 1: Divide a two-digit number by a one-digit divisor with a remainder in the tens place.
5 tens 7 ones ÷ 3
T: (Write 57 ÷ 3.) Let’s divide 57 into 3 equal
groups. Break 57 into tens and ones.
S: 5 tens 7 ones.
T: Let’s divide 5 tens first. Why?
S: When we divide, we always start with the
larger units. We divide the tens first
because we may have to change tens for
ones.
T: 5 tens divided by 3…?
S: (Record the steps of the algorithm.) 1 ten in
each group, with 2 tens remaining.
T: We’ve distributed 3 tens. Let’s write 3 in the tens NOTES ON
place. We also write that there are 2 tens remaining MULTIPLE MEANS
because 5 tens minus 3 tens is 2 tens. OF ACTION AND
T: How do we divide the remaining 2 tens? EXPRESSION:
Scaffold long division with the
S: We unbundle the 2 tens as 20 ones.
following options:
T: Yes. So, how many ones do we have altogether? Provide graph paper for easy
S: 27. alignment of tens and ones.
T: Yes, 20 ones plus 7 ones is 27 ones. Label the tens and ones places.
T: You know your threes facts. Get ready for some Write zeros as place holders.
mental math. What’s 27 ones divided by 3?
S: 9 ones!
T: 9 ones in each group is recorded above, in the ones place. Record the remaining steps. Read the
quotient.
Problem 2: Divide with a remainder in the tens and ones places using the
division algorithm.
8 tens 6 ones ÷ 5
T: (Write 86 ÷ 5.) You solved 57 divided by 3 by unbundling tens.
Let’s try a more challenging problem. How many groups will
we divide 86 into?
S: 5.
T: What is the first step?
S: Start with the tens. Divide 8 tens into 5
groups. That’s 1 ten in each group with 3 tens
remaining.
T: Show me on your personal white board using long
division, or the division algorithm, how you recorded
the distributed tens and the remaining tens.
T: What will you do with the 3 remaining tens?
S: Unbundle 3 tens as 30 ones.
T: How many ones altogether?
S: 36.
T: Next step?
S: Divide 36 ones into 5 groups. That’s 7 ones in each group, with 1 one remaining.
T: How did you record what you distributed? What remains? Check your neighbor’s work. Thumbs up
if you agree.
T: I see you’ve written 35 ones distributed under the 36 ones you had at first. Did you write R1? Read
your quotient. Read your remainder. What is 86 divided by 5?
S: 17 with a remainder of 1.
T: How could you prove your division is correct?
S: Multiply 17 by 5, and then add 1 more.
T: Work with your partner to check with multiplication.
Any combination of the questions below may be used to lead the discussion.
Compare the remainders to the divisors on the
Problem Set. What do you find is true? Which
always has a larger value? Why is that?
How did the zero effect your division in
Problem 9?
What did you notice about the divisor, the whole,
and quotients in Problems 9 and 10?
Can you predict whether or not there will be a
remainder? How?
The whole is the same in Problems 11 and 12.
Why is the quotient smaller in Problem 11?
Name Date
Solve using the standard algorithm. Check your quotient and remainder by using multiplication and addition.
1. 46 ÷ 2 2. 96 ÷ 3
3. 85 ÷ 5 4. 52 ÷ 4
5. 53 ÷ 3 6. 95 ÷ 4
7. 89 ÷ 6 8. 96 ÷ 6
9. 60 ÷ 3 10. 60 ÷ 4
11. 95 ÷ 8 12. 95 ÷ 7
Name Date
Solve using the standard algorithm. Check your quotient and remainder by using multiplication and addition.
1. 93 ÷ 7 2. 99 ÷ 8
Name Date
Solve using the standard algorithm. Check your quotient and remainder by using multiplication and addition.
1. 84 ÷ 2 2. 84 ÷ 4
3. 48 ÷ 3 4. 80 ÷ 5
5. 79 ÷ 5 6. 91 ÷ 4
7. 91 ÷ 6 8. 91 ÷ 7
9. 87 ÷ 3 10. 87 ÷ 6
11. 94 ÷ 8 12. 94 ÷ 6
Lesson 19
Objective: Explain remainders by using place value understanding and
models.
Note: This Sprint reviews content from previous lessons and reinforces place value used in the division
algorithm.
Lesson 19: Explain remainders by using place value understanding and models.
267
Materials: (T) Tens place value chart (Lesson 16 Template) (S) Personal white board, tens place value chart
(Lesson 16 Template)
Problem 1: Model division with remainders in the tens and ones places using place value disks.
41 ÷ 3
T: (Write 41 ÷ 3.) What disks will you draw to represent 41?
S: 4 tens 1 one.
T: How many equal groups will we divide 41 into?
S: 3.
T: Draw 3 groups, and let’s share 4 tens equally. How many
tens in each group? Draw place value disks as you
distribute 4 tens into 3 groups like you’re dealing cards to
3 players.
S: 1 ten in each group, with 1 ten remaining.
T: How can we divide the remaining ten?
S: Unbundle 1 ten as 10 ones.
T: Let’s see you draw that. (Allow students time to
draw.) What did you do?
S: I drew an arrow from the remaining tens disk in the
tens place and drew 10 ones in the ones place.
T: How many ones do you have now?
S: 11 ones. NOTES ON
T: Let’s divide those 11 ones equally into 3 groups. Divide MULTIPLE MEANS
11 ones into 3 groups by distributing 1 to each group. OF ACTION AND
How many ones are remaining? EXPRESSION:
S: 8. Some learners may need less guidance
T: Are there enough to distribute again? to model 41 ÷ 3 and, after solving
quickly and independently, may benefit
S: Yes. We can distribute another one to each group. more from writing a step-by-step script
T: How many are left now? for solving 41 ÷ 3 in preparation for
S: Five. We can distribute again. We will have 2 Problem 5 of the Problem Set. This
script might be used in a video of the
remaining.
student supporting his peers as they
T: Explain what happened. learn long division.
S: 2 ones are left after distributing the rest equally. We
had to keep distributing until we didn’t have enough to
distribute evenly again.
T: Now, your place value disks clearly show the solution for 41 ÷ 3. Tell me the quotient. Tell me the
remainder.
Lesson 19: Explain remainders by using place value understanding and models.
268
Lesson 19: Explain remainders by using place value understanding and models.
269
Lesson Objective: Explain remainders by using place value cross out distribute share draw
understanding and models. tens ones four five
three unbundle divide equal
The Student Debrief is intended to invite reflection and active
processing of the total lesson experience. fairly next then last
Lesson 19: Explain remainders by using place value understanding and models.
270
Lesson 19: Explain remainders by using place value understanding and models.
271
A
Number Correct: _______
Mental Division
1. 20 ÷ 2 = 23. 68 ÷ 2 =
2. 4÷2= 24. 96 ÷ 3 =
3. 24 ÷ 2 = 25. 86 ÷ 2 =
4. 30 ÷ 3 = 26. 93 ÷ 3 =
5. 6÷3= 27. 88 ÷ 4 =
6. 36 ÷ 3 = 28. 99 ÷ 3 =
7. 40 ÷ 4 = 29. 66 ÷ 3 =
8. 8÷4= 30. 66 ÷ 2 =
9. 48 ÷ 4 = 31. 40 ÷ 4 =
11. 40 ÷ 2 = 33. 60 ÷ 4 =
12. 42 ÷ 2 = 34. 68 ÷ 4 =
14. 60 ÷ 3 = 36. 40 ÷ 2 =
15. 63 ÷ 3 = 37. 30 ÷ 2 =
17. 80 ÷ 4 = 39. 30 ÷ 3 =
18. 84 ÷ 4 = 40. 39 ÷ 3 =
19. 40 ÷ 5 = 41. 45 ÷ 3 =
20. 50 ÷ 5 = 42. 60 ÷ 3 =
21. 60 ÷ 5 = 43. 57 ÷ 3 =
22. 70 ÷ 5 = 44. 51 ÷ 3 =
Lesson 19: Explain remainders by using place value understanding and models.
272
B
Number Correct: _______
Improvement: _______
Mental Division
1. 30 ÷ 3 = 23. 86 ÷ 2 =
2. 9÷3= 24. 69 ÷ 3 =
3. 39 ÷ 3 = 25. 68 ÷ 2 =
4. 20 ÷ 2 = 26. 96 ÷ 3 =
5. 6÷2= 27. 66 ÷ 3 =
6. 26 ÷ 2 = 28. 99 ÷ 3 =
7. 80 ÷ 4 = 29. 88 ÷ 4 =
8. 4÷4= 30. 88 ÷ 2 =
9. 84 ÷ 4 = 31. 40 ÷ 4 =
11. 60 ÷ 2 = 33. 60 ÷ 4 =
12. 62 ÷ 2 = 34. 64 ÷ 4 =
14. 90 ÷ 3 = 36. 40 ÷ 2 =
15. 93 ÷ 3 = 37. 30 ÷ 2 =
17. 40 ÷ 4 = 39. 30 ÷ 3 =
18. 48 ÷ 4 = 40. 36 ÷ 3 =
19. 50 ÷ 5 = 41. 42 ÷ 3 =
20. 60 ÷ 5 = 42. 60 ÷ 3 =
21. 70 ÷ 5 = 43. 54 ÷ 3 =
22. 80 ÷ 5 = 44. 48 ÷ 3 =
Lesson 19: Explain remainders by using place value understanding and models.
273
Name Date
1. When you divide 94 by 3, there is a remainder of 1. Model this problem with place value disks. In the
place value disk model, how did you show the remainder?
2. Cayman says that 94 ÷ 3 is 30 with a remainder of 4. He reasons this is correct because (3 × 30) + 4 = 94.
What mistake has Cayman made? Explain how he can correct his work.
Lesson 19: Explain remainders by using place value understanding and models.
274
a. They have 5 ten-dollar bills and 6 one-dollar bills. Draw a picture to show how the bills will be
shared. Will they have to make change at any stage?
Lesson 19: Explain remainders by using place value understanding and models.
275
5. Imagine you are filming a video explaining the problem 45 ÷ 3 to new fourth graders. Create a script to
explain how you can keep dividing after getting a remainder of 1 ten in the first step.
Lesson 19: Explain remainders by using place value understanding and models.
276
Name Date
1. Molly’s photo album has a total of 97 pictures. Each page of the album holds 6 pictures. How many
pages can Molly fill? Will there be any pictures left? If so, how many? Use place value disks to solve.
2. Marti’s photo album has a total of 45 pictures. Each page holds 4 pictures. She said she can only fill
10 pages completely. Do you agree? Explain why or why not.
Lesson 19: Explain remainders by using place value understanding and models.
277
Name Date
1. When you divide 86 by 4, there is a remainder of 2. Model this problem with place value disks. In the
place value disk model, how can you see that there is a remainder?
2. Francine says that 86 ÷ 4 is 20 with a remainder of 6. She reasons this is correct because
(4 × 20) + 6 = 86. What mistake has Francine made? Explain how she can correct her work.
Lesson 19: Explain remainders by using place value understanding and models.
278
a. To count the blueberries, they put them into small bowls of 10 blueberries. Draw a picture to show
how the blueberries can be shared equally. Will they have to split apart any of the bowls
of 10 blueberries when they share them?
Lesson 19: Explain remainders by using place value understanding and models.
279
5. Imagine you are drawing a comic strip showing how to solve the problem 72 ÷ 4 to new fourth graders.
Create a script to explain how you can keep dividing after getting a remainder of 3 tens in the first step.
Lesson 19: Explain remainders by using place value understanding and models.
280
Lesson 20
Objective: Solve division problems without remainders using the area
model.
Note: This fluency activity prepares students for Lesson 22’s Concept Development
T: (Write 5 × ___ = 15.) Say the unknown factor.
S: 3.
T: (Write 15 ÷ 5.) On your personal white board, write the division problem.
S: (Write 15 ÷ 5 = 3.)
Continue with the following possible sequence: 3 × ___ = 12, 4 × ___ = 12, 5 × ___ = 35, 6 × ___ = 36,
7 × ___ = 49, 9 × ___ = 81, 6 × ___ = 48, 7 × ___ = 42, and 9 × ___ = 54.
Lesson 20: Solve division problems without remainders using the area model.
281
Write an expression to find the unknown length of each rectangle. Then, find the sum of the two unknown
lengths.
a. 8
4 cm 40 square cm square
cm
b.
4 cm 80 square cm 16 square cm
Note: This Application Problem serves as an introduction to today’s Concept Development, in which students
find the total unknown length of a rectangle with an area of 48, corresponding to Part (a), and 96,
corresponding to Part (b).
Lesson 20: Solve division problems without remainders using the area model.
282
T: Let’s find the unknown side lengths of the smaller rectangles and add them. (Show as the
distribution of the quotients shown above.) What is 40 ÷ 4?
S: 10.
T: What is 8 ÷ 4?
S: 2.
T: What is 10 and 2?
S: 12.
T: What is 48 divided by 4?
S: 12.
T: What is the length of the unknown side?
S: 12 units.
T: Take a moment to record the number sentences,
reviewing with your partner their connection to both
the number bond and the area model.
T: Work with your partner to partition the same area of
48 as 2 twenties and 8. When you are finished, try to
find another way to partition the area of 48 so it’s easy
to divide.
Lesson 20: Solve division problems without remainders using the area model.
283
T: (Allow students to work for about four minutes.) Did anyone find another way to partition the area
of 48 so it’s easy to divide?
S: Yes! 24 + 24. 24 divided by 4 is 6. 6 + 6 is 12. 30 and 18 don’t work well because 30 has a
remainder when you divide it by 4. I did it by using 4 rectangles, each with an area of 12 square
units. Oh, yeah, 12 + 12 + 12 + 12.
T: Explain to your partner why different ways of partitioning give us the same correct side length.
S: You are starting with the same amount of area but just chopping it up differently. The sum of the
lengths is the same as the whole length. You can take a total, break it into parts, and divide each
of them separately. I use the same break apart and distribute strategy to find the answer to
56 ÷ 8. 40 ÷ 8 is 5. 16 ÷ 8 is 2. 5 and 2 makes 7.
Repeat the same process with Part (b) from the Application Problem.
Lesson 20: Solve division problems without remainders using the area model.
284
Lesson 20: Solve division problems without remainders using the area model.
285
Lesson 20: Solve division problems without remainders using the area model.
286
Lesson 20: Solve division problems without remainders using the area model.
287
Name Date
a. Look at the area model. What division problem did Alfonso solve?
b. Show a number bond to represent Alfonso’s area model. Start with the total, and then show how
the total is split into two parts. Below the two parts, represent the total length using the distributive
property, and then solve.
(___÷___) + (___÷___)
= ____ + ____
= _____
2. Solve 45 ÷ 3 using an area model. Draw a number bond, and use the distributive property to solve for
the unknown length.
Lesson 20: Solve division problems without remainders using the area model.
288
3. Solve 64 ÷ 4 using an area model. Draw a number bond to show how you partitioned the area, and
represent the division with a written method.
4. Solve 92 ÷ 4 using an area model. Explain, using words, pictures, or numbers, the connection of the
distributive property to the area model.
Lesson 20: Solve division problems without remainders using the area model.
289
Name Date
1. Tony drew the following area model to find an unknown length. What division equation did he model?
2. Solve 42 ÷ 3 using the area model, a number bond, and a written method.
Lesson 20: Solve division problems without remainders using the area model.
290
Name Date
a. Look at the area model. What division problem did Maria solve?
b. Show a number bond to represent Maria’s area model. Start with the total, and then show how the
total is split into two parts. Below the two parts, represent the total length using the distributive
property, and then solve.
(___÷___) + (___÷___)
= ____ + ____
= _____
2. Solve 42 ÷ 3 using an area model. Draw a number bond, and use the distributive property to solve for
the unknown length.
Lesson 20: Solve division problems without remainders using the area model.
291
3. Solve 60 ÷ 4 using an area model. Draw a number bond to show how you partitioned the area, and
represent the division with a written method.
4. Solve 72 ÷ 4 using an area model. Explain, using words, pictures, or numbers, the connection of the
distributive property to the area model.
Lesson 20: Solve division problems without remainders using the area model.
292
Lesson 21
Objective: Solve division problems with remainders using the area model.
Note: This Sprint reviews content from Topic E, including division with and without remainders.
Note: This fluency activity prepares students for Lesson 22’s Concept Development
T: (Write 6 × ___ = 18.) Say the unknown factor.
S: 3.
T: (Write 18 ÷ 6.) On your personal white board, complete the division sentence.
S: (Write 18 ÷ 6 = 3.)
Continue with the following possible sequence: 3 × ___ = 21, 4 × ___ = 20, 5 × ___ = 25, 6 × ___ = 42,
7 × ___ = 56, 9 × ___ = 72, 6 × ___ = 54, 7 × ___ = 63, and 9 × ___ = 63.
Lesson 21: Solve division problems with remainders using the area model.
293
A rectangle has an area of 36 square units and a width of 2 units. What is the unknown side length?
Method 1: Method 2:
Note: This Application Problem serves as an introduction to Problem 1 in the Concept Development, in
which students find the total unknown length of a rectangle with an area of 37 and a width of 2. In today’s
Concept Development, students move on to the complexity of using the area model when there is a
remainder.
Note: Use the Problem Set for Lesson 21 to record work for Problems 1 and 2 of this Concept Development.
Use the remaining problems on the Problem Set for class instruction or independent practice.
Problem 1: 37 ÷ 2
T: (Display the Application Problem with an area
of 36 square units on grid paper.) This
rectangle has a side length of 18. What would
be the area of a rectangle with a width of 2
units and a length of 19 units? (Draw on grid
paper.)
S: 38 square units.
T: So, we cannot represent a rectangle with a width of 2 and an area of 37 square units. Let’s get as
close as we can to 37 square units by building a rectangle part to whole as we did yesterday.
Lesson 21: Solve division problems with remainders using the area model.
294
Lesson 21: Solve division problems with remainders using the area model.
295
Problem 2: 76 ÷ 3
T: (Write 76 ÷ 3.) I’m going to represent this
with an area model moving from part to
whole by place value, just as we did with
37 ÷ 2. What should the total area be?
S: 76 square units.
T: (Draw a rectangle.) What is the width or the
known side length?
S: 3 length units.
T: (Label a width of 3.) Three times how many
tens gets us as close as possible to an area of
7 tens? (Point to the 7 tens of the dividend.)
S: 2 tens.
T: Let’s give 2 tens to the length. (Write the
length on the area model.) Let’s record 2 tens
in the tens place.
T: What is 2 tens times 3?
S: 6 tens. (Record 6 tens below the 7 tens.)
T: How many square units of area is that?
S: 60 square units. (Record in the rectangle.)
T: How many tens remain?
S: 1 ten. (Record 1 ten below the
6 tens.)
T: Let’s add the remaining ten to the
6 ones. What is 1 ten + 6 ones?
(Record the 6 ones to the right of the
1 ten.)
S: 16 ones.
T: We have an area of 16 square units remaining
with a width of 3. (Point to the 16 in the
algorithm.) Three times how many ones gets us
as close as possible to an area of 16?
S: 5 ones.
T: Let’s give 5 ones to the length. (Label the length.)
T: This rectangle has an area of…?
S: 15 square units.
T: How many square units remain?
S: 1 square unit.
T: What is the unknown length, and how
many square units remain?
Lesson 21: Solve division problems with remainders using the area model.
296
Lesson 21: Solve division problems with remainders using the area model.
297
Lesson 21: Solve division problems with remainders using the area model.
298
Lesson 21: Solve division problems with remainders using the area model.
299
A
Number Correct: _______
Lesson 21: Solve division problems with remainders using the area model.
300
B
Number Correct: _______
Improvement: _______
Division with Remainders
Lesson 21: Solve division problems with remainders using the area model.
301
Name Date
1. Solve 37 ÷ 2 using an area model. Use long division and the distributive property to record your work.
2. Solve 76 ÷ 3 using an area model. Use long division and the distributive property to record your work.
b. Show how Carolina’s model can be represented using the distributive property.
Lesson 21: Solve division problems with remainders using the area model.
302
Solve the following problems using the area model. Support the area model with long division or the
distributive property.
4. 48 ÷ 3 5. 49 ÷ 3
6. 56 ÷ 4 7. 58 ÷ 4
8. 66 ÷ 5 9. 79 ÷ 3
Lesson 21: Solve division problems with remainders using the area model.
303
10. Seventy-three students are divided into groups of 6 students each. How many groups of 6 students are
there? How many students will not be in a group of 6?
Lesson 21: Solve division problems with remainders using the area model.
304
Name Date
1. Kyle drew the following area model to find an unknown length. What division equation did he model?
2. Solve 93 ÷ 4 using the area model, long division, and the distributive property.
Lesson 21: Solve division problems with remainders using the area model.
305
Name Date
1. Solve 35 ÷ 2 using an area model. Use long division and the distributive property to record your work.
2. Solve 79 ÷ 3 using an area model. Use long division and the distributive property to record your work.
b. Show how Paulina’s model can be represented using the distributive property.
Lesson 21: Solve division problems with remainders using the area model.
306
Solve the following problems using the area model. Support the area model with long division or the
distributive property.
4. 42 ÷ 3 5. 43 ÷ 3
6. 52 ÷ 4 7. 54 ÷ 4
8. 61 ÷ 5 9. 73 ÷ 3
Lesson 21: Solve division problems with remainders using the area model.
307
10. Ninety-seven lunch trays were placed equally in 4 stacks. How many lunch trays were in each stack?
How many lunch trays will be left over?
Lesson 21: Solve division problems with remainders using the area model.
308
4
GRADE
Mathematics Curriculum
GRADE 4 • MODULE 3
Topic F
Reasoning with Divisibility
4.OA.4
Focus Standard: 4.OA.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole
number is a multiple of each of its factors. Determine whether a given whole number
in the range 1–100 is a multiple of a given one-digit number. Determine whether a
given whole number in the range 1–100 is prime or composite.
Instructional Days: 4
Coherence -Links from: G3–M1 Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10
G3–M3 Multiplication and Division with Units of 0, 1, 6–9, and Multiples of 10
-Links to: G5–M2 Multi-Digit Whole Number and Decimal Fraction Operations
G5–M3 Addition and Subtraction of Fractions
In Topic F, armed with an understanding of remainders, students explore factors, multiples, and prime and
composite numbers within 100 (4.OA.4). Students gain valuable insights into patterns of divisibility as they
test for primes and find factors and multiples, at times using their new skill of dividing double-digit dividends.
This prepares them for Topic G’s work with dividends of up to four digits.
Lesson 22 has students find factor pairs for numbers
to 100 and then use their understanding of factors to
determine whether numbers are prime or composite.
In Lesson 23, students use division to examine
numbers to 100 for factors and make observations
about patterns they observe, for example, “When 2 is
a factor, the numbers are even.” Lesson 24 transitions
the work with factors into a study of multiples,
encouraging students to notice that the set of
multiples of a number is infinite while the set of
factors is finite.
In Lesson 25, the Sieve of Eratosthenes uses multiples
to enable students to identify and explore the
properties of prime and composite numbers to 100.
Objective 2: Use division and the associative property to test for factors and observe patterns.
(Lesson 23)
Objective 4: Explore properties of prime and composite numbers to 100 by using multiples.
(Lesson 25)
Lesson 22
Objective: Find factor pairs for numbers to 100, and use understanding of
factors to define prime and composite.
Lesson 22: Find factor pairs for numbers to 100, and use understanding of factors
to define prime and composite. 311
8 × ____ = 96. Find the unknown side length, or factor. Use an area model to solve the problem.
Note: This Application Problem applies the Topic E skill of dividing a two-digit dividend using an area model
and serves as a lead-in to this lesson’s Concept Development by using area models to illustrate the concept of
factor pairs.
factors product
factors product
Lesson 22: Find factor pairs for numbers to 100, and use understanding of factors
to define prime and composite. 312
Lesson 22: Find factor pairs for numbers to 100, and use understanding of factors
to define prime and composite. 313
Lesson 22: Find factor pairs for numbers to 100, and use understanding of factors
to define prime and composite. 314
Problem 3: Identify factors of numbers and determine if they are prime or composite.
Display the numbers 23, 35, and 48.
T: Let’s use a table to record the factor pairs for 35. Say the first
factor pair.
Guide students to complete the table as a class, using their multiplication
facts.
T: Is 35 a prime or composite number? Why?
S: Composite, because it has more than one factor pair.
T: With your partner, use a table to list factors of 23 and 48 and tell if
each one is prime or composite.
Allow three minutes for students to work.
T: Are any of these numbers prime numbers? How do you know?
S: Twenty-three is prime because we thought about all the possible factors, other than one, up to 11
and none worked.
T: Why can we stop at the number 11?
S: Eleven is the closest whole number to half of 23. Once I get halfway, I have found all of the factor
pairs. After that, they just keep repeating.
T: Why is 48 composite?
S: There are more than two factors. It has 6 and 8 as factors. It also has 4 and 12. I found
10 factors! It sure isn’t prime!
Lesson 22: Find factor pairs for numbers to 100, and use understanding of factors
to define prime and composite. 315
Lesson 22: Find factor pairs for numbers to 100, and use understanding of factors
to define prime and composite. 316
Lesson 22: Find factor pairs for numbers to 100, and use understanding of factors
to define prime and composite. 317
Name Date
1. Record the factors of the given numbers as multiplication sentences and as a list in order from least to
greatest. Classify each as prime (P) or composite (C). The first problem is done for you.
Lesson 22: Find factor pairs for numbers to 100, and use understanding of factors
to define prime and composite. 318
2. Find all factors for the following numbers, and classify each number as prime or composite. Explain your
classification of each as prime or composite.
4. Sheila has 28 stickers to divide evenly among 3 friends. She thinks there will be no leftovers. Use what
you know about factor pairs to explain if Sheila is correct.
Lesson 22: Find factor pairs for numbers to 100, and use understanding of factors
to define prime and composite. 319
Name Date
Record the factors of the given numbers as multiplication sentences and as a list in order from least to
greatest. Classify each as prime (P) or composite (C).
Prime (P)
Multiplication Sentences Factors or
Composite (C)
a. 9 The factors of 9 are:
Lesson 22: Find factor pairs for numbers to 100, and use understanding of factors
to define prime and composite. 320
Name Date
1. Record the factors of the given numbers as multiplication sentences and as a list in order from least to
greatest. Classify each as prime (P) or composite (C). The first problem is done for you.
Lesson 22: Find factor pairs for numbers to 100, and use understanding of factors
to define prime and composite. 321
2. Find all factors for the following numbers, and classify each number as prime or composite. Explain your
classification of each as prime or composite.
4. Julie has 27 grapes to divide evenly among 3 friends. She thinks there will be no leftovers. Use what you
know about factor pairs to explain whether or not Julie is correct.
Lesson 22: Find factor pairs for numbers to 100, and use understanding of factors
to define prime and composite. 322
Lesson 23
Objective: Use division and the associative property to test for factors and
observe patterns.
Note: This fluency activity reviews Lesson 22’s content. To challenge students, have them construct the
arrays instead of having them projected.
T: (Project a 1 × 8 array.) What is the width of the array?
S: 1 unit.
T: (Write 1.) What’s the length of the array?
S: 8 units.
T: (Write 8.) Write the multiplication sentence.
S: (Write 1 × 8 = 8.)
Repeat process for a 2 × 4 array.
T: List the factors of 8.
S: (Write factors of 8: 1, 2, 4, 8.)
Continue with following possible sequence: factors of 12, factors of 16, and factors of 18.
Lesson 23: Use division and the associative property to test for factors and
observe patterns. 323
Note: This fluency activity reviews the Concept Development from Lessons 9, 10, and 22.
T: (Write 174 × 2 = .) On your personal white board, solve the multiplication sentence using the
standard algorithm.
S: What are 4 factors of 348 you know right away?
S: 1 and 348, 2 and 174.
Repeat the process using the following possible sequence: 348 × 2, 696 × 2, and 1,392 × 2. Students may
realize that if 348 is a factor of 696, then 174 is, too!
Lesson 23: Use division and the associative property to test for factors and
observe patterns. 324
Problem 2: Use the associative property to find additional factors of larger numbers.
T: Talk to your partner. Is it necessary to divide to figure out if 5 is a factor of 54?
S: Fifty-four can’t be divided by 5 exactly. There is a remainder. When you count by fives, each
number ends with 5 or 0. Fifty-four does not end with 0 or 5. Five isn’t a factor of 54.
T: We divided to determine if 3 was a factor of 54, but for 2 and 5 we don’t need to divide. Explain to
your partner why not.
S: The even numbers all have 2 for a factor. If the digit in the ones place is odd, the number doesn’t
have 2 as a factor. Numbers with 5 as a factor have 0 or 5 as a digit in the ones place. We can
use patterns for 2 and 5.
T: How can we know if 6 a factor of 54?
S: Six times 9 equals 54.
Lesson 23: Use division and the associative property to test for factors and
observe patterns. 325
T: Earlier we saw that 2 and 3 are both factors of 54. Talk to your partner. Is this number sentence
true?
T: (Write 54 = 6 × 9 = (2 × 3) × 9.)
S: (Share ideas.)
T: Let’s write it vertically so that it is very easy to see how the factor 6 is
related to 2 times 3.
T: (Write the problem as modeled to the right.) Now let’s move the
parentheses so that 3 associates with 9 rather than 2. Three times 9 is?
S: 27.
T: Find the product of 2 and 27. (Pause.) Is it true that 2 times 27 equals 54?
S: Yes!
T: We used the associative property to show that both 2 and 3 are factors of 54.
T: Let’s test this method to see if it works with a number other than 54. Forty-two is 6 times…?
S: 7.
T: Let’s use the associative property to see if 2 and 3 are also factors of
42.
T: (Write 42 = 6 × 7.) How will we rewrite 6?
S: 2 × 3.
T: (Beneath 6 × 7, write = (2 × 3) × 7.) Let’s now move the parentheses
to first multiply 3 times 7, to associate 3 with 7 rather than 2. 3 times
7 is?
S: 21.
T: Find the product of 2 and 21. (Pause.) Is it true that 2 times 21 equals 42?
S: Yes!
Record the thought process as shown to the right. Have students use the associative property to prove that
since 6 is a factor of 60, both 2 and 3 are also factors.
Lesson 23: Use division and the associative property to test for factors and
observe patterns. 326
Lesson 23: Use division and the associative property to test for factors and
observe patterns. 327
We can use number patterns to determine if 2 and 5 are factors of other numbers. What other
numbers do you think have patterns? Do you see a pattern for determining which numbers 3 is a
factor of? Can you describe one?
If 8 is a factor of 96, what other numbers must also be factors of 96? How can we use the
associative property to prove this?
Once someone tried to tell me that the two statements in Problem 4 say the same thing. How would
you explain that the two statements are different?
Lesson 23: Use division and the associative property to test for factors and
observe patterns. 328
Name Date
Lesson 23: Use division and the associative property to test for factors and
observe patterns. 329
a. 24 = 12 × 2 b. 36 = ____ × 4
= ( ___ × 3) × 2 = ( ____ × 3) × 4
= ___ × (3 × 2) = ____ × (3 × 4)
= ___ × 6 = ____ × 12
= ___ = ____
3. In class, we used the associative property to show that when 6 is a factor, then 2 and 3 are factors,
because 6 = 2 × 3. Use the fact that 8 = 4 × 2 to show that 2 and 4 are factors of 56, 72, and 80.
56 = 8 × 7 72 = 8 × 9 80 = 8 × 10
4. The first statement is false. The second statement is true. Explain why, using words, pictures, or
numbers.
If a number has 2 and 4 as factors, then it has 8 as a factor.
If a number has 8 as a factor, then both 2 and 4 are factors.
Lesson 23: Use division and the associative property to test for factors and
observe patterns. 330
Name Date
2. Use the associative property to explain why the following statement is true.
Any number that has 9 as a factor also has 3 as a factor.
Lesson 23: Use division and the associative property to test for factors and
observe patterns. 331
Name Date
Lesson 23: Use division and the associative property to test for factors and
observe patterns. 332
a. 12 = 6 × 2 b. 30 = ____ × 5
= ( ___ × 2) × 2 = ( ____ × 3) × 5
= ___ × (2 × 2) = ____ × (3 × 5)
= ___ = ____
3. In class, we used the associative property to show that when 6 is a factor, then 2 and 3 are factors,
because 6 = 2 × 3. Use the fact that 10 = 5 × 2 to show that 2 and 5 are factors of 70, 80, and 90.
70 = 10 × 7 80 = 10 × 8 90 = 10 × 9
4. The first statement is false. The second statement is true. Explain why, using words, pictures, or
numbers.
If a number has 2 and 6 as factors, then it has 12 as a factor.
If a number has 12 as a factor, then both 2 and 6 are factors.
Lesson 23: Use division and the associative property to test for factors and
observe patterns. 333
Lesson 24
Objective: Determine if a whole number is a multiple of another number.
8 cm × 12 cm = 96 square centimeters. Imagine a rectangle with an area of 96 square centimeters and a side
length of 4 centimeters. What is the length of its unknown side? How will it look when compared to the
8 centimeter by 12 centimeter rectangle? Draw and label both rectangles.
Note: This Application Problem relates finding factors (Lessons 22 and 23) to
multiples (Lesson 24). Consider leading students to visualize the columns of 4 or
8 square centimeters. When counting by the number of squares in those columns,
will the count arrive exactly at 96? When counting by the number of squares in one
row, 24 or 12, will the count also arrive exactly at 96? (Consider using graph paper
to demonstrate for those students who would benefit from pictorial representation.)
Also consider showing students how the associative property beautifully illustrates
how as the 8 is split in two, the 12 doubles (pictured to the right).
Problem 2: Determine if one number is a multiple of another number, and list multiples of given numbers.
T: Why is 24 a multiple of 4?
S: When we count by fours we get to 24. 4 times 6 is 24. Four is a factor of 24.
T: Is 24 a multiple of 5?
S: No, because we can’t skip-count by five to 24.
No, because 24 divided by 5 has a remainder.
No, because 5 is a not a factor of 24.
T: What about 8? Is 24 a multiple of 8?
S: Yes! Eight times 3 is 24. Well, 8 is a factor of
24, so 24 must be a multiple of 8.
T: We know 96 is a multiple of 4 from our Application
Problem, since 4 times 24 is 96. What did we do to
figure that out?
S: I used long division. I used the associative
property.
T: Yes, because for some it is beyond mental math.
How can we find out if 96 is a multiple of 3?
S: We can divide to see if 96 is divisible by 3. We
might use the associative property since we know that
8 times 12 and 4 times 24 are 96 from the Application NOTES ON
Problem. MULTIPLE MEANS
T: Try that. OF REPRESENTATION:
Students who struggle with the
Allow time for students to divide or use the associative
difference between a factor and a
property. multiple might benefit from creating a
T: What did you discover? three-column chart that lists numbers
in the first column, factors in the
S: There was no remainder, so 3 is a factor of 96.
second, and then multiples in the third,
That makes 96 a multiple of 3. always followed by an ellipsis to
T: What is the factor pair of 3? remember the infinite number of
S: 32. multiples of any number. Students can
refer to this visual representation as
T: If you count by 32 three times, will you get to 96? they complete the lesson and as they
S: Yes. think about how factors and multiples
T: Is 96 a multiple of both 3 and 32? are related.
S: Yes!
T: List the first five multiples of 3.
S: (Write 3, 6, 9, 12, 15.)
T: What number did you begin with?
S: 3.
T: But isn’t 0 a multiple of 3? Should we start with 0 first?
S: No. It’s less. But 0 times 3 is 0, so maybe.
T: Since zero times any number is zero, zero is a multiple of every number, so we could consider it the
first multiple of every number. However, when we skip-count, we usually start with the number
we’re counting by. So, we usually think of the number itself, in this case 3, as the first multiple,
instead of 0. That way, the first multiple is 1 × 3, the second is 2 × 3, and so on.
(Optional) Problem 3: Use the associative property to see that any multiple of 6 is also a multiple of 3 and 2.
T: Shout out a multiple of 6.
S: 12. 30. 60. 24. 600.
T: Is any multiple of 6 also a multiple of 2 and 3?
T: Let’s use the associative property (and commutative property) to find out. (Write the following.)
T: 60 = 10 × 6
= 10 × (2 × 3)
= (10 × 2) × 3= 20 × 3
Yes, 60 is a multiple of 3. If we count by 3 twenty NOTES ON
times, we get to 60. MULTIPLE MEANS
= (10 × 3) × 2= 30 × 2 OF REPRESENTATION:
Yes, 60 is a multiple of 2. If we count by 2 thirty times, When using the associative property,
we get to 60. consider bringing the word to life by
asking three students to stand at the
T: Let’s use a letter to represent the number of sixes to
front of the class in a line. Ask the
see if this is true for all sixes. (Write the following person in the middle to associate with
three equations on the board.) the person on the right. Ask them to
n × 6 = n × (2 × 3) associate with the person on their left.
n × 6 = (n × 2) × 3 Ask those on the ends to associate.
What changed? (The associations.)
n × 6 = (n × 3 ) × 2
Next, give each person an identity as a
T: Discuss with your partner why these equations are
factor, perhaps 9, 2, and 5 respectively.
true. You might try plugging in 4 or 5 as the number of Have the factor of 2 first associate with
sixes, n, to help you understand. the 9 and then with the five. Then,
S: Wow! These equations are true. It’s just that it takes have the 9 and 5 associate.
twice as many threes to get to the multiple as sixes. Which is easiest: 18 times 5, 9 times
Yeah, it’s double the number of multiples of six, 10, or 45 times 2?
2 × n. And it’s three times as many twos to get
there! It’s because twos are smaller units so it takes
more.
T: So, maybe the multiples of a number are also the multiples of its factors.
If there is time, consider repeating the process with the multiples of 8 being multiples of both 2 and 4.
Students might approach the generalization that the multiples of a given number include the multiples of the
number’s factors.
Name Date
1. For each of the following, time yourself for 1 minute. See how many multiples you can write.
3. Use mental math, division, or the associative property to solve. (Use scratch paper if you like.)
4. Can a prime number be a multiple of any other number except itself? Explain why or why not.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
a. Circle in red the multiples of 2. When a number is a multiple of 2, what are the possible values for
the ones digit?
b. Shade in green the multiples of 3. Choose one. What do you notice about the sum of the digits?
Choose another. What do you notice about the sum of the digits?
c. Circle in blue the multiples of 5. When a number is a multiple of 5, what are the possible values for
the ones digit?
d. Draw an X over the multiples of 10. What digit do all multiples of 10 have in common?
Name Date
5 × 11 = _____
6 × 11 = _____
7 × 11 = _____
8 × 11 = _____
9 × 11 = _____
Name Date
1. For each of the following, time yourself for 1 minute. See how many multiples you can write.
3. Use mental math, division, or the associative property to solve. (Use scratch paper if you like.)
4. Can a prime number be a multiple of any other number except itself? Explain why or why not.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
a. Underline the multiples of 6. When a number is a multiple of 6, what are the possible values for the
ones digit?
b. Draw a square around the multiples of 4. Look at the multiples of 4 that have an odd number in the
tens place. What values do they have in the ones place?
c. Look at the multiples of 4 that have an even number in the tens place. What values do they have in
the ones place? Do you think this pattern would continue with multiples of 4 that are larger than
100?
d. Circle the multiples of 9. Choose one. What do you notice about the sum of the digits?
Choose another one. What do you notice about the sum of the digits?
Lesson 25
Objective: Explore properties of prime and composite numbers to 100 by
using multiples.
Lesson 25: Explore properties of prime and composite numbers to 100 by using
multiples. 346
Note: This fluency activity gives students practice in remembering the difference between factors and
multiples.
T: (Write 3.) List as many multiples of 3 as you can in the next 20 seconds. Take your mark. Get set.
Go.
S: (Write 3, 6, 9, 12, 15, 18, 21, 24, ….)
T: List the factors of 3.
S: (Write 1, 3.)
Continue with the following possible sequence: multiples of 4, factors of 4; multiples of 5, factors of 5.
Materials: (T) Sieve (for the Student Debrief) (S) Problem Set, orange crayon, red crayon
Lesson 25: Explore properties of prime and composite numbers to 100 by using
multiples. 347
As students are charged with determining multiples that are greater than those in the times tables, some will
choose to continue adding on, while others will choose to divide, and some will begin to rely on number
patterns they have noticed. Encourage partners to compare strategies.
Note: At a certain point, the majority of students will have finished marking off multiples of 7. A few may
have begun to notice that the multiples of the remaining numbers have already been crossed off. Interrupt
the class at this point. Below is the suggested midpoint dialogue.
T: After you marked off multiples of 7, what was the next number that you circled?
S: 11.
T: Were there any multiples of 11 that hadn’t been crossed out already?
S: No.
T: What about 13? Are there any multiples of 13 that still need to be crossed off?
S: No, they’re already crossed off from before.
T: I wonder if that’s true of the rest? Go back to 11. Let’s see if we can figure out what happened.
Count by elevens within 100 using the chart.
S: 11, 22, 33, 44, 55, …, 99.
T: Ninety-nine is how many elevens?
S: 9 elevens.
T: So, by the time we circled 11, is it true that we’d already marked all of the multiples of 2, 3, all the
way up to 10?
S: Well, yeah, we circled 2, 3, 5, and 7, and crossed off their multiples. We didn’t have to do fours,
because the fours got crossed out when we crossed out multiples of 2. The same thing happened
with the sixes, eights, nines, and tens.
T: Interesting, so we had already crossed out 2 × 11, 3 × 11, all the way up to 9 × 11. I wonder if the
same thing happens with 13. Discuss with a partner: Will there be more or fewer groups of 13 than
groups of 11 within a hundred?
S: More, because it is a bigger number. Fewer, because it is a larger number so fewer will fit in 100.
Fewer because 9 × 11 is 99, so maybe 7 or 8 times 13 will be less than 100. 9 × 13 is more than
100, so fewer groups.
T: Take a moment to figure out how many multiples of 13 are within 100.
S: (Might count by 13 or multiply.)
T: How many multiples of 13 are less than 100?
S: 7.
T: 7 times 13 is…?
S: 91.
T: We already marked off 91 because it is a multiple of 7. The same is true for 6 × 13, 5 × 13, and so on.
Do we need to mark off multiples of 17?
S: No, because there will be even fewer groups, and we already marked off those factors.
T: Exactly. The highest multiple of 17 on the hundreds chart is 85. 5 seventeens is 85. We already
marked 2 × 17 up to 5 × 17.
Lesson 25: Explore properties of prime and composite numbers to 100 by using
multiples. 348
Following this dialogue, have students return to work. Once students have correctly completed page 1, have
them continue to page 2. Allow students time to thoroughly discuss and answer each question. Circulate and
offer assistance as needed. Be ready to initiate or prompt discussions when students seem unsure.
Answer questions with questions to keep students thinking and analyzing.
Regroup, as the class completes page 2, to share responses to the Student Debrief questions.
Problem Set
Please note that the Problem Set comprises only questions
used in the Concept Development. No additional time is
allotted here since all problems are completed during the
lesson. The Student Debrief has additional time allotted for
the purpose of whole-class discussion of questions raised
and discoveries made by the students during the Concept
Development.
Lesson 25: Explore properties of prime and composite numbers to 100 by using
multiples. 349
Lesson 25: Explore properties of prime and composite numbers to 100 by using
multiples. 350
Name Date
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Lesson 25: Explore properties of prime and composite numbers to 100 by using
multiples. 351
b. Why were the circled numbers not crossed off along the way?
c. Except for the number 1, what is similar about all of the numbers that were crossed off?
Lesson 25: Explore properties of prime and composite numbers to 100 by using
multiples. 352
Name Date
Lesson 25: Explore properties of prime and composite numbers to 100 by using
multiples. 353
Name Date
1. A student used the sieve of Eratosthenes to find all prime numbers less than 100. Create a step-by-step
set of directions to show how it was completed. Use the word bank to help guide your thinking as you
write the directions. Some words may be used just once, more than once, or not at all.
Word Bank
number shade
circle X
multiple prime
Lesson 25: Explore properties of prime and composite numbers to 100 by using
multiples. 354
2. What do all of the numbers that are crossed out have in common?
4. There is one number that is neither crossed out nor circled. Why is it treated differently?
Lesson 25: Explore properties of prime and composite numbers to 100 by using
multiples. 355
4
GRADE
Mathematics Curriculum
GRADE 4 • MODULE 3
Topic G
Division of Thousands, Hundreds,
Tens, and Ones
4.OA.3, 4.NBT.6, 4.NBT.1
Focus Standards: 4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number
answers using the four operations, including problems in which remainders must be
interpreted. Represent these problems using equations with a letter standing for the
unknown quantity. Assess the reasonableness of answers using mental computation
and estimation strategies including rounding.
4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and
one-digit divisors, using strategies based on place value, the properties of operations,
and/or the relationship between multiplication and division. Illustrate and explain the
calculation by using equations, rectangular arrays, and/or area models.
Instructional Days: 8
Coherence -Links from: G3–M1 Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10
G3–M3 Multiplication and Division with Units of 0, 1, 6–9, and Multiples of 10
-Links to: G5–M2 Multi-Digit Whole Number and Decimal Fraction Operations
Topic G extends to division with three- and four-digit dividends using place value understanding. Students
begin the topic by connecting multiplication of 10, 100, and 1,000 by single-digit numbers from Topic B to
division of multiples of 10, 100, and 1,000 in Lesson 26. Using unit language, students find their division facts
allow them to divide much larger numbers.
Students then move to the abstract level in Lessons 28 and 29, recording long division with place value
understanding, first of three-digit, then four-digit numbers using small divisors. In Lesson 30, students
practice dividing when zeros are in the dividend or in the quotient.
Lessons 31 and 32 give students opportunities to apply their understanding of division by solving word
problems (4.OA.3). In Lesson 31, students identify word problems as number of groups unknown or group
size unknown, modeled using tape diagrams. Lesson 32 allows students to apply their place value
understanding of solving long division using larger divisors of 6, 7, 8, and 9. Concluding this topic, Lesson 33
has students make connections between the area model and the standard algorithm for long division.
A Teaching Sequence Toward Mastery of Division of Thousands, Hundreds, Tens, and Ones
Objective 1: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
(Lesson 26)
Objective 2: Represent and solve division problems with up to a three-digit dividend numerically and
with place value disks requiring decomposing a remainder in the hundreds place.
(Lesson 27)
Objective 3: Represent and solve three-digit dividend division with divisors of 2, 3, 4, and 5 numerically.
(Lesson 28)
Objective 5: Solve division problems with a zero in the dividend or with a zero in the quotient.
(Lesson 30)
Objective 6: Interpret division word problems as either number of groups unknown or group size
unknown.
(Lesson 31)
Objective 7: Interpret and find whole number quotients and remainders to solve one-step division word
problems with larger divisors of 6, 7, 8, and 9.
(Lesson 32)
Objective 8: Explain the connection of the area model of division to the long division algorithm for
three- and four-digit dividends.
(Lesson 33)
Lesson 26
Objective: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
Note: This fluency activity prepares students for this lesson’s Concept Development.
Repeat the process from Lesson 15 with the following possible sequence (projected or drawn).
1 hundreds disk, 2 tens disks, and 3 ones disks
4 hundreds disks, 1 tens disk, and 3 ones disks
3 hundreds disks, 15 tens disks, and 2 ones disks
2 hundreds disks, 15 tens disks, and 3 ones disks
Follow by having students draw disks for 524, 231, and 513.
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
358
Note: This fluency activity reviews Topic F’s content and gives students practice in remembering the
difference between factors and multiples.
Repeat the process from Lesson 25 with the following possible sequence: 4 multiples of 6 starting from 60,
the 4 factors of 6, the 4 factors of 8, 4 multiples of 8 starting at 80, the 3 factors of 9, and 4 multiples of 9
starting at 90.
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
359
A coffee shop uses 8-ounce mugs to make all of its coffee drinks. In one week, they served 30 mugs of
espresso, 400 lattes, and 5,000 mugs of coffee. How many ounces of coffee drinks did they make in that one
week?
Note: By reviewing multiplication of 10, 100, and 1,000, this Application Problem leads up to today’s Concept
Development, which will explore division of multiples of 10, 100, and 1,000.
Display 9 ÷ 3 and 90 ÷ 3.
T: Let’s draw place value disks to represent
these expressions. Solve. Compare your
models to your partner’s.
T: Give me a number sentence for each in
unit form.
S: 9 ones ÷ 3 = 3 ones. 9 tens ÷ 3 = 3 tens.
Display 900 ÷ 3 and 9,000 ÷ 3.
T: Tell your partner how you might model
these two expressions.
S: It’s just like we did for the last problems.
We represented 9 disks and divided them
into 3 groups. Our disks will be in the
hundreds or in the thousands. We won’t
have a remainder because 3 is a factor of
9.
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
360
T: Model these expressions, using place value disks, with your partner.
S: (Draw disks and divide.)
T: What do you notice?
S: All 9 disks were split into 3 groups of 3, but they are groups of different units.
T: Write these number sentences in unit form. Turn and talk with your partner about what you notice.
S: They all look similar. They are the same with different units. They are all solved with 9 divided
by 3; they just have different units.
Problem 2
500 ÷ 5
350 ÷ 5
3,000 ÷ 5
Display 500 ÷ 5.
T: On your personal white board, rewrite the expression
500 ÷ 5 in unit form.
S: (Write 5 hundreds ÷ 5.)
T: Why don’t you need a pencil and paper to solve this problem?
S: Because 5 divided by 5 is 1, and the unit is hundreds.
The answer is 1 hundred. Five of anything divided by 5 is 1.
Yeah, 5 bananas divided by 5 is 1 banana.
Display 350 ÷ 5.
T: Now, let’s look at 350 divided by 5. Rewrite this expression in
unit form. Talk to your partner about how representing this
expression is different from the last one.
S: This time we have two units, hundreds and tens. I can
rename 3 hundreds and 5 tens as 35 tens. 35 tens divided by 5.
We didn’t have to decompose 5 hundreds, but now we do
have to change 3 hundreds for tens since we can’t divide
3 hundreds by 5.
T: Let’s use 35 tens. Say the number sentence you will use to solve in
unit form.
S: 35 tens ÷ 5 = 7 tens.
T: What is the quotient of 350 divided by 5?
S: 70.
T: Let’s model this on the place value chart just to be sure you really
understand. Draw 3 hundreds and 5 tens and change the hundreds
into smaller units.
S: It’s true. When I decomposed each hundred, I got 10 more tens.
5 tens + 10 tens + 10 tens + 10 tens is 35 tens. Each 10 tens is 1 hundred.
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
361
Display 3,000 ÷ 5.
T: Discuss with your partner a way to solve this problem. NOTES ON
T: (Allow one minute for students to discuss.) Solve. MULTIPLE MEANS
Compare your solution with a pair near you. Discuss OF ACTION AND
the strategy you used. EXPRESSION:
T: (Allow time for sharing.) Is there a pair that would like Support English language learners and
to share their solution? others as they transcribe number form
S: We had to decompose 3,000 into 30 hundreds because to unit form. If helpful, guide students
there weren’t enough thousands to divide. 30 to whisper-say the number before
writing. Depending on students’
hundreds divided by 5 is easy because we know 30
proficiency, provide the spelling of
divided by 5 is 6. Then, we just had to divide
hundreds and thousands.
30 hundreds by 5 and got a quotient of 6 hundreds, or
Help students understand how to
600.
determine the appropriate unit form.
T How is this problem related to 350 ÷ 5? Say, “If the divisor is greater than the
S: 3 hundreds got changed for 30 tens, and 3 thousands first digit, try a smaller unit form.”
got changed for 30 hundreds. In both problems, we Give multiple examples.
had to change 3 larger units for 30 of the next smaller
units. It’s like when we are subtracting and we
don’t have enough units—we have to change a larger
unit for smaller units, too.
T: Good connections. Turn and restate the ideas of your
peers to your partner in your own words.
T: (Allow time for talk.) Let me fire some quick problems at
you. Tell me the first expression you would solve.
For example, if I say 250 ÷ 2, you say 2 hundreds divided by
2. If I say 250 ÷ 5, you say 25 tens divided by 5. Ready?
Give students a sequence of problems such as the following:
120 ÷ 2; 400 ÷ 2; 6,200 ÷ 2; 1,800 ÷ 2; 210 ÷ 3; 360 ÷ 3; 1,200 ÷ 3;
and 4,200 ÷ 3.
Problem 3
Display: The Hometown Hotel has a total of
480 guest rooms. That is 6 times as many rooms as
the Travelers Hotel down the street. How many
rooms are there in the Travelers Hotel?
T: Let’s read this problem together. Draw a
tape diagram to model this problem. When
you have drawn and labeled your diagram,
compare it with your partner’s.
T: How can we determine the value of 1 unit?
(Point to the unit representing the number
of rooms at the Travelers Hotel.)
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
362
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
363
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
364
Name Date
1. Draw place value disks to represent the following problems. Rewrite each in unit form and solve.
a. 6 ÷ 2 = ________ 1 1 1 1 1 1
6 ones ÷ 2 = _________ ones
b. 60 ÷ 2 = ________
6 tens ÷ 2 = ______________
c. 600 ÷ 2 = ________
___________________________ ÷ 2 = ___________________________
d. 6,000 ÷ 2 = ________
___________________________ ÷ 2 = ___________________________
2. Draw place value disks to represent each problem. Rewrite each in unit form and solve.
a. 12 ÷ 3 = ________
b. 120 ÷ 3 = ________
___________________________ ÷ 3 = ___________________________
c. 1,200 ÷ 3 = ________
___________________________ ÷ 3 = ___________________________
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
365
8 hundreds ÷ 2 =
4 hundreds
30 tens ÷ 6 = ____
tens
36 hundreds ÷ 4 =
____ hundreds
4. Some sand weighs 2,800 kilograms. It is divided equally among 4 trucks. How many kilograms of sand are
in each truck?
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
366
5. Ivy has 5 times as many stickers as Adrian has. Ivy has 350 stickers. How many stickers does Adrian have?
6. An ice cream stand sold $1,600 worth of ice cream on Saturday, which was 4 times the amount sold on
Friday. How much money did the ice cream stand collect on Friday?
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
367
Name Date
6 hundreds ÷ 3 =
____ hundreds
2. Hudson and 7 of his friends found a bag of pennies. There were 320 pennies, which they shared equally.
How many pennies did each person get?
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
368
Name Date
1. Draw place value disks to represent the following problems. Rewrite each in unit form and solve.
a. 6 ÷ 3 = ________ 1 1 1 1 1 1
6 ones ÷ 3 = _________ones
b. 60 ÷ 3 = ________
6 tens ÷ 3 = ______________
c. 600 ÷ 3 = ________
___________________________ ÷ 3 =___________________________
d. 6,000 ÷ 3 = ________
___________________________ ÷ 3 = ___________________________
2. Draw place value disks to represent each problem. Rewrite each in unit form and solve.
a. 12 ÷ 4 = ________
12 ones ÷ 4 = _________ones
b. 120 ÷ 4 = ________
___________________________ ÷ 4 = ___________________________
c. 1,200 ÷ 4 = ________
___________________________ ÷ 4 = ___________________________
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
369
8 hundreds ÷ 4 =
2 hundreds
20 tens ÷ 4 = ____
tens
12 hundreds ÷ 3 =
____ hundreds
4. A fleet of 5 fire engines carries a total of 20,000 liters of water. If each truck holds the same amount of
water, how many liters of water does each truck carry?
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
370
5. Jamie drank 4 times as much juice as Brodie. Jamie drank 280 milliliters of juice. How much juice did
Brodie drink?
6. A diner sold $2,400 worth of French fries in June, which was 4 times as much as was sold in May.
How many dollars’ worth of French fries were sold at the diner in May?
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
371
Lesson 26: Divide multiples of 10, 100, and 1,000 by single-digit numbers.
372
Lesson 27
Objective: Represent and solve division problems with up to a three-digit
dividend numerically and with place value disks requiring decomposing a
remainder in the hundreds place.
Note: This fluency activity reviews Lesson 26’s Concept Development and strengthens students’
understanding of place value’s role in the long division algorithm.
T: (Display 6 ÷ 2.) On your personal white board, draw place value disks to represent the expression.
S: (Draw 6 ones disks and divide them into 2 groups of 3.)
T: Say the division sentence in unit form.
S: 6 ones ÷ 2 = 3 ones.
Repeat the process using the following possible sequence: 60 ÷ 2; 600 ÷ 2; 6,000 ÷ 2; 80 ÷ 2; 1,200 ÷ 3, and
1,200 ÷ 4.
Emma takes 57 stickers from her collection and divides them up equally between 4 of her friends. How many
stickers will each friend receive? Emma puts the remaining stickers back in her collection. How many stickers
will Emma return to her collection?
Note: This Application Problem reviews work with two-digit dividends from Lesson 17.
Materials: (T) Thousands place value chart for dividing (Lesson 26 Template) (S) Personal white board,
thousands place value chart for dividing (Lesson 26 Template)
Problem 1: Divide a three-digit number by a one-digit number using place value disks, regrouping in the
hundreds.
Display 423 ÷ 3.
T: Let’s find the quotient. Represent 423 on the place
value chart. Tell your partner how many groups
below will be needed.
S: (Draw disks on chart.) Three groups.
T: Four hundreds divided by 3. Distribute your disks
and cross off what you’ve used. What is the
quotient?
S: 1 hundred with a remainder of 1 hundred.
T: Tell me how to decompose the remaining 1 hundred.
S: Change 1 hundred for 10 tens.
T: Let’s decompose 1 hundred. Turn to your partner and decide together what to do next.
S: 10 tens and 2 tens makes 12 tens. Now, we have 12 tens to divide by 3.
T: Why didn’t we stop when we had a remainder of 1 hundred?
S: Because 1 hundred is just 10 tens, so you can keep dividing.
Problem 2
Display 783 ÷ 3.
T: Let’s solve 783 ÷ 3 using a place value chart and long
division side by side. Represent 783 in a place value
chart and prepare for long division. (Allow time for
students to draw disks and write the problem.)
Starting with the largest unit, tell me what to divide.
S: We divide 7 hundreds by 3.
T: Do that on your chart. 7 hundreds divided by 3.
What is the quotient?
S: 2 hundreds, with 1 hundred remaining.
T: (Record 2 hundreds. Point to the place value
chart.) In your place value chart, you recorded
2 hundreds three times. Say a multiplication
sentence that tells that.
S: 2 hundreds times 3 equals 6 hundreds.
As students say the multiplication equation, refer to the
algorithm, point to the 2 hundreds and the divisor, and
finally, record 6 hundreds.
T: (Point to the place value chart.) We started with 7 hundreds, distributed 6 hundreds, and have
1 hundred remaining. Tell me a subtraction sentence for that.
S: 7 hundreds minus 6 hundreds equals 1 hundred.
As students say the subtraction sentence, refer to the algorithm, point to the hundreds column, record a
subtraction line and symbol, and record 1 hundred.
T: (Point to the place value chart.) How many tens
remain to be divided?
S: 8 tens.
T: (Record an 8 next to the 1 hundred remainder.)
We decompose the remaining 1 hundred for 10 tens
and add on the 8 tens. Decompose the 1 hundred.
Say a division sentence for how we should distribute
18 tens.
S: 18 tens divided by 3 equals 6 tens.
As students say the division sentence, refer to the algorithm,
point to the 18 tens and the divisor, and then record 6 tens in
the quotient. Likewise, distribute the 18 tens in the place value
chart.
T: (Point to the place value chart.) You recorded 6 tens,
three times. Say a multiplication sentence that tells
that.
S: 6 tens times 3 equals 18 tens.
As students say the multiplication equation, refer to the algorithm, point to 6 tens, then the divisor, and
finally, record 18 tens.
T: (Point to the place value chart.) We renamed
10 tens, distributed all 18 tens, and have no tens
remaining. Say a subtraction sentence for that.
S: 18 tens minus 18 tens equals 0 tens.
As students say the subtraction equation, refer to the
algorithm, record a subtraction line and symbol, and 0 tens.
T: What is left to distribute?
S: The ones.
T: (Point to the place value chart.) How many ones
remain to be divided?
S: 3 ones.
T: (Record a 3 next to the 0 in the tens column.) Say a
division sentence for how we should distribute
3 ones.
S: 3 ones divided by 3 equals 1 one.
As students say the division sentence, refer to the algorithm, point to the 3 ones and the divisor, and then
record 1 one in the quotient.
T: (Point to the place value chart.) You recorded
1 one, three times. Say a multiplication sentence
that describes that.
S: 1 one times 3 equals 3 ones.
As students say the multiplication equation, refer to the
algorithm, point to 1 one, then the divisor, and finally,
record 3 ones.
T: (Point to the place value chart.) We have 3 ones, and we distributed 3 ones. Say a subtraction
sentence for that.
S: 3 ones minus 3 ones equals 0 ones.
Have students share with a partner how the model matches the algorithm. Note that both show equal
groups, as well as how both can be used to check their work using multiplication.
Problem 3
Display 546 ÷ 3.
T: Work together with a partner to solve 546 ÷ 3 using place value disks and long division. One partner
solves the problem using a place value chart and disks, while the other partner uses long division.
Work at the same pace, matching the action of the disks with the written method, and, of course,
compare your quotients.
Circulate as students are working to offer assistance as needed.
T: How was this problem unlike the others we solved today?
S: There were more hundreds left after we distributed them. We had to decompose 2 hundreds
this time.
A
Number Correct: _______
1. 4 3 23. 40 41 42
2. 6 3 24. 42 43 44
3. 8 3 25. 49 47 45
4. 5 10 26. 53 50 55
5. 5 12 27. 54 56 59
6. 5 14 28. 99 97 95
7. 8 7 29. 90 92 91
8. 9 11 30. 95 96 97
9. 11 15 31. 88 89 90
10. 15 17 32. 60 61 62
11. 19 16 33. 63 65 67
12. 14 11 34. 71 70 69
13. 13 12 35. 73 75 77
14. 18 17 36. 49 79 99
15. 19 20 37. 63 93 83
16. 21 23 38. 22 2 12
17. 25 19 39. 17 27 57
18. 29 27 40. 5 15 25
19. 31 30 41. 39 49 59
20. 33 37 42. 1 21 31
21. 9 2 43. 51 57 2
22. 51 2 44. 84 95 43
B
Number Correct: _______
Improvement: _______
Circle the Prime Number
1. 4 5 23. 42 41 40
2. 6 5 24. 44 43 42
3. 8 5 25. 45 47 49
4. 7 10 26. 53 55 50
5. 7 12 27. 56 54 59
6. 7 14 28. 95 97 99
7. 4 3 29. 90 91 92
8. 11 10 30. 99 98 97
9. 15 11 31. 90 89 88
10. 17 15 32. 67 65 63
11. 19 20 33. 62 61 60
12. 14 13 34. 72 71 70
13. 11 12 35. 77 75 73
14. 16 17 36. 27 67 77
15. 19 18 37. 39 49 59
16. 22 23 38. 32 2 22
17. 21 19 39. 19 49 69
18. 29 28 40. 5 15 55
19. 31 33 41. 99 49 59
20. 35 37 42. 1 21 41
21. 2 9 43. 45 51 2
22. 57 2 44. 48 85 67
Name Date
a. 324 ÷ 2
b. 344 ÷ 2
c. 483 ÷ 3
d. 549 ÷ 3
2. Model using place value disks and record using the algorithm.
a. 655 ÷ 5
Disks Algorithm
b. 726÷ 3
Disks Algorithm
c. 688 ÷ 4
Disks Algorithm
Name Date
Divide. Use place value disks to model each problem. Then, solve using the algorithm.
1. 423 ÷ 3
Disks Algorithm
2. 564 ÷ 4
Disks Algorithm
Name Date
a. 346 ÷ 2
b. 528 ÷ 2
c. 516 ÷ 3
d. 729 ÷ 3
2. Model using place value disks, and record using the algorithm.
a. 648 ÷ 4
Disks Algorithm
b. 755 ÷ 5
Disks Algorithm
c. 964 ÷ 4
Disks Algorithm
Lesson 28
Objective: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically.
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 389
T: (Beneath 3 tens × 1 ten = 3 hundreds, write 30 × 10 = 300. Project area model of 3 tens × 2 tens.
Beneath it, write 3 tens × 2 tens.) Say the number sentence in unit form.
S: 3 tens × 2 tens = 6 hundreds.
T: (Write 3 tens × 2 tens = 6 hundreds.) Write the number sentence in standard form.
S: (Write 30 × 20 = 600.)
T: Beneath 3 tens × 2 tens = 6 hundreds, write 30 × 20 = 600.
Continue with the following possible sequence: 3 tens × 3 tens, 3 tens × 5 tens, 2 tens × 1, 2 tens × 1 ten,
2 tens × 2 tens, 2 tens × 4 tens, and 3 tens × 4 tens.
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 390
Use 846 ÷ 2 to write a word problem. Then, draw an accompanying tape diagram and solve.
Note: This Application Problem connects to Lesson 27’s halving discussion in the Student Debrief. It also
reinforces the use of inverse operations to check calculations. It uses the division problem from the fluency
activity Divide Three-Digit Numbers. Encourage students to revise their word problems to use the word half.
Materials: (T) Thousands place value chart for dividing (Lesson 26 Template) (S) Personal white board,
thousands place value chart for dividing (Lesson 26 Template)
Problem 1: 297 ÷ 4
T: (Write 297 ÷ 4.) Set up 297 ÷ 4 in your thousands place value chart, and write the problem to solve
using long division.
T: Divide 2 hundreds by 4.
S: There aren’t enough hundreds to put them into 4 groups. I need to break them apart.
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 391
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 392
Problem 2
How many weeks are there in one year?
T: What do we need to know in order to solve this problem?
S: The number of days in one year.
T: How many days are in one year?
S: 365. Sometimes 366.
T: Good! Let’s use 365 days. What other information is
necessary?
S: There are 7 days in a week.
T: Okay, use a tape diagram to represent this problem. Show
your partner how you set up your tape diagram. Solve and
then check your work.
Allow students time to work independently. Circulate
and offer assistance as necessary.
T: Did you find that 365 could be divided by 7 evenly?
S: No, there was a remainder of 1.
T: In this problem, what does the remainder mean?
S: It means that there is one extra day.
T: Talk to your partner. How did you know it was an extra day?
S: Our whole, or total, represented the number of days in a
year, 365, so our remainder is days. 365 minus 52 groups
of 7 leaves 1 day remaining. 1 one is one day. 365 ones,
or days, is one year.
T: So, what would be a good sentence to write?
S: We can say, “There are 52 weeks and 1 day in one year.”
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 393
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 394
Name Date
1. Divide. Check your work by multiplying. Draw disks on a place value chart as needed.
a. 574 ÷ 2
b. 861 ÷ 3
c. 354 ÷ 2
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 395
d. 354 ÷ 3
e. 873 ÷ 4
f. 591 ÷ 5
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 396
g. 275 ÷ 3
h. 459 ÷ 5
i. 678 ÷ 4
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 397
j. 955 ÷ 4
2. Zach filled 581 one-liter bottles with apple cider. He distributed the bottles to 4 stores. Each store
received the same number of bottles. How many liter bottles did each of the stores receive?
Were there any bottles left over? If so, how many?
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 398
Name Date
1. Divide. Check your work by multiplying. Draw disks on a place value chart as needed.
a. 776 ÷ 2 b. 596 ÷ 3
2. A carton of milk contains 128 ounces. Sara’s son drinks 4 ounces of milk at each meal. How many
4-ounce servings will one carton of milk provide?
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 399
Name Date
1. Divide. Check your work by multiplying. Draw disks on a place value chart as needed.
a. 378 ÷ 2
b. 795 ÷ 3
c. 512 ÷ 4
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 400
d. 492 ÷ 4
e. 539 ÷ 3
f. 862 ÷ 5
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 401
g. 498 ÷ 3
h. 783 ÷ 5
i. 621 ÷ 4
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 402
j. 531 ÷ 4
2. Selena’s dog completed an obstacle course that was 932 meters long. There were 4 parts to the course,
all equal in length. How long was 1 part of the course?
Lesson 28: Represent and solve three-digit dividend division with divisors
of 2, 3, 4, and 5 numerically. 403
Lesson 29
Objective: Represent numerically four-digit dividend division with divisors
of 2, 3, 4, and 5, decomposing a remainder up to three times.
Note: This fluency activity reviews Lesson 26’s Concept Development and strengthens students’
understanding of place value’s role in the long division algorithm.
Repeat the process from Lesson 28 using the following possible sequence: 9 ones ÷ 3, 9 tens ÷ 3,
9 hundreds ÷ 3, 9 thousands ÷ 3, 16 tens ÷ 4, 15 hundreds ÷ 5, 27 hundreds ÷ 3, 24 tens ÷ 3, 32 tens ÷ 4,
40 tens ÷ 5, and 20 hundreds ÷ 5.
Problem 2
Ellie bought two packs of beads. Altogether, she has 1,254 beads. If the number of beads in each bag is the
same, how many beads are in three packs?
T: Draw something to help you solve this problem.
(Pause.) What did you draw?
S: (Method A) I drew a tape diagram. I made
2 units and labeled the whole as 1,254, since we
know that there are 1,254 beads in two packs.
Then, I just drew a third unit. I labeled all 3 units
with a question mark to represent how many
beads are in three packs.
S: (Method B) Not me. After I drew two
equal parts, I drew a second tape diagram
below with three equal parts.
T: What conclusions did you make from your
drawing?
S: We need to divide 1,254 by 2 to find out how
many beads are in each bag. This helps because
if we know how many beads are in one bag, we
can multiply by 3 to find out how many beads
are in three bags.
T: 1,254 divided by 2 is …?
S: 1,254 divided by 2 is 627.
T: Are we done?
S: No! We needed to multiply 627 by 3 to find the
total number of beads in three packs.
S: 627 times 3 equals 1,881. There are 1,881 beads in three packs.
Note: Clearly this is scripted to reflect a classroom where students have confidence with the tape diagram.
If students need a more guided approach, it should be provided.
Name Date
a. 1,672 ÷ 4
b. 1,578 ÷ 4
c. 6,948 ÷ 2
d. 8,949 ÷ 4
e. 7,569 ÷ 2
f. 7,569 ÷ 3
g. 7,955 ÷ 5
h. 7,574 ÷ 5
i. 7,469 ÷ 3
j. 9,956 ÷ 4
2. There are twice as many cows as goats on a farm. All the cows and goats have a total of 1,116 legs.
How many goats are there?
Name Date
a. 1,773 ÷ 3 b. 8,472 ÷ 5
2. The post office had an equal number of each of 4 types of stamps. There was a total of 1,784 stamps.
How many of each type of stamp did the post office have?
Name Date
a. 2,464 ÷ 4
b. 1,848 ÷ 3
c. 9,426 ÷ 3
d. 6,587 ÷ 2
e. 5,445 ÷ 3
f. 5,425 ÷ 2
g. 8,467 ÷ 3
h. 8,456 ÷ 3
i. 4,937 ÷ 4
j. 6,173 ÷ 5
2. A truck has 4 crates of apples. Each crate has an equal number of apples. Altogether, the truck is carrying
1,728 apples. How many apples are in 3 crates?
Lesson 30
Objective: Solve division problems with a zero in the dividend or with a
zero in the quotient.
Note: This fluency activity reviews the Concept Development from Lessons 10 and 11, in anticipation of
Topic H.
T: (Write 773 × 2 = .) On your personal white board, find the product using the standard algorithm.
S: (Solve.)
Repeat the process for the following possible sequence: 147 × 3, 1,605 × 3, and 5,741 × 5.
Note: This fluency activity reviews Lesson 26’s Concept Development and strengthens students’
understanding of place value’s role in the long division algorithm.
Repeat the process from Lesson 28 using the following possible sequence: 15 ones ÷ 3, 15 tens ÷ 3,
25 hundreds ÷ 5, 21 hundreds ÷ 3, 28 tens ÷ 4, 30 tens ÷ 5, and 40 hundreds ÷ 5.
Lesson 30: Solve division problems with a zero in the dividend or with a zero in
the quotient. 418
The store wanted to put 1,455 bottles of juice into packs of 4. How many complete packs can they make?
How many more bottles do they need to make another pack?
Note: This problem is a review of Lesson 29, which bridges dividing with remainders to the current lesson.
Lesson 30: Solve division problems with a zero in the dividend or with a zero in
the quotient. 419
Lesson 30: Solve division problems with a zero in the dividend or with a zero in
the quotient. 420
Lesson 30: Solve division problems with a zero in the dividend or with a zero in
the quotient. 421
Lesson 30: Solve division problems with a zero in the dividend or with a zero in
the quotient. 422
Name Date
1. 204 ÷ 4 2. 704 ÷ 3
3. 627 ÷ 3 4. 407 ÷ 2
Lesson 30: Solve division problems with a zero in the dividend or with a zero in
the quotient. 423
5. 760 ÷ 4 6. 5,120 ÷ 4
7. 3,070 ÷ 5 8. 6,706 ÷ 5
Lesson 30: Solve division problems with a zero in the dividend or with a zero in
the quotient. 424
b. How could you change the digit in the ones place of the whole so that there would be no remainder?
Explain how you determined your answer.
Lesson 30: Solve division problems with a zero in the dividend or with a zero in
the quotient. 425
Name Date
1. 380 ÷ 4 2. 7,040 ÷ 3
Lesson 30: Solve division problems with a zero in the dividend or with a zero in
the quotient. 426
Name Date
1. 409 ÷ 5 2. 503 ÷ 2
3. 831 ÷ 4 4. 602 ÷ 3
Lesson 30: Solve division problems with a zero in the dividend or with a zero in
the quotient. 427
5. 720 ÷ 3 6. 6,250 ÷ 5
7. 2,060 ÷ 5 8. 9,031 ÷ 2
Lesson 30: Solve division problems with a zero in the dividend or with a zero in
the quotient. 428
Lesson 30: Solve division problems with a zero in the dividend or with a zero in
the quotient. 429
Lesson 31
Objective: Interpret division word problems as either number of groups
unknown or group size unknown.
Note: This Sprint reviews Lesson 26’s Concept Development and strengthens students’ understanding of
place value’s role in the long division algorithm.
Lesson 31: Interpret division word problems as either number of groups unknown
or group size unknown. 430
1,624 shirts need to be sorted into 4 equal groups. How many shirts will be in each group?
NOTES ON
MULTIPLE MEANS
OF ENGAGEMENT:
Differentiate the difficulty of the
Application Problem by adjusting the
numbers.
Extend for students working above
Note: This Application Problem is a review of Lesson 30, grade level with these questions:
practicing with a zero in the quotient. In Problem 1 of the
How or why might the shirts be
Concept Development, students discuss whether the unknown sorted?
in this problem is the group size or the number of groups.
Were you able to predict that a zero
would be in the quotient? How?
Problem 1
Dr. Casey has 1,868 milliliters of Medicine T. She pours equal amounts of the medicine into 4 containers.
How many milliliters of medicine are in each container?
T: Can you draw something to help you solve this
problem? What can you draw? Go ahead and do so.
S: (Draw.)
T: What did you draw?
S: I drew a tape diagram with the whole labeled as
1,868 milliliters. I made the whole into 4 equal parts
because she poured the medicine into four containers.
MP.5 T: What are we trying to find out?
S: We need to find out how many milliliters are in each
container. We need to find the size of the group.
T: Right, we are finding the size of the group. We already
know how many groups there are, four.
T: Let’s label the unknown with t for Medicine T.
T: Solve for how much medicine will be in each container. (Allow time for students to work.)
S: There will be 467 milliliters in each container.
Lesson 31: Interpret division word problems as either number of groups unknown
or group size unknown. 431
T: Compare this tape diagram to the one you drew in the Application Problem. Discuss the similarities.
Were you solving for the number of groups or the size of the group?
S: Both tape diagrams are broken into four groups. Both show we were solving for the size of each
group.
Problem 2
T: (Draw or project the tape diagram shown below.) With your partner, discuss the tape diagram.
Then, create your own word problem to match. Remember to determine if you are finding the size
of the group or the number of groups. (We might also express this choice as the number of
measurements or the size of the measurements.)
Guide students to see that the equal partitioned parts of the tape diagram tell how many groups there are.
Students need to write a problem that asks for the number in each group or the size of the measurement.
Suggest the context of 168 liters of cleaning solution to be poured equally into 3 containers. Have a few sets
of partners share their word problems to verify students are writing to solve for group size unknown.
Problem 3
Two hundred thirty-two people are driving to a conference. If each car holds 4 people, including the driver,
how many cars will be needed?
T: Can you draw something to help you solve this problem? Go ahead. (Pause while students draw.)
What did you draw?
S: I drew a tape diagram with the whole labeled as 232 people.
T: Tell your partner how you partitioned the tape diagram. Are you finding the size of each group or
the number of groups?
S: We made 4 equal parts because each car has
4 people in it. We know the size of the
group. Each car has 4 people. We don’t
know how many groups or how many cars.
We showed that 4 are in each car, but we
don’t know the number of cars. Each unit
of the tape diagram shows there are 4 people
in each car, but we didn’t know how many
cars to draw.
T: We labeled the tape diagram to show 4 people in each car and used a question mark to show we
didn’t know how many cars were needed. Solve.
S: (Solve.)
Lesson 31: Interpret division word problems as either number of groups unknown
or group size unknown. 432
Guide students to see the first partitioned part of the tape diagram tells how many are in each group.
Students need to write a problem that asks for the number of groups and account for a remainder.
Suggest the context of 138 feet of rope cut into 3-foot segments, solving for the number of ropes, or groups
(measurements). Have a few sets of partners share their word problems to verify students are writing to find
the unknown number of groups. Have students compare and contrast the tape diagrams and word problems
for this problem and Problem 2 of the Concept Development.
Lesson 31: Interpret division word problems as either number of groups unknown
or group size unknown. 433
Invite students to review their solutions for the Problem Set. They should check work by comparing answers
with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can
be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the
lesson.
Any combination of the questions below may be used to
lead the discussion .
How and why are the tape diagrams in Problems
1 and 2 different?
Share your tape diagram for Problem 3. What led
you to draw a tape diagram to solve for the
number of groups?
For Problem 3, if our tape diagram shows the
whole divided into 3 equal groups instead, would
we get the wrong quotient?
Compare your tape diagrams for Problem 2 and
Problem 4. Describe how your tape diagrams
differ between one- and two-step problems.
If there are two unknowns, how do you
determine which one to solve first?
If, for Problem 5, the tape diagram was drawn to
show groups of 5, instead of 5 equal groups, how
might that lead to challenges when solving the
second part of the problem?
Lesson 31: Interpret division word problems as either number of groups unknown
or group size unknown. 434
A
Number Correct: _______
Divide.
2. 60 ÷ 2 = 24. 3,000 ÷ 5 =
3. 600 ÷ 2 = 25. 16 ÷ 4 =
5. 9÷3= 27. 18 ÷ 6 =
6. 90 ÷ 3 = 28. 1,800 ÷ 6 =
7. 900 ÷ 3 = 29. 28 ÷ 7 =
9. 10 ÷ 5 = 31. 48 ÷ 8 =
Lesson 31: Interpret division word problems as either number of groups unknown
or group size unknown. 435
B
Number Correct: _______
Improvement: _______
Divide.
2. 40 ÷ 2 = 24. 2,000 ÷ 5 =
3. 400 ÷ 2 = 25. 12 ÷ 4 =
5. 6÷3= 27. 21 ÷ 7 =
6. 60 ÷ 3 = 28. 2,100 ÷ 7 =
7. 600 ÷ 3 = 29. 18 ÷ 6 =
9. 10 ÷ 5 = 31. 54 ÷ 9 =
Lesson 31: Interpret division word problems as either number of groups unknown
or group size unknown. 436
Name Date
Draw a tape diagram and solve. The first two tape diagrams have been drawn for you. Identify if the group
size or the number of groups is unknown.
1. Monique needs exactly 4 plates on each table for the banquet. If she has 312 plates, how many tables is
she able to prepare?
312
4 …?...
2. 2,365 books were donated to an elementary school. If 5 classrooms shared the books equally, how many
books did each class receive?
2,365
3. If 1,503 kilograms of rice was packed in sacks weighing 3 kilograms each, how many sacks were packed?
Lesson 31: Interpret division word problems as either number of groups unknown
or group size unknown. 437
4. Rita made 5 batches of cookies. There was a total of 2,400 cookies. If each batch contained the same
number of cookies, how many cookies were in 4 batches?
5. Every day, Sarah drives the same distance to work and back home. If Sarah drove 1,005 miles in 5 days,
how far did Sarah drive in 3 days?
Lesson 31: Interpret division word problems as either number of groups unknown
or group size unknown. 438
Name Date
Solve the following problems. Draw tape diagrams to help you solve. Identify if the group size or the number
of groups is unknown.
1. 572 cars were parked in a parking garage. The same number of cars was parked on each floor. If there
were 4 floors, how many cars were parked on each floor?
2. 356 kilograms of flour were packed into sacks holding 2 kilograms each. How many sacks were packed?
Lesson 31: Interpret division word problems as either number of groups unknown
or group size unknown. 439
Name Date
Solve the following problems. Draw tape diagrams to help you solve. Identify if the group size or the number
of groups is unknown.
1. 500 milliliters of juice was shared equally by 4 children. How many milliliters of juice did each child get?
2. Kelly separated 618 cookies into baggies. Each baggie contained 3 cookies. How many baggies of cookies
did Kelly make?
3. Jeff biked the same distance each day for 5 days. If he traveled 350 miles altogether, how many miles did
he travel each day?
Lesson 31: Interpret division word problems as either number of groups unknown
or group size unknown. 440
4. A piece of ribbon 876 inches long was cut by a machine into 4-inch long strips to be made into bows.
How many strips were cut?
5. Five Martians equally share 1,940 Groblarx fruits. How many Groblarx fruits will 3 of the Martians
receive?
Lesson 31: Interpret division word problems as either number of groups unknown
or group size unknown. 441
Lesson 32
Objective: Interpret and find whole number quotients and remainders to
solve one-step division word problems with larger divisors of 6, 7, 8, and 9.
Quadrilaterals (4 minutes)
Materials: (T) Shapes (Fluency Template)
Attributes
Note: This fluency activity reviews Grade 3
Number of Sides
geometry concepts in anticipation of Module 4
Length of Sides
content. The sheet can be duplicated for
Size of Angle
students, if you prefer. Right Angle
T: (Project the shapes template and the list
of attributes.) Take one minute to
discuss the attributes of the shapes you Shapes
see. You can use the list to help.
S: Some have right angles. All have Quadrilateral
straight sides. They all have 4 sides. Rhombus
B and G and maybe H and K have all Square
equal sides. I’m not really sure. Rectangle
Parallelogram
T: If we wanted to verify whether the sides
Trapezoid
are equal, what would we do?
S: Measure!
T: What about the angles? How could you verify that they’re right angles?
S: I could compare it to something that I know is a right angle.
Lesson 32: Interpret and find whole number quotients and remainders to solve
one-step division word problems with larger divisors of 6, 7, 8, and 9. 442
T: (Post the shape names.) Now, look at the shape names. Determine, to the best of your ability,
which shapes might fall into each category.
S: B and G might be squares. All of them are quadrilaterals. H and K might be rhombuses.
It’s hard to know if their sides are equal. D and I are rectangles. Oh yeah, and B and G are, too.
L and A look like trapezoids.
T: Which are quadrilaterals?
S: All of them.
T: Which shapes appear to be rectangles?
S: B, D, G, and I.
T: Which appear to have opposite sides of equal length but are not rectangles?
S: C, H, K. A and L have one pair of opposite sides that look the same.
T: Squares are rhombuses with right angles. Do you see any other shapes that might have four equal
sides without right angles?
S: H and K.
Lesson 32: Interpret and find whole number quotients and remainders to solve
one-step division word problems with larger divisors of 6, 7, 8, and 9. 443
Problem 1
We all know there are 7 days in a week. How many weeks are in 259 days?
T: Draw what we know and what we need to know
on a tape diagram.
S: I labeled the whole as 259 days. Then, I put a 7 in one
part because there are 7 days in each week. We don’t
know how many groups of 7 days there are.
T: How did you represent the number of weeks that are
unknown?
S: I labeled the rest of the tape diagram with a question
mark.
There are 37 weeks in 259 days.
T: Solve for how many weeks there are in 259 days.
S: There are 37 weeks in 259 days.
Lesson 32: Interpret and find whole number quotients and remainders to solve
one-step division word problems with larger divisors of 6, 7, 8, and 9. 444
T: The divisor in this problem is larger than in many division problems we have solved. Tell your
partner a strategy you can use to find the quotient when dividing by 7.
S: 25 tens divided by 7 is easy. It’s 3 tens with 4 tens left over. I counted by sevens, 10 at a time:
10 sevens is 70, 20 sevens is 140, 30 sevens is 210, and 40 sevens would be too big. So, I got
30 sevens with 49 left over. That still means you get 3 tens in the quotient. One way is like we
did with place value disks. The other is like we did with the area model. But they’ll both give the
same answer.
T: Either way of thinking will work for finding the quotient. When our divisor is large, how do I check
to see if my quotient and remainder are correct?
S: The same way we always do! It’s no different for big divisors than for small divisors—multiply the
number of groups times the size of each group. And, don’t forget to add the remainder.
Multiply the divisor by the quotient, and add the remainder.
T: So, what we learned about small divisors still helps us now!
Problem 2
Everyone is given the same number of colored pencils in art class. If there are 249 colored pencils and
8 students, how many pencils does each student receive?
T: Draw a tape diagram to represent the problem. Describe the parts of your tape diagram to your
partner.
S: I recorded and labeled the total of 249 pencils. Then, I made 8 equal parts because there are
8 students. I need to solve for how many in each group, so I put a question mark in one part to
show that I need to solve for how many pencils each student will get.
T: Solve for how many pencils each student will receive. (Allow students time to work.)
S: Each student will receive 31 colored pencils. There will be 1 pencil left over.
T: Does your drawing of the tape diagram account for the remaining pencil? Let’s revise our tape
diagram to show the remainder.
S: I can shade a small portion at the end of the tape diagram to represent the remaining pencil. I will
have to resize each of the eight parts to make them equal.
T: Discuss a strategy you might have used when dividing by a larger divisor, like 8.
S: I counted by 8 tens. 8 tens, 16 tens, 24 tens. I know there are 2 fours in each eight. There are
6 fours in 24. So, half of 6 is 3. There are 3 eights in 24. I used my facts. I know 8 times 3 tens is
24 tens.
Lesson 32: Interpret and find whole number quotients and remainders to solve
one-step division word problems with larger divisors of 6, 7, 8, and 9. 445
Problem 3
Mr. Hughes has 155 meters of volleyball netting. How many
nets can he make if each court requires 9 meters of netting? NOTES ON
T: Draw a tape diagram to represent the problem. MULTIPLE MEANS
Describe the parts of your tape diagram to your OF REPRESENTATION:
partner. English language learners and others
S: My tape diagram shows a total of 155. I partitioned may benefit from a brief explanation of
one section for 9 meters. I don’t know how many nets the terms volleyball, netting, and court.
he can make, but I do know the length of each.
T: Solve for how many nets can be made using long division.
S: Seventeen nets can be made, but 2 meters of netting will be left over.
T: Does your drawing of the tape diagram account for the remaining netting? Let’s revise our tape
diagram to show the remainder.
S: I can shade a small portion at the end of the tape diagram to represent the remaining 2 meters.
T: What strategy did you use for dividing with the divisor of 9?
S: I counted by 9 tens. 9 tens, 18 tens. One hundred eighty was too big. I used a special strategy.
I made 10 nets, which meant I used 90 meters of netting. That left 65 meters. Nine times 7 is 63 so
that meant 7 more nets and 2 meters left over. I used my nines facts.
Lesson 32: Interpret and find whole number quotients and remainders to solve
one-step division word problems with larger divisors of 6, 7, 8, and 9. 446
Lesson 32: Interpret and find whole number quotients and remainders to solve
one-step division word problems with larger divisors of 6, 7, 8, and 9. 447
Name Date
Solve the following problems. Draw tape diagrams to help you solve. If there is a remainder, shade in a small
portion of the tape diagram to represent that portion of the whole.
1. A concert hall contains 8 sections of seats with the same number of seats in each section. If there are
248 seats, how many seats are in each section?
2. In one day, the bakery made 719 bagels. The bagels were divided into 9 equal shipments. A few bagels
were left over and given to the baker. How many bagels did the baker get?
3. The sweet shop has 614 pieces of candy. They packed the candy into bags with 7 pieces in each bag.
How many bags of candy did they fill? How many pieces of candy were left?
Lesson 32: Interpret and find whole number quotients and remainders to solve
one-step division word problems with larger divisors of 6, 7, 8, and 9. 448
4. There were 904 children signed up for the relay race. If there were 6 children on each team, how many
teams were made? The remaining children served as referees. How many children served as referees?
5. 1,188 kilograms of rice are divided into 7 sacks. How many kilograms of rice are in 6 sacks of rice?
How many kilograms of rice remain?
Lesson 32: Interpret and find whole number quotients and remainders to solve
one-step division word problems with larger divisors of 6, 7, 8, and 9. 449
Name Date
Solve the following problems. Draw tape diagrams to help you solve. If there is a remainder, shade in a small
portion of the tape diagram to represent that portion of the whole.
1. Mr. Foote needs exactly 6 folders for each fourth-grade student at Hoover Elementary School. If he
bought 726 folders, to how many students can he supply folders?
2. Mrs. Terrance has a large bin of 236 crayons. She divides them equally among four containers.
How many crayons does Mrs. Terrance have in each container?
Lesson 32: Interpret and find whole number quotients and remainders to solve
one-step division word problems with larger divisors of 6, 7, 8, and 9. 450
Name Date
Solve the following problems. Draw tape diagrams to help you solve. If there is a remainder, shade in a small
portion of the tape diagram to represent that portion of the whole.
1. Meneca bought a package of 435 party favors to give to the guests at her birthday party. She calculated
that she could give 9 party favors to each guest. How many guests is she expecting?
2. 4,000 pencils were donated to an elementary school. If 8 classrooms shared the pencils equally, how
many pencils did each class receive?
3. 2,008 kilograms of potatoes were packed into sacks weighing 8 kilograms each. How many sacks were
packed?
Lesson 32: Interpret and find whole number quotients and remainders to solve
one-step division word problems with larger divisors of 6, 7, 8, and 9. 451
4. A baker made 7 batches of muffins. There was a total of 252 muffins. If there was the same number of
muffins in each batch, how many muffins were in a batch?
5. Samantha ran 3,003 meters in 7 days. If she ran the same distance each day, how far did Samantha run in
3 days?
Lesson 32: Interpret and find whole number quotients and remainders to solve
one-step division word problems with larger divisors of 6, 7, 8, and 9. 452
shapes
Lesson 32: Interpret and find whole number quotients and remainders to solve
one-step division word problems with larger divisors of 6, 7, 8, and 9. 453
Lesson 33
Objective: Explain the connection of the area model of division to the long
division algorithm for three- and four-digit dividends.
Quadrilaterals (4 minutes)
Materials: (T) Shapes (Lesson 32 Fluency Template) (S) Personal white board
Lesson 33: Explain the connection of the area model of division to the long
division algorithm for three- and four-digit dividends. 454
T: Rhombuses are quadrilaterals with four equal sides. Is this polygon a rhombus?
S: Yes.
T: Is it a rectangle?
S: Yes.
T: (Point to the rhombus that is not a square.) This polygon has four equal sides, but the angles are not
the same. Write the name of this quadrilateral.
S: (Write rhombus.)
T: Is the square also a rhombus?
S: Yes!
T: (Point to the rectangle that is not a square.) This polygon has four equal angles, but the sides are not
equal. Write the name of this quadrilateral.
S: (Write rectangle.)
T: Draw a quadrilateral that is not a square, rhombus, or rectangle.
Lesson 33: Explain the connection of the area model of division to the long
division algorithm for three- and four-digit dividends. 455
Write an equation to find the unknown length of each rectangle. Then, find the sum of the two unknown
lengths.
3m 72 square
600 square m 3m
m
Note: This Application Problem serves as an introduction to today’s Concept Development, in which students
find the total unknown length of a rectangle with an area of 672 square meters.
Problem 1
672 ÷ 3 and 1,344 ÷ 6
T: Draw a rectangle with an area of 672 square
inches and a width of 3 inches.
S: (Draw.)
T: Draw a new rectangle with the same area directly
below, but partitioned to make it easy for you to
divide each part using mental math and your
knowledge of place value. (Allow time for students
to work.)
T: Share with a partner how you
partitioned your new rectangle.
S: I made one part 6 hundred, two parts
of 3 tens, and one part 12 ones. I made
two parts of 3 hundreds, one part of 6 tens, and
one part 12 ones. I made mine one part
6 hundred and two parts 36.
T: Draw a number bond to match the whole and
parts of your rectangles.
S: (Draw bonds as pictured to the right.)
Lesson 33: Explain the connection of the area model of division to the long
division algorithm for three- and four-digit dividends. 456
T: Find the unknown side lengths of the smaller rectangles, and add them to find the length of the
largest rectangle.
T: Take a moment to record the number sentences, reviewing with your partner the connection to
both the number bond and the area model.
Those who finish early can find other ways to decompose the rectangle or work with 1,344 ÷ 6.
T: (Allow students to work for about four minutes.)
T: What were some ways you found to partition 1,344 to divide it easily by 6?
S: We chopped it into 12 hundreds, 12 tens, and 24 ones. We decomposed it as 2 six hundreds,
2 sixties, and 24. I realized 1,344 is double 672. But, 6 is double 3 and that’s like the associative
property 224 × 2 × 3, so 1,344 ÷ 6 equals 672 ÷ 3.
T: How can we see from our bonds that 1,344 is double 672?
S: When we chopped up the rectangles, I saw 600, 60, and 12 made 672, and the chopped up rectangle
for 1,344 had two of all those!
T: Explain to your partner why different ways of partitioning give us the same correct side length.
S: You are starting with the same amount of area but just chopping it up differently. The sum of the
lengths is the same as the whole length. You can take a total, break it into two or more parts, and
divide each of them separately.
Lesson 33: Explain the connection of the area model of division to the long
division algorithm for three- and four-digit dividends. 457
Problem 2
672 ÷ 3
T: (Write 672 ÷ 3.) This expression can describe a rectangle with an area of 672 square units. We are
trying to find out the length of the unknown side.
T: What is the known side length?
S: 3.
T: (Draw a rectangle with a width of 3.) Three times
how many hundreds gets us as close as possible
to an area of 6 hundred square units? (Point to
the 6 hundreds of the dividend.)
S: 2 hundreds.
T: Let’s give 2 hundreds to the length. (Label 2
lengths of hundreds.) Let’s record the 2 hundreds
in the hundreds place.
T: What is 3 times 2 hundreds?
S: 6 hundreds. (Record 6 below the
6 hundreds.)
T: How many square units is that?
S: 600 square units. (Record 600 square units
in the rectangle.)
T: How many hundreds remain?
S: Zero.
T: (Record 0 hundreds below the
6 hundreds.) 0 hundreds and 7 tens is…?
(Record the 7 tens to the right of the
0 hundreds.)
S: 7 tens.
T: We have 70 square units left with a width of
3. (Point to the 7 tens in the algorithm.)
Three times how many tens gets us as close
as possible to 7 tens?
S: 2 tens.
T: Let’s give 2 tens to the length.
T: 3 times 2 tens is?
S: 6 tens.
T: How many square units?
S: 60 square units.
T: 7 tens minus 6 tens is?
S: 1 ten.
Lesson 33: Explain the connection of the area model of division to the long
division algorithm for three- and four-digit dividends. 458
T: That is 10 square units of area to add to 2 square units. The remaining area is…?
S: 12 square units!
T: Three times how many ones gets us as close as possible to 12 ones?
S: 4 ones.
T: Let’s give 4 ones to the length.
T: Three times 4 ones is…?
S: 12 ones.
T: Do we have any remaining area?
S: No!
T: What is the length of the unknown side?
S: 224 length units.
T: Review our drawings and our process with your partner. Try to reconstruct what we did step by
step before we try another one. (Allow students time to review.)
T: We solved 672 divided by 3 in two very different ways using the area model. First we started with
the whole rectangle and partitioned it. The second way was to go one place value at a time and
make the whole rectangle from parts.
Give students the chance to try the following problems in partners, in a small group with the teacher, or
independently, as they are able.
539 ÷ 2
This first practice problem has an easy divisor and a remainder in the ones. Guide students to determine the
greatest length possible first for the remaining area at each place value.
438 ÷ 5
This next practice problem involves seeing the first area as
40 tens and having a remainder of 3 in the ones. NOTES ON
1,216 ÷ 4 MULTIPLE MEANS
OF ACTION AND
The final practice problem involves a four-digit number.
EXPRESSION:
Like the previous example, students must see the first area as
Guide English language learners and
12 hundreds and the next area as 16 ones.
students working below grade level
who may not complete the Problem
Problem Set (13 minutes) Set in the allotted 13 minutes to set
specific goals for their work. After
Students should do their personal best to complete the Problem
briefly considering their progress,
Set within the allotted 13 minutes. For some classes, it may be strengths, and weaknesses, have
appropriate to modify the assignment by specifying which students choose the problems they will
problems they work on first. Some problems do not specify a solve strategically. For example, a
method for solving. Students should solve these problems using learner who is perfecting sequencing
the RDW approach used for Application Problems. his written explanations might choose
Problem 2. Connect this short-term
goal to long-term goals.
Lesson 33: Explain the connection of the area model of division to the long
division algorithm for three- and four-digit dividends. 459
Lesson 33: Explain the connection of the area model of division to the long
division algorithm for three- and four-digit dividends. 460
Name Date
b. Show a number bond to represent Ursula’s area model, and represent the total length using the
distributive property.
2. a. Solve 960 ÷ 4 using the area model. There is no remainder in this problem.
b. Draw a number bond and use the long division algorithm to record your work from Part (a).
Lesson 33: Explain the connection of the area model of division to the long
division algorithm for three- and four-digit dividends. 461
b. Draw a number bond to represent this c. Record your work using the long division
problem. algorithm.
b. Draw a number bond to represent this c. Record your work using the long division
problem. algorithm.
Lesson 33: Explain the connection of the area model of division to the long
division algorithm for three- and four-digit dividends. 462
Name Date
b. Show a number bond to represent Anna’s area model, and represent the total length using the
distributive property.
b. Draw a number bond to represent this c. Record your work using the long division
problem. algorithm.
Lesson 33: Explain the connection of the area model of division to the long
division algorithm for three- and four-digit dividends. 463
Name Date
b. Show a number bond to represent Arabelle’s area model, and represent the total length using the
distributive property.
2. a. Solve 816 ÷ 4 using the area model. There is no remainder in this problem.
b. Draw a number bond and use a written method to record your work from Part (a).
Lesson 33: Explain the connection of the area model of division to the long
division algorithm for three- and four-digit dividends. 464
b. Draw a number bond to represent this c. Record your work using the long division
problem. algorithm.
b. Draw a number bond to represent this c. Record your work using the long division
problem. algorithm.
Lesson 33: Explain the connection of the area model of division to the long
division algorithm for three- and four-digit dividends. 465
4
GRADE
Mathematics Curriculum
GRADE 4 • MODULE 3
Topic H
Multiplication of Two-Digit by Two-
Digit Numbers
4.NBT.5, 4.OA.3, 4.MD.3
Focus Standard: 4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and
multiply two two-digit numbers, using strategies based on place value and the
properties of operations. Illustrate and explain the calculation by using equations,
rectangular arrays, and/or area models.
Instructional Days: 5
Coherence -Links from: G3–M1 Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10
G3–M3 Multiplication and Division with Units of 0, 1, 6–9, and Multiples of 10
-Links to: G5–M2 Multi-Digit Whole Number and Decimal Fraction Operations
Topic H culminates at the most abstract level with Lesson 38 as students are introduced to the multiplication
algorithm for two-digit by two-digit numbers. Knowledge from Lessons 34–37 provides a firm foundation for
understanding the process of the algorithm as students make connections from the area model to partial
products to the standard algorithm (4.NBT.5). Students see that partial products written vertically are the
same as those obtained via the distributive property: 4 twenty-sixes + 30 twenty-sixes = 104 + 780 = 884.
Objective 2: Multiply two-digit multiples of 10 by two-digit numbers using the area model.
(Lesson 35)
Objective 4: Transition from four partial products to the standard algorithm for two-digit by two-digit
multiplication.
(Lessons 37–38)
Lesson 34
Objective: Multiply two-digit multiples of 10 by two-digit numbers using a
place value chart.
Note: This fluency activity reviews Grade 3 geometry and fraction concepts in anticipation of Modules 4 and 5.
Accept reasonable drawings. Using rulers and protractors is not necessary to review the concept and takes too
long.
T: On your personal white boards, draw a quadrilateral with 4 equal sides and 4 right angles.
S: (Draw.)
T: What’s the name of a quadrilateral with 4 equal sides and 4 right angles?
S: Square.
T: Partition the square into 3 equal parts.
S: (Partition.)
T: Shade in 1 part of 3.
S: (Shade.)
T: Write the fraction of the square that’s shaded.
1
S: (Write .)
3
Repeat the process, partitioning a rhombus into fourths, a rectangle into fifths, and a rectangle into eighths.
Note: This fluency activity reviews content from Lessons 32 and 33.
T: (Write 732 ÷ 6.) Solve this problem by drawing place value disks.
S: (Solve.)
T: Solve 732 ÷ 6 using the area model.
S: (Solve.)
T: Solve 732 ÷ 6 using the standard algorithm.
S: (Solve.)
Continue with this possible suggestion: 970 ÷ 8.
Mr. Goggins planted 10 rows of beans, 10 rows of squash, 10 rows of tomatoes, and 10 rows of cucumbers in
his garden. He put 22 plants in each row. Draw an area model, label each part, and then write an expression
that represents the total number of plants in the garden.
(4 × 10) × 22
Note: This Application Problem builds on Topic B, where students learned to multiply by multiples of 10, and
Topic C, where students learned to multiply two-digit by one-digit numbers using an area model. This
Application Problem helps bridge to today’s lesson as students learn to multiply multiples of 10 by two-digit
numbers.
Materials: (S) Personal white board, thousands place value chart (Lesson 4 Template)
Write:
40 × 22 = (4 × 10) × 22
40 × 22 = 4 × (10 × 22)
40 × 22 = 10 × (4 × 22)
T: Show 22 on your place value chart.
T: Show 10 times as many. 10 × 22 is?
S: (Draw disks to show 22, and then draw arrows to show 10 times that amount.)
T: How many hundreds? How many tens?
S: 2 hundreds and 2 tens.
T: Show 4 times as many.
S: (Draw disks to show 3 more groups of 2 hundreds 2 tens.)
T: Tell how many you have now.
S: 8 hundreds and 8 tens.
T: What number does that represent? Say the number sentence.
S: 4 × (10 × 22) = 880. 40 × 22 = 880.
Repeat the process, this time beginning by multiplying 22 by 4 as in the model above and to the right.
(Write 40 × 22 = (10 × 4) × 22 = 10 × (4 × 22).)
Next, have students see that they can conceive of the problem as 40 times 22, as pictured below, without
breaking the process into the two steps of multiplying by 4 and 10 in whatever order.
Name Date
1. Use the associative property to rewrite each expression. Solve using disks, and then complete the
number sentences.
= ( ____ × 10) × 24
= _______
= (4 × 10) × _____
= 4 × (10 × ___ )
= _______
= (3 × ____ ) × _____
= 3 × (10 × _____ )
= _______
b. 40 × 31
4. Use the distributive property to solve the following problems. Distribute the second factor.
a. 40 × 34 b. 60 × 25
Name Date
1. Use the associative property to rewrite each expression. Solve using disks, and then complete the
number sentences.
20 × 41
hundreds tens ones
____ × ____ × ____ = ____
60 × 32
Name Date
1. Use the associative property to rewrite each expression. Solve using disks, and then complete the
number sentences.
= (____ × 10) × 34
= _______
= (3 × 10) × _____
= 3 × (10 × ___)
= _______
= (3 × ____) × _____
= 3 × (10 × _____)
= _______
4. Use the distributive property to solve the following. Distribute the second factor.
a. 40 × 43 b. 70 × 23
Lesson 35
Objective: Multiply two-digit multiples of 10 by two-digit numbers using
the area model.
Notes: This fluency activity reviews Grade 3 geometry and fraction concepts in anticipation of Modules 4 and 5.
Accept reasonable drawings. Using rulers and protractors is not necessary to review the concept and takes too
long.
T: On your personal white boards, write the name for any four-sided figure.
S: (Write quadrilateral.)
T: Draw a quadrilateral that has 4 right angles but not 4 equal sides.
S: (Draw a rectangle that is not a square.)
T: Partition the rectangle into 3 equal parts.
S: (Partition.)
T: Label the whole rectangle as 1. Write the unit fraction in each part.
Continue partitioning and labeling with the following possible sequence: a square as 4 fourths, a rhombus as
2 halves, a square as 5 fifths, and a rectangle as 6 sixths.
Lesson 35: Multiply two-digit multiples of 10 by two-digit numbers using the area
model. 480
Note: This fluency activity reviews content from Lessons 32 and 33.
T: (Write 348 ÷ 6.) Find the quotient using place value disks.
S: (Solve.)
T: Find the quotient using the area model.
S: (Solve.)
T: Find the quotient using the standard algorithm.
S: (Solve.)
Continue for 2,816 ÷ 8.
For 30 days out of one month, Katie exercised for 25 minutes a day.
What is the total number of minutes that Katie exercised? Solve using a
place value chart.
Note: This Application Problem builds on the content of Lesson 34 by
using a place value chart to represent and then multiply a multiple of 10
by a two-digit number. Although some students may easily solve this problem
using mental math, encourage them to see that the model verifies their mental
math skills. Students can use their mental math and place value chart solution
to verify their answer in Problem 1 of the Concept Development.
Lesson 35: Multiply two-digit multiples of 10 by two-digit numbers using the area
model. 481
Lesson 35: Multiply two-digit multiples of 10 by two-digit numbers using the area
model. 482
T: I noticed this time you gave me the units of both factors. Why?
S: They were both tens. This way, I can just think of 3 × 2, and all I have to do is figure out what the
new unit will be. Tens times tens gives me hundreds.
T: Find the product for 30 × 25, and discuss with your partner how the two products, (3 × 25) and
(30 × 25), are related.
S: One was 75 and the other was 750. That’s 10 times as much. The first was 6 tens plus 15 ones.
The other was 6 hundreds plus 15 tens. For the first one, we did 3 × 5 and 3 × 20. On the second,
we just multiplied the 3 by 10 and got 30 × 5 and 30 × 20. That’s 150 + 600, or 750. The only
difference was the unit on the 3. 3 ones were changed to 3
tens.
Lesson 35: Multiply two-digit multiples of 10 by two-digit numbers using the area
model. 483
Lesson 35: Multiply two-digit multiples of 10 by two-digit numbers using the area
model. 484
Lesson 35: Multiply two-digit multiples of 10 by two-digit numbers using the area
model. 485
Name Date
Use an area model to represent the following expressions. Then, record the partial products and solve.
1. 20 × 22
2 2
× 20
2. 50 × 41
4 1
× 50
3. 60 × 73
7 3
× 60
Lesson 35: Multiply two-digit multiples of 10 by two-digit numbers using the area
model. 486
Draw an area model to represent the following expressions. Then, record the partial products vertically and
solve.
4. 80 × 32 5. 70 × 54
Visualize the area model, and solve the following expressions numerically.
6. 30 × 68 7. 60 × 34
8. 40 × 55 9. 80 × 55
Lesson 35: Multiply two-digit multiples of 10 by two-digit numbers using the area
model. 487
Name Date
Use an area model to represent the following expressions. Then, record the partial products and solve.
1. 30 × 93
9 3
× 30
2. 40 × 76
7 6
× 40
Lesson 35: Multiply two-digit multiples of 10 by two-digit numbers using the area
model. 488
Name Date
Use an area model to represent the following expressions. Then, record the partial products and solve.
1. 30 × 17
1 7
× 30
2. 40 × 58
5 8
× 40
3. 50 × 38
3 8
× 50
Lesson 35: Multiply two-digit multiples of 10 by two-digit numbers using the area
model. 489
Draw an area model to represent the following expressions. Then, record the partial products vertically and
solve.
4. 60 × 19 5. 20 × 44
Visualize the area model, and solve the following expressions numerically.
6. 20 × 88 7. 30 × 88
8. 70 × 47 9. 80 × 65
Lesson 35: Multiply two-digit multiples of 10 by two-digit numbers using the area
model. 490
Lesson 36
Objective: Multiply two-digit by two-digit numbers using four partial
products.
Note: This fluency activity reviews Grade 3 geometry and fraction concepts in anticipation of Modules 4 and
5. Accept reasonable drawings. Using rulers is not necessary to review the concept and takes too long.
T: On your personal white boards, write the name for any four-sided figure.
S: (Write quadrilateral.)
T: Draw a quadrilateral that has 4 right angles and 4 equal sides.
S: (Draw a square.)
T: Partition the square into 4 equal parts.
S: (Partition.)
T: Shade in 1 of the parts.
S: (Shade.)
T: Write the fraction of the square that you shaded.
1
S: (Write .)
4
1
Continue with the following possible sequence: Partition a rectangle into 5 equal parts, shading ; partition
5
1 1
a rhombus into 2 equal parts, shading ; partition a square into 12 equal parts, shading ; and partition a
2 12
1
rectangle into 8 equal parts, shading .
8
Lesson 36: Multiply two-digit by two-digit numbers using four partial products.
491
Lesson 36: Multiply two-digit by two-digit numbers using four partial products.
492
Lesson 36: Multiply two-digit by two-digit numbers using four partial products.
493
Problem 2: Find the product of 23 and 31 using an area model and partial products to solve.
T: Let’s solve 23 × 31 using area to model the
product.
T: (Draw a rectangle.) Break down the length and
width according to place value units.
S: 2 tens 3 ones and 3 tens 1 one. 20 and 3.
30 and 1.
T: (Draw one vertical and one horizontal line
subdividing the rectangle.) Turn and tell your
partner the length and width of each of the 4
smaller rectangles we just created.
S: 3 and 1, 3 and 30, 20 and 1, and 20 and 30.
T: Using the area model that you just drew, write an
equation that represents the product of 23 and 31 as
the sum of those four areas. NOTES ON
S: 23 × 31 = (3 × 1) + (3 × 30) + (20 × 1) + (20 × 30). MULTIPLE MEANS
T: Now, we are ready to solve! OF REPRESENTATION:
T: Let’s look at a way to record the partial products. Students working below grade level
(Write 23 × 31 vertically.) Recall that when we may benefit from continuing to write
multiplied a one-digit number by a two-, three-, or out the expressions used to find each
four-digit number, we recorded the partial products. of the partial products. Students may
We also recorded partial products when we multiplied write the expressions in numerical
a two-digit number by a multiple of 10. Let’s put it all form or in unit form.
together and do precisely the same thing here. To help solidify place value, it might
also be helpful to have students shade,
T: (Point to the area model and the expression showing
in different colors, the rectangles that
the distributive property.) What is the product of 3 represent the ones, tens, and
ones and 1 one? hundreds.
S: 3 ones. Students working above grade level
T: Record the product below. Draw an arrow connecting may be ready to use the four partial
the rectangle with the corresponding partial product. product algorithm and can be
How about 3 ones times 3 tens? encouraged to do so.
S: 9 tens.
T: Record the product below the first partial product. Draw an arrow connecting the rectangle with
the corresponding partial product. What is 2 tens times 1 one?
S: 2 tens or 20.
T: As before, record the partial product below the other two and do the same with 2 tens times 3 tens.
T: Draw arrows to connect the new partial products with the corresponding rectangles. Now, let’s add
the partial products together. What is the sum?
S: The sum is 713. That means that 23 × 31 = 713.
Lesson 36: Multiply two-digit by two-digit numbers using four partial products.
494
Problem 3: Find the product of 26 and 34 using partial products. Verify partial products using the area
model.
T: Draw an area model to represent 26 × 34.
T: How do I find the area of the smallest rectangle?
S: Multiply 6 ones times 4 ones.
T: Point to 6 ones times 4 ones in the algorithm. What is 6 ones times 4 ones?
S: 24 ones.
T: Record 24 beneath the expression and in the
corresponding area.
T: Point to the next area to solve for. Tell me the
expression.
S: 6 ones times 3 tens.
T: Locate those numbers in the algorithm. Solve
for 6 ones times 3 tens.
S: 18 tens.
T: Record 18 tens under the expression.
S: We can also record 18 tens in this rectangle.
Continue connecting the width and length of each rectangle in the model to the location of those units in the
algorithm. Record the partial products first under the expression and then inside the area.
T: What is the last step?
S: Add together all of the partial products. 24 + 180 + 80 + 600 = 884. 26 × 34 = 884.
Problem 4: Find the product of 26 and 34 without using an area model. Record the partial products to solve.
T: Take a mental picture of your area model before you
erase it, the partial products, and the final product.
T: When we multiplied these numbers before, with
what did we start?
S: 6 ones × 4 ones.
T: Do you see 6 ones × 4 ones?
S: Yes.
Students point to 6 ones × 4 ones. You might model on the
board as students also record.
T: What is 6 ones × 4 ones?
S: 24 ones.
T: Record 24 ones as a partial product.
T: What did we multiply next?
S: 6 ones × 3 tens. That’s 18 tens or 180.
T: Where do we record 180?
S: Below the 24.
Lesson 36: Multiply two-digit by two-digit numbers using four partial products.
495
T: Now what?
S: We multiply the tens. 2 tens × 4 ones and then 2 tens × 3 tens.
T: What are 2 tens × 4 ones and 2 tens × 3 tens?
S: 8 tens and 6 hundreds.
T: Let’s record these as partial products. Notice that we have four partial products. Let’s again identify
from where they came. (Point to each part of the algorithm as students chorally read the
expressions used to solve the two-digit by two-digit multiplication.)
S: 6 ones × 4 ones. 6 ones × 3 tens = 18 tens. 2 tens × 4 tens.
2 tens × 3 tens = 6 hundreds.
T: What is their sum?
S: 24 + 180 + 80 + 600 = 884. 26 × 34 = 884.
T: Visualize to relate this back to the area model that we drew earlier.
Repeat for 38 × 43. You might first draw the area model (without multiplying out the partial products) and
then erase it so that students again visualize the connection.
Lesson 36: Multiply two-digit by two-digit numbers using four partial products.
496
Lesson 36: Multiply two-digit by two-digit numbers using four partial products.
497
Name Date
1. a. In each of the two models pictured below, write the expressions that determine the area of each of
the four smaller rectangles.
10 2
10
b. Using the distributive property, rewrite the area of the large rectangle as the sum of the areas of the
four smaller rectangles. Express first in number form, and then read in unit form.
2. Use an area model to represent the following expression. Record the partial products and solve.
14 × 22
2 2
× 14
Lesson 36: Multiply two-digit by two-digit numbers using four partial products.
498
Draw an area model to represent the following expressions. Record the partial products vertically and solve.
3. 25 × 32
4. 35 × 42
Visualize the area model and solve the following numerically using four partial products. (You may sketch an
area model if it helps.)
5. 42 × 11 6. 46 × 11
Lesson 36: Multiply two-digit by two-digit numbers using four partial products.
499
Name Date
Draw an area model first to support your work, or draw the area model last to check your work.
1. 26 × 43
2. 17 × 55
Lesson 36: Multiply two-digit by two-digit numbers using four partial products.
500
Name Date
1. a. In each of the two models pictured below, write the expressions that determine the area of each of
the four smaller rectangles.
10 2
10
b. Using the distributive property, rewrite the area of the large rectangle as the sum of the areas of the
four smaller rectangles. Express first in number form, and then read in unit form.
Use an area model to represent the following expression. Record the partial products and solve.
2. 17 × 34
3 4
× 17
Lesson 36: Multiply two-digit by two-digit numbers using four partial products.
501
Draw an area model to represent the following expressions. Record the partial products vertically and solve.
3. 45 × 18 4. 45 × 19
Visualize the area model and solve the following numerically using four partial products. (You may sketch an
area model if it helps.)
5. 12 × 47 6. 23 × 93
7. 23 × 11 8. 23 × 22
Lesson 36: Multiply two-digit by two-digit numbers using four partial products.
502
Lesson 37
Objective: Transition from four partial products to the standard algorithm
for two-digit by two-digit multiplication.
Note: This fluency activity prepares students for composing and decomposing benchmark angles of 90 and
180 degrees in Module 4.
T: (Project a number bond with a whole of 90 and a part of 10.) On your personal white boards, fill in
the unknown part in the number bond.
S: (Fill in 80.)
T: (Write 90 – 10 = .) Say the subtraction sentence.
S: 90 – 10 = 80.
Continue decomposing 90, taking away the following possible suggested parts: 20, 30, 85, 40, 45, 25, 35,
and 15.
Using the same process, take away the following possible suggested parts from 180: 10, 100, 90, 70, 150, 60,
5, 15, 75, 65, and 45.
Lesson 37: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 503
Lesson 37: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 504
Problem 1: Solve 26 × 35 using four partial products and two partial products.
T: Work with a partner:
1. Draw an area model for 26 × 35.
2. Record the partial products within each of four smaller rectangles.
3. Write the expression 26 × 35 vertically.
4. Write the four partial products under the expression.
5. Find their sum.
6. Connect the rectangles in the area model to the partial products using arrows.
S: (Draw area model and solve.) 26 × 35 = 910.
T: Shade the top half of the area model with the side of your pencil. Shade the corresponding partial
products as well.
T: Use mental math to add the two partial products that you just shaded.
S: 30 + 180 = 210.
T: What multiplication expression can be used to represent the entire shaded area?
S: 6 × 35.
T: Find the total for 6 thirty-fives.
S: (Solve.)
S: 6 × 35 = 210. Hey, that’s the same as when we added the two partial products that are shaded.
T: Explain why they are the same.
S: The two smaller rectangles in the shaded portion take up the same amount of space as the larger
rectangle in the shaded portion.
T: Use mental math to add the two partial products that are not shaded.
S: 100 + 600 = 700.
T: What expression can be used to represent the area of the larger rectangle that is not shaded?
Lesson 37: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 505
S: 20 × 35.
T: Solve for 20 thirty-fives.
S: (Solve.)
S: 700. It’s the same!
T: (Draw an area model to show two partial products.) Say an
addition sentence for the sum of the two parts.
S: 210 + 700 = 910. That’s the same answer as when we added
the four partial products.
T: We can solve by finding two partial products instead of four!
Problem 2: Solve 43 × 67 using four partial products and two partial products.
T: Work with a partner to draw and label an area model for 43 × 67 and solve.
T: Draw arrows to show how the parts of the area model relate to the partial products.
T: Draw and label another area model, as we did in Problem 1, which shows how we can combine the
rectangles in the top portion and the rectangles in the bottom portion. (Guide students as they draw
and label.) What expressions do the rectangles represent? Write the expressions in the rectangles.
Solve for each expression.
MP.4
Lesson 37: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 506
Lesson 37: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 507
Lesson 37: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 508
Lesson 37: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 509
Name Date
1. Solve 14 × 12 using 4 partial products and 2 partial products. Remember to think in terms of units as you
solve. Write an expression to find the area of each smaller rectangle in the area model.
1 2 1 2
× 1 4 × 1 4
10 2 12
4 ones × 2 ones 4 ones × 12 ones
4 4
4 ones × 1 ten
1 ten × 12 ones
10 1 ten × 2 ones 10
1 ten × 1 ten
2. Solve 32 × 43 using 4 partial products and 2 partial products. Match each partial product to its area on
the models. Remember to think in terms of units as you solve.
4 3
4 3
× 3 2
40 3 43 × 3 2
2 ones × 3 ones
2 2
2 ones × 4 tens 2 ones × 43 ones
3 tens × 3 ones
30 30 3 tens × 43 ones
3 tens × 4 tens
Lesson 37: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 510
3. Solve 57 × 15 using 2 partial products. Match each partial product to its rectangle on the area model.
4. Solve the following using 2 partial products. Visualize the area model to help you.
a. 2 5 b. 1 8
× 4 6 × 6 2
c. 3 9 d. 7 8
× 4 6 × 2 3
Lesson 37: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 511
Name Date
1. Solve 43 × 22 using 4 partial products and 2 partial products. Remember to think in terms of units as you
solve. Write an expression to find the area of each smaller rectangle in the area model.
2 2
2 2
20 2 × 4 3
22 × 4 3
3 ones × 2 ones 3
3 3 ones × 22 ones
3 ones × 2 tens
4 tens × 2 tens
64
× 15
5 ones × 64 ones
1 ten × 64 ones
Lesson 37: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 512
Name Date
1. Solve 26 × 34 using 4 partial products and 2 partial products. Remember to think in terms of units as you
solve. Write an expression to find the area of each smaller rectangle in the area model.
3 4 3 4
× 2 6 × 2 6
30 4 3 4
6 ones × 3 tens
2 tens × 34 ones
20 2 tens × 4 ones 20
2 tens × 3 tens
2. Solve using 4 partial products and 2 partial products. Remember to think in terms of units as you solve.
Write an expression to find the area of each smaller rectangle in the area model.
4 1 4 1
× 8 2 × 8 2
40 1 4 1
2 ones × 4 tens
8 tens × 41 ones
80 8 tens × 1 one 80
8 tens × 4 tens
Lesson 37: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 513
3. Solve 52 × 26 using 2 partial products and an area model. Match each partial product to its area on the
model.
4. Solve the following using 2 partial products. Visualize the area model to help you.
a. 6 8 b. 4 9
× 2 3 × 3 3
c. 1 6 d. 5 4
× 2 5 × 7 1
Lesson 37: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 514
Lesson 38
Objective: Transition from four partial products to the standard algorithm
for two-digit by two-digit multiplication.
Note: This fluency activity prepares students for composing and decomposing benchmark angles of 90 and
180 degrees in Module 4.
T: (Project a number bond with a whole of 90 and a part of 10.) On your personal white boards, fill in
the unknown part in the number bond.
S: (Fill in 80.)
T: (Write 90 – 10 = .) Say the subtraction sentence.
S: 90 – 10 = 80.
Continue decomposing 90, taking away the following possible suggested parts:
20, 30, 85, 40, 45, 25, 35, and 15.
Repeat the process, taking away the following possible suggested parts from 180:
10, 100, 90, 70, 150, 60, 5, 15, 75, 65, and 45.
Lesson 38: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 515
Lesson 38: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 516
Lesson 38: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 517
S: 54 tens.
T: (Point to the regrouped ten.) 54 tens plus 1 ten is…?
S: 55 tens. Now we need to cross off the 1 ten that we regrouped.
T: What is 9 × 62?
S: 558.
T: Now let’s find the value of the second partial product,
20 sixty-twos.
T: 2 tens times 2 ones is…?
S: 4 tens.
T: Record the 4 tens as 40 ones. 2 tens times 6 tens is…?
S: 12 hundreds.
T: Record 12 hundreds in the second partial product.
What is our second partial product?
S: 1,240.
T: What is the sum of our partial products?
S: 1,798. NOTES ON
MULTIPLE MEANS
T: What is 29 × 62? Say the complete equation.
OF REPRESENTATION:
S: 29 × 62 = 1,798.
Use graph paper or a template that
T: Yes, 9 sixty-twos plus 20 sixty-twos is 29 sixty-twos. allows for wide rows to show how the
The product is 1,798. regrouping is within the same partial
product and how it relates to the value
Problem 3: Solve 46 x 63 involving a regrouping in the second of that row. Students can then see the
partial product. regrouped number is intentionally
placed in the next column within the
T: Let’s find the value of 46 sixty-threes. Write the same partial product. Relate back to
multiplication expression. the place value disk model of
S: (Write 46 × 63.) representation as needed.
T: Which partial product do we find first?
S: 6 × 63.
T: 6 ones times 3 ones is…?
S: 18 ones.
MP.8 T: Let’s record. (Write the 1 on the line under the tens place first
and the 8 in the ones place second.)
T: What do we multiply next?
S: 6 ones times 6 tens. That’s 36 tens. When I add the 1 ten, I get
37 tens.
T: Record 37 tens. Did you remember to cross off the 1 ten? The
value of 6 sixty-threes is…?
S: 378.
T: Now, let’s find the value of 40 sixty-threes. What do we do first?
Lesson 38: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 518
Lesson 38: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 519
Lesson 38: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 520
Name Date
54 5 4
3 × 2 3
23 × 54 = (___ fifty-fours) + (___ fifty-fours)
3 × _____
20
20 × _____
54 5 4
6 × 4 6
46 × 54 = (___ fifty-fours) + (___ fifty-fours)
_____ × _____
40 _____ × _____
4 7
55 × 47 = ( ____ × _____ ) + ( ____ × _____ ) × 5 5
_____ × _____
_____ × _____
Lesson 38: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 521
5 8
× 4 5
____ × ____
____ × _____
8 2
× 5 5
_____ × _____
_____ × _____
6. 53 × 63 7. 84 × 73
Lesson 38: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 522
Name Date
1.
7 2
× 4 3
____ × ____
____ × _____
2. 35 × 53
Lesson 38: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 523
Name Date
43
4 3
6
× 2 6
26 × 43 = (_____ forty-threes) + (____ forty-threes)
6 × _____
20
20 × _____
63
6 3
7
47 × 63 = (____ sixty-threes) + (____ sixty-threes) × 4 7
_____ × _____
40
_____ × _____
6 7
54 × 67 = (___ × ____) + (___ × ____)
× 5 4
_____ × _____
_____ × _____
Lesson 38: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 524
5 2
× 3 4
____ × _____
____ × _____
8 6
× 5 6
_____ × _____
_____ × _____
6. 54 × 52 7. 44 × 76
Lesson 38: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 525
8. 63 × 63 9. 68 × 79
Lesson 38: Transition from four partial products to the standard algorithm for
two-digit by two-digit multiplication. 526
Name Date
2. Identify each number as prime or composite. Then, list all of its factors.
a. 3 ______________________ _______________________________________
b. 6 ______________________ _______________________________________
c. 15 ______________________ _______________________________________
d. 24 ______________________ _______________________________________
e. 29 ______________________ _______________________________________
a. 3,600 ÷ 9
b. 96 pencils come in a box. If 4 teachers share 3 boxes equally, how many pencils does each teacher
receive?
4. 427 ÷ 3
c. 29 × 56 d. 17 × 43
Solve using a model or equation. Show your work, and write your answer as a statement.
a. The store’s rectangular floor is 42 meters long and 39 meters wide. How many square meters of
flooring do they need? Use estimation to assess the reasonableness of your answer.
b. The store ordered small posters and large posters to promote their opening. 12 times as many small
posters were ordered as large posters. If there were 48 large posters, how many more small posters
were ordered than large posters?
c. Uniforms are sold in packages of 8. The store’s 127 employees will each be given 3 uniforms. How
many packages will the store need to order?
d. There are three numbers for the combination to the store’s safe. The first number is 17. The other
two numbers can be multiplied together to give a product of 28. What are all of the possibilities for
the other two numbers? Write your answers as multiplication equations, and then write all of the
possible combinations to the safe.
2 The student is unable The student correctly The student correctly The student correctly
to complete the answers prime or answers prime or answers:
4.OA.4 majority of Parts (a)– composite for three composite for four of a. Prime;
(e). parts and misses more the five parts and
1, 3
than a total of three misses three or fewer
factors. factors. b. Composite; 1, 2, 3,
6
c. Composite; 1, 3, 5,
15
d. Composite;
1, 2, 3, 4, 6, 8, 12,
24
e. Prime;
1, 29
3 The student incorrectly The student correctly The student answers The student correctly
answers both parts and answers one part and one part correctly but answers using any
4.OA.3 shows no reasoning. shows little reasoning. shows solid reasoning place value strategy:
4.NBT.5 in both problems, or a. 400
4.NBT.6 the student shows
b. Each teacher
some reasoning with
received 72
correct answers for
pencils.
both parts.
4 The student incorrectly The student incorrectly The student The student correctly
represents division solves the numeric decomposes incorrectly decomposes and
4.NBT.6 using place value disks equation but shows in one place value or divides using the place
and incorrectly solves some understanding of does not include the value disks and
numerically. the place value chart remainder. provides a numerical
and use of the answer of 142 with a
algorithm. remainder of 1.
5 The student answers The student correctly The student correctly The student solves all
fewer than two parts solves two parts, solves three parts with parts correctly using
4.NBT.6 correctly, showing little showing little evidence understanding of place any place value
to no evidence of place of place value value strategies, or the strategy:
value strategies. strategies. student correctly solves a. 1,772
all four parts but does
b. 761 with a
not show solid
remainder of 4
evidence of place value
understanding. c. 1,624
d. 731
6 The student incorrectly The student correctly The student answers all The student correctly
answers two or more of answers two of four four parts correctly but answers all four parts,
4.MD.3 the four parts, showing parts, showing some shows little reasoning showing solid evidence
4.OA.1 little to no reasoning. reasoning. in Part (a), or the of place value
4.OA.2 student answers three understanding:
of four parts correctly
4.OA.3 a. 1,638 square
showing solid reasoning meters of flooring
4.NBT.5
and understanding (estimate 40 × 40
4.NBT.6 mathematically. = 1,600 square m).
It is a reasonable
because the
answer and
estimate have a
difference of only
38 square meters.
b. 528 more small
posters than large
posters.
c. 48 packages.
d. Equations of
1 × 28 = 28
28 × 1 = 28
2 × 14 = 28
14 × 2 = 28
4 × 7 = 28
7 × 4 = 28
Combinations of
17, 1, 28
17, 28, 1
17, 2, 14
17, 14, 2
17, 4, 7
17, 7, 4
4
GRADE
Mathematics Curriculum
GRADE 4 • MODULE 3
Answer Key
GRADE 4 • MODULE 3
Multi-Digit Multiplication and Division
Lesson 1
Problem Set
1. a. 63 sq units; 32 units 4. a. 10 cm
b. 54 sq units; 30 units b. 7 cm
2. a. 22 cm; 30 sq cm 5. a. 40 cm
b. 22 cm; 24 sq cm b. 250 cm
3. a. 530 m 6. a. 6 cm; 4 cm
b. 450 cm or 4 m 50 cm b. 12 m; 2 m
Exit Ticket
1. 16 sq cm; 20 cm
2. 892 m
Homework
1. a. 40 sq units; 26 units 4. a. 10 cm
b. 35 sq units; 24 units b. 5 m
2. a. 20 cm; 21 sq cm 5. a. 50 cm
b. 26 cm; 36 sq cm b. 350 m
3. a. 450 m 6. a. 8 cm; 4 cm
b. 510 cm or 5 m 10 cm b. 3 m; 12 m
Lesson 2
Problem Set
1. a. Width 4 ft, length 12 ft 4. a. Diagram drawn and labeled; 18 ft
b. 32 ft b. Diagram drawn and labeled; 36 ft
2. a. Diagram drawn; width 5 in, length 30 in c. The perimeter of the second rectangle is
b. 70 in; 150 sq in twice the first rectangle.
3. a. 6 cm d. 80 sq ft
b. Diagram drawn; width 18 cm, length 7 cm e. 4
c. 50 cm f. When the side lengths are doubled, the
perimeter will double but the area will
quadruple.
Exit Ticket
1. a. Width 2 ft, length 12 ft
b. 28 ft
2. a. Diagram drawn; width 4 ft, length 12 ft
b. 32 ft; 48 sq ft
Homework
1. a. Width 7 ft, length 21 ft 4. a. Diagram drawn and labeled; 16 ft
b. 56 ft b. Diagram drawn and labeled; 32 ft
2. a. Diagram drawn; width 3 in, length 12 in c. The perimeter of the living room rug is
b. 30 in; 36 sq in double the perimeter of the bedroom rug.
3. a. 4 cm d. 60 sq ft
b. Diagram drawn; width 9 cm, length 12 cm e. 4
c. 42 cm f. When the side lengths are doubled, the
perimeter will double but the area will
quadruple.
Lesson 3
Sprint
Side A
1. 4 12. 49 23. 7 34. 4
2. 2 13. 64 24. 9 35. 8
3. 9 14. 8 25. 5 36. 49
4. 3 15. 100 26. 8 37. 3
5. 25 16. 10 27. 16 38. 9
6. 5 17. 9 28. 4 39. 64
7. 1 18. 81 29. 8 40. 4
8. 1 19. 5 30. 25 41. 7
9. 4 20. 9 31. 3 42. 81
10. 16 21. 4 32. 9 43. 6
11. 7 22. 4 33. 36 44. 100
Side B
1. 25 12. 36 23. 8 34. 3
2. 5 13. 81 24. 9 35. 9
3. 4 14. 9 25. 3 36. 49
4. 2 15. 100 26. 7 37. 4
5. 9 16. 10 27. 16 38. 7
6. 3 17. 7 28. 2 39. 64
7. 1 18. 49 29. 7 40. 3
8. 1 19. 4 30. 25 41. 8
9. 4 20. 8 31. 4 42. 81
10. 16 21. 4 32. 8 43. 7
11. 6 22. 5 33. 36 44. 100
Problem Set
1. 70 ft
2. 32 sq ft
3. 5 ft
4. 36 sq ft
Exit Ticket
Poster: Length 9 in, width 3 in
Banner: Length 10 in, width 2 in
Homework
1. 44 in
2. 11 sq cm
3. 3 ft
4. 32 sq in
Lesson 4
Problem Set
1. Disks drawn; 500; 500; 5 hundreds 4. Disks drawn; 120; 12 tens
2. Disks drawn; 5,000; 5,000; 5 thousands 5. Disks drawn; 1,800; 1,800; 18 hundreds
3. a. 60 6. Disks drawn; 25,000; 25,000; 25 thousands
b. 100 7. 10; 10; 120
c. 6 8. 2, 100; 6, 100; 600
d. 40 9. 4, 4, 1,000; 16, 1,000; 16,000
e. 100 10. 5, 4, 1,000; 20, 1,000; 20,000
f. 1,000
g. 9,000
h. 90
i. 9
Exit Ticket
a. 50
b. 100
c. 5
d. 20
e. 100
f. 200
g. 1,800
h. 320
i. 48
j. 240
k. 3,000
l. 40,000
Homework
1. Disks drawn; 700; 700; 7 hundreds 4. Disks drawn; 150; 15 tens
2. Disks drawn; 7,000; 7,000; 7 thousands 5. Disks drawn; 1,700; 1,700; 17 hundreds
3. a. 80 6. Disks drawn; 36,000; 36,000; 36 thousands
b. 100 7. 10; 10; 160
c. 8 8. 4, 100; 8, 100; 800
d. 30 9. 5, 5, 1,000; 25, 1,000; 25,000
e. 1,000 10. 7, 6, 1,000; 42, 1,000; 42,000
f. 100
g. 4,000
h. 40
i. 4
Lesson 5
Problem Set
1. Disks drawn; 6; 3, 6; 6 6. 180 balloons
2. Disks drawn; 60; 3, 6 tens; 60 7. 180 baseball cards
3. Disks drawn; 600; 3 hundreds, 6 hundreds; 600 8. 3 fish
4. Disks drawn; 6,000; 2, 3 thousands, 6 thousands; 6,000
5. a. 140
b. 180
c. 1,200
d. 1,600
e. 210
f. 360
g. 1,600
h. 32,000
i. 150
j. 300
k. 2,000
l. 40,000
Exit Ticket
1. Disks drawn; 800; 2 hundreds, 8 hundreds; 800
2. Disks drawn; 8,000; 4, 2 thousands, 8 thousands; 8,000
3. a. 90
b. 160
c. 2,400
d. 1,800
e. 640
f. 120
g. 3,000
h. 40,000
4. 210 hours
Homework
1. Disks drawn; 10; 2,10; 10 5. a. 180
2. Disks drawn; 100; 2, 10 tens; 100 b. 420
3. Disks drawn; 1,000; 2 hundreds, 10 hundreds; c. 4,900
1,000 d. 2,700
4. Disks drawn; 10,000; 5, 2 thousands, e. 810
10 thousands; 10,000 f. 280
g. 3,600
h. 48,000
i. 350
j. 400
k. 1,000
l. 30,000
6. 1,800 chicken nuggets
7. 240 stickers
8. 3 flowers
Lesson 6
Problem Set
1. Disks drawn; 800; 800; 800 6. 1,200; tens, 12
2. Area model drawn; 8 hundreds 7. 1,400; 7, 2, hundreds
3. Area model drawn; 12 hundreds; 1,200 8. 2,100; 7 tens, 3 tens, 21
4. Area model drawn; 10 hundreds; 1,000 9. 3,600 seats
5. 400; 4 10. $4,000
Exit Ticket
1. Disks drawn; 600; 600; 600
2. Area model drawn; 6 hundreds
3. 1,200 pages
Homework
1. Disks drawn; 1,800; 1,800; 1,800
2. Area model drawn; 18 hundreds
3. Area model drawn; 4 hundreds; 400
4. Area model drawn; 24 hundreds; 2,400
5. 1,000; 10
6. 1,500; tens, 15
7. 1,200; 6, 2, hundreds
8. 2,800; 4 tens, 7 tens, 28
9. 3,600 seconds
10. 2,000 pieces of paper
Lesson 7
Sprint
Side A
1. 6 12. 900 23. 35 34. 54,000
2. 60 13. 9,000 24. 3,500 35. 8,100
3. 600 14. 12,000 25. 24 36. 64,000
4. 6,000 15. 1,200 26. 240 37. 490
5. 6,000 16. 120 27. 36 38. 3,600
6. 8 17. 15 28. 36,000 39. 5,600
7. 80 18. 1,500 29. 42 40. 63,000
8. 800 19. 14 30. 4,200 41. 1,000
9. 8,000 20. 140 31. 72 42. 300
10. 9 21. 16 32. 720 43. 20,000
11. 90 22. 16,000 33. 54 44. 4,000
Side B
1. 8 12. 600 23. 45 34. 54,000
2. 80 13. 6,000 24. 4,500 35. 6,400
3. 800 14. 12,000 25. 32 36. 81,000
4. 8,000 15. 1,200 26. 320 37. 4,900
5. 8,000 16. 120 27. 27 38. 360
6. 9 17. 15 28. 27,000 39. 5,600
7. 90 18. 150 29. 42 40. 63,000
8. 900 19. 12 30. 4,200 41. 100
9. 9,000 20. 120 31. 56 42. 3,000
10. 6 21. 16 32. 560 43. 2,000
11. 60 22. 1,600 33. 54 44. 40,000
Problem Set
1. Disks drawn and partial products recorded 2. Disks drawn and partial products recorded
a. Answer provided a. 72
b. 2 × 4 tens + 2 × 3 ones; 86 b. 183
c. 3 × 4 tens + 3 × 3 ones; 129 c. 336
d. 4 × 4 tens + 4 × 3 ones; 172
Exit Ticket
1. Disks drawn and partial products recorded; 246
2. Disks drawn and partial products recorded; 217
Homework
1. Disks drawn and partial products recorded 2. Disks drawn and partial products recorded
a. 3 × 2 tens + 3 × 4 ones; 72 a. 108
b. 3 × 4 tens + 3 × 2 ones; 126 b. 210
c. 4 × 3 tens + 4 × 4 ones; 136 3. No; explanations will vary.
Lesson 8
Problem Set
1. Disks drawn and partial products recorded 2. Disks drawn and partial products recorded
a. 2, 1, 3; 213 a. 636
b. 2 × 2 hundreds + 2 × 1 ten + 2 × 3 ones; 426 b. 8,072
c. 3 × 2 hundreds + 3 × 1 ten + 3 × 4 ones; 642 c. 7,638
d. 3 × 1 thousand + 3 × 2 hundreds + 3 × 5 d. 4,221
tens + 3 × 4 ones; 3,762 3. 720 bagels
Exit Ticket
1. Disks drawn and partial products recorded; 2,052
2. Disks drawn and partial products recorded; 3,162
Homework
1. Disks drawn and partial products recorded 2. Disks drawn and partial products recorded
a. 4 hundreds, 2 tens, 4; 848 a. 1,234
b. 3 × 4 hundreds + 3 × 2 tens + 3 × 4 ones; b. 3,210
1,272 c. 9,102
c. 4 × 1 thousand + 4 × 4 hundreds + 4 × 2 3. a. 966 m
tens + 4 × 4 ones; 5,696 b. 2,898 m
Lesson 9
Problem Set
1. a. 136; 136 3. 602
b. 672; 672 4. 4,113
2. a. 753 5. 90 cm
b. 810 6. $952
c. 2,736 7. 1,008 g
d. 1,620
e. 1,580
f. 2,352
Exit Ticket
1. a. 5,472
b. 4,018
2. 92 years old
Homework
1. a. 92; 92 3. 432
b. 1,260; 1,260 4. 1,050 points
2. a. 928 5. $477
b. 852 6. $1,316
c. 2,198 7. 512 pages
d. 1,320
e. 4,056
f. 3,456
Lesson 10
Problem Set
1. a. 126 2. 1,095 days
b. 252 3. 1,848 m
c. 2,586 4. 42,240 ft
d. 1,293
e. 18,636
f. 9,318
g. 17,236
h. 34,472
Exit Ticket
1. a. 14,088
b. 11,753
2. 4,820 sunflowers
Homework
1. a. 123 2. 768 fluid oz
b. 369 3. 2,748 days
c. 1,001 4. 8,192 megabytes
d. 2,002
e. 8,192
f. 16,384
g. 32,768
h. 32,768
Lesson 11
Problem Set
1. Standard algorithm, partial products method, 2. 774; partial products method used
and area model used 3. 1,868; tape diagram drawn
a. 1,700; 400, 20, 5 4. 35,917
b. 3,738; 500, 30, 4; 7, 500, 7, 30, 7, 4 5. 8,415
c. 1,672; 8, 200, 9; 8, 200, 8, 9 6. 23,850 pounds
Exit Ticket
1. 11,236
2. 6,075 pages
Homework
1. Standard algorithm, partial products method 2. 1,900 people; partial products method used
and area model used 3. 2,304; tape diagram drawn
a. 2,416; 300, 2 4. 18,759
b. 1,080; 200, 10, 6; 5, 200, 5, 10, 5, 6 5. 21,511
c. 5,337; 9, 500, 90, 3; 9, 500, 9, 90, 9, 3 6. 18,744 pounds
Lesson 12
Problem Set
1. 657¢ or $6.57
2. 11,508 L
3. 589 marbles
4. a. Equations will vary.
b. Word problems and units will vary; 3,262
Exit Ticket
872 beads
Homework
1. 644 stickers
2. 12,236 copies
3. 285 bars
4. a. Equations will vary.
b. Word problems will vary; 3,142 m
Lesson 13
Sprint
Side A
1. 4 12. 115 23. 63 34. 6,339
2. 40 13. 9 24. 363 35. 6,393
3. 44 14. 120 25. 84 36. 6,933
4. 2 15. 129 26. 284 37. 96
5. 40 16. 8 27. 484 38. 175
6. 42 17. 140 28. 684 39. 162
7. 6 18. 148 29. 884 40. 378
8. 90 19. 6 30. 9 41. 500
9. 96 20. 180 31. 39 42. 642
10. 15 21. 186 32. 639 43. 10,426
11. 100 22. 189 33. 3,639 44. 8,540
Side B
1. 6 12. 125 23. 84 34. 4,226
2. 60 13. 16 24. 484 35. 4,262
3. 66 14. 120 25. 48 36. 4,622
4. 2 15. 136 26. 248 37. 92
5. 60 16. 8 27. 448 38. 265
6. 62 17. 180 28. 648 39. 135
7. 9 18. 188 29. 848 40. 216
8. 60 19. 6 30. 6 41. 645
9. 69 20. 120 31. 26 42. 500
10. 25 21. 126 32. 426 43. 10,624
11. 100 22. 129 33. 2,426 44. 4,940
Problem Set
1. $748
2. 216 lb
3. 1,362 lb
4. 7,183 pages
Exit Ticket
1. $1,512
2. $1,920
3. David; $408
Homework
1. $534
2. $245
3. 1,972 seats
4. 5,191 reams of paper
Lesson 14
Problem Set
1. 9 pairs; yes; 1 sock
2. 4 bows; yes; 4 in
3. 5 chairs; yes; 2 chairs
4. 5 days
5. 72 apples; 4 apples
6. 7 vans
Exit Ticket
8 groups; 9 chaperones
Homework
1. 8 booklets; yes; 1 sheet
2. 8 booklets; yes; 2 in
3. 4 groups; 5 students
4. 8 days; Day 9
5. 8 rows; 3 soldiers
6. 9 groups; 6 students
Lesson 15
Problem Set
Array and area model drawn for each solution
1. 3, 0; yes
2. 3, 1; no, one small square outside of the larger rectangle
3. Quotient 9, Remainder 2
4. Quotient 4, Remainder 2
5. Quotient 10, Remainder 3
6. Quotient 8, Remainder 3
Exit Ticket
Array and area model drawn for each solution
1. Quotient 5, Remainder 2
2. Quotient 5, Remainder 2
Homework
Array and area model drawn for each solution
1. 6, 0; yes
2. 6, 1; no, one small square outside of the larger rectangle
3. Quotient 6, Remainder 2
4. Quotient 5, Remainder 4
5. Quotient 6, Remainder 1
6. Quotient 5, Remainder 6
Lesson 16
Problem Set
1. Disks drawn 3R 1; 3; 1; 6; 7 4. Disks drawn 12R 2; 12; 2; 12 × 3 = 36, 36 + 2 = 38
2. Disks drawn 13R 1; 13; 1; 13 × 2 = 26, 26 + 1 = 27 5. Disks drawn 1R 2; 1; 2; 4 × 1 = 4, 4 + 2 = 6
3. Disks drawn 3R 2; 2; 2; 2 × 3 = 6, 6 + 2 = 8 6. Disks drawn 21R 2; 21; 2; 4 × 21 = 84, 84 + 2 = 86
Exit Ticket
1. Disks drawn 1R 2; 1; 2; 1 × 3 = 3, 3 + 2 = 5
2. Disks drawn 21R 2; 21; 2; 3 × 21 = 63, 63 + 2 = 65
Homework
1. Disks drawn 2R 1; 2; 1; 6, 6 + 1 = 7 4. Disks drawn 42R 1; 42; 1; 42 × 2 = 84, 84 + 1 = 85
2. Disks drawn 22R 1; 22; 1; 22 × 3 = 66, 66 + 1 = 67 5. Disks drawn 1R 1; 1; 1; 1 × 4 = 4, 4 + 1 = 5
3. Disks drawn 2R 1; 2; 1; 2 × 2 = 4, 4 + 1 = 5 6. Disks drawn 21R 1; 21; 1; 4 × 21 = 84, 84 + 1 = 85
Lesson 17
Problem Set
1. Disks drawn; 2; 1; 4, 2 × 2 = 4, 4 + 1 = 5 4. Disks drawn; 25; 0; 3 × 25 = 75
2. Disks drawn; 25; 0; 2 × 25 = 50 5. Disks drawn; 2; 1; 4 × 2 = 8, 8 + 1 = 9
3. Disks drawn; 2; 1; 3 × 2 = 6, 6 + 1 = 7 6. Disks drawn; 23; 0; 23 × 4 = 92
Exit Ticket
1. Disks drawn; 1; 1; 4 × 1 = 4, 4 + 1 = 5
2. Disks drawn; 14; 0; 14 × 4 = 56
Homework
1. Disks drawn; 3; 1; 3 × 2 = 6, 6 + 1 = 7 4. Disks drawn; 15; 2; 4 × 15 = 60, 60 + 2 = 62
2. Disks drawn; 36; 1; 2 × 36 = 72, 72 + 1 = 73 5. Disks drawn; 2; 2; 3 × 2 = 6, 6 + 2 = 8
3. Disks drawn; 1; 2; 1 × 4 = 4, 4 + 2 = 6 6. Disks drawn; 28; 0; 3 × 28 = 84
Lesson 18
Problem Set
1. 23; 23 × 2 = 46 7. 14 R5; 14 × 6 = 84, 84 + 5 = 89
2. 32; 32 × 3 = 96 8. 16; 16 × 6 = 96
3. 17; 17 × 5 = 85 9. 20; 20 × 3 = 60
4. 13; 13 × 4 = 52 10. 15; 15 × 4 = 60
5. 17 R2; 17 × 3 = 51, 51 + 2 = 53 11. 11 R7; 11 × 8 = 88, 88 + 7 = 95
6. 23 R3; 23 × 4 = 92, 92 + 3 = 95 12. 13 R4; 13 × 7 = 91, 91 + 4 = 95
Exit Ticket
1. 13 R2; 13 × 7 = 91, 91 + 2 = 93
2. 12 R3; 12 × 8 = 96, 96 + 3 = 99
Homework
1. 42; 42 × 2 = 84 7. 15 R1; 15 × 6 = 90, 90 + 1 = 91
2. 21; 21 × 4 = 84 8. 13; 13 × 7 = 91
3. 16; 16 × 3 = 48 9. 29; 29 × 3 = 87
4. 16; 16 × 5 = 80 10. 14 R3; 14 × 6 = 84, 84 + 3 = 87
5. 15 R4; 15 × 5 = 75, 75 + 4 = 79 11. 11 R6; 11 × 8 = 88, 88 + 6 = 94
6. 22 R3; 22 × 4 = 88, 88 + 3 = 91 12. 15 R4; 15 × 6 = 90, 90 + 4 = 94
Lesson 19
Sprint
Side A
1. 10 12. 21 23. 34 34. 17
2. 2 13. 1 24. 32 35. 10
3. 12 14. 20 25. 43 36. 20
4. 10 15. 21 26. 31 37. 15
5. 2 16. 1 27. 22 38. 18
6. 12 17. 20 28. 33 39. 10
7. 10 18. 21 29. 22 40. 13
8. 2 19. 8 30. 33 41. 15
9. 12 20. 10 31. 10 42. 20
10. 1 21. 12 32. 20 43. 19
11. 20 22. 14 33. 15 44. 17
Side B
1. 10 12. 31 23. 43 34. 16
2. 3 13. 1 24. 23 35. 10
3. 13 14. 30 25. 34 36. 20
4. 10 15. 31 26. 32 37. 15
5. 3 16. 2 27. 22 38. 19
6. 13 17. 10 28. 33 39. 10
7. 20 18. 12 29. 22 40. 12
8. 1 19. 10 30. 44 41. 14
9. 21 20. 12 31. 10 42. 20
10. 1 21. 14 32. 20 43. 18
11. 30 22. 16 33. 15 44. 16
Problem Set
1. Equation accurately modeled; remainder circled
2. Remainder is greater than divisor; explanations will vary.
3. Equation accurately modeled; 1 remaining ten is decomposed into 10 ones.
4. a. Picture accurately models division; yes
b. Explanations will vary.
5. Answers will vary.
Exit Ticket
1. Disks drawn; 16; yes; 1
2. No; she can fill 11 pages completely; explanations will vary.
Homework
1. Equation accurately modeled; remainder circled
2. Remainder is greater than divisor; explanations will vary.
3. Equation accurately modeled; 2 remaining tens are decomposed into 20 ones.
4. a. Picture accurately models division; yes
b. Explanations will vary.
5. Answers will vary.
Lesson 20
Problem Set
1. a. 72 ÷ 4 = 18
b. Whole: 72; parts: 40 and 32; 40, 4, 32, 4, 10, 8, 18
2. 15; whole: 45; parts: 30 and 15; (30 ÷ 3) + (15 ÷ 3) = 10 + 5 = 15; area model and number bond drawn
3. 16; whole: 64; parts: 40 and 24; area model and number bond drawn; solved with distributive property
or standard algorithm
4. 23; solved with area model; explanations will vary.
5. 12; solved with area model and standard algorithm
Exit Ticket
1. 72 ÷3 = 24
2. 14; solved with area model, number bond, and written method
Homework
1. a. 54 ÷ 3 = 18
b. Whole: 54; parts: 30 and 24; 30, 3, 24, 3, 10, 8, 18
2. 14; whole: 42; parts: 30 and 12; (30 ÷ 3) + (12 ÷ 3) = 10 + 4 = 14; area model and number bond drawn
3. 15; whole: 60; part: 40; part: 20; area model and number bond drawn; solved with distributive
property or standard algorithm
4. 18; solved with area model; explanations will vary.
5. 16; solved with area model and standard algorithm
Lesson 21
Sprint
Side A
1. 4 12. 1 R1 23. 3 34. 1 R4
2. 4 R1 13. 2 R1 24. 3 R1 35. 1
3. 1 14. 2 R2 25. 1 36. 1
4. 1 R1 15. 3 26. 1 R1 37. 3 R1
5. 1 R2 16. 1 R2 27. 1 R2 38. 3 R3
6. 1 R3 17. 1 R3 28. 1 R3 39. 3 R3
7. 1 R2 18. 1 29. 1 40. 3 R3
8. 2 19. 1 R1 30. 1 R1 41. 3 R5
9. 2 20. 1 31. 2 42. 7 R1
10. 2 R1 21. 1 R1 32. 2 R1 43. 8 R5
11. 1 22. 1 33. 3 44. 9 R1
Side B
1. 1 R1 12. 1 R1 23. 2 34. 4 R1
2. 1 13. 1 24. 2 R1 35. 4
3. 1 R3 14. 1 R1 25. 2 36. 4 R5
4. 1 R2 15. 1 R1 26. 2 R1 37. 5 R5
5. 1 16. 1 R2 27. 3 38. 2 R5
6. 1 R1 17. 1 28. 2 R2 39. 6 R6
7. 1 R3 18. 1 R1 29. 1 R4 40. 6 R3
8. 1 R2 19. 3 30. 1 41. 7 R5
9. 1 R2 20. 3 R1 31. 1 R1 42. 8 R5
10. 2 21. 1 R3 32. 1 43. 7 R5
11. 1 22. 1 R2 33. 1 44. 7 R7
Problem Set
1. 18 R1; answer includes area model, long division, and distributive property
2. 25 R1; answer includes area model, long division, and distributive property
3. a. 53 ÷ 4 = 13 R1
b. (40 ÷ 4) + (12 ÷ 4) = 10 + 3 = 13
4. 16; answer includes area model and long division or distributive property
5. 16 R1; answer includes area model and long division or distributive property
6. 14; answer includes area model and long division or distributive property
7. 14 R2; answer includes area model and long division or distributive property
8. 13 R1; answer includes area model and long division or distributive property
9. 26 R1; answer includes area model and long division or distributive property
10. 12 groups; 1 student
Exit Ticket
1. 59 ÷ 2 = 29 R1
2. 23 R1; answer includes area model, long division, and distributive property
Homework
1. 17 R1; answer includes area model, long division, and distributive property
2. 26 R1; answer includes area model, long division, and distributive property
3. a. 98 ÷ 4 = 24 R2
b. (40 ÷ 4) + (40 ÷ 4) + (16 ÷ 4) = 10 + 10 + 4 = 24
4. 14; answer includes area model and long division or distributive property
5. 14 R1; answer includes area model and long division or distributive property
6. 13; answer includes area model and long division or distributive property
7. 13 R2; answer includes area model and long division or distributive property
8. 12 R1; answer includes area model and long division or distributive property
9. 24 R1; answer includes area model and long division or distributive property
10. 24 lunch trays; 1 lunch tray
Lesson 22
Problem Set
1. a. Answer provided
b. 1 × 6 = 6, 2 × 3 = 6; 1, 2, 3, 6; C
c. 1 × 7= 7; 1, 7; P
d. 1 × 9 = 9, 3 × 3 = 9; 1, 3, 9; C
e. 1 × 12 = 12, 2 × 6 = 12; 3 × 4 = 12; 1, 2, 3, 4, 6, 12; C
f. 1 × 13 = 13; 1, 13; P
g. 1 × 15 = 15, 3 × 5 = 15; 1, 3, 5, 15; C
h. 1 × 16 = 16, 2 × 8 = 16, 4 × 4 = 16; 1, 2, 4, 8, 16; C
i. 1 × 18 = 18, 2 × 9 = 18, 3 × 6 = 18; 1, 2, 3, 6, 9, 18; C
j. 1 × 19 = 19; 1, 19; P
k. 1 × 21 = 21; 3 × 7 = 21; 1, 3, 7, 21; C
l. 1 × 24 = 24, 2 × 12 = 24, 3 × 8 = 24, 4 × 6 = 24; 1, 2, 3, 4, 6, 8, 12, 24; C
2. For 25: (1, 25); (5, 5); composite; more than 2 factors
For 28: (1, 28); (2, 14); (4, 7); composite; more than 2 factors
For 29: (1, 29); prime; only 2 factors
3. a. 2, 3, 5, 7, 11, 13, 17, 19
b. 2 is a prime and even number.
4. Incorrect; 3 is not a factor of 28.
Exit Ticket
a. 1 × 9 = 9, 3 × 3 = 9; 1, 3, 9; C
b. 1 × 12 = 12, 2 × 6 = 12; 3 × 4 = 12; 1, 2, 3, 4, 6, 12; C
c. 1 × 19 = 19; 1, 19; P
Homework
1. a. Answer provided
b. 1 × 10 = 10, 2 × 5 = 10; 1,2, 5, 10; C
c. 1 × 11 = 11; 1, 11; P
d. 1 × 14 = 14, 2 × 7 = 14; 1, 2, 7, 14; C
e. 1 × 17 = 17; 1, 17; P
f. 1 × 20 = 20, 2 × 10 = 20, 4 × 5 = 20; 1, 2, 4, 5, 10, 20; C
g. 1 × 22 = 22, 2 × 11 = 22; 1, 2, 11, 22; C
h. 1 × 23 = 23; 1, 23; P
i. 1 × 25 = 25, 5 × 5 = 25; 1, 5, 25; C
j. 1 × 26 = 26; 2 × 13 = 26; 1, 2, 13, 26; C
k. 1 × 27 = 27, 3 × 9 = 27; 1, 3, 9, 27; C
l. 1 × 28 = 28, 2 × 14 = 28, 4 × 7 = 28; 1, 2, 4, 7, 14, 28; C
2. For 19: (1, 19); prime; only 2 factors
For 21: (1, 21); (3, 7); composite; more than 2 factors
For 24: (1, 24); (2, 12); (3, 8); (4, 6); composite; more than 2 factors
3. a. 1, 3, 5, 7, 9, 11, 13, 15, 17, 19
b. 9 and 15 are odd and composite
4. Correct; 3 is a factor of 27
Lesson 23
Problem Set
1. Explanations may vary. 2. a. 4; 4; 4; 24
a. Yes b. 9; 3; 3; 3; 36
b. No 3. (4 × 2) × 7 = 4 × (2 × 7) = 4 × 14 = 56
c. Yes (4 × 2) × 9 = 4 × (2 × 9) = 4 × 18 = 72
d. Yes (4 × 2) × 10 = 4 × (2 × 10) = 4 × 20 = 80
e. Yes 4. Explanations may vary.
f. Yes
g. No
h. No
Exit Ticket
1. Explanations may vary.
a. Yes
b. No
c. Yes
d. Yes
2. Explanations may vary.
Homework
1. Explanations may vary. 2. a. 3; 3; 3; 4; 12
a. Yes b. 6; 2; 2; 2; 30
b. No 3. (5 × 2) × 7 = 5 × (2 × 7) = 5 × 14 = 70
c. Yes (5 × 2) × 8 = 5 × (2 × 8) = 5 × 16 = 80
d. Yes (5 × 2) × 9 = 5 × (2 × 9) = 5 × 18 = 90
e. Yes 4. Explanations may vary.
f. Yes
g. No
h. No
Lesson 24
Problem Set
1. a. 100, 105, 110, 115, 120, 125, 130, 135, 4. Yes; explanations will vary.
140, 145, 150, 155, 160, 165, 170, 175, 5. a. Multiples of 2 circled red; 0, 2, 4, 6, 8
180, 185, 190, 195, 200, 205, 210, etc. b. Multiples of 3 shaded green; answers will
b. 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, vary; sums are multiples of 3 or divisible by 3.
68, 72, 76, 80, 84, 88, 92, 96, 100, 104, c. Multiples of 5 circled blue; 0, 5
108, 112, 116, 120, etc. d. Multiples of 10 crossed out; zero in the
c. 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, ones place
102, 108, 114, 120, 126, 132, 138, 144,
150, 156, 162, 168, 174, 180, etc.
2. 1, 2, 3, 4, 6, 8, 12, 24
3. a. Yes; yes
b. No; no
c. Yes; yes
Exit Ticket
1. 55; 66; 77; 88; 99
2. 21, 35, 42, 49, 56, 63, 70
3. a. 1, 2, 3, 6, 9, 18
b. 1, 2, 3, 6, 9, 18
c. Yes; explanations will vary.
Homework
1. a. 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 4. No; explanations will vary.
130, 135, 140, 145, 150, 155, 160, 165, 170, 5. a. Multiples of 6 underlined; 0, 2, 4, 6, 8
175, 180, 185, etc. b. Multiples of 4 identified; 2, 6
b. 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, c. 0, 4, 8; answers will vary.
92, 96, 100, 104, 108, 112, 116, 120, 124, 128, d. Multiples of 9 circled; sum is 9.
132, 136, 140, etc.
c. 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96,
102, 108, 114, 120, 126, 132, 138, 144, 150, etc.
2. 1, 2, 3, 5, 6, 10, 30
3. a. Yes; yes
b. Yes; no
c. No; no
Lesson 25
Problem Set
1. Chart completed per directions
2. a. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
b. Not multiples of any numbers except one and themselves
c. Composite numbers
d. Prime numbers
Exit Ticket
1. 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30 crossed off
2. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 circled
3. 1
Homework
1. Answers will vary.
2. Composite
3. Prime
4. 1; neither prime nor composite
Lesson 26
Problem Set
1. Disks accurately drawn 3. a. Answer provided
a. 3; 3 b. 300; 6 hundreds ÷ 2 = 3 hundreds
b. 30; 3 tens c. 200; 8 hundreds ÷ 4 = 2 hundreds
c. 300; 6 hundreds, 3 hundreds d. 300; 9 hundreds ÷ 3 = 3 hundreds
d. 3,000; 6 thousands, 3 thousands e. 50; 5
2. Disks accurately drawn f. 60; 24 tens ÷ 4 = 6 tens
a. 4; 4 g. 90; 45 tens ÷ 5 = 9 tens
b. 40; 12 tens, 4 tens h. 40; 20 tens ÷ 5 = 4 tens
c. 400; 12 hundreds, 4 hundreds i. 900; 9
j. 600; 24 hundreds ÷ 4 = 6 hundreds
k. 800; 24 hundreds ÷ 3 = 8 hundreds
l. 800; 40 hundreds ÷ 5 = 8 hundreds
4. 700 kg
5. 70 stickers
6. $400
Exit Ticket
1. a. 2
b. 200; 12 hundreds ÷ 6 = 2 hundreds
c. 300; 21 hundreds ÷ 7 = 3 hundreds
d. 400; 32 hundreds ÷ 8 = 4 hundreds
2. 40 pennies
Homework
1. Disks accurately drawn
a. 2; 2
b. 20; 2 tens
c. 200; 6 hundreds, 2 hundreds
d. 2,000; 6 thousands, 2 thousands
2. Disks accurately drawn
a. 3; 3
b. 30; 12 tens, 3 tens
c. 300; 12 hundreds, 3 hundreds
3. a. Answer provided
b. 300; 9 hundreds ÷ 3 = 3 hundreds
c. 200; 4 hundreds ÷ 2 = 2 hundreds
d. 100; 3 hundreds ÷ 3 = 1 hundred
e. 50; 5
f. 80; 16 tens ÷ 2 = 8 tens
g. 80; 40 tens ÷ 5 = 8 tens
h. 60; 30 tens ÷ 5 = 6 tens
i. 400; 4
j. 400; 16 hundreds ÷ 4 = 4 hundreds
k. 600; 24 hundreds ÷ 4 = 6 hundreds
l. 600; 30 hundreds ÷ 5 = 6 hundreds
4. 4,000 L
5. 70 mL
6. $600
Lesson 27
Sprint
Side A
1. 3 12. 11 23. 41 34. 71
2. 3 13. 13 24. 43 35. 73
3. 3 14. 17 25. 47 36. 79
4. 5 15. 19 26. 53 37. 83
5. 5 16. 23 27. 59 38. 2
6. 5 17. 19 28. 97 39. 17
7. 7 18. 29 29. 91 40. 5
8. 11 19. 31 30. 97 41. 59
9. 11 20. 37 31. 89 42. 31
10. 17 21. 2 32. 61 43. 2
11. 19 22. 2 33. 67 44. 43
Side B
1. 5 12. 13 23. 41 34. 71
2. 5 13. 11 24. 43 35. 73
3. 5 14. 17 25. 47 36. 67
4. 7 15. 19 26. 53 37. 59
5. 7 16. 23 27. 59 38. 2
6. 7 17. 19 28. 97 39. 19
7. 3 18. 29 29. 91 40. 5
8. 11 19. 31 30. 97 41. 59
9. 11 20. 37 31. 89 42. 41
10. 17 21. 2 32. 67 43. 2
11. 19 22. 2 33. 61 44. 67
Problem Set
1. Disks accurately drawn
a. 162
b. 172
c. 161
d. 183
2. Disks accurately drawn; algorithm accurately recorded
a. 131
b. 242
c. 172
Exit Ticket
1. 141; Disks accurately drawn; algorithm accurately recorded
2. 141; Disks accurately drawn; algorithm accurately recorded
Homework
1. Disks accurately drawn
a. 173
b. 264
c. 172
d. 243
2. Disks accurately drawn; algorithm accurately recorded
a. 162
b. 151
c. 241
Lesson 28
Problem Set
1. a. 287
b. 287
c. 177
d. 118
e. 218 R1
f. 118 R1
g. 91 R2
h. 91 R4
i. 169 R2
j. 238 R3
2. 145 bottles; yes; 1 bottle
Exit Ticket
1. a. 388 2. 32 servings
b. 198 R2
Homework
1. a. 189
b. 265
c. 128
d. 123
e. 179 R2
f. 172 R2
g. 166
h. 156 R3
i. 155 R1
j. 132 R3
2. 233 m
Lesson 29
Problem Set
1. a. 418
b. 394 R2
c. 3474
d. 2,237 R1
e. 3,784 R1
f. 2,523
g. 1,591
h. 1,514 R4
i. 2,489 R2
j. 2,489
2. 93 goats
Exit Ticket
1. a. 591 2. 446 stamps
b. 1,694 R2
Homework
1. a. 616
b. 616
c. 3,142
d. 3,293 R1
e. 1,815
f. 2,712 R1
g. 2,822 R1
h. 2,818 R2
i. 1,234 R1
j. 1,234 R3
2. 1,296 apples
Lesson 30
Problem Set
1. 51
2. 234 R2
3. 209
4. 203 R1
5. 190
6. 1,280
7. 614
8. 1,341 R1
9. 2,078 R1
10. 3,002 R2
11. a. 1,043 R2
b. Answers will vary.
Exit Ticket
1. 95
2. 2,346 R2
Homework
1. 81 R4
2. 251 R1
3. 207 R3
4. 200 R2
5. 240
6. 1,250
7. 412
8. 4,515 R1
9. 1,554 R2
10. 2,000
Lesson 31
Sprint
Side A
1. 3 12. 300 23. 60 34. 40
2. 30 13. 500 24. 600 35. 80
3. 300 14. 700 25. 4 36. 800
4. 3,000 15. 900 26. 40 37. 80
5. 3 16. 90 27. 3 38. 700
6. 30 17. 2 28. 300 39. 80
7. 300 18. 3 29. 4 40. 900
8. 3,000 19. 30 30. 40 41. 90
9. 2 20. 300 31. 6 42. 80
10. 3 21. 5 32. 600 43. 900
11. 30 22. 6 33. 700 44. 800
Side B
1. 2 12. 50 23. 40 34. 60
2. 20 13. 70 24. 400 35. 70
3. 200 14. 700 25. 3 36. 700
4. 2,000 15. 900 26. 30 37. 70
5. 2 16. 90 27. 3 38. 600
6. 20 17. 3 28. 300 39. 800
7. 200 18. 4 29. 3 40. 70
8. 2,000 19. 40 30. 30 41. 800
9. 2 20. 400 31. 6 42. 90
10. 3 21. 5 32. 600 43. 800
11. 30 22. 4 33. 700 44. 80
Problem Set
1. 78 tables; number of groups unknown
2. 473 books; group size unknown
3. 501 sacks; number of groups unknown
4. 1,920 cookies; group size unknown
5. 603 miles; group size unknown
Exit Ticket
1. 143 cars; group size unknown
2. 178 sacks; number of groups unknown
Homework
1. 125 mL; group size unknown
2. 206 baggies; number of groups unknown
3. 70 miles; group size unknown
4. 219 strips; number of groups unknown
5. 1,164 Groblarx fruits; group size unknown
Lesson 32
Problem Set
1. 31 seats
2. 8 bagels
3. 87 bags; 5 pieces of candy
4. 150 teams; 4 children
5. 1,014 kg; 5 kg
Exit Ticket
1. 121 students
2. 59 crayons
Homework
1. 48 guests
2. 500 pencils
3. 251 sacks
4. 36 muffins
5. 1,287 m
Lesson 33
Problem Set
1. a. 892 ÷ 4 = 223
b. Whole: 892; parts: 400, 400, 80, 12
(400 ÷ 4)+(400 ÷ 4)+(80 ÷ 4)+(12 ÷ 4)= 100 + 100 + 20 + 3 = 223
2. a. 240; area model accurately drawn
b. Answers will vary.
3. a. 258; area model accurately drawn
b. Answers will vary.
c. Algorithm accurately recorded
4. a. 792; area model accurately drawn
b. Answers will vary.
c. Algorithm accurately recorded
Exit Ticket
1. a. 747 ÷ 3 = 249
b. Whole: 747; parts: 600, 120, 27
(600 ÷ 3) + (120 ÷ 3) + (27 ÷ 3) = 200 + 40 + 9 = 249
2. a. 684; area model accurately drawn
b. Answers will vary.
c. Algorithm accurately recorded
Homework
1. a. 1,828 ÷ 4 = 457
b. Whole: 1,828; parts: 1,600, 200, 28
(1,600 ÷ 4) + (200 ÷ 4) + (28 ÷ 4) = 400 + 50 + 7 = 457
2. a. 204; area model accurately drawn
b. Answers will vary.
3. a. 183; area model accurately drawn
b. Answers will vary.
c. Algorithm accurately recorded
4. a. 1,381; area model accurately drawn
b. Answers will vary.
c. Algorithm accurately recorded
Lesson 34
Problem Set
1. Disks drawn accurately
a. 3; 3; 720
b. 43; 43; 1,720
c. 10, 37; 37; 1,110
2. Disks drawn accurately
a. 540
b. 1,240
3. a. 1,360
b. 2,150
4. a. 1,360
b. 1,500
Exit Ticket
1. 2, 10, 41, 820
2. 1,920
Homework
1. Disks drawn accurately
a. 2; 2; 680
b. 34; 34; 1,020
c. 10, 42; 42; 1,260
2. Disks drawn accurately
a. 320
b. 1,280
3. a. 630
b. 2,520
4. a. 1,720
b. 1,610
Lesson 35
Problem Set
1. 40; 400; 440
2. 50; 2,000; 2,050
3. 180; 4,200; 4,380
4. Area model drawn; 2,560
5. Area model drawn; 3,780
6. 2,040
7. 2,040
8. 2,200
9. 4,400
Exit Ticket
1. 90; 2,700; 2,790
2. 240; 2,800; 3,040
Homework
1. 210; 300; 510;
2. 320; 2,000; 2,320
3. 400; 1,500; 1,900
4. Area model drawn; 1,140
5. Area model drawn; 880
6. 1,760
7. 2,640
8. 3,290
9. 5,200
Lesson 36
Problem Set
1. a. 4 × 2 , 4 × 10, 10 × 2, 10 × 10
b. 2, 10, 2, 10
2. 308; area model and partial products accurately recorded
3. 800; area model and partial products accurately recorded
4. 1,470; area model and partial products accurately recorded
5. 462; partial products accurately recorded
6. 506; partial products accurately recorded
Exit Ticket
1. 1,118; area model and partial products accurately recorded
2. 935; area model and partial products accurately recorded
Homework
1. a. 3 × 2, 3 × 10, 10 × 2, 10 × 10
b. 2, 10, 2, 10
2. 578; area model and partial products accurately recorded
3. 810; area model and partial products accurately recorded
4. 855; area model and partial products accurately recorded
5. 564; partial products accurately recorded
6. 2,139; partial products accurately recorded
7. 253; partial products accurately recorded
8. 506; partial products accurately recorded
Lesson 37
Problem Set
1. 4 × 2, 4 × 10, 10 × 2, 10 × 10; 8, 40, 20, 100, 168; 4 × 12, 10 × 12; 48, 120, 168
2. 2 × 3, 2 × 40, 30 × 3, 30 × 40; 6, 80, 90, 1,200, 1,376; 2 × 43, 30 × 43; 86, 1,290, 1,376
3. 7 × 15, 50 × 15; 105, 750, 855
4. a. 150, 6, 25; 1,000, 40, 25; 1,150
b. 36, 2, 18; 1,080, 60, 18; 1,116
c. 234, 1,560, 1,794
d. 234, 1,560, 1,794
Exit Ticket
1. 3 × 2, 3 × 20, 40 × 2, 40 × 20; 6, 60, 80, 800, 946; 3 × 22, 40 × 22; 66, 880, 946
2. 5 × 64, 10 × 64; 320, 640, 960
Homework
1. 6 × 4, 6 × 30, 20 × 4, 20 × 30; 24, 180, 80, 600, 884; 6 × 34, 20 × 34; 204, 680, 884
2. 2 × 1, 2 × 40, 80 × 1, 80 × 40; 2, 80, 80, 3,200, 3,362; 2 × 41, 80 × 41; 82, 3,280, 3,362
3. 2 × 26, 50 × 26; 52, 1,300, 1,352
4. a. 204, 3, 68; 1,360, 20, 68; 1,564
b. 147, 3, 49; 1,470, 30, 49; 1,617
c. 80, 320, 400
d. 54, 3,780, 3,834
Lesson 38
Problem Set
1. 3 × 54, 20 × 54; 3, 20; 162, 54; 1,080, 54; 1,242
2. 6 × 54, 40 × 54; 6, 40; 324, 6, 54; 2,160, 40, 54; 2,484
3. 5 × 47, 50 × 47; 5, 47, 50, 47; 235, 5, 47; 2,350, 50, 47; 2,585
4. 290, 5, 58; 2,320, 40, 58; 2,610
5. 410, 5, 82; 4,100, 50, 82; 4,510
6. 3,339
7. 6,132
Exit Ticket
1. 216, 3, 72; 2,880, 40, 72; 3,096
2. 1,855
Homework
1. 6 × 43, 20 × 43; 6, 20; 258, 43; 860, 43; 1,118
2. 7 × 63, 40 × 63; 7, 40; 441, 7, 63; 2,520, 40, 63; 2,961
3. 4 × 67, 50 × 67; 4, 67, 50, 67; 268, 4, 67; 3,350, 50, 67; 3,618
4. 208, 4, 52; 1,560, 30, 52; 1,768
5. 516, 6, 86; 4,300, 50, 86; 4,816
6. 2,808
7. 3,344
8. 3,969
9. 5,372