Differentiated Lesson

MIAA 340

Cassie McLemore

Teachers College of San Joaquin



























Cassie McLemore
Math Concept: Geometry
Grade 1
CCSS.Math.Content.1.G.1 Distinguish between defining attributes (e.g. triangles are close and three-
sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to
possess defining attributes.

Big Ideas: Two- and three-dimensional objects can be described, classified and
analyzed by their attributes. Two-dimensional objects are plane shapes. Three-
dimensional figures are solid shapes.

Open Question: How many different kinds of shapes can your group draw and
identify?

Expected Student Responses:
Circle
Square
Rectangle

Management
Students are working in groups of 3 or 4 to identify as many shapes as
they know. They are using a worksheet that is divided into two
sections (shapes they know the name of and ones they dont) and
each shape must be drawn whether they know the name or not.
What other shapes can you draw that you dont know the name of?
How would you describe this shape to someone?
Record on chart paper the shapes students identify.
Students will then discuss in their groups how to describe each shape
without telling what the name is. What attributes does the shape have
that makes it unique?
Record again on the same chart paper the attributes of each shape.

Academic Language Check: Students need to use their academic or scholarly
language to describe shapes. Students will need to learn the words plane shape,
hexagon, trapezoid, straight line, corner and curved edge.

Debrief:
Small groups will share out their findings with the whole group in sentences using
their geometric/academic language with attributes unique to each shape.

1. This shape is a ___________________________. I know this because it

Summarize what we have learned:

Lesson Goal #1: To activated students prior knowledge of plane shapes
and their attributes.
There are many different plane shapes and we identify them by their attributes. A
_________________ is a plane shape and I know it is a ______________ because it has
_____________________________________________________________________. Another plane shape is
a _____________________ and I know it is a ____________________ because it has ________________
_________________________________________________________.

Formative Assessment (Exit Ticket, Ticket Out the Door, Show What You
Know):

A shape is identified by its ___________________. (Attributes this answer will be given
orally because of the grade level).

Tiered Lesson:
Group #1: Students will be with teacher to review what a plane shape is and
how to determine what attributes classify that shape. We will begin with listing the
types of shapes discussed in lesson #1 individually and their attributes. We will
practice our academic language by utilizing the sentence frame:

This is a ________________ I know this because its attributes are

After students can identify all of the familiar shapes I will have them list the two
new shapes (hexagon and trapezoid) and their attributes. Then they will practice
the sentence frame again for all of the familiar and new shapes.

Group #2: Students will be on their own in groups of 3 to 4 to extend their
knowledge of shapes. Students will be given the two new shapes and asked to
practice their name and identify their attributes. Students will also discuss and
come up with a working definition of the difference between a square and rectangle.
Students will work on practice coloring only the specified shape requested based on
its attributes. Some three-dimensional shapes will be added.

Academic Language Check: Students need to use their academic or scholarly
language to describe shapes. Students will need to learn the words plane shape,
hexagon, trapezoid, straight line, corner and curved edge. Students will need to
demonstrate their knowledge of hexagon or trapezoid and their attributes today.

Debrief: Students first share with their partner the attributes of as many figures as
they can in 2 minutes. Each partner will have to include at least one of the new
shapes introduced today (hexagon or trapezoid). Students will then as a whole
group practice choral response for all of the shapes and their attributes that have
been covered in the last two days.

This shape is a ____________________. I know this because its attributes are

Lesson Goal #2: To activated student knowledge of plane and solid
shapes in everyday objects.
Summarize what we have learned: We have learned many shapes and their
attributes (square, rectangle, circle, hexagon, trapezoid).

Formative Assessment: I can determine what a shape is by its attributes (this
response will be given orally because of the grade level).











































Cassie McLemore
Math Concept: Ratio and Proportional Relationships
Grade 7
CCSS.Math.Content. 7.RP Analyze proportional relationships and use them to solve real-world and
mathematical problems. Compute unit rates associated with ratios and fractions, including ratios of lengths,
areas and other quantities measured in like or different units.

Big Ideas: Unit rates are ratios that are defined by their relationship in a one or per
quantity representation in the denominator.

Open Question: What is the difference between a fraction and a ratio?

Expected Student Responses:
There is no difference or they are the same.
A fraction is a part of something
A ratio tells the rate that things change together
A ratio is a relationship between two numbers
I dont know

Management
Students are working in groups of 3 or 4 to identify what the
differences between fractions and ratios are. I will write one fraction
and one ratio on the board for students to use to analyze the
differences.
How are these two numbers different? What could each number
represent if you assigned it to something?
Record on chart paper the differences students identify.
Students will then discuss in their groups how to describe the
difference between a fraction and a ratio. What attributes does the
fraction or ratio have that makes it unique? What does each number
tell you about itself?
Record again on the same chart paper the attributes of a fraction or
ratio.
What does the word unit mean? How about the prefix uni-? Can you
think of a word that starts with uni- that represents what a unit is?
Students will work in their small groups to determine words that start
with uni- that represent what unit is.
Record again on the same chart paper words that start with uni that
represent what a unit is.

Academic Language Check: Students need to use their academic or scholarly
language to describe fractions and ratios. Students will need to know numerator,
denominator, fraction, ratio, unit, rate, and per.

Lesson Goal #1: To activated students prior knowledge of the
attributes of ratios and fractions and define the requirements of
computing unit rates.
Debrief:
Small groups will share out their findings with the whole group in sentences using
their academic language with attributes unique to fractions and ratios.

2. Fractions are _____________________ of a _________________.
3. Ratios are _________________ between __________ quantities.
4. A Unit rate is a _________________ that is represented as a __________ unit quantity.

Summarize what we have learned:

There are many differences between fractions, ratios and unit rates. One attribute
of a _________________ is that they are parts of a _________________. One attribute of ratios
is that they establish a _______________________ between _________ quantities. Finally, unit
rates are ratios that are __________________ as a ____________ unit quantity.

Formative Assessment (Exit Ticket, Ticket Out the Door, Show What You
Know): Define the attributes of fractions, ratios and unit rates in your own words
or with a graphic.


Tiered Lesson:
Group #1: Students will work to review the differences between fractions,
ratios and unit rates. Working with manipulatives, students will build fractions,
ratios and determine unit rate. Activities include:
Find the fraction of yellow circles to all circles (9/12)
Find the fraction of red circles to all circles (3/12)
Find the ratio of yellow circles to red circles (9:3)
Find the unit rate of yellow circles to red circles (3:1)

Students will practice sentence frames and academic language to identify these
answers.

The fraction of yellow circles to all circles is nine twelfths
The fraction of red circles to all circles is three twelfths
The ratio of yellow circles to red circles is nine to three
The unit rate of yellow circles to red circles is three to one.
The unit rate is three yellow circles per red circle.

Group #2: Students will work in groups of three to four to review, identify
and group numbers that are fractions, ratios and unit rates. Students will then
discuss with their groups each numbers attributes and why they were categorized
as a fraction ratio or unit rate. This activity will reinforce the students ability to
determine whether a number is a fraction, ratio or unit rate based on their
attributes. If time permits students will be asked to pick 6 of the numbers (two
Lesson Goal #2: To understand the difference between fractions,
ratios and unit rates.
from each type) and create a scenario in which the fraction, ratio or unit rate could
occur in the real world. (Ex. Mike gave me of his candy bar, 4:2 for every 4
boys on the basketball team there are 2 girls, and 3:1 candy bars are on sale 3 per
dollar).

Academic Language Check: Students will demonstrate their ability to categorize
fractions, ratios and unit rates by describing their attributes in academic language in
group and recitation activities.

Debrief: Small groups will share out their findings with the whole group and
discuss the attributes of fractions, ratios and unit rates. Each group will utilize the
sentence frames from day or create their own sentence to demonstrate their
knowledge of the attributes of fractions, ratios and unit rates.

Summarize what we have learned:

There are many differences between fractions, ratios and unit rates. One attribute
of a _________________ is that they are parts of a _________________. One attribute of ratios
is that they establish a _______________________ between _________ quantities. Finally, unit
rates are ratios that are __________________ as a ____________ unit quantity.

Formative Assessment:
Define the attributes of fractions, ratios and unit rates in your own words or with a
graphic. Provide a real world scenario where each could be applied.
























Cassie McLemore
Math Concept: Number System
Grade 8
CCSS.Math.Content. 8.NS Know that there are numbers that are not rational, and approximate
them by rational numbers. 8.NS.1 Know that numbers that are not rational are called irrational.
Understand informally that every number has a decimal expansion; for rational numbers show that
the decimal expansion repeats eventually, and convert a decimal expansion, which repeats eventually
into a rational number.

Big Ideas: Rational and irrational numbers have attributes that define them.

Open Question: What does rational and irrational mean?

Expected Student Responses:
Rational is opposite of irrational
Rational means something that makes sense
Irrational means something that doesnt make sense
I dont know
It has something to do with ratios

Management
In groups of 3 to 4 discuss what irrational and rational mean. Decide on a
definition or example to explain the difference between the two.
Record on chart paper the attributes of rational and irrational. Give
students the definitions of the two words (reasonable, logical;
unreasonable, without logic).
In groups of 3 to 4 discuss how a number could be rational or irrational.
Decide on a definition or example to explain the difference between
rational and irrational numbers.
Record on chart paper each groups definition or example of rational and
irrational numbers. Define the two words for the groups in terms of
numbers (all numbers than can reasonably be represented by a fraction,
terminating or repeating decimal; all numbers that cannot be reasonably
represented by a fraction, terminating or repeating decimal
In groups of 3 or 4 list at least 5 rational and irrational numbers.
Record on Chart paper rational and irrational number examples. Discuss
if necessary any examples that were misclassified.

Academic Language Check: Students will need to understand the meaning of the
words rational, irrational, fraction, terminating decimal, nonterminating decimal,
reasonable, and logical.

Debrief: Students will discuss the attributes of rational and irrational numbers and
a spokesperson from each group will share their working definition of rational and
Lesson Goal #1: Students will understand the attributes that make a
number rational or irrational.
irrational numbers including their attributes. Students will use one of the sentence
frames provided or create their own using academic language.

1. The attributes that make a number rational are
2. The attributes that make a number irrational are


Summarize what we have learned: Students will create a paragraph using their
academic language to demonstrate their knowledge of the differences between
rational and irrational numbers.

There are many differenced between _____________________________________________________
numbers. The ______________________ of a rational number are ____________________________
________________________________________________________. The ________________________ of an
irrational number are _______________________________________________________________________
__________________________________________.

Formative Assessment (Exit Ticket, Ticket Out the Door, Show What You
Know): List at least 2 irrational numbers and explain using your academic language
the attributes that make these numbers irrational.


Tiered Lesson:
Group #1: Students will work with teacher to review the attributes of
rational and irrational numbers. Students will evaluate numbers individually and
determine whether they can be converted to a simple fraction, terminating or
repeating decimal. Students will create a T-chart of their findings and write an
explanation for each including the attributes that determined how the numbers
were placed on the T-chart.

Rational Numbers Irrational Numbers
Number:


Explanation:




Number:


Explanation:




Group #2: Students will work independently in small groups to determine
whether numbers are rational or irrational. Students will evaluate numbers
individually and determine whether they can be converted to a simple fraction,
terminating or repeating decimal. Students will create a T-chart of their findings
Lesson Goal #2: Determine whether a number is rational or
irrational.
and write an explanation for each including the attributes that determined how the
numbers were placed on the T-chart.

Academic Language Check: Students will need to understand the meaning of the
words rational, irrational, fraction, terminating decimal, nonterminating decimal,
reasonable, and logical.

Debrief: Students will discuss and share with the group their number choices and
the attributes that determined why they chose rational or irrational.

Summarize what we have learned: Each student will choose one number from
each category and explain what attributes determined their choice.

Formative Assessment: Students T-chart with numbers and explanations as well
as their verbal number choices with explanations.

































Annotated Bibliography
Gavin., M.K., Casa, T.M., Adelson, J.L., & Firmender, J.M. (2013). The impact of
challenging geometry and measurement units on the achievement of grade 2
students. Journal for Research in Mathematics, 44(3), 478-509. Retrieved
from http://www.nctm.org

Gavin, Casa, Adelson and Firmender studied the effect of teaching geometry and
measurement to second graders as a means of effecting higher-level mathematics
achievement. This study included an experimental and non-experimental group to
determine if the designed program (called M squared) was more effectual in
achieving higher test scores in mathematics. The experimental program had
teaching components that included communication and nurturing environments as
well as then normed depth, complexity and differentiation. The results showed
minimal gains by the experimental group on standardized tests scores (only a one
point difference overall) but did show a significant gain for the experimental group
on the verbal and written response assessments. This program (M squared) could
show significant gains in both categories if the program were extended throughout
the grade levels. It was an interesting but challenging read and required extensive
knowledge of quantitative analysis structure to navigate.


Kastberg, S.E., DAmbrosio, B.S., Lynch-Davis, K., Mintos, A., & Krawczyk, K. (2013).
CCSSM challenge: Graphing ratio and proportion. Mathematics Teaching in
the Middle School 19(5) 294-300. Retrieved from http://www.nctm.org

The CCSS placed ratio and proportional relationships in the standards for 7
th
grade.
It is one of the lesson that I focused on for this assignments. In this overview of
graphing ratios and proportions Kastberg and associates focus on the challenge that
teachers have in satisfying this standard. The difficulty for students comes when
trying to take a graph and determine where there is a ratio or proportional
relationship. Students when given the initial problem of solving a proportional
comparison employed purely mathematical computations, often times
misinterpreting the data. When students were given a second chance to solve by
graphing they werent sure what to graph, how to define the axes, and how to
interpret what they were seeing (p. 296). Teachers moving forward are going to
have to be creative in their discussions of mathematics and allow students to
interpret graphs in ratio and proportional reasoning. Looking at student work and
analyzing their errors will assist teachers in designing better lessons to address this
misconceptions.


Kramarski, B., & Mevarech, Z.R. (2003). Enhancing mathematical reasoning in the
classroom: The effects of cooperative learning and metacognitive training.
American Educational Research Journal 40(1), 281-310. DOI:
10.3102/00028312040001281

This study looked at 8
th
grade students and their ability to improve their
mathematics knowledge through cooperative learning and metacognitive training.
Four different groups were studied with the following conditions (1) cooperative
and metacognitive, (2) individual and metacognitive, (3) cooperative without
metacognitive and (4)individual only. The study found that the cooperative and
metacognitive group outperformed all other groups in all but one of the categories.
Students were taught and assessed in mathematical reasoning, graph interpretation,
graph construction and metacognitive knowledge. This study support the theory
that students working in a cooperative environment receiving metacognitive
training are more prepared to defend and argue their reasoning in a mathematical
environment. Even though this article was written before the CCSS standards were
created it has current implications for a restructuring of our classrooms to meet
these standards.


Szilagyi, J., Clements, D.H., & Sarama, J. (2013). Young childrens understanding of
length measurement: Evaluating a learning trajectory. Journal for Research
in Mathematics Education 44(3), 581-620. Retrieved from
http://www.nctm.org

In this study the authors look at the effects of designing a standardized Learning
Trajectory (LT), which is normally individualized, on the learning of length
measurement. The students sampled were pulled from two countries (US and
Hungary), had diverse socioeconomic conditions (to establish generalizability) and
ranged from Pre-K to 2
nd
grade. The researchers only studies students responses to
assessments. No instruction was given. This research was conduced to see if there
was a similar sequencing of students process thinking in the way that they handled
length measurement so that a standardized LT model could be designed. The
researchers found that there was a significant sequencing. This developmental
progression has implications for teachers and curriculum designers. It was not that
useful in my own lesson planning.

Vispoel, W.P. & Austin, J.R. (1995). Success and failure in junior high school: A
critical incident approach to understanding students attributional beliefs.
American Educational Research Journal 32, 377-412. DOI:
10.3102/00028312032002377

This study analyzes the attributional (what students think they are capable of)
beliefs in situational, dispositional and critical incident approaches to learning
(p.378). Students were asked to rate responses to failures on academic tests and
situations. Their responses were used to determine how they actually felt when in
similar situation. Students who are low achievers often cited their lack of ability for
failure, whereas high achievers cited lack of effort for failure (p.381). This study
was not limited to mathematics it also included, English, music and physical
education. The study found that most students perceived effort as the highest
indicator of success or failure with ability (at least in mathematics) coming in a close
second in the failure category. These findings suggest that attribution is a very
important aspect of the learning process and teachers need to focus more on
motivation and success. Students need to feel successful in order for them to
believe that they are capable of success. By the middle grades students have
experienced years of failure and reversing that damage requires meaningful and
thoughtful classroom structure. There are several implications for teachers from
page 405-407 that can help in moving your classroom in the right direction. This
article from 1995 is still relevant to todays struggling learners.


Webb, J.M., & Farivar, S. (1994). Promoting helping behavior in cooperative small
groups in middle school mathematics. American Educational Research
Journal 31(2) 369-395. DOI: 10.3102/00028312031002369

This study tests the research that reports that cooperative small groups increase learning
and understanding of mathematics. It studying the effects that small group cooperative
learning has on explanation, disagreement and argumentation. It has been proven in
many research studies that students respond to their peers in positive ways, in respect to
learning (explanations, misconceptions and language). In this study two teachers with 3
general mathematics classes each, with a comparable achievement level and diverse
cultures. Students were given instruction of two types (1) received instruction in
cooperative groups and elaborative techniques and (2) only work in cooperative groups.
The students in the experimental group (type 1) showed higher achievement and
demonstrated a higher level of elaboration in-group activities. The comparison group
(type 2) showed lower levels of elaboration and were mainly focused on the answer only.
This study shows that students must be trained in both cooperative group work and
elaboration techniques in order for dialogue and not just answer sharing to occur. It also
showed that the largest gains in achievement were with the Latino and African American
groups. Another finding of this study showed that there was a difference in the groups by
teacher. One teacher was more effective in raising the elaboration techniques. As in
most direct instruction style is as important as content. Questioning techniques were
cited as the primary difference in this study. Another finding was that the experimental
group was most effective when the tasks required were higher level reasoning skills.