NCMath 2 Unpack

The Math Resource for Instruction for
North Carolina
Math 2
Link for: Feedback for NC’s Math Link to: Suggest Resources for NC’s Math
Resource for Instruction Resource for Instruction
North Carolina Course of Study - Math 2 Standards
Number Algebra Functions Geometry Statistics & Probability
The real number system Overview Overview Overview Overview
Extended the properties of Seeing structure in Interpreting functions Congruence Making Inference and
exponents to rational expressions Understand the concept of a Experiment with Justifying Conclusions
exponents Interpret the structure of function and use function transformations in the plane Understand and evaluate
NC.M2.N-RN.1 expressions notation NC.M1.G-CO.2 random processes underlying
NC.M2.N-RN.2 NC.M2.A-SSE.1a NC.M2.F-IF.1 NC.M1.G-CO.3 statistical experiments
Use properties of rational and NC.M2.A-SSE.1b NC.M2.F-IF.2 NC.M1.G-CO.4 NC.M1.S-IC.2
irrational numbers NC.M2.A-SSE.3 Interpret functions that arise NC.M1.G-CO.5
NC.M2.N-RN.3 in applications in terms of a Understand congruence in Conditional probability and
Perform arithmetic context terms of rigid motions the rules for probability
The complex number system operations on polynomials NC.M2.F-IF.4 NC.M1.G-CO.6 Understand independence and
Defining complex numbers Perform arithmetic operations Analyze functions using NC.M1.G-CO.7 conditional probability and
NC.M2.N-CN.1 on polynomials different representations NC.M1.G-CO.8 use them to interpret data
NC.M2.A-APR.1 NC.M2.F-IF.7 Prove geometric theorems NC.M1.S-CP.1
NC.M2.F-IF.8 NC.M1.G-CO.9 NC.M1.S-CP.3a
Creating equations NC.M2.F-IF.9 NC.M1.G-CO.10 NC.M1.S-CP.3b
Create equations that describe NC.M1.S-CP.4
numbers or relationships Building functions Similarity, right triangles, NC.M1.S-CP.5
NC.M2.A-CED.1 Build a function that models a and trigonometry Use the rules of probability to
NC.M2.A-CED.2 relationship between two Understand similarity in terms compute probabilities of
NC.M2.A-CED.3 quantities of similarity transformations compound events in a uniform
NC.M2.A-CED.4 NC.M2.F-BF.1 NC.M1.G-SRT.1a probability model
Build new functions from NC.M1.G-SRT.1b NC.M1.S-CP.6
Reasoning with equations existing functions NC.M1.G-SRT.1c NC.M1.S-CP.7
and inequalities NC.M2.F-BF.3 NC.M1.G-SRT.1d NC.M1.S-CP.8
Understand solving equations NC.M1.G-SRT.2a
as a process of reasoning and NC.M1.G-SRT.2b
explain the reasoning NC.M1.G-SRT.3
NC.M2.A-REI.1 Prove theorems involving
NC.M2.A-REI.2 similarity
Solve equations and NC.M1.G-SRT.4
inequalities in one variable Define trigonometric ratios
NC.M2.A-REI.4a and solve problems involving
NC.M2.A-REI.4b right triangles
Solve systems of equations NC.M1.G-SRT.6
NC.M2.A-REI.7 NC.M1.G-SRT.8
Represent and solve equations NC.M1.G-SRT.12
and inequalities graphically
NC.M2.A-REI.11
The Math Resource for Instruction for NC Math 2 Tuesday, February 7, 2017
Number – The Real Number System
NC.M2.N-RN.1
Extend the properties of exponents to rational exponents.
Explain how expressions with rational exponents can be rewritten as radical expressions.
Concepts and Skills The Standards for Mathematical Practices

Pre-requisite Connections
 Rewrite algebraic expressions using the properties of exponents (NC.M1.N- Generally, all SMPs can be applied in every standard. The following SMPs can be
RN.1) highlighted for this standard.
6 – Attend to precision
7 – Look for and make use of structure
7 – Look for and express regularity in repeated reasoning
Connections Disciplinary Literacy
 Rewrite expressions with radicals and rational exponents using the properties of As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
exponents (NC.M2.N-RN.2) all oral and written communication
 Justify the step in a solving process (NC.M2.A-REI.1)
Students should be able to explain with mathematical reasoning how expressions with
rational exponents can be rewritten as radical expressions.
Mastering the Standard

Comprehending the Standard Assessing for Understanding
The meaning of an exponent relates the frequency with which a number is used as a Students should be able to use their understanding of rational exponents to solve
factor. So 53 indicates the product where 5 is a factor 3 times. Extend this meaning to a problems.
1
rational exponent, then 125 ⁄3 indicates one of three equal factors whose product is 125. Example: Determine the value of x
1
a. 642 = 8𝑥
Students recognize that a fractional exponent can be expressed as a radical or a root. b. (125 )𝑥 = 12
1 1
For example, an exponent of is equivalent to a cube root; an exponent of is
3 4
equivalent to a fourth root. Students should be able to explain their reasoning when rewriting expressions with
rational exponents.
𝑛 𝑚 𝑛𝑚 Examples:
Students extend the use of the power rule, (𝑏 ) = 𝑏 from whole number exponents 1
2 3 6
i.e., (7 ) = 7 to rational exponents. a. Write 𝑥 5 as a radical expression.
1⁄ 2 1⁄ ∗2 1
They compare examples, such as (7 2) =7 2 = 71 = 7 to (√7)2 = 7 to establish a b. Write (𝑥 2 𝑦)2 as a radical expression.
1⁄ 1⁄ c. Explain how the power rule of exponents, (𝑏 𝑛 )𝑚 = 𝑏 𝑚𝑛 , can be
connection between radicals and rational exponents: 7 2 = √7 and, in general, 𝑏 2 = 3
used to justify why ( √𝑏)3 = 𝑏.
√𝑏. 2
3 3
d. Explain why 𝑥 3 is equivalent to √𝑥 2 and ( √𝑥 )2 .
Students can then extend their understanding to exponents where the numerator of the
1 3
rational exponent is a number greater than 1. For example 72∗3 = 72 = √73 = (√7)3 .
Instructional Resources
Tasks Additional Resources
Back to: Table of Contents
NC.M2.N-RN.2
Extend the properties of exponents to rational exponents.
Rewrite expressions with radicals and rational exponents into equivalent expressions using the properties of exponents.

 Rewrite algebraic expressions using the properties of exponents (NC.M1.N- Generally, all SMPs can be applied in every standard. The following SMPs can be
RN.1) highlighted for this standard.
 Explain how expressions with rational expressions can be written as radical 6 – Attend to precision
expressions (NC.M2.N-RN.1) 7 – Look for and make use of structure
 Operations with polynomials (NC.M2.A-APR.1) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
 Solve one variable square root equations (NC.M2.A-REI.2) all oral and written communication.
 Solve quadratic equations in one variable (NC.M2.A-REI.4a, NC.M2.A-REI.b)
Students should be able to explain their reasoning while simplifying expressions with
rational exponents and radicals.

Students should be able to simplify expressions Students should be able to rewrite expressions with rational expression into forms that are more simple or useful.
with radicals and with rational exponents. Example: Using the properties of exponents, simplify
Students should be able to rewrite expressions 4 2
a. ( √323 )
involving rational exponents as expressions 5
√𝑏 3
involving radicals and simplify those b. 4
expressions. 𝑏3
3
Students should be able to rewrite expressions Example: Write √27𝑥 2 𝑦 6 𝑧 3 as an expression with rational exponents.
2
involving radicals as expressions using rational Example: Write an equivalent exponential expression for 83 ? Explain how they are equivalent.
exponents and use the properties of exponents to 2 1 1 2
simplify the expressions. Solution: 83 = (82 )3 = (83 ) = 22 In the first expression, the base number is 8 and the exponent is 2/3. This
means that the expression represents 2 of the 3 equal factors whose product is 8, thus the value is 4, since
Students should be able to explain their (2 × 2 × 2) = 8; there are three factors of 2; and two of these factors multiply to be 4. In the second
reasoning while simplifying expressions with expression, there are 2 equal factors of 8 or 64. The exponent 1/3 represents 1 of the 3 equal factors of 64.
rational exponents and radicals. Since 4 × 4 × 4 = 64 then one of the three factors is 4. The last expression there is 1 of 3 equal factors of 8
which is 2 since 2 × 2 × 2 = 8. Then there are 2 of the equal factors of 2, which is 4.
3 3
4 4
Example: Given 814 = √813 = ( √81) , which form would be easiest to calculate without using a calculator. Justify
your answer?
Example: Determine whether each equation is true or false using the properties of exponents. If false, describe at
least one way to make the math statement true.
5
a. √32 = 22
3
b. 162 = 82
1
4
c. 42 = √64
3 6
d. 28 = ( √16)
1 1
e. (√64)3 = 86
NC.M2.N-RN.3
Use properties of rational and irrational numbers.
Use the properties of rational and irrational numbers to explain why:
 the sum or product of two rational numbers is rational;
 the sum of a rational number and an irrational number is irrational;
 the product of a nonzero rational number and an irrational number is irrational.

 Understand rational numbers (8.NS.1) Generally, all SMPs can be applied in every standard. The following SMPs can be
highlighted for this standard.
2 – Reason abstractly and quantitatively
3 – Construct viable arguments and critique the reasoning of others

 These concepts close out the learning about the real number system. As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
all oral and written communication

Students know and justify that when Students should be able to explain the properties of rational and irrational numbers.
 adding or multiplying two rational numbers the result is a Example: Explain why the number 2π must be irrational.
rational number. Sample Response: If 2π were rational, then half of 2π would also be rational, so π would have to be
 adding a rational number and an irrational number the result is rational as well.
irrational.
 multiplying of a nonzero rational number and an irrational Example: Explain why the sum of 3 + 2𝜋 must be irrational.
number the result is irrational.
Example: Explain why the product of 3 ∙ √2 must be irrational.
Note: Since every difference is a sum and every quotient is a
𝑎 𝑟 𝑎 𝑟
product, this includes differences and quotients as well. Example: Given one rational number and another rational number , find the product of ∙ . Use
𝑏 𝑠 𝑏 𝑠
Explaining why the four operations on rational numbers produce this product to justify why the product of two rational numbers must be a rational number. Include in
rational numbers can be a review of students understanding of 𝑎 𝑟
your justification why the number or could represent any rational number.
fractions and negative numbers. Explaining why the sum of a 𝑏 𝑠
rational and an irrational number is irrational, or why the product
is irrational, includes reasoning about the inverse relationship
between addition and subtraction and the relationship between
multiplication and addition.
FAL: Evaluating Statements About Rational and Irrational Numbers (Mathematics

Assessment Project)
Number – The Complex Number System
NC.M2.N-CN.1
Defining complex numbers.
Know there is a complex number i such that 𝑖 2 = – 1, and every complex number has the form 𝑎 + 𝑏𝑖 where 𝑎 and 𝑏 are real numbers.

 The understanding of number systems is developed through middle school Generally, all SMPs can be applied in every standard. The following SMPs can be
(8.NS.1) highlighted for this standard.
 Solve quadratic equations in one variable (NC.M2.A-REI.4b) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
Complex Number
Imaginary
Students should be able to define a complex number and identify when they are likely
to use them.

When students solve quadratic equations they should Students should be able to
understand that there is a solution to an equation when rewrite expressions using Answers
a negative appears in the radicand. This solution does what they know about Problem Solution
1
not produce x-intercepts for the function and is not complex numbers. 𝑖2
𝑖 = (√−1)2 = (−1)2∗2 = −1
2
included in the real number system. This means that it Example: Simplify.
√−36 √−36 = √−1 ∙ √36 = 6𝑖
is now time to introduce students to a broader a. 𝑖 2
classification of numbers so that we have a way to b. √−36 2√−49 2√−49 = 2√−1 ∙ √49 = 2 ∙ 7𝑖 = 14𝑖
express these solutions. c. 2√−49 −3√−10 −3√−10 = −3√−1 ∙ √10 = −3 ∙ 𝑖 ∙ √10 = −3𝑖√10
d. −3√−10 5√−7 5√−7 = 5√−1 ∙ √7 = 5 ∙ 𝑖 ∙ √7 = 5𝑖√7
Students should know that every number can be
e. 5√−7 −3 + √9 − 4 ∗ 2 ∗ 5 −3 + √9 − 4 ∗ 2 ∗ 5 −3 + √−31 −3 + 𝑖√31
written in the form 𝑎 + 𝑏𝑖 ,where a and b are real = =
−3+√9−4∗2∗5 4 4 4 4
numbers and 𝑖 = √−1, f.
4 −3 √31
are classified as complex numbers. If 𝑎 = 0, then the Which can be written in the form 𝑎 + 𝑏𝑖 as + 𝑖
4 4
number is a pure imaginary number. If 𝑏 = 0 the
number is a real number. This means that all real
numbers are included in the complex number system
and that the square root of a negative number is a
complex number.
Students should connect what they have learned

regarding properties of exponents to understand that
1
(√−1)2 = (−1)2∗2 = −1.
Students should be able to express solutions to a

quadratic equation as a complex number.
Algebra, Functions & Function Families
NC Math 1 NC Math 2 NC Math 3
Functions represented as graphs, tables or verbal descriptions in context

Focus on comparing properties of linear Focus on properties of quadratic functions A focus on more complex functions
function to specific non-linear functions and and an introduction to inverse functions • Exponential
rate of change. through the inverse relationship between • Logarithm
• Linear quadratic and square root functions. • Rational functions w/ linear denominator
• Exponential • Quadratic • Polynomial w/ degree < three
• Quadratic • Square Root • Absolute Value and Piecewise
• Inverse Variation • Intro to Trigonometric Functions
A Progression of Learning of Functions through Algebraic Reasoning

The conceptual categories of Algebra and Functions are inter-related. Functions describe situations in which one quantity varies with
another. The difference between the Function standards and the Algebra standards is that the Function standards focus more on the
characteristics of functions (e.g. domain/range or max/min points), function definition, etc. whereas the Algebra standards provide the
computational tools and understandings that students need to explore specific instances of functions. As students progress through high
school, the coursework with specific families of functions and algebraic manipulation evolve. Rewriting algebraic expressions to create
equivalent expressions relates to how the symbolic representation can be manipulated to reveal features of the graphical representation of
a function.
Note: The Numbers conceptual category also relates to the Algebra and Functions conceptual categories. As students become more fluent
with their work within particular function families, they explore more of the number system. For example, as students continue the study of
quadratic equations and functions in Math 2, they begin to explore the complex solutions. Additionally, algebraic manipulation within the
real number system is an important skill to creating equivalent expressions from existing functions.
Algebra – Seeing Structure in Expressions
NC.M2.A-SSE.1a
Interpret the structure of expressions.
Interpret expressions that represent a quantity in terms of its context.
a. Identify and interpret parts of a quadratic, square root, inverse variation, or right triangle trigonometric expression, including terms, factors,
coefficients, radicands, and exponents.

 Interpreting parts of expressions in context (NC.M1.A-SSE.1a, NC.M1.A- Generally, all SMPs can be applied in every standard. The following SMPs can be
SSE.1b) highlighted for this standard.
2 – Reason abstractly and quantitatively.
4 – Model with mathematics
7 – Look for and make use of structure.
 Creating equation to solve, graph, and make systems (NC.M2.A-CED.1, As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
NC.M2.A-CED.2, NC.M2.A-CED.3) all oral and written communication
 Solve and interpret one variable inverse variation and square root equations
(NC.M2.A-REI.2) New Vocabulary: inverse variation, right triangle trigonometry
 Interpreting functions (NC.M2.F-IF.4, NC.M2.F-IF.7, NC.M2.F-IF.9)
 Understand the effect of transformations on functions (NC.M2.F-BF.3)

When given an expression with a context, Students should be able to identify and interpret parts of an expression in its context.
students should be able to explain how the parts Example: The expression −4.9𝑡 2 + 17𝑡 + 0.6 describes the height in meters of a basketball t seconds after it has been
of the expression relate to the context of the thrown vertically into the air. Interpret the terms and coefficients of the expression in the context of this situation.
problem.
Example: The area of a rectangle can be represent by the expression 𝑥 2 + 8𝑥 + 12. What do the factors of this
Students should be able to write equivalent expression represent in the context of this problem?
forms of an expression to be able to identify
parts of the expression that can relate to the Example: The stopping distance in feet of a car is directly proportional to the square of its speed. The formula that
context of the problem. relates the stopping distance and speed of the car is 𝐷 = 𝑘 ∙ 𝑉 2 , where 𝐷 represents the stopping distance in feet, k
represents a constant that depends on the frictional force of the pavement on the wheels of a specific car, and V
The parts of expressions that students should be represents the speed the car was traveling in miles per hour.
able to interpret include any terms, factors, 𝐷
coefficients, radicands, and exponents. When there is a car accident it is important to figure out how fast the cars involved were traveling. The expression √
𝑘
can be evaluated to find the speed that a car was traveling. What does the radicand represent in this expression?
Students should be given contexts that can be
modeled with quadratic, square root, inverse
variation, or right triangle trigonometric Example: Ohm’s Law explains the relationship between current, resistance, and voltage. To determine the current
expressions. 𝑉
passing through a conductor you would need to evaluate the expression , where V represents voltage and R represents
𝑅
resistance. If the resistance is increased, what must happen to the voltage so that the current passing through the
conductor remains constant?
𝑦
Example: The tangent ratio is where (𝑥, 𝑦) is a coordinate on the terminal side of the angle in standard position. Use
𝑥
the diagram to justify why the tangent of 45° is always 1. Then, expand that reasoning to justify why every individual
angle measure has exactly one value for tangent.
Use similar reasoning to justify why every angle has exactly one value of sine and one value of cosine.
The Physics Professor (Illustrative Mathematics)

Quadrupling leads to Halving (Illustrative Mathematics)
NC.M2.A-SSE.1b
Interpret expressions that represent a quantity in terms of its context.
b. Interpret quadratic and square root expressions made of multiple parts as a combination of single entities to give meaning in terms of a
context.

 Interpreting parts of expressions in context (NC.M1.A-SSE.1a, NC.M1.A- Generally, all SMPs can be applied in every standard. The following SMPs can be
2 – Reason abstractly and quantitatively.
7 – Look for and make use of structure.
 Use completing the square to write equivalent form of quadratic expressions to As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
reveal extrema (NC.M2.A-SSE.3) all oral and written communication
 Creating equation to solve, graph, and make systems (NC.M2.A-CED.1,
NC.M2.A-CED.2, NC.M2.A-CED.3) Students should be able to describe their interpretation of an expression.
 Solve and interpret one variable inverse variation and square root equations
(NC.M2.A-REI.2)
 Interpreting functions (NC.M2.F-IF.4, NC.M2.F-IF.7, NC.M2.F-IF.9)
 Understand the effect of transformations on functions (NC.M2.F-IF.2,
NC.M2.F-BF.3)

When given an expression with a context that Students should be able to see parts of an expression as a single quantity that has a meaning based on context.
has multiple parts, students should be able to Example: If the volume of a rectangular prism is represented by 𝑥(𝑥 + 3)(𝑥 + 2), what can (𝑥 + 3)(𝑥 + 2) represent?
explain how combinations of those parts of the
expression relate to the context of the problem. Example: Sylvia is organizing a small concert as a charity event at her school. She has done a little research and found
that the expression −10𝑥 + 180 represents the number of tickets that will sell, given that x represents the price of a
Students should be able to write equivalent ticket. Explain why the income for this event can be represented by the expression −10𝑥 2 + 180𝑥. If all of the expenses
forms of an expression to be able to identify will add up to $150, explain why the expression −10𝑥 2 + 180𝑥 − 150 represents the profit.
combinations of parts of the expression that can
represent a quantity in the context of the Example: When calculating the standard deviation of a population you must first find the mean of the data, subtract the
problem. mean from each value in the data set, square each difference, add all of the squared differences together, divide by the
number of terms in the data set and then take the square root. The expression used for calculating standard deviation of a
Students should be given contexts that can be ∑(𝑥−𝜇)2
modeled with quadratic and square root population is √ . Given the above description of the process of calculating standard deviation and what you have
𝑛
expressions. learned in a previous course about standard deviation being a measure of spread, answer the following questions.
a. Describe what you are finding when you calculate 𝑥 − 𝜇.
b. Describe how the formula for standard deviation is similar to the formula for finding mean.
c. What part of the radicand would have to increase so that the value of the standard deviation would also
increase: the numerator (∑(𝑥 − 𝜇)2 ) or the denominator (n)? Justify your answer.
NC.M2.A-SSE.3
Write an equivalent form of a quadratic expression by completing the square, where 𝑎 is an integer of a quadratic expression, 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, to
reveal the maximum or minimum value of the function the expression defines.

 Rewrite quadratic expression to reveal zeros and solutions (NC.M1.A-SSE.3) Generally, all SMPs can be applied in every standard. The following SMPs can be
 Interpret parts of a function as single entities in context (NC.M2.A-SSE.1b) highlighted for this standard.
 Understand the relationship between the quadratic formula and the process of As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
completing the square (NC.M2.A-REI.4a) all oral and written communication
 Find and compare key features of quadratic functions (NC.M2.F-IF.7,
NC.M2.F-IF.8, NC.M2.F-IF.9) Students should be able to explain when the process of completing the square is
necessary.
New Vocabulary: completing the square

When given an equation in the form 𝑎𝑥 2 + 𝑏𝑥 + Students should be able to reveal the vertex of a quadratic expression using the process of completing the square.
𝑐 students should be able to complete the square Example: Write each expression in vertex form and identify the minimum or maximum value of the function.
to write a quadratic equation in vertex form: a) 𝑥 2 − 4𝑥 + 5
𝑎(𝑥 − ℎ)2 + 𝑘. b) 𝑥 2 + 5𝑥 + 8
c) 2𝑥 2 + 12𝑥 − 18
Students should be able to determine that if 𝑎 > d) 3𝑥 2 − 12𝑥 − 1
0 there is a minimum and if 𝑎 < 0 there is a e) 2𝑥 2 − 15𝑥 + 3
maximum.
Example: The picture at the right demonstrates the process of completing
Students should be able to identify the the square using algebra tiles. Looking at the picture, why might this
maximum or minimum point (ℎ, 𝑘) from an process be called “completing the square”?
equation in vertex form. Note: There are at least two good answers to this question. First the
product must form a square, so you must arrange and complete this missing
Algebra Tiles are a great way to demonstrate parts using zero pairs to make the square. The second, completing the
this process. You can demonstrate the reasoning square is about finding the “new C” which in the process will be a square
for all of the steps in the process. as seen in the yellow blocks in this picture.
This process also links previous learning of the
area model for multiplication.
Seeing Dots (Illustrative Mathematics)
Algebra – Arithmetic with Polynomial Expressions
NC.M2.A-APR.1
Perform arithmetic operations on polynomials.
Extend the understanding that operations with polynomials are comparable to operations with integers by adding, subtracting, and multiplying
polynomials.

 Operations with polynomials (NC.M1.A-APR.1) Generally, all SMPs can be applied in every standard. The following SMPs can be
 Rewrite expressions with radicals and rational exponents using the properties of highlighted for this standard.
exponents (NC.M2.N-RN.2) 6 – Attend to precision

 Solving systems of linear and quadratic equations (NC.M2.A-REI.7) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
 Use equivalent expression to develop completing the square (NC.M2.F-IF.8) all oral and written communication
 Understand the effect of transformations on functions (NC.M2.F-BF.3)
Students should be able to describe their process to multiply polynomials.

The primary strategy for this cluster is to make Students should be able to rewrite polynomials into equivalent forms through addition, subtraction and multiplication.
connections between arithmetic of integers and Example: Simplify and explain the properties of operations apply.
arithmetic of polynomials. In order to a) (𝑥 3 + 3𝑥 2 − 2𝑥 + 5)(𝑥 − 7)
understand this standard, students need to work b) 4𝑏(𝑐𝑏 − 𝑧𝑑)
toward both understanding and fluency with c) (4𝑥 2 − 3𝑦 2 + 5𝑥𝑦) − (8𝑥𝑦 + 3𝑦 2 )
polynomial arithmetic. Furthermore, to talk d) (4𝑥 2 − 3𝑦 2 + 5𝑥𝑦) + (8𝑥𝑦 + 3𝑦 2 )
about their work, students will need to use e) (𝑥 + 4)(𝑥 − 2)(3𝑥 + 5)
correct vocabulary, such as integer, monomial,
binomial, trinomial, polynomial, factor, and
term.
Algebra – Creating Equations
NC.M2.A-CED.1
Create equations that describe numbers or relationships.
Create equations and inequalities in one variable that represent quadratic, square root, inverse variation, and right triangle trigonometric
relationships and use them to solve problems.

 Create and solve equations in one variable (NC.M1.A-CED.1) Generally, all SMPs can be applied in every standard. The following SMPs can be
 Interpret parts of an expression in context (NC.M2.A-SSE.1a, NC.M2.A- highlighted for this standard.
SSE.1b) 1 – Make sense of problems and persevere in solving them
 Justify solving methods and each step (NC.M2.A-REI.1) 2 – Reason abstractly and quantitatively
5 – Use appropriate tools strategically

 Solve inverse variation, square root and quadratic equations (NC.M2.A-REI.2, As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
NC.M2.A-REI.4a, NC.M2.A-REI.4b) all oral and written communication
 Use trig ratios to solve problems (NC.M2.G-SRT.8)
 Solve systems of equations (NC.M2.A-REI.7) Students should be able to explain their reasoning behind their created equation.
 Write a system of equations as an equation or write an equations as a system of New Vocabulary: inverse variation, right triangle trigonometry
equations to solve (NC.M2.A-REI.11)

Students should be able to Students should be able to create one variable equations from multiple representations, including from functions.
determine a correct equation or Example: Lava ejected from a caldera in a volcano during an eruption follows a parabolic path. The formula to find the height
inequality to model a given context of the lava can be found by combining three terms that represent the different forces effecting the lava. The first term is the
and use the model to solve original height of the volcano. The second term concerns the speed at which the lava is ejected. The third term is the effect of
problems. gravity on the lava.
1
Focus on contexts that can be ℎ𝑒𝑖𝑔ℎ𝑡(𝑡) = 𝑜𝑟𝑖𝑔𝑛𝑎𝑙 ℎ𝑒𝑖𝑔ℎ𝑡 + (𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑎𝑣𝑎) ∙ 𝑡 + (𝑒𝑓𝑓𝑒𝑐𝑡𝑠 𝑜𝑓 𝑔𝑟𝑎𝑣𝑖𝑡𝑦) ∙ 𝑡 2
2
modeled with quadratic, square The original height of the caldera is 936𝑓𝑡. The lava was ejected at a speed of 64𝑓𝑡/𝑠. The effect of gravity on any object on
root, inverse variation, and right earth is approximately −32𝑓𝑡/𝑠 2 . Write and solve an equation that will find how long (in seconds) it will take for the lava to
triangle trigonometric equations and reach a height of 1000ft.
inequalities.
Example: The function ℎ(𝑥) = 0.04𝑥 2 − 3.5𝑥 + 100 defines the height (in feet) of a major support cable on a suspension bridge
Students need to be familiar with from the bridge surface where x is the horizontal distance (in feet) from the left end of the bridge. Write an inequality or equation for
algebraic, tabular, and graphic each of the following problems and then find the solutions.
methods of solving equations and a. Where is the cable less than 40 feet above the bridge surface?
inequalities. b. Where is the cable at least 60 feet above the bridge surface?
Example: Jamie is selling key chains that he has made to raise money for school trip. He has done a little research and found that the
expression −20𝑥 + 140 represents the number of keychains that he will be able to sell, given that x represents the price of one
keychain. Each key chain costs Jamie $.50 to make. Write and solve an inequality that he can use to determine the range of prices he
could charge make sure that he earns at least $150 in profit.
Example: In kickboxing, it is found that the force, f, needed to break a board, varies inversely with the length, l, of the board. Write
and solve an equation to answer the following question:
If it takes 5 lbs. of pressure to break a board 2 feet long, how many pounds of pressure will it take to break a board that is 6 feet
long?
Example: To be considered a ‘fuel efficient’ vehicle, a car must get more than 30 miles per gallon. Consider a test run of 200 miles.
How many gallons of fuel can a car use and be considered ‘fuel-efficient’?
Example: The centripetal force 𝐹 exerted on a passenger by a spinning amusement park ride is related to the number of seconds 𝑡
155𝜋2
the ride takes to complete one revolution by the equation 𝑡 = √ . Write and solve an equation to find the centripetal force
𝐹
exerted on a passenger when it takes 12 seconds for the ride to complete one revolution.
Students should be able to create equations using right triangle trigonometry.

Example: Write and solve an equation to find the hypotenuse of the following triangle.
Example: John has a 20-foot ladder leaning against a wall. If the height of the wall that the ladder needs to
reach is at least 15ft, create and solve an inequality to find the angle the ladder needs to make with the
ground.
Throwing a Ball (Illustrative Mathematics)
NC.M2.A-CED.2
Create and graph equations in two variables to represent quadratic, square root and inverse variation relationships between quantities.

 Create and graph equations in two variables (NC.M1.A-CED.2) Generally, all SMPs can be applied in every standard. The following SMPs can be
SSE.1b) 2 – Reason abstractly and quantitatively

 Write equations for a system (NC.M2.A-CED.3) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
 Solve systems of equations (NC.M2.A-REI.7) all oral and written communication
 Write a system of equations as an equation or write an equation as a system of
equations to solve (NC.M2.A-REI.11) New Vocabulary: inverse variation, constant of proportionality
 Analyze functions for key features (NC.M2.F-IF.7)
 Build quadratic and inverse variation functions (NC.M2.F-BF.1)

In this standard students are creating equations and Students should be able to create an equation from a context or representation and graph the equation.
graphs in two variables. Example: The area of a rectangle is 40 in2. Write an equation for the length of the rectangle related to the width.
Graph the length as it relates to the width of the rectangle. Interpret the meaning of the graph.
Focus on contexts that can be modeled with quadratic,
square root and inverse variation relationships. Example: The formula for the volume of a cylinder is given by 𝑉 = 𝜋𝑟 2 ℎ, where r represents the radius of the
circular cross-section of the cylinder and h represents the height. Given that ℎ = 10𝑖𝑛…
This standard needs to be connected with other a. Graph the volume as it relates to the radius.
standards where students interpret functions, generate b. Graph the radius as it relates to the volume.
multiple representations, solve problems, and compare c. Compare the graphs. Be sure to label your graphs and use an appropriate scale.
functions.
Example: Justin and his parents are having a discussion about driving fast. Justin’s parents argue that driving
faster does not save as much time as he thinks. Justin lives 10 miles from school. Using the formula 𝑟 ∙ 𝑡 = 𝑑,
where r is speed in miles per hour and d is the distance from school, rewrite the formula for t and graph. Do
Justin’s parents have a point?
Marbleslides: Parabolas (Desmos.com) NEW
NC.M2.A-CED.3
Create systems of linear, quadratic, square root, and inverse variation equations to model situations in context.

 Create equations for a system of equations in context (NC.M1.A-CED.3) Generally, all SMPs can be applied in every standard. The following SMPs can be
1 – Make sense of problems and persevere in solving them
SSE.1b)
 Create equations in two variables (NC.M2.A-CED.2) 4 – Model with mathematics

 Solve systems of equations (NC.M2.A-REI.7) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
 Write a system of equations as an equation or write an equations as a system of all oral and written communication
Students should be able to justify their created equations through unit analysis.
New Vocabulary: inverse variation, constant of proportionality

Students create systems of equations to model Students should be able to recognize when a context requires a system of equations and create the equations of that system.
situations in contexts. Example: In making a business plan for a pizza sale fundraiser, students determined that both the income and the
expenses would depend on the number of pizzas sold. They predicted that 𝐼(𝑛) = −0.05𝑛2 + 20𝑛 and 𝐸(𝑛) = 5𝑛 +
Contexts should be limited to linear, quadratic, 250. Determine values for which 𝐼(𝑛) = 𝐸(𝑛) and explain what the solution(s) reveal about the prospects of the pizza
square root and inverse variation equations. sale fundraiser.
This standard should be connected with Example: The FFA has $2400 in a fund to raise money for a new tractor. They are selling trees and have determined
NC.M2.A-REI.7 where students solve and 2400
that the number of trees they can buy to sell depends on the price of the tree p, according to the function 𝑛(𝑝) = .
interpret systems and with NC.M2.A-REI.11 𝑝
where students understand the representation of Also, after allowing for profit, the number of trees that customers will purchase depends on the price which the group
the solutions of systems graphically. purchased the trees with function 𝑐(𝑝) = 300 − 6𝑝. For what price per tree will the number of trees that can be equal
the number of trees that will be sold?
Example: Susan is designing wall paper that is made of several different sized squares. She is using a drawing tool for
the square where she can adjust the area and the computer program automatically adjusts the side length by using the
formula 𝑠 = √𝐴. The perimeter of the square can also be inputted into the computer so that the computer will
𝑃
automatically adjust the side length with the formula 𝑠 = . Susan wants to see what the design would look like if the
4
perimeter and area of one of the squares was the same. Create a system of equations that Susan could solve so that she
knows what to input into the computer to see her design. What is the side length that produces the same area and
perimeter?
Algebra – Reasoning with Equations and Inequalities
NC.M2.A-REI.1
Understand solving equations as a process of reasoning and explain the reasoning.
Justify a chosen solution method and each step of the solving process for quadratic, square root and inverse variation equations using mathematical
reasoning.

 Justify a solving method and each step in the process (NC.M1.A-REI.1) Generally, all SMPs can be applied in every standard. The following SMPs can be
 Explain how expressions with rational exponents can be rewritten as radical highlighted for this standard.
expressions (NC.M2.N-RN.1)
 Use equivalent expressions to explain the process of completing the square 6 – Attend to precision
(NC.M2.F-IF.8) 7 – Look for and make use of structure
 Create and solve one variable equations (NC.M2.A-CED.1) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
 Solve inverse variation, square root and quadratic equations (NC.M2.A-REI.2, all oral and written communication
NC.M2.A-REI.4a, NC.M2.A-REI.4b)
 Use trig ratios to solve problems (NC.M2.G-SRT.8) Students should be able to predict the justifications of another student’s solving
 Solve systems of equations (NC.M2.A-REI.7) process.

Students need to be able to explain Students should be able to justify each step in a solving process.
why they choose a specific method to Example: Explain why the equation 𝑥 2 + 14 = 9𝑥 can be solved by determining values of x such that 𝑥 − 7 = 0 and 𝑥 − 2 = 0.
solve an equation.
For example, with a quadratic Example: Solve 3𝑥 2 = −4𝑥 + 8. Did you chose to solve by factoring, taking the square root, completing the square, using the
equation, students could choose to quadratic formula, or some other method? Why did you chose that method? Explain each step in your solving process.
factor, use the quadratic formula, take
the square root, complete the square to 2
Example: Solve = 𝑥 + 1. Did you chose to solve by factoring, taking the square root, completing the square, using the quadratic
take the square root, solve by graphing 𝑥
formula, or some other method? Why did you chose that method? Explain each step in your solving process.
or with a table. Students should be
able to look at the structure of the
quadratic to make this decision. With Example: Solve √𝑥 + 3 = 3𝑥 − 1 using algebraic methods and justify your steps. Solve graphically and compare your solutions.
a square root equation, students could
choose to square both sides, solve by Example: If a, b, c, and d are real numbers, explain how to solve how to solve 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 𝑑 in 2 different methods. Discuss
graphing or with a table. the pros and cons of each method.
Students should be able to chose and justify solution Method A Method B
Discussions on the solving processes methods. 2
5𝑥 + 10 = 90 5𝑥 2 + 10 = 90
and the benefits and drawbacks of Example: To the right are two methods for
−10 = −10 −90 = −90
each method should lead students to solving the equation 5𝑥 2 + 10 = 90. Select one 2 2
not rely on one solving process. of the solution methods and construct a viable 5𝑥 = 80 5𝑥 − 80 = 0
2 2
Students should make determinations argument for the use of the method. 5𝑥 80 5(𝑥 − 16) = 0
=
on the solving process based on the 5 5 5(𝑥 + 4)(𝑥 − 4) = 0
context of the problem, the nature and 𝑥 2 = 16 𝑥 + 4 = 0 𝑜𝑟 𝑥 − 4 = 0
structure of the equation, and 𝑥 = ±√16 𝑥 = 4 𝑜𝑟 𝑥 = −4
efficiency.
𝑥 = 4 𝑜𝑟 𝑥 = −4
While solving algebraically, students
need to use the properties of equality
Method A Method B
to justify and explain each step Example: To the right are two methods for
2𝑥 2 − 3𝑥 + 4 = 0 2𝑥 2 − 3𝑥 + 4 = 0
obtained from the previous step, solving the equation 2𝑥 2 − 3𝑥 + 4 = 0.
assuming the original equation has a Select one of the solution methods and 3
3 ± √(−3)2 − 4(2)(4) 𝑥2 − 𝑥 + 2 = 0
solution. construct a viable argument for the use of 𝑥= 2
the method. 2(2) 3 9 9
Students need to solve quadratic, 𝑥2 − 𝑥 + = −2 +
2 16 16
square root and inverse variation 3 ± √−23
equations. 𝑥= 3 2 −23
4 (𝑥 − ) =
4 16
3 ± 𝑖√23
𝑥= 3 −23
4 𝑥− = ±√
4 16
3 𝑖√23
𝑥= ± 3 𝑖√23
4 4 𝑥= ±
4 4
NC.M2.A-REI.2
Understand solving equations as a process of reasoning and explain the reasoning.
Solve and interpret one variable inverse variation and square root equations arising from a context, and explain how extraneous solutions may be
produced.

 Solve quadratic equations by taking square roots (NC.M1.A-REI.4) Generally, all SMPs can be applied in every standard. The following SMPs can be
 Interpret a function in context be relating it domain and range (NC.M1.F-IF.5) highlighted for this standard.
 Rewrite expressions with radicals and rational exponents using the properties
of exponents (NC.M2.N-RN.2) 8 – Look for and express regularity in repeated reasoning
 Interpret parts of an expression in context (NC.M2.A-SSE.1a, NC.M2.A-
SSE.1b)
 Know there is a complex number and the form of complex numbers As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
(NC.M2.N-NC.1) all oral and written communication
 Create and solve one variable equations (NC.M2.A-CED.1)
New Vocabulary: inverse variation, extraneous solutions
 Justify the solving method and each step in the solving process (NC.M2.A-
REI.1)
 Solve quadratic equations (NC.M2.A-REI.4a, NC.M2.A-REI.4b)
 Use trig ratios and the Pythagorean Theorem to solve problems (NC.M2.G-
SRT.8)

Solve one variable inverse variations and square root Students should be able to solve inverse variation equations.
equations that arise from a context. Example: Tamara is looking to purchase a new outdoor storage shed. She sees an advertisement for a
custom built shed that fits into her budget. In this advertisement, the builder offers a 90 square foot shed
Students should be familiar with direct variation, learned in with any dimensions. Tamara would like the shed to fit into her a corner of her backyard, but the width will
7th and 8th grades. Direct variations occur when two quantities be restricted by a tree. She remembers the formula for the area of a rectangle is 𝑙 ∙ 𝑤 = 𝑎 and solves for the
𝑦 𝑎
are divided to produce a constant, 𝑘 = . This is why direct width to get 𝑤 = . She then measures the restricted width to be 12 feet. What can be the dimensions of
𝑥 𝑙
variation is linked to proportional reasoning. Indirect the shed?
variations occur when two quantities are multiplied to
produce a constant, 𝑘 = 𝑦 ∙ 𝑥.
Students should understand that the process of algebraically 𝑑
Example: The relationship between rate, distance and time can be calculated with the equation 𝑟 = ,
𝑡
solving an equation can produce extraneous solutions.
where 𝑟 is the rate (speed), 𝑑 represents the distance traveled, and 𝑡 represents the time. If the speed of a
Students study this in Math 2 in connection mainly to square
wave from a tsunami is 150 m/s and the distance from the disturbance in the ocean to the shore is 35
root functions. When teaching this standard, it will be
kilometers, how long will it take for the wave to reach the shore?
important to link to the concept of having a limited domain,
not only by the context of a problem, but also by the nature of
Students should be able to solve square root equations and identify extraneous solutions.
the equation.
Example: Solve algebraically: √𝑥 − 1 = 𝑥 − 7
Interpret solutions in terms of the context. a) Now solve by graphing.
b) What do you notice?
c) Check the solutions in the original equation.
d) Why was an “extra” answer produced?
Example: The speed of a wave during a tsunami can be calculated with the formula 𝑠 = √9.81𝑑 where
𝑠 represents speed in meters per second, 𝑑 represents the depth of the water in meters where the
disturbance (for example earthquake) takes place, and 9.81 m/s2 is the acceleration due to gravity. If the
speed of the wave is 150 m/s, what is depth of the water where the disturbance took place?
NC.M2.A-REI.4a
Solve equations and inequalities in one variable.
Solve for all solutions of quadratic equations in one variable.
a. Understand that the quadratic formula is the generalization of solving 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 by using the process of completing the square.

 Rewrite expressions with radicals and rational exponents using the properties of Generally, all SMPs can be applied in every standard. The following SMPs can be
exponents (NC.M2.N-RN.2) highlighted for this standard.
 Use completing the square to write equivalent form of quadratic expressions to 2 – Reason abstractly and quantitatively
reveal extrema (NC.M2.A-SSE.3) 7 – Look for and make use of structure
REI.1)
 Solve inverse variation and square root equations (NC.M2.A-REI.2) all oral and written communication
 Explain that quadratic equations have complex solutions (NC.M2.A-REI.4b)
Students should be able to discuss the relationship between the quadratic formula and
 Solve systems of equations (NC.M2.A-REI.7) the process of completing the square.
 Write a system of equations as an equation or write an equation as a system of New Vocabulary: completing the square, quadratic formula
 Analyze and compare functions (NC.M2.F-IF.7, NC.M2.F-IF.9)

Students have used the method of completing the Students should be able to explain the process of completing the square and be able to generalize it into the quadratic
square to rewrite a quadratic expression in formula.
standard NC.M2.A-SSE.3. In this standard Example: Solve − 2𝑥² − 16𝑥 = 20 by completing the square and the quadratic formula. How are the two methods
students are extending the method to solve a related?
quadratic equation. Complete the square
Example: We often see the need to create a formula when the same steps are
Some students may set the quadratic equal to repeated in the same type of problems. This is true for completing the square.
zero, rewrite into vertex form 𝑎 (𝑥 − ℎ) ² + 𝑘 = Recall the steps for completing the square using a visual model, like algebra
0, and then begin solving to get the equation into tiles. A completed example is provided to the right.
−𝑘 To make a formula, we need to generalize the process. To do this, we replace
the form (𝑥 − ℎ) ² = 𝑞 where 𝑞 = . Other
𝑎 each coefficient with a variable and then solve with those variables in place
students may adapt the method (i.e. not having to
and we treat those variables same as a numbers.
start with the quadratic equal to 0) to get the
Below are two columns. In the left is an example, similar to those you have
equation into the same form.
been asked to solve. On the right is a generalized form of the problem. For the
Students who write vertex form first left column, provide a mathematical reason for each step as you have done before. (Refer back to a visual model as
−2𝑥 2 − 16𝑥 − 20 = 0 needed.) One the right side, identify how you can see that mathematical reasoning in the generalized form. When
−2(𝑥 2 − 8𝑥) − 20 = 0 complete, try out the new formula with the example problem from the left column.
−2(𝑥 2 − 8𝑥 + 16) − 20 − 32 = 0
Completing the Square Completing the Square
−2(𝑥 − 4)2 − 52 = 0
(Example) (Generalized)
−2(𝑥 − 4)2 = 52
3𝑥 2 + 5𝑥 + 4 = 0 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
(𝑥 − 4)2 = 26
𝑥 − 4 = ±√26 5 4 𝑏 𝑐
𝑥2 + 𝑥 + = 0 𝑥2 + 𝑥 + = 0
𝑥 = 4 ± √26 3 3 𝑎 𝑎
5 52 52 4 𝑏 𝑏2 𝑏2 𝑐
Students who adapts method 𝑥2 + 𝑥 + 2 2 = 2 2 − 𝑥2 + 𝑥 + 2 2 = 2 2 −
3 2 ∙3 2 ∙3 3 𝑎 2 ∙𝑎 2 ∙𝑎 𝑎
−2(𝑥 2 − 8𝑥) = 20
5 25 25 4 𝑏 𝑏2 𝑏2 𝑐
−2(𝑥 2 − 8𝑥 + 16) = 20 + 32 𝑥2 + 𝑥 + = − 𝑥2 + 𝑥 + = −
−2(𝑥 − 4)2 = 52 3 36 36 3 𝑎 4 ∙ 𝑎2 4 ∙ 𝑎2 𝑎
(𝑥 − 4)2 = 26 5 25 25 4 12 𝑏 𝑏2 𝑏2 𝑐 4𝑎
𝑥2 + 𝑥 + = − ∙ 𝑥2 + 𝑥 + = − ∙
𝑥 − 4 = ±√26 3 36 36 3 12 𝑎 4∙𝑎 2 4∙𝑎 2 𝑎 4𝑎
𝑥 = 4 ± √26 5 25 −23
𝑥2 + 𝑥 + = 𝑏 𝑏2 𝑏 2 − 4𝑎𝑐
3 36 36 𝑥2 + 𝑥 + 2
=
This standard is about understanding that the 𝑎 4∙𝑎 4 ∙ 𝑎2
quadratic formula is derived from the process of 5 2 −23 𝑏 2 𝑏 2 − 4𝑎𝑐
(𝑥 + ) = (𝑥 + ) =
completing the square. Students should become 6 36 2𝑎 4 ∙ 𝑎2
very familiar with this process before introducing
the quadratic formula. Students should 5 −23
𝑥+ = ±√ 𝑏 𝑏 2 − 4𝑎𝑐
understand completing the square both visually 6 36 𝑥+ = ±√
and symbolically. Algebra titles are a great way 2𝑎 4 ∙ 𝑎2
for students to understand the reasoning behind −5 √−23
𝑥= ± −𝑏 √𝑏 2 − 4𝑎𝑐
the process of completing the square. 6 6 𝑥= ±
It is not the expectation for students to memorize 2𝑎 2𝑎
the steps in deriving the quadratic formula. −5 ± 𝑖√23
𝑥= −𝑏 ± √𝑏 2 − 4𝑎𝑐
(Remember that students have no experience with 6 𝑥=
2𝑎
rational expressions which is required as part of
completing the derivation on their own!)
NC.M2.A-REI.4b
Solve equations and inequalities in one variable.
Solve for all solutions of quadratic equations in one variable.
b. Explain when quadratic equations will have non-real solutions and express complex solutions as 𝑎 ± 𝑏𝑖 for real numbers 𝑎 and 𝑏.

 Rewrite expressions with radicals and rational exponents using the properties of Generally, all SMPs can be applied in every standard. The following SMPs can be
exponents (NC.M2.N-RN.2) highlighted for this standard.
 Know there is a complex number and the form of complex numbers (NC.M2.N-
NC.1) 6 – Attend to precision
 Solve quadratic equations (NC.M2.A-REI.4a)
 Justify the solving method and each step in the solving process (NC.M2.A- all oral and written communication
REI.1)
Students should be able to identify the number of real number solutions of a quadratic
 Solve inverse variation and square root equations (NC.M2.A-REI.2) equation and justify their assertion.
 Solve systems of equations (NC.M2.A-REI.7) New Vocabulary: complex solutions
 Analyze and compare functions (NC.M2.F-IF.7, NC.M2.F-IF.9)

Students recognize when the quadratic formula gives complex solutions and are able to Students should be able to identify the number and type of solution(s) of a quadratic
write them as 𝑎 ± 𝑏𝑖. equation.
Students relate the value of the discriminant to the type of roots expected. A natural Example: How many real roots does 2𝑥² + 5 = 2𝑥 have? Find all solutions of
extension would be to relate the type of solutions to 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 to the the equation.
behavior of the graph of 𝑦 = 𝑎𝑥² + 𝑏𝑥 + 𝑐.
Students are not required to use the word discriminant, but should be familiar with the Example: What is the nature of the roots of 𝑥² + 6𝑥 + 10 = 0? How do you
concepts of the discriminant. know?
Students should develop these concepts through experience and reasoning.
Value of Nature Nature of Examples: Solve each quadratic using the method indicated and explain when in
Discriminant of Roots Graph the solving process you knew the nature of the roots.
𝑏 2 − 4𝑎𝑐 = 0 1 real Intersects x- a) Square root 3𝑥 2 + 9 = 72
root axis once b) Quadratic formula 4𝑥 2 + 13𝑥 − 7 = 0
2
𝑏 − 4𝑎𝑐 > 0 2 real Intersects x- c) Factoring 6𝑥 2 + 13𝑥 = 5
2
roots axis twice d) Complete the square 𝑥 + 12𝑥 − 2 = 0
𝑏 2 − 4𝑎𝑐 < 0 2 Does not Example: Ryan used the quadratic formula to solve an equation and his result was
complex intersect x-axis 8±√(−8)2 −4(1)(−2)
𝑥= .
solutions 2(1)
a) Write the quadratic equation Ryan started with in standard form.
b) What is the nature of the roots?
c) What are the x-intercepts of the graph of the corresponding quadratic
function?
Example: Solve 𝑥 2 + 8𝑥 = −17 for x.
NC.M2.A-REI.7
Solve systems of equations.
Use tables, graphs, and algebraic methods to approximate or find exact solutions of systems of linear and quadratic equations, and interpret the
solutions in terms of a context.

 Use tables, graphs and algebraic methods to find solutions to systems of linear Generally, all SMPs can be applied in every standard. The following SMPs can be
equations (NC.M1.A-REI.6) highlighted for this standard.
 Operations with polynomials (NC.M2.A-APR.1)
REI.1)
 Create equations (NC.M2.A-CED.1, NC.M2.A-CED.2, NC.M2.A-CED.3) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
 Write a system of equations as an equation or write an equation as a system of all oral and written communication
Students should be able to discuss the number of solutions possible in a system with a
 Analyze and compare functions (NC.M2.F-IF.7, NC.M2.F-IF.9) linear and quadratic function and a system with two quadratic functions.

Students solve a system containing a Students should be able to efficiently solve systems of equations with various methods.
linear equation and a quadratic equation Example: In a gymnasium a support wire for the overhead score board slopes down to a point behind the basket. The function
1
in two-variables. Students solve 𝑤(𝑥) = − 𝑥 + 38 describes the height of the wire above the court, 𝑤(𝑥), and the distance in feet from the edge of the score
5
graphically and algebraically.
board, x. During a game, a player must shoot a last second shot while standing under the edge of score board. The trajectory of
the shot is 𝑏(𝑥) = −.08𝑥 2 + 3𝑥 + 6, where 𝑏(𝑥) is the height of the basketball and x is the distance from the player. Describe
Students interpret solutions of a system of
what could have happened to the shot. (All measurements are in feet.)
linear and quadratic equations in terms of
a context.
Example: The area of a square can be calculated with the formula 𝐴𝑟𝑒𝑎 = 𝑠 2 and the perimeter can be calculated with the
formula 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 4𝑠 where 𝑠 is the length of a side of the square. If the area of the square is the same as its perimeter,
what is the length of the side? Demonstrate how you can find the side length using algebraic methods, a table and with a graph.
Example: The student council is planning a dance for their high school. They did some research and found that the relationship
between the ticket price and income that they will receive from the dance can be modeled by the function 𝑓(𝑥) =
−100(𝑥 − 4)2 + 1500. They also calculated their expenses and found that their expenses can be modeled by the function
𝑔(𝑥) = 300 + 10𝑥. What ticket price(s) could the student council charge for the dance if they wanted to break-even (the
expenses are equal to the income)? Demonstrate how you can find the answer using algebraic methods, a table and with a
graph.
NC.M2.A-REI.11
Represent and solve equations and inequalities graphically
Extend the understanding that the 𝑥-coordinates of the points where the graphs of two square root and/or inverse variation equations 𝑦 =
𝑓(𝑥) and 𝑦 = 𝑔(𝑥) intersect are the solutions of the equation 𝑓(𝑥) = 𝑔(𝑥) and approximate solutions using graphing technology or successive
approximations with a table of values.

 Understand the mathematical reasoning behind the methods of graphing, using Generally, all SMPs can be applied in every standard. The following SMPs can be
tables and technology to solve systems and equations (NC.M1.A-REI.11) highlighted for this standard.
 Create equations (NC.M2.A-CED.1, NC.M2.A-CED.3) 4 – Model with mathematics

 Solve systems of equations (NC.M2.A-REI.7) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
Students should be able to discuss how technology impacts their ability to solve more
complex equations or unfamiliar equation types.

Students understand that they can solve a system of equations by Students should be able to solve complex equations and systems of equations.
graphing and finding the point of intersection of the graphs. At Example: Given the following equations determine the x-value that results in an equal output for both
this point of intersection the outputs 𝑓(𝑥) and 𝑔(𝑥) are the same functions.
when both graphs have the same input, 𝑥. 𝑓(𝑥) = √3𝑥 – 2
𝑔(𝑥) = √𝑥 + 2
Students also understand why they can solve any equation by
graphing both sides separately and looking for the point of Example: Solve for x by graphing or by using a table of values.
intersection. 1
= √2𝑥 + 3
𝑥
In addition to graphing, students can look at tables to find the
value of 𝑥 that makes 𝑓(𝑥) = 𝑔(𝑥).
Algebra, Functions & Function Families
Functions represented as graphs, tables or verbal descriptions in context

Focus on comparing properties of linear Focus on properties of quadratic functions A focus on more complex functions
function to specific non-linear functions and and an introduction to inverse functions • Exponential
rate of change. through the inverse relationship between • Logarithm
• Linear quadratic and square root functions. • Rational functions w/ linear denominator
• Exponential • Quadratic • Polynomial w/ degree < three
• Quadratic • Square Root • Absolute Value and Piecewise
• Inverse Variation • Intro to Trigonometric Functions
A Progression of Learning of Functions through Algebraic Reasoning

The conceptual categories of Algebra and Functions are inter-related. Functions describe situations in which one quantity varies with
another. The difference between the Function standards and the Algebra standards is that the Function standards focus more on the
characteristics of functions (e.g. domain/range or max/min points), function definition, etc. whereas the Algebra standards provide the
computational tools and understandings that students need to explore specific instances of functions. As students progress through high
school, the coursework with specific families of functions and algebraic manipulation evolve. Rewriting algebraic expressions to create
equivalent expressions relates to how the symbolic representation can be manipulated to reveal features of the graphical representation of
a function.
Note: The Numbers conceptual category also relates to the Algebra and Functions conceptual categories. As students become more fluent
with their work within particular function families, they explore more of the number system. For example, as students continue the study of
quadratic equations and functions in Math 2, they begin to explore the complex solutions. Additionally, algebraic manipulation within the
real number system is an important skill to creating equivalent expressions from existing functions.
Functions – Interpreting Functions
NC.M2.F-IF.1
Understand the concept of a function and use function notation.
Extend the concept of a function to include geometric transformations in the plane by recognizing that:
 the domain and range of a transformation function f are sets of points in the plane;
 the image of a transformation is a function of its pre-image.

 Formally define a function (NC.M1.F-IF.1) Generally, all SMPs can be applied in every standard. The following SMPs can be

 Extend the use of a function to express transformed geometric figures As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
(NC.M2.F-IF.2) all oral and written communication
 Understand the effects of transformations on functions (NC.M2.F-BF.3)
Students should discuss how an ordered pair can be the domain of a function.
 Experiment with transformations on the plane (NC.M2.G-CO.2) New Vocabulary: preimage, image

Students need to understand that coordinate In previous courses, the x-coordinates were the domain and the y-coordinates were the range. As the students understanding
transformations are functions that have a domain is extended, students should be able to view an entire ordered pair as the domain and another ordered pair as the range.
and range that are points on the coordinate Example: If the domain of a function that is reflected over the x-axis is (3,4), (2,-1), (-1,2), what is the range?
plane.
Example: If the domain of the coordinate transformation 𝑓(𝑥, 𝑦) = (𝑦 + 1, −𝑥 − 4) is (1,4), (−3,2), (−1, −1), what is
The domain consists of the points of the pre- the range?
image and the range consists of points from the
transformed image. Example: If the range of the coordinate transformation 𝑓(𝑥, 𝑦) = (−2𝑥, −3𝑦 + 1) is (10, −2), (8, −5), (−2,4), what is
the domain?
This means that the transformed image is a
function of its pre-image.
Example: Using the graph below, if this transformation was written as a function, identify the domain and range.
NC.M2.F-IF.2
Understand the concept of a function and use function notation.
Extend the use of function notation to express the image of a geometric figure in the plane resulting from a translation, rotation by multiples of 90
degrees about the origin, reflection across an axis, or dilation as a function of its pre-image.

 Describe the effects of dilations, translations, rotations, and reflections on Generally, all SMPs can be applied in every standard. The following SMPs can be
geometric figure using coordinates (8.G.3) highlighted for this standard.
 Interpret parts of a function as single entities in context (NC.M2.A-SSE.1b)
 Extend the concept of functions to include geometric transformations
(NC.M2.F-IF.1)
 Interpret key features of functions from graphs, tables, and descriptions As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
(NC.M2.F-IF.4) all oral and written communication
 Understand the effects of the transformation of functions on other
Students should explain with mathematical reasoning how a dilation, rotation,
representations (NC.M2.F-BF.3) reflection, and translation can be represented as a function.

Students use function notation to express a geometric Students should be able to identify the type of transformation through the function notation.
transformation when performing the following operations: Example: Evaluate the function 𝑓(𝑥, 𝑦) = (−𝑥, −𝑦) for the coordinates (4,5), (3,1), and (-1,4). Graph the
 Translation 𝑓(𝑥, 𝑦) = (𝑥 + ℎ, 𝑦 + 𝑘), where h is a image of the transformation and describe the transformation with words.
horizontal translation and k is a vertical translation.
 Rotation 90° counterclockwise or 270° clockwise Students should be able to use function notation to describe a geometric transformation.
𝑓(𝑥, 𝑦) = (−𝑦, 𝑥) Example: Write a function rule using function notation that will transform a geometric figure by rotating
 Rotation 180° 𝑓(𝑥, 𝑦) = (−𝑥, −𝑦) the figure 90° counterclockwise.
 Rotation 90° clockwise or 270° counterclockwise
𝑓(𝑥, 𝑦) = (𝑦, −𝑥) Example: Write a function rule using function notation that will translate a geometric figure 3 units to the
 Reflection over the x-axis 𝑓(𝑥, 𝑦) = (𝑥, −𝑦) right and 4 units down.
 Reflection over the y-axis 𝑓(𝑥, 𝑦) = (−𝑥, 𝑦)
 Dilation 𝑓(𝑥, 𝑦) = (𝑘𝑥, 𝑘𝑦) where 𝑘 is the scale factor
Students should also continue to use function notation with all

functions introduced in this course and Math 1.
Transformations (Geogebra) NEW
NC.M2.F-IF.4
Interpret functions that arise in applications in terms of the context.
Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities,
including: domain and range, rate of change, symmetries, and end behavior.

 Interpret key features of graphs, tables and verbal descriptions (NC.M1.F-IF.4) Generally, all SMPs can be applied in every standard. The following SMPs can be
SSE.1b)
 Extend the use of function notation to geometric transformations (NC.M2.F-
IF.2)
 Analyze and compare functions (NC.M2.F-IF.7, 8, 9) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
 Build a quadratic and inverse variation function given a graph, description, or all oral and written communication
ordered pairs (NC.M2.F-BF.1)
Students should be able to describe how they identified key features of graph, table, or
 Understand the effects of transformations on functions (NC.M2.F-BF.3) verbal description and interpret those key features in context.

When given a table, graph, or verbal descrition Students should be able to interpret key features of a function from a verbal description.
of a function that models a real-life situation, Example: Jason kicked a soccerball that was laying on the ground. It was in the air for 3 seconds before it hit the
explain the meaning of the key features in the ground again. While the soccer ball was in the air it reached a height of aproximately 30ft. Assuming that the soccer
context of the problem. balls height (in feet) is a function of time (in seconds), interpret the domain, range, rate of change, line of symmertry,
and end behavior in this context.
Key features include: domain and range, rate of
change, symmetries, and end behavior. Students should be able to interpret key features of a function from a table.
Example: Julia was experimenting with a toy car and 4ft ramp. She found that as she increased the height of one end of
When interpreting rate of change students the ramp, the time that the car took to reach the end of the ramp decreased. She collected data to try to figure out the
should be able to describe the rate at which the relationship between ramp height and time and came up with the following table.
function is increasing or decreasing. For Height (ft) .25 .5 .75 1 1.25
example, a linear function with a positive slope Time (sec) 3.9 2.1 1.4 1.1 .9
is increasing at a constant rate. A quadratic with Assuming that time is a function of height, interpret the domain, range, rate of change, and end behavior in terms of this
a maximum point is increasing at a decreasing context.
rate, reaching the maximum, and then
decreasing at an increasing rate. An inverse
variation function in the first quadrant is Students should be able to interpret key features of a function from a graph.
decreasing at a decreasing rate.
Example: The graph to the right is the voltage, v, in a given circuit as a
Connect this standard with NC.M2.F-IF.7. This function of the the time (in seconds). What was the maximum voltage and
standard focuses on interpretation from various for how long did it take to complete the circuit?
representations whereas NC.M2.F-IF.7 focuses
on generating different representations. Also,
this standard is not limited by function type and
can include functions that students do not have
the algebraic skills to manipulate. NC.M2.F-IF.7
lists specific function types for which students
can use algebra to analyze key features of the
function.
NC.M2.F-IF.7
Analyze functions using different representations.
Analyze quadratic, square root, and inverse variation functions by generating different representations, by hand in simple cases and using
technology for more complicated cases, to show key features, including: domain and range; intercepts; intervals where the function is increasing,
decreasing, positive, or negative; rate of change; maximums and minimums; symmetries; and end behavior.

 Interpret parts of an expression in context (NC.M2.A-SSE.1a, NC.M2.A- Generally, all SMPs can be applied in every standard. The following SMPs can be
 Use completing the square to write equivalent form of quadratic expressions to 2 – Reason abstractly and quantitatively
reveal extrema (NC.M2.A-SSE.3) 4 – Model with mathematics
 Interpret key features of functions from graphs, tables, and descriptions
(NC.M2.F-IF.4)
 Create and graph two variable equations (NC.M2.A-CED.2) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
 Analyze quadratic functions rewritten into vertex form (NC.M2.F-IF.8) all oral and written communication
 Compare functions (NC.M2.F-IF.8)
Students should explain which key features are necessary to find given the context of
 Build a quadratic and inverse variation function given a graph, description, or the problem.
ordered pairs (NC.M2.F-BF.1) New Vocabulary: inverse variation, constant of proportionality

Students need to be able to represent a function Students should be able to find the appropriate key feature to solve problems by analyzing the given function.
with an equation, table, graph, and 3ℎ
verbal/written description. Example: The distance a person can see to the horizon can be found using the function 𝑑(ℎ) = √ , where 𝑑(ℎ)
2
represents the distance in miles and h represents the height the person is above sea level. Create a table and graph to
When given one representation students need to represent this function. Use a table, graph, and the equation to find the domain and range, intercepts, end behavior and
be able to generate the other representations and intervals where the function is increasing, decreasing, positive, or negative.
use those representations to identify key
features. Example: Represent the function 𝑓(𝑥) = 2(𝑥 + 3)2 − 2 with a table and graph. Identify the following key features:
domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change;
Key features include: domain and range; maximums and minimums; symmetries; and end behavior.
intercepts; intervals where the function is
increasing, decreasing, positive, or negative; rate
2
of change; maximums and minimums; Example: Represent the function 𝑓(𝑥) = with a table and graph. Identify the following key features: domain and
𝑥
symmetries; and end behavior.
range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; maximums
and minimums; symmetries; and end behavior.
In Math 2 students should focus on quadratic,
square root, and inverse variation functions.
Egg Launch Contest NEW Card Sort: Parabolas (Desmos.com) NEW
NC.M2.F-IF.8
Use equivalent expressions to reveal and explain different properties of a function by developing and using the process of completing the square to
identify the zeros, extreme values, and symmetry in graphs and tables representing quadratic functions, and interpret these in terms of a context.

 Rewrite a quadratic function to reveal key features (NC.M1.F-IF.8a) Generally, all SMPs can be applied in every standard. The following SMPs can be
SSE.1b) 7 – Look for and make use of structure
 Use completing the square to write equivalent form of quadratic expressions to
reveal extrema (NC.M2.A-SSE.3)
 Creating and graphing equations in two variables (NC.M2.A-CED.2) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
 Interpret key features of functions from graphs, tables, and descriptions all oral and written communication
(NC.M2.F-IF.4)
 Analyze and compare functions for key features (NC.M2.F-IF.7, NC.M2.F-IF.9) Students should be able to explain which key features can be found from each form of
a quadratic function.
 Build a quadratic and inverse variation function given a graph, description, or
New Vocabulary: completing the square

Students look at equivalent expressions of Students should be able use the process of completing the square to identify key features of the function.
functions to identify key features on the graph Example: Coyote was chasing roadrunner, seeing no easy escape, Roadrunner jumped off a cliff towering above the
and in a table of the function. roaring river below. Molly Mathematician was observing the chase and obtained a digital picture of this fall. Using her
mathematical knowledge, Molly modeled the Road Runner’s fall with the following quadratic functions:
For example, students should factor quadratics ℎ(𝑡) = −16𝑡 2 + 32𝑡 + 48
to identify the zeros, complete the square to ℎ(𝑡) = −16(𝑡 + 1)(𝑡 – 3)
reveal extreme values and the line of symmetry, ℎ(𝑡) = −16(𝑡 − 1)2 + 64
and look at the standard form of the equation to a) How can Molly have three equations?
reveal the y-intercept. b) Which of the rules would be most helpful in answering each of these questions? Explain.
i. What is the maximum height the Road Runner reaches and when will it occur?
Students could also argue that by factoring and ii. When would the Road Runner splash into the river?
finding the zeros they could easily find the line iii. At what height was the Road Runner when he jumped off the cliff?
of symmetry by finding the midpoint between
the zeros.
Once identifying the key features students

should interpret them in terms of the context.
Students should be able to identify the key features able to be found in each form of a quadratic function.
Example: Which of the following equations could describe the function of the given
graph to the right? Explain.
𝑓1 (𝑥) = (𝑥 + 12)2 + 4 𝑓5 (𝑥) = −4(𝑥 + 2)(𝑥 + 3)
𝑓2 (𝑥) = −(𝑥 − 2)2 − 1 𝑓6 (𝑥) = (𝑥 + 4)(𝑥 − 6)
2
𝑓3 (𝑥) = (𝑥 + 18) − 40 𝑓7 (𝑥) = (𝑥 − 12)(−𝑥 + 18)
2
𝑓4 (𝑥) = (𝑥 + 12) + 4 𝑓8 (𝑥) = (20 − 𝑥)(30 − 𝑥)
Throwing Horseshoes (Illustrative Mathematics)
FAL: Representing Quadratics Graphically (Mathematics Assessment Project)
Profit of a Company (Illustrative Mathematics)
NC.M2.F-IF.9
Compare key features of two functions (linear, quadratic, square root, or inverse variation functions) each with a different representation
(symbolically, graphically, numerically in tables, or by verbal descriptions).

 Compare key features of two functions (NC.M1.F-IF.9) Generally, all SMPs can be applied in every standard. The following SMPs can be
SSE.1b)
 Use completing the square to write equivalent form of quadratic expressions to
reveal extrema (NC.M2.A-SSE.3)
(NC.M2.F-IF.4)
 Analyze functions for key features (NC.M2.F-IF.7, NC.M2.F-IF.8)
 Build a quadratic and inverse variation function given a graph, description, or
As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in

Students need to compare characteristics of two Students should be able to compare key features of two functions in different representations.
functions. The representations of the functions Example: Compare the constant of proportionality for each of the following inverse
should vary: table, graph, algebraically, or variation models and list them in order from least to greatest.
verbal description. x y
5 36
In this standard students are comparing any two 90 10 18
of the following functions: 𝑦= 𝑥
15 12
 Linear 20 9
 Quadratic 25 7.2
 Square root
 Inverse variation
This means that students need to be able to Example: Compare and contrast the domain and range, rate of change and intercepts of the two functions below
compare functions that are in the same function represented below.
family (for example quadratic vs quadratic) and
functions that are in different function families Meredith runs at a constant
(for example square root vs inverse variation). rate of 6 miles per hour when
she runs on her treadmill. The
The representations of the functions that are distance that she runs on her
being compared needs to be different. For treadmill is a function of the
example compare a graph of one function to an time that she is runs.
equation of another.
Example: Compare and contrast the end behavior and symmetries of the two functions represented below.
𝑥 𝑓(𝑥)
-2 4
-1 1
0 0
1 1
2 4
Throwing Baseballs (Illustrative Mathematics)
Functions – Building Functions
NC.M2.F-BF.1
Build a function that models a relationship between two quantities.
Write a function that describes a relationship between two quantities by building quadratic functions with real solution(s) and inverse variation
functions given a graph, a description of a relationship, or ordered pairs (include reading these from a table).

 Build linear and exponential functions from tables, graphs, and descriptions Generally, all SMPs can be applied in every standard. The following SMPs can be
(NC.M1.F-BF.1a) highlighted for this standard.
 Creating and graphing equations in two variables (NC.M2.A-CED.2)
 Interpret key features of functions from graphs, tables, and descriptions 5 – Use appropriate tools strategically
(NC.M2.F-IF.4)

 Analyze and compare functions for key features (NC.M2.F-IF.7, NC.M2.F-IF.8, As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
NC.M2.F-IF.9) all oral and written communication
Students should be able to justify their chosen model with mathematical reasoning.

Given a graph, ordered pairs (including a table), Students should be able to build functions that model a given situation using the context and information available from
or description of a relationship, students need to various representations.
be able to write an equation of a function that Example: Write an equation of the function given the table.
describes a quadratic or inverse variation 𝑥 -3 -2 -1 0 1 2 3
relationship. 𝑓(𝑥) -4 -6 -12 undefined 12 6 4
Make sure that quadratic functions have real
solutions. (Operations with complex numbers
are not part of the standards.) Example: Write an equation to represent the following relationship: y varies inversely with x. When 𝑥 = 3 then 𝑦 = 5.
Student should realize that in an inverse

variation relationship they can multiply the x
and y coordinates of an ordered pair together to
get the constant of proportionality.
When given the x-intercepts and a point on a

quadratic students can solve the equation
𝑓(𝑥) = 𝑎(𝑥 − 𝑚)(𝑥 − 𝑛) for 𝑎 after
substituting the x-intercpets for 𝑚 and 𝑛, and the
𝑥 and 𝑦 coordinates from the point for 𝑥 and
𝑓(𝑥). Once the student has solved for 𝑎 they can Example: Write an equation of the function given the graph.
plug 𝑎, 𝑚, and 𝑛 into the equation so that their
equation is written in factored form.
When given a maximum or minimum point on a

quadratic and another point students can use the
equation 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘 to solve for 𝑎
so that their function equation is written in
vertex form.
Functions – Building Functions
NC.M2.F-BF.3
Build new functions from existing functions.
Understand the effects of the graphical and tabular representations of a linear, quadratic, square root, and inverse variation function f with 𝑘 ∙ 𝑓(𝑥),
𝑓(𝑥) + 𝑘, 𝑓(𝑥 + 𝑘) for specific values of 𝑘 (both positive and negative).

 Interpret parts of an expression in context (NC.M2.A-SSE.1a, NC.M2.A- Generally, all SMPs can be applied in every standard. The following SMPs can be
 Operations with polynomials (NC.M2.A-APR.1) 7 – Look for and make sense of structure
 Extend the concept of functions to include geometric transformations
(NC.M2.F-IF.1)
 Extend the use of function notation to express the transformation of geometric As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
figures (NC.M2.F-IF.2) all oral and written communication
Students should be able to compare and contrast the transformation of geometric
(NC.M2.F-IF.4) figures and two variable equations expressed as functions.
 Analyze and compare functions for key features (NC.M2.F-IF.7, NC.M2.F-IF.9) New Vocabulary: inverse variation, constant of proportionality, vertical compression,
vertical stretch

It is important to note that this standard is under Students should be able to describe the effect of transformations on algebraic functions.
the domain of building functions. The functions Example: Compare the shape and position of the graphs of 𝑓(𝑥) = 𝑥 2 and 𝑔(𝑥) =
are being built for a purpose, to solve a problem 2𝑥 2 and explain the differences in terms of the algebraic expressions for the
or to offer insight. functions.
Students should conceptually understand the
transformations of functions and refrain from
blindly memorizing patterns of functions.
Students should be able to explain why 𝑓(𝑥 +
𝑘) moves the graph of the function left or right
depending on the value of k.
Students should understand how changes in the Example: Describe the effect of varying the parameters 𝑎, ℎ, and 𝑘 on the shape and position of the graph of the
equation effect changes in graphs and tables of equation𝑓 (𝑥) = 𝑎 (𝑥 − ℎ) 2 + 𝑘. Then compare that to the effect of varying the parameters 𝑎, ℎ, and 𝑘 on the shape
values. and position of the graph of the equation 𝑔(𝑥) = 𝑎√𝑥 − ℎ + 𝑘.
 𝑘 ∙ 𝑓(𝑥) If 0 < 𝑘 < 1 there is a vertical
compression meaning that the outputs of the
function have been reduced since they were Example: Describe the transformation that took place with the function transformation where 𝑓(𝑥) = √𝑥 is transformed
multiplied by a number between 0 and 1. If to 𝑔(𝑥) = 2√𝑥 + 3 − 4.
𝑘 > 1 there is a vertical stretch meaning
that the outputs have all been multiplied by 1
Example: Write an equation for the transformation of 𝑓(𝑥) = after it has been translated 3 units to the right and reflected
the same value. If 𝑘 is negative, then all of 𝑥
the outputs will change signs and this will over the x-axis.
result in a reflection over the x-axis.
 𝑓(𝑥) + 𝑘 If 𝑘 is positive all of the outputs Example: A computer game uses functions to simulate the paths of an archer’s arrows. The x-axis represents the level
are being increased by the same value and ground on which the archer stands, and the coordinate pair (2,5) represents the top of a castle wall over which he is
the graph of the function will move up. If 𝑘 trying to fire an arrow.
is negative, all of the outputs are being In response to user input, the first arrow followed a path defined by the function 𝑓(𝑥) = 6 − 𝑥 2 failing to clear the castle
decreased by the same value and the graph wall.
of the function will move down.
The next arrow must be launched with the same force and
 𝑓(𝑥 + 𝑘) If 𝑘 is positive then all of the
trajectory, so the user must reposition the archer in order for
inputs are increasing by the same value.
his next arrow to have any chance of clearing the wall.
Since they are increasing before they are
plugged into the operations of the function,
the graph will move to the left. If 𝑘 is a) How much closer to the wall must the archer stand in order
negative, then all of the inputs are for the arrow to clear the wall by the greatest possible
decreasing by the same value. Since they distance?
are decreasing before they are plugged into b) What function must the user enter in order to accomplish
the operations of the function the graph will this?
move to the right. c) If the user can only enter functions of the form 𝑓(𝑥 + 𝑘),
what are all the values of k that would result in the arrow
Students should focus on linear, quadratic, clearing the castle wall?
square root, and inverse variation functions in https://www.illustrativemathematics.org/content-
this course. standards/HSF/BF/B/3/tasks/695
Medieval Archer (Illustrative Mathematics)
Geometry
Analytic & Euclidean
Focus on coordinate geometry Focus on triangles Focus on circles and continuing the work
• Distance on the coordinate plane • Congruence with triangles
• Midpoint of line segments • Similarity • Introduce the concept of radian
• Slopes of parallel and perpendicular lines • Right triangle trigonometry • Angles and segments in circles
• Prove geometric theorems algebraically o Special right triangles • Centers of triangles
• Parallelograms
A Progression of Learning
Integration of Algebra and Geometry Geometric proof and SMP3 Geometric Modeling
• Building off of what students know from • An extension of transformational • Connecting analytic geometry, algebra,
th th
5 – 8 grade with work in the geometry concepts, lines, angles, and functions, and geometric measurement to
coordinate plane, the Pythagorean triangles from 7th and 8th grade modeling.
theorem and functions. mathematics. • Building from the study of triangles in
• Students will integrate the work of • Connecting proportional reasoning from Math 2, students will verify the properties
algebra and functions to prove geometric 7th grade to work with right triangle of the centers of triangles and
theorems algebraically. trigonometry. parallelograms.
• Algebraic reasoning as a means of proof • Students should use geometric reasoning
will help students to build a foundation to prove theorems related to lines, angles,
to prepare them for further work with and triangles.
geometric proofs.
It is important to note that proofs here are not limited
to the traditional two-column proof. Paragraph, flow
proofs and other forms of argumentation should be
encouraged.
Geometry – Congruence
NC.M2.G-CO.2
Experiment with transformations in the plane.
 Represent transformations in the plane.
 Compare rigid motions that preserve distance and angle measure (translations, reflections, rotations) to transformations that do not preserve
both distance and angle measure (e.g. stretches, dilations).
 Understand that rigid motions produce congruent figures while dilations produce similar figures.

 Verify experimentally the properties of rotations, reflections and translations. Generally, all SMPs can be applied in every standard. The following SMPs can be
(8.G.1) highlighted for this standard.
 Understand congruence through rotations, reflections and translations (8.G.2)
 Use coordinates to describe the effects of transformations on 2-D figures (8.G.3) 6 – Attend to precision
 Verify experimentally properties of rigid motions in terms of angles, circles,  As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
and || lines and line segments (NC.M2.G-CO.4) in all oral and written communication
 Verify experimentally the properties of dilations given center and scale factor
New Vocabulary: rigid motion, non-rigid motion
(NC.M2.G-SRT.1)
 Geometric transformations as functions (NC.M2.F-IF.1)
 Using function notation to express transformations (NC.M2.F-IF.2)
 Given a regular polygon, identify reflections/rotations that carry the image onto
itself (NC.M2.G-CO.3)
 Given a geometric figure and a rigid motion, find the image of the figure/Given
a figure and its image, describe a sequence of rigid motions between preimage
and image (NC.M2.G-CO.5)

In 8th grade, students understand transformations and their Students describe and compare function transformations on a set of points as inputs to produce another
relationship to congruence and similarity through the use of set of points as outputs.
physical models, transparencies, and geometry software. Example: A plane figure is translated 3 units right and 2 units down. The translated figure is then
dilated with a scale factor of 4, centered at the origin.
In Math 2, students begin to formalize these ideas and connect a. Draw a plane figure and represent the described transformation of the figure in the plane.
transformations to the algebraic concept of function. A b. Explain how the transformation is a function with inputs and outputs.
transformation is a new type of function that maps two numbers c. Write a mapping rule for this function.
(an ordered pair) to another pair of numbers. d. Determine what type of relationship, if any, exists between the pre-image and the image after
this series of transformations. Provide evidence to support your thinking.
Transformations that are rigid (preserve distance and angle
measure: reflections, rotations, translations, or combinations of Example: Transform 𝛥𝐴𝐵𝐶 with vertices 𝐴 (1,1), 𝐵 (6,3) and 𝐶 (2,13) using the function rule
these) and those that are not (stretches, dilations or rigid motions (𝑥, 𝑦)  (−𝑦, 𝑥). Describe the transformation as completely as possible.
followed by stretches or dilations). Translations, rotations and
reflections produce congruent figures while dilations produce
similar figures.
Note: It is not intended for students to memorize transformation

rules and thus be able to identify the transformation from the
rule. Students should understand the structure of the rule and how
to use it as a function to generate outputs from the provided
inputs.
Horizontal Stretch of the Plane (Illustrative Mathematics) Transforming 2D Figures (Mathematics Assessment Project)
Marcellus the Giant (Desmos.com) NEW
NC.M2.G-CO.3
Given a triangle, quadrilateral, or regular polygon, describe any reflection or rotation symmetry i.e., actions that carry the figure onto itself. Identify
center and angle(s) of rotation symmetry. Identify line(s) of reflection symmetry. Represent transformations in the plane.

 Understand congruence through rotations, reflections and translations (8.G.2) Generally, all SMPs can be applied in every standard. The following SMPs can be
 Use coordinates to describe the effects of transformations on 2-D figures (8.G.3) highlighted for this standard.
 Geometric transformations as functions (NC.M2.F-IF.1) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
 Using function notation to express transformations (NC.M2.F-IF.2) in all oral and written communication
 Understand that rigid motions produce congruent figures (NC.M2.G-CO.2)  What kinds of figures have only rotational symmetry? What kinds of figures
have only reflection symmetry? What kind have both? Why do you think this
 Verify experimentally properties of rigid motions in terms of angles, circles and
happens?
lines (NC.M2.G-CO.4)
 Given a geometric figure and a rigid motion, find the image of the figure/Given
a figure and its image, describe a sequence of rigid motions between preimage
and image (NC.M2.G-CO.5)

“The concepts of congruence, similarity, and Students describe and illustrate how figures such as an isosceles triangle, equilateral triangle, rectangle, parallelogram,
symmetry can be understood from the kite, isosceles trapezoid or regular polygon are mapped onto themselves using transformations.
perspective of geometric transformation. Example: For each of the following figures, describe and illustrate the rotations and/or reflections that carry the
Fundamental are the rigid motions: translations, figure onto itself.
rotations, reflections, and combinations of
these, all of which are here assumed to preserve
distance and angles (and therefore shapes
generally). Reflections and rotations each
explain a particular type of symmetry, and the
symmetries of an object offer insight into its
attributes—as when the reflective symmetry of
an isosceles triangle assures that its base angles
are congruent.” (Intro of HS Geometry strand of the
CCSS-M) Students should make connections between the symmetries of a geometric figure and its properties. In addition to the
example of an isosceles triangle noted above, figures with 180 rotation symmetry have opposite sides that are congruent.
Example: What connections can you make between a particular type of symmetry and the properties of a figure?
Students can describe and illustrate the center of rotation and angle(s) of rotation symmetry and line(s) of reflection
symmetry.
Transforming 2D Figures (Mathematics Assessment Project)
NC.M2.G-CO.4
Verify experimentally properties of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line
segments.

 Using coordinates to solve geometric problems algebraically (NC.M1.G-GPE.4) Generally, all SMPs can be applied in every standard. The following SMPs can be
 Using slope to determine parallelism and perpendicularity (NC.M1.G-GPE.5) highlighted for this standard.
 Finding midpoint/endpoint of a line segment, given either (NC.M1.G-GPE.6)
As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
in all oral and written communication

This standard is intended to help students Students develop the definition of each transformation in regards to the characteristics between pre-image and image
develop the definition of each rigid motion in points.
regards to the characteristics between pre- Example: Triangle A’B’C’ is a translation of triangle ABC. Write the rule for the translation. Draw line segments
image and image points through connecting corresponding vertices. What do you notice?
experimentation.
 For translations: connecting points on the
pre-image to corresponding points on the
image produces line segments that are Productive answers:
congruent and parallel. (𝑥, 𝑦) → (𝑥 + 5, 𝑦 − 2)
 For reflections: the line of reflection is the ̅̅̅̅̅ ∥ 𝐵𝐵′
𝐴𝐴′ ̅̅̅̅̅ ∥ 𝐶𝐶′
̅̅̅̅̅
perpendicular bisector of any line segment ̅̅̅̅̅
𝐴𝐴′ ≅ 𝐵𝐵′ ≅ ̅̅̅̅̅
̅̅̅̅̅ 𝐶𝐶′
joining a point on the pre-image to the
corresponding point on the image.
Therefore, corresponding points on the
pre-image and the image are equidistant
from the line of reflection.
 For rotations: a point on the pre-image and
its corresponding point on the image lie on
a circle whose center is the center of
rotation. Therefore, line segments
connecting corresponding points on the
pre-image and the image to the center of
rotation are congruent and form an angle Example: Quadrilateral A’B’C’D’ is a reflection of quadrilateral ABCD Productive answers:
equal to the angle of rotation. across the given line. Draw line segments connecting A to A’ and C to C’. ̅̅̅̅̅ ∥ 𝐶𝐶′
𝐴𝐴′ ̅̅̅̅̅
Label the points of intersection with the line of reflection as E and F. What ̅̅̅̅ ≅ 𝐴
𝐴𝐸 ̅̅̅̅̅
′𝐸
There are two approaches – both that should be do you notice? ̅̅̅̅
𝐶𝐹 ≅ 𝐶′𝐹̅̅̅̅̅
used when teaching this standard. First, work ̅̅̅̅̅
𝐴𝐴′ ⊥ ̅̅̅̅
𝐸𝐹
with transformations on the coordinate plane. ̅̅̅̅̅
𝐶𝐶′ ⊥ ̅̅̅̅
𝐸𝐹
For this, students need to have some reasoning A and A’ are equidistant from the
skills with figures on the coordinate plane. line of reflection.
Calculating distances on the coordinate plane C and C’ are equidistant from the
line of reflection.
can help achieve this:
 show that the line of symmetry bisects the
segment connecting image to preimage for
a reflection;
 show that the segments connecting the E
image to center and preimage to center are
the same length and represent the radius of
the circle whose central angle is the angle F
of rotation
 show line segments are parallel for
translations Example: Triangle 𝐴’𝐵’𝐶’ is a rotation of
 show line segments are perpendicular for triangle 𝐴𝐵𝐶. Describe the rotation,
reflection indicating center, angle, and direction. Draw line segments connecting
corresponding vertices to the center. What do you notice?
The second approach is to work with the
Triangle ABC is rotated 90 CW around
transformations on the Euclidean plane. point D.
Students should use tools (patty paper, mirrors, Corresponding vertices lie on the same
rulers, protractors, string, technology, etc) to circle. The circles all have center D.
measure and reason. ̅̅̅̅ ≅ ̅̅̅̅̅
𝐶𝐷 𝐶 ′ 𝐷 and 𝑚∠𝐶𝐷𝐶 ′ = 90°.
𝐴𝐷 ≅ ̅̅̅̅̅
̅̅̅̅ 𝐴′ 𝐷 and 𝑚∠𝐴𝐷𝐴′ = 90°.
̅̅̅̅
𝐵𝐷 ≅ 𝐵 ̅̅̅̅̅
′ 𝐷 and 𝑚∠𝐵𝐷𝐵 ′ = 90°.
NC.M2.G-CO.5
Given a geometric figure and a rigid motion, find the image of the figure. Given a geometric figure and its image, specify a rigid motion or
sequence of rigid motions that will transform the pre-image to its image.

 Understand congruence through rotations, reflections and translations (8.G.2) Generally, all SMPs can be applied in every standard. The following SMPs can be

 Geometric transformations as functions (NC.M2.F-IF.1) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
 Using function notation to express transformations (NC.M2.F-IF.2) in all oral and written communication
 Understand that rigid motions produce congruent figures (NC.M2.G-CO.2)
 Verify experimentally properties of rigid motions in terms of angles, circles and
lines (NC.M2.G-CO.4)
 Given a regular polygon, identify reflections/rotations that carry the image onto
itself (NC.M2.G-CO.3)
 Determining congruence through a sequence of rigid motions (NC.M2.G-CO.6)
In 8th grade, students build an understanding of Students transform a geometric figure given a rotation, reflection, or translation, using graph paper, tracing paper and/or
congruence through translations, reflections geometry software.
and rotation informally and in terms of Example: Using the figure on the right:
coordinates. Students in MS verify that images Part 1: Draw the shaded triangle after:
transformed in the plane with rigid motions a. It has been translated −7 units horizontally and +1 units
keep the same property as the preimage. They vertically. Label your answer A.
also note the effect of the rigid motion on the b. It has been reflected over the x-axis. Label your answer B.
coordinates of the image and preimage. This c. It has been rotated 90° clockwise about the origin. Label
standard extends the work in MS by requiring your answer C.
students to give precise descriptions of d. It has been reflected over the line 𝑦 = 6. Label your answer
sequences of rigid motions where they specify D.
exact points, lines and angles with coordinates
and/or equations. Analytically, each rigid
motion should be specified as follows:
 For each rotation, students should specify
a point (𝑥, 𝑦) and angle.
 For each translation, specific pairs of
points (𝑥, 𝑦) should be identified;
 For each reflection, the equation of the Students predict and verify the sequence of transformations (a composition) that will map a figure onto another.
line (𝑦 = 𝑚𝑥 + 𝑏) should be identified. Part 2: Describe fully the transformation or sequence of transformations that:
a. Takes the shaded triangle onto the triangle labeled E.
These specificities hold true whether working b. Takes the shaded triangle onto the triangle labeled F.
in the coordinate or Euclidean plane. Students
must specify all points, lines of
reflection/symmetry and angles of rotation.
NC.M2.G-CO.6
Understand congruence in terms of rigid motions.
Determine whether two figures are congruent by specifying a rigid motion or sequence of rigid motions that will transform one figure onto the
other.

 Given a geometric figure and a rigid motion, find the image of the figure/Given Generally, all SMPs can be applied in every standard. The following SMPs can be
a figure and its image, describe a sequence of rigid motions between preimage highlighted for this standard.
and image (NC.M2.G-CO.5) 3 – Construct viable arguments and critique the reasoning of others
 Use the properties of rigid motions to show that two triangles are congruent if As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
their corresponding sides and angles are congruent (NC.M2.G-CO.7) in all oral and written communication

This standard connects to the 8th grade standard Students use descriptions of rigid motion and transformed geometric figures to predict the effects rigid motion has on
where students informally addressed figures in the coordinate plane.
congruency of figures through rigid motions to Example: Consider parallelogram ABCD with coordinates 𝐴(2, −2), 𝐵(4,4), 𝐶(12, 4) and 𝐷(10, −2). Consider the
the formalized HS standard where students following transformations. Make predictions about how the lengths, perimeter, area and angle measures will change
specifically defined points, lines, planes and under each transformation below:
angles of rigid motion transformations. a. A reflection over the x-axis.
b. A rotation of 270° counter clockwise about the origin.
Students recognize rigid transformations c. A dilation of scale factor 3 about the origin.
preserve size and shape (or distance and angle)
d. A translation to the right 5 and down 3.
and develop the definition of congruence. This
standard goes beyond the assumption of mere
correspondence of points, lines and angles and Verify your predictions by performing the transformations. Compare and contrast which transformations preserved
thus establishing the properties of congruent the size and/or shape with those that did not preserve size and/or shape. Generalize: which types of
figures. transformation(s) will produce congruent figures?
Students determine if two figures are congruent by determining if rigid motions will map one figure onto the other.
Example: Determine if the figures are congruent. If so, describe and
demonstrate a sequence of rigid motions that maps one figure onto the
other.
NC.M2.G-CO.7
Use the properties of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs
of angles are congruent.
 Determining congruence through a sequence of rigid motions (NC.M2.G-CO.6) Generally, all SMPs can be applied in every standard. The following SMPs can be
 Use and justify criteria to determine triangle congruence (NC.M2.G-CO.8) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation

A rigid motion is a transformation of points in Students identify corresponding sides and corresponding angles of congruent triangles. Explain that in a pair of congruent
space consisting of a sequence of one or more triangles, corresponding sides are congruent (distance is preserved) and corresponding angles are congruent (angle
translations, reflections, and/or rotations. Rigid measure is preserved). They demonstrate that when distance is preserved (corresponding sides are congruent) and angle
motions are assumed: measure is preserved (corresponding angles are congruent) the triangles must also be
 to map lines to lines, rays to rays, and congruent.
segments to segments and Example: Illustrative Mathematics Task – Properties of Congruent Triangles
To the right is a picture of two triangles:
 to preserve distances and angle measures.
Two triangles are said to be congruent if one
a. Suppose there is a sequence of rigid motions which maps ∆𝐴𝐵𝐶 to
can be exactly superimposed on the other by a
rigid motion, and the congruence theorems ∆𝐷𝐸𝐹. Explain why corresponding sides and angles of these triangles
specify the conditions under which this can are congruent.
occur. b. Suppose instead that corresponding sides and angles of ∆𝐴𝐵𝐶 to ∆𝐷𝐸𝐹 are congruent. Show that there is a
sequence of rigid motions which maps ∆𝐴𝐵𝐶 to ∆𝐷𝐸𝐹’
This standard connects the establishment of
congruence to congruent triangle proofs based
on corresponding sides and angles.
NC.M2.G-CO.8
Use congruence in terms of rigid motion.
Justify the ASA, SAS, and SSS criteria for triangle congruence. Use criteria for triangle congruence (ASA, SAS, SSS, HL) to determine whether two
triangles are congruent.

 Use the properties of rigid motions to show that two triangles are congruent if Generally, all SMPs can be applied in every standard. The following SMPs can be
their corresponding sides and angles are congruent (NC.M2.G-CO.7) highlighted for this standard.
 Use triangle congruence to prove theorems about lines, angles, and segments for As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
relationships in geometric figures (NC.M2.G-CO.9) in all oral and written communication
 Use triangle congruence to prove theorems about triangles (NC.M2.G-CO.10)

Extending from the 7th grade standard where Students list the sufficient conditions to prove triangles are congruent: ASA, SAS, and SSS. They map a triangle with one
students examine the conditions required to of the sufficient conditions (e.g., SSS) onto the original triangle and show that corresponding sides and corresponding
determine a unique triangle, students come to angles are congruent.
understand the specific characteristics of Example: Josh is told that two triangles 𝛥𝐴𝐵𝐶 and 𝛥𝐷𝐸𝐹 share two sets of congruent sides and one set of congruent
congruent triangles which lays the groundwork angles: ̅̅̅̅ 𝐷𝐸 , ̅̅̅̅
𝐴𝐵 is congruent to ̅̅̅̅ 𝐵𝐶 is congruent to ̅̅̅̅
𝐸𝐹 , and ∠𝐵 is congruent to ∠𝐸. He is asked if these two triangles
for geometric proof. Proving these theorems must be congruent. Josh draws the two triangles marking congruent sides and angles. Then he says, “They are
helps students to then prove theorems about definitely congruent because two pairs of sides are congruent and the angle between them is congruent!”
lines and angles in other geometric figures and a. Draw the two triangles. Explain whether Josh’s reasoning is correct using triangle congruence criteria.
other triangle proofs. b. Given two triangles Δ𝐴𝐵𝐶 and Δ𝐷𝐸𝐹, give an example of three sets of congruent parts that will not always
Videos of Transformation Proofs: guarentee that the two triangles are congruent. Explain your thinking.
Animated Proof of SAS (YouTube)
Animated Proof of ASA (YouTube)
Why Does SAS Work? (Illustrative Mathematics)
Why Does ASA Work? (Illustrative Mathematics)
Why Does SSS Work? (Illustrative Mathematics)
NC.M2.G-CO.9
Prove geometric theorems.
Prove theorems about lines and angles and use them to prove relationships in geometric figures including:
 Vertical angles are congruent.
 When a transversal crosses parallel lines, alternate interior angles are congruent.
 When a transversal crosses parallel lines, corresponding angles are congruent.
 Points are on a perpendicular bisector of a line segment if and only if they are equidistant from the endpoints of the segment.
 Use congruent triangles to justify why the bisector of an angle is equidistant from the sides of the angle.

 Use informal arguments to establish facts about angle sums and exterior angles Generally, all SMPs can be applied in every standard. The following SMPs can be
in triangles and angles created by parallel lines cut by a transversal (8.G.5) highlighted for this standard.
 Verify experimentally properties of rigid motions in terms of angles, circles, 
and // lines and line segments (NC.M2.G-CO.4) 6 – Attend to precision
 Use and justify criteria to determine triangle congruence (NC.M2.G-CO.8) 7 – Look for and make use of structure

 Use triangle congruence to prove theorems about triangles (NC.M2.G-CO.10) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
 Apply properties, definitions, and theorems of 2-D figures to prove geometric all oral and written communication
theorems (NC.M3.G-CO.14)

In 8th grade, students experimented with the properties of Students can prove theorems about intersecting lines and their angles.
angles and lines. The focus in this standard is on proving Example: Prove that any point equidistant from the endpoints of a line segment lies on the perpendicular
the properties; not just knowing and applying them. bisector of the line. [Example YouTube Proof: Point equidistant from segment end points is on perpendicular
bisector]
Students should use transformations and tactile
experiences to gain an intuitive understanding of these Students can prove theorems about parallel lines cut by a transversal and the angles formed by the lines.
theorems, before moving to a formal proof. For example, Example: A carpenter is framing a wall and wants to make sure the edges of his wall are parallel. He is using
vertical angles can be shown to be equal using a a cross-brace as show in the diagram.
reflection across a line passing through the vertex or a a. What are some different ways that he could verify that the
180 rotation around the vertex. Alternate interior edges are parallel?
angles can be matched up using a rotation around a b. Write a formal argument to show that the walls are parallel.
point midway between the parallel lines on the c. Pair up with another student who created a different argument
transversal. Corresponding angles can be matched up than yours, and critique their reasoning. Did you modify your
using a translation. diagram as a result of the collaboration? How? Why?
Expose students to multiple formats for writing proofs, Example: The diagram below depicts the construction of a parallel line, above the ruler. The steps in the
such as narrative paragraphs, bulleted lists of statements, construction result in a line through the given point that is parallel to the given line. Which statement
flow diagrams, two-column format, and using diagrams below justifies why the constructed line is parallel to the given line?
without words. Students should be encouraged to focus a. When two lines are each perpendicular to a third line,
on the validity of the underlying reasoning while the lines are parallel.
exploring a variety of formats for expressing that b. When two lines are each parallel to a third line, the
reasoning. Students should not be required to master all lines are parallel.
formats, but to be able to read and analyze proofs in c. When two lines are intersected by a transversal and
different formats, choosing a format (or formats) that best alternate interior angles are congruent, the lines are
suit their learning style for writing proofs. parallel.
d. When two lines are intersected by a transversal and
corresponding angles are congruent, the lines are parallel.
Points equidistant from two points in the plane (Illustrative Mathematics) Videos of Angle and Line Proofs:
 Vertical angles are congruent. (Khan Academy)
Congruent angles made by parallel lines and a transverse (Illustrative Mathematics)  Alternate interior angles congruent (YouTube)
 Corresponding Angle Proof (YouTube)
Proving the Alternate Interior Angles Theorem (CPalms)  Corresponding Angle Proofs – by contradiction (YouTube)
NC.M2.G-CO.10
Prove geometric theorems.
Prove theorems about triangles and use them to prove relationships in geometric figures including:
 The sum of the measures of the interior angles of a triangle is 180º.
 An exterior angle of a triangle is equal to the sum of its remote interior angles.
 The base angles of an isosceles triangle are congruent.
 The segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length.

 Verify experimentally properties of rigid motions in terms of angles, circles,  Generally, all SMPs can be applied in every standard. The following SMPs can be
and // lines and line segments (NC.M2.G-CO.4) highlighted for this standard.
 Use and justify criteria to determine triangle congruence (NC.M2.G-CO.8)
 Use triangle congruence to prove theorems about lines, angles, and segments for 6 – Attend to precision
relationships in geometric figures (NC.M2.G-CO.9) 7 – Look for and make use of structure
 Verify experimentally, properties of the centers of triangles (NC.M3.G-CO.10) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
 Prove theorems about parallelograms (NC.M3.G-CO.11) all oral and written communication
 Apply properties, definitions, and theorems of 2-D figures to prove geometric

Encourage multiple ways of writing proofs, such as narrative paragraphs and flow diagrams. Students should be encouraged Students can prove theorems about triangles.
to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Example: Prove the Converse of the
Isosceles Triangle Theorem: If two angles
Geometry is visual and should be taught in ways that leverage this aspect. Sketching, drawing and constructing figures and of a triangle are congruent, then the sides
relationships between and within geometric objects should be central to any geometric study and certainly to proof. The use opposite them are congruent.
of transparencies and dynamic geometry software can be important tools for helping students conceptually understand
important geometric concepts. Example: Prove that an equilateral
triangle is also equiangular.
Example Proofs:
Triangle Angle Sum Theorem
Given 𝚫𝑨𝑩𝑪, prove that the 𝒎∠𝑨 + 𝒎∠𝑩 + 𝒎∠𝑪 = 𝟏𝟖𝟎°.
Draw ⃡𝐸𝐷 through point A, parallel to ⃡𝐵𝐶 . Since ⃡𝐸𝐷 and ⃡𝐵𝐶 are parallel,
alternate interior angles are congruent. Therefore, ∠𝐷𝐴𝐶 ≅ ∠𝐴𝐶𝐵 and ∠𝐸𝐴𝐵 ≅
∠𝐴𝐵𝐶. By Angle Addition Postulate, ∠𝐸𝐴𝐵 + ∠𝐵𝐴𝐶 + ∠𝐷𝐴𝐶 = ∠𝐸𝐴𝐷. Since
∠𝐸𝐴𝐷 is a straight angle, its measure is 180. Therefore 𝑚∠𝐸𝐴𝐵 + 𝑚∠𝐵𝐴𝐶 +
𝑚∠𝐷𝐴𝐶 = 180. Thus, the sum of the measures of the interior angles of a
triangle is 180.
Exterior Angle Theorem

Statement Reason
Given the figure on the right, prove
mDEG + mGEF = 180º Two angles that form a straight
mEFG + mFGE = mDEG line are supplementary.
mEFG+mFGE+mGEF=180º Sum of angles in a triangle is
180º
mEFG+mFGE+mGEF= Substitution as both sums equal
mDEG+mGEF 180º.
mEFG+mFGE=mDEG Subtract mGEF from both sides
of equation
Triangle Midsegment Theorem

Given that D is the midpoint of ̅̅̅̅
𝑨𝑩,
̅̅̅̅
and E is the midpoint of 𝑨𝑪, prove
̅̅̅̅||𝑩𝑪
𝑫𝑬 ̅̅̅̅ = 𝟏 𝑩𝑪
̅̅̅̅ and 𝑫𝑬 ̅̅̅̅.
𝟐
Seven Circles (Illustrative Mathematics) Exterior Angle Theorem (YouTube video)
Base Angles Congruent (Khan Academy Video)
Triangle Midsegment Theorem (Proof using dilations)
Geometry – Similarity, Right Triangles, and Trigonometry

NC.M2.G-SRT.1
Understand similarity in terms of similarity transformations.
Verify experimentally the properties of dilations with given center and scale factor:
a. When a line segment passes through the center of dilation, the line segment and its image lie on the same line. When a line segment does not
pass through the center of dilation, the line segment and its image are parallel.
b. Verify experimentally the properties of dilations with given center and scale factor: The length of the image of a line segment is equal to the
length of the line segment multiplied by the scale factor.
c. The distance between the center of a dilation and any point on the image is equal to the scale factor multiplied by the distance between the
dilation center and the corresponding point on the pre-image.
d. Dilations preserve angle measure.

 Use coordinates to describe the effects of transformations on 2-D figures (8.G.3) Generally, all SMPs can be applied in every standard. The following SMPs can be
 Understand similarity through transformations (8.G.4) highlighted for this standard.
 Finding the distance between points in the coordinate plane (8.G.8)
 Using slope to determine parallelism and perpendicularity (NC.M1.G-GPE.5)
 Understand that dilations produce similar figures (NC.M2.G-CO.2)
 Using coordinates to solve geometric problems algebraically (NC.M1.G-GPE.4) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
 Determining similarity by a sequence of transformations; use the properties of in all oral and written communication
dilations to show that two triangles are similar if their corresponding sides
proportional and corresponding angles are congruent (NC.M2.G-SRT.2)
 Verify experimentally properties of rigid motions in terms of angles, circles, 
and // lines and line segments (NC.M2.G-CO.4)

Students use hands-on techniques (graph paper) Students verify that a side length of the image is equal to the scale factor multiplied by the corresponding side length of
and/or technology (geometry software) to the pre-image.
experiment with dilations. This standard Example: Given 𝛥𝐴𝐵𝐶 with 𝐴 (−2, −4), 𝐵 (1, 2) and 𝐶 (4, −3).
extends to the observance of the basic a. Perform a dilation from the origin using the following function rule 𝑓(𝑥, 𝑦) → (3𝑥, 3𝑦). What is the scale
properties of dilations as they build a deeper factor of the dilation?
understanding of similarity.
b. Using 𝛥𝐴𝐵𝐶 and its image 𝛥𝐴’𝐵’𝐶’, connect the corresponding pre-image and image points. Describe how
Students should understand that a dilation is a the corresponding sides are related.
transformation that moves each point along the c. Determine the length of each side of the triangle. How do the side lengths compare? How is this comparison
ray through the point emanating from a fixed related to the scale factor?
center, and multiplies distances from the center d. Determine the distance between the origin and point A and the distance between the origin and point A’. Do
by a common scale factor. the same for the other two vertices. What do you notice?
e. Determine the angle measures for each angle of 𝛥𝐴𝐵𝐶 and 𝛥𝐴’𝐵’𝐶’. What do you notice?
Students perform a dilation with a given center and scale factor on a figure in the coordinate plane.
Example: Suppose we apply a dilation by a factor of 2,
centered at the point P to the figure below.
a. In the picture, locate the images A’, B’, and C’ of the
points A, B, C under this dilation.
b. What is the relationship between ⃡𝐴𝐶 and ⃡𝐴′𝐶′?
c. What is the relationship between the length of A’B’ and
the length of AB? Justify your thinking.
Students verify that when a side passes through the center of dilation, the side and its image lie on the same line and the
remaining corresponding sides of the pre-image and images are parallel.
NC.M2.G-SRT.2
Understand similarity in terms of transformations.
a. Determine whether two figures are similar by specifying a sequence of transformations that will transform one figure into the other.
b. Use the properties of dilations to show that two triangles are similar when all corresponding pairs of sides are proportional and all
corresponding pairs of angles are congruent

 Given a geometric figure and a rigid motion, find the image of the figure/Given Generally, all SMPs can be applied in every standard. The following SMPs can be
a figure and its image, describe a sequence of rigid motions between preimage highlighted for this standard.
and image (NC.M2.G-CO.5) 3 – Construct viable arguments and critique the reasoning of others
4 – Model with Mathematics
 Verify experimentally properties of dilations with given center and scale factor
(NC.M2.G-SRT.1)
 Use the properties of dilations to show that two triangles are similar if their As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
corresponding sides proportional and corresponding angles are congruent in all oral and written communication
Determining similarity by a sequence of transformations (NC.M2.G-SRT.2b)
 Use transformations for the AA criterion for triangle similarity
(NC.M2.G-SRT.3)
 Verify experimentally that side ratios in similar right triangles are properties of
the angle measures and use to define trig ratios (NC.M2.G-SRT.6)

Students use the idea of dilation Students use the idea of dilation transformations to develop the definition of similarity.
transformations to develop the definition of Example: In the picture to the right, line segments AD and BC intersect at X. Line
similarity. They understand that a similarity segments AB and CD are drawn, forming two triangles ΔAXB and ΔCXD.
transformation is a combination of a rigid In each part a-d below, some additional assumptions about the picture are given. For each
motion and a dilation. assumption:
I. Determine whether the given assumptions are enough to prove that the two triangles are
Students demonstrate that in a pair of similar similar. If so, what is the correct correspondence of vertices. If not, explain why not.
triangles, corresponding angles are congruent II. If the two triangles must be similar, prove this result by describing a sequence of
(angle measure is preserved) and similarity transformations that maps one variable to the other.
corresponding sides are proportional. They a. The lengths of AX and AD satisfy the equation 2𝐴𝑋 = 3𝑋𝐷.
determine that two figures are similar by 𝐴𝑋 𝐷𝑋
b. The lengths AX, BX, CX, and DX satisfy the equation = (From Illustrative Mathematics)
𝐵𝑋 𝐶𝑋
verifying that angle measure is preserved and
c. Lines AB and CD are parallel.
corresponding sides are proportional.
d. ∠ XAB is congruent to angle ∠XCD.
Similar Triangles (Illustrative Mathematics)
NC.M2.G-SRT.3
Understand similarity in terms of transformations.
Use transformations (rigid motions and dilations) to justify the AA criterion for triangle similarity.

 Verify experimentally properties of dilations with given center and scale factor Generally, all SMPs can be applied in every standard. The following SMPs can be
(NC.M2.G-SRT.1) highlighted for this standard.
 Determining similarity by a sequence of transformations; use the properties of 5 – Use appropriate tools strategically
dilations to show that two triangles are similar if their corresponding sides
proportional and corresponding angles are congruent (NC.M2.G-SRT.2)
 Use similarity to prove The Triangle Proportionality Theorem and the As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
Pythagorean Theorem (NC.M2.G-SRT.4) in all oral and written communication

Given two triangles for which 𝐴𝐴 holds, students use rigid Students can use the properties of dialations to show that two triangles are similar based on the 𝐴𝐴
motions to map a vertex of one triangle onto the corresponding criterion.
vertex of the other in such a way that their corresponding sides are Example: Given that 𝛥𝑀𝑁𝑃 is a dialation of 𝛥𝐴𝐵𝐶 with scale factor k, use properties of dilations
in line. Then show that the dilation will complete the mapping of to show that the 𝐴𝐴 criterion is sufficient to prove similarity.
one triangle onto the other. See p. 98 of Dr. Wu, Teaching
Geometry According to the Common Core Standards.
Dynamic geometry software visual of this process. (Geogebra.org)
Informal Proof of AA Criterion for Similarity (EngageNY)
The AA Criterion for Two Triangles to Be Similar (EngageNY)
NC.M2.G-SRT.4
Prove theorems involving similarity.
Use similarity to solve problems and to prove theorems about triangles. Use theorems about triangles to prove relationships in geometric figures.
 A line parallel to one side of a triangle divides the other two sides proportionally and its converse.
 The Pythagorean Theorem

 Use transformations for the AA criterion for triangle similarity (NC.M2.G- Generally, all SMPs can be applied in every standard. The following SMPs can be
SRT.3) highlighted for this standard.
 Use trig ratios and the Pythagorean Theorem in right triangles (NC.M2.G- As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
SRT.8) in all oral and written communication
 Derive the equation of a circle given center and radius using the Pythagorean
Theorem (NC.M3.G-GPE.1)
 Prove theorems about parallelograms (NC.M3.G-CO.11)
 Apply properties, definitions, and theorems of 2-D figures to prove geometric
 Understand apply theorems about circles (NC.M3.G-C.2)
 Use similarity to demonstrate that the length of the arc is proportional to the
radius of the circle (NC.M3.G-C.5)

Students use the concept of similarity to solve Students use similarity to prove the Pythagorean Theorem.
problem situations (e.g., indirect measurement, Example: Calculate the distance across the river, AB.
missing side(s)/angle measure(s)). Students
use the properties of dilations to prove that a
line parallel to one side of a triangle divides the
other two sides proportionally (often referred to
as side-splitter theorem) and its converse.
The altitude from the right angle is drawn to the

hypotenuse, which creates three similar
triangles. The proportional relationships
among the sides of these three triangles can be
used to derive the Pythagorean relationship. Students can use triangle theorems to prove relationships in geometric figures.
Example: In the diagram, quadrilateral PQRS is a parallelogram, SQ is a
diagonal, and SQ || XY.
a. Prove that ΔXYR~ΔSQR.
b. Prove that ΔXYR~ΔQSP.
Example: Parade Route Problem

The parade committee has come up with the Beacon County
Homecoming Parade route for next year. They want to start at the
intersection of 17th Street and Beacon Road. The parade will proceed
south on Beacon Road, turning left onto 20 th Street. Then the parade will
turn left onto Pine Avenue and finish back at 17 th Street. For planning
purposes, the committee needs to know approximately how long the
parade will last. Can you help them? Justify your estimate. What
assumptions did you make?
(adapted from http://www.math.uakron.edu/amc/Geometry/HSGeometryLessons/SideSplitterTheorem.pdf)
Example: Use similarity to prove the slope criteria for similar triangles.
(https://www.illustrativemathematics.org/content-standards/HSG/SRT/B/5/tasks/1876)
Example proofs:
Bank Shot Task (Illustrative Mathematics)
Proof of Pythagorean Theorem using similar triangles (YouTube video)
Side-Splitter Theorem (YouTube video)
NC.M2.G-SRT.6
Define trigonometric ratios and solve problems involving right triangles.
Verify experimentally that the side ratios in similar right triangles are properties of the angle measures in the triangle, due to the preservation of angle
measure in similarity. Use this discovery to develop definitions of the trigonometric ratios for acute angles.

 Determining similarity by a sequence of transformations; use the properties of Generally, all SMPs can be applied in every standard. The following SMPs can be
dilations to show that two triangles are similar if their corresponding sides are highlighted for this standard.
proportional and their corresponding angles are congruent (NC.M2.G-SRT.2) 2 – Reason abstractly and quantitatively
 Develop properties of special right triangles (NC.M2.G-SRT.12) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
New Vocabulary: sine, cosine, tangent

Students establish that the side ratios of a right triangle are Students can use proportional reasoning to develop definitions of the trigonometric ratios of acute
equivalent to the corresponding side ratios of similar right triangles angles.
and are a function of the acute angle(s). Because all right triangles Example: Find the sine, cosine, and tangent of x.
have a common angle, the right angle, if two right triangles have an
acute angle in common (i.e. of the same measure), then they are
similar by the AA criterion. Therefore, their sides are proportional.
We define the ratio of the length of the side opposite the acute angle Example: Explain why the sine of x is the same regardless of which triangle is used to find it in
to the length of the side adjacent to the acute angle as the tangent the figure below.
ratio. Note that the tangent ratio corresponds to the slope of a line
passing through the origin at an angle to the x-axis that equals the
measure of the acute angle. For example, in the diagram below,
students can see that the tangent of 45 is 1, since the slope of a line
passing through the origin at a 45 angle is 1. Using this visual, it is
also easy to see that the slope of lines making an angle less than 45
will be less than 1; therefore the tangent ratio for angles between 0
and 45 is less than 1. Similarly, the slope of lines making an angle
greater than 45 will be greater than 1; therefore, the tangent ratio for
angles between 45 and 90 will be greater than 1.
Connect with 8.EE.6 “Use similar triangles to explain why the slope
m is the same between any two distinct points on a non-vertical line
in the coordinate plane.”
We define the ratio of the length of the side opposite the acute angle
to the length of the hypotenuse as the sine ratio.
We define the ratio of the length of the side adjacent to the acute
angle to the length of the hypotenuse as the cosine ratio.
NC.M2.G-SRT.8
Use trigonometric ratios and the Pythagorean Theorem to solve problems involving right triangles in terms of a context.

 Use similarity to prove The Triangle Proportionality Theorem and the Generally, all SMPs can be applied in every standard. The following SMPs can be
Pythagorean Theorem (NC.M2.G-SRT.4) highlighted for this standard.
4 - Model with mathematics (contextual situations are required)
 Develop properties of special right triangles (NC.M2.G-SRT.12) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
 Understand apply theorems about circles (NC.M3.G-C.2) in all oral and written communication
 Build an understanding of trigonometric functions (NC.M3.F-TF.2)

This standard is an application standard where Students can use trig ratios and the Pythagorean theorem to find side lengths and angle measures in right triangles.
students use the Pythagorean Theorem, learned Example: Find the height of a flagpole to the nearest tenth if the angle of elevation of the sun is 28° and the shadow
in MS, and trigonometric ratios to solve of the flagpole is 50 feet.
application problems involving right triangles,
including angle of elevation and depression, Example: A new house is 32 feet wide. The rafters will rise at a 36 angle and meet above the centerline of the
navigation, and surveying. house. Each rafter also needs to overhang the side of the house by 2 feet. How long should the carpenter make each
rafter?
NC.M2.G-SRT.12
Develop properties of special right triangles (45-45-90 and 30-60-90) and use them to solve problems.

 Use similarity to prove The Triangle Proportionality Theorem and the Generally, all SMPs can be applied in every standard. The following SMPs can be
Pythagorean Theorem (NC.M2.G-SRT.4) highlighted for this standard.
 Verify experimentally that side ratios in similar right triangles are properties of As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
the angle measures and use to define trig ratios (NC.M2.G-SRT.6) in all oral and written communication
 Use trig ratios and the Pythagorean Thm to solve problems (NC.M2.G-SRT.8)
 Understand apply theorems about circles (NC.M3.G-C.2)
 Build an understanding of trigonometric functions (NC.M3.F-TF.2)

By drawing the altitude to one side of an equilateral triangle, students form two Students can solve problems involving special right triangles.
congruent 30° − 60° − 90° triangles. Starting with an initial side length of 2𝑥, Example: The Garden Club at Heritage High wants to build a flower garden
students use the Pythagorean Theorem to develop relationships between the sides of a near the outdoor seating at the back of the school. The design is a square with
30° − 60° − 90° triangle. diagonal walkways. The length of each side of the garden is 50 ft. How long is
each walkway?
Example: If 𝐴𝐵 = 8√3, find AE.

A
30
E
B
Students begin by drawing an isosceles right triangle with leg length of x. Using the 60
Isosceles Triangle Theorem, the Triangle Angle Sum Theorem, and the Pythagorean 45
Theorem students develop and justify relationships between the sides of a 45° − 45° − C D
90°triangle.
In Math 3, this relationship can be revisited with quadrilaterals by drawing the

diagonal of a square to create two congruent 45° − 45° − 90° triangles. Using the
properties of the diagonal and the Pythagorean Theorem, these relationships can be
established in a different manner.
Statistics & Probability
A statistical process is a problem-solving process consisting of four steps:
1. Formulating a statistical question that anticipates variability and can be answered by data.
2. Designing and implementing a plan that collects appropriate data.
3. Analyzing the data by graphical and/or numerical methods.
4. Interpreting the analysis in the context of the original question.

Focus on analysis of univariate and Focus on probability Focus on the use of sample data to
bivariate data • Categorical data and two-way tables represent a population
• Use of technology to represent, analyze • Understanding and application of the • Random sampling
and interpret data Addition and Multiplication Rules of • Simulation as it relates to sampling and
• Shape, center and spread of univariate Probability randomization
numerical data • Conditional Probabilities • Sample statistics
• Scatter plots of bivariate data • Independent Events • Introduction to inference
• Linear and exponential regression • Experimental vs. theoretical probability
• Interpreting linear models in context.
A Progression of Learning
• A continuation of the work from middle • A continuation of the work from 7th • Bringing it all back together
grades mathematics on summarizing grade where students are introduced to • Sampling and variability
and describing quantitative data the concept of probability models, • Collecting unbiased samples
distributions of univariate (6th grade) chance processes and sample space; and • Decision making based on analysis of
and bivariate (8th grade) data. 8th grade where students create and data
interpret relative frequency tables.
• The work of MS probability is extended
to develop understanding of conditional
probability, independence and rules of
probability to determine probabilities of
compound events.
Statistics and Probability – Making Inference and Justifying Conclusion
NC.M2.S-IC.2
Understand and evaluate random processes underlying statistical experiments
Use simulation to determine whether the experimental probability generated by sample data is consistent with the theoretical probability based on
known information about the population.

 Random sampling can be used to support valid inferences if the sample is Generally, all SMPs can be applied in every standard. The following SMPs can be
representative of the population (7.SP.1) highlighted for this standard.
 Approximate probabilities by collecting data and observing long-run frequencies 2 – Reason abstractly and quantitatively
(7.SP.6) 4 – Model with Mathematics
 Use simulation to understand how samples are used to estimate population As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
means/proportions and how to determine margin of error (NC.M3.S-IC.4) all oral and written communication.
 Use simulation to determine whether observed differences between samples New vocabulary – simulation, experimental probability, theoretical probability
indicates actual differences in terms of the parameter of interest (NC.M3.S-IC.5)

This standard is an expansion of MS (7th grade) where students approximate the Students explain how well and why a sample represents the variable of interest from a
probability of a chance event by collecting data and observing long-run relative population.
frequencies of chance phenomenon. In the middle grades work, students understand Example: Multiple groups flip coins. One group flips a coin 5 times, one group
that increasing the size of the trial yields results that are pretty consistent with the flips a coin 20 times, and one group flips a coin 100 times. Which group’s results
theoretical probability model. They also understand that randomization is an important will most likely approach the theoretical probability?
element of sampling and that samples that reflect the population can be used to make
inferences about the population.
This standard is extended to the idea of increasing the number of samples collected
and examining the results of more samples opposed to larger sample sizes. This
standard uses simulation to build an understanding of how taking more samples of the
same size can be used to make predictions about the population of interest.
Simulation can be used to mock real-world experiments. It is time saving and provides
a way for students to conceptually understand and explain random phenomenon.
It is suggested at this level for students to conduct simulation using tactile tools and
methods. Cards, number cubes, spinners, colored tiles and other common items are
excellent tools for performing simulation. Technology can be used to compile and
analyze the results, but should not be used to perform simulations at this level.
Statistics and Probability – Conditional Probability and the Rules for Probability
NC.M2.S-CP.1
Understand independence and conditional probability and use them to interpret data.
Describe events as subsets of the outcomes in a sample space using characteristics of the outcomes or as unions, intersections and complements of
other events.

 Find probabilities of compound events using lists, tables, tree diagrams and Generally, all SMPs can be applied in every standard. The following SMPs can be
simulations (7.SP.8) highlighted for this standard.
 Develop and understand independence and conditional probability (NC.M2.S- As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
CP.3a, NC.M2.S-CP.3b) in all oral and written communication.
 Use the rules of probability to compute probabilities (NC.M2.S-CP.6, NC.M2.S- New vocabulary – subset, union, intersections, complements
CP.7, NC.M2.S-CP.8)

In MS (7th grade) students collect data to Students define a sample space and events within the sample space.
approximate relative frequencies of probable events. Example: Describe the sample space for rolling two number cubes.
They use the information to understand theoretical For the teacher: This may be modeled well with a 6x6 table with the rows labeled for the first event and the columns labeled for the second event.
probability models based on long-run relative
Example: Describe the sample space for picking a colored marble from a bag with red and black marbles.
frequency. This allows students to assign probability For the teacher: This may be modeled with set notation.
to simple events, therefore students develop the
understanding for sample space as the collection of Example: Andrea is shopping for a new cellphone. She is either going to contract with Verizon (60% chance)
all possible outcomes. Additionally, MS students or with Sprint (40% chance). She must choose between an Android phone (25% chance) or an IPhone (75%
develop probability models for compound events chance). Describe the sample space. For the teacher: This may be modeled well with an area model.
using lists tables, tree diagrams and simulations.
Example: The 4 aces are removed from a deck of cards. A coin is tossed and one of the aces is
This standard builds on the MS work by formalizing chosen. Describe the sample space. For the teacher: This may be modeled well with a tree diagram.
probability terminology associated with simple and
compound events and using characteristics of the Students establish events as subsets of a sample space. An event is a subset of a sample space.
outcomes: Example: Describe the event of rolling two number cubes and getting evens.
Example: Describe the event of pulling two marbles from a bag of red/black marbles.
Example: Describe the event that the summing of two rolled number cubes is larger than 7 and even, and
contrast it with the event that the sum is larger than 7 or even.
 The intersection of two For sets A and B:
sets A and B is the set of 𝐴∩𝐵 Example: If the subset of outcomes for choosing one card from a standard deck of cards is the intersection of
elements that are common two events: {queen of hearts, queen of diamonds}.
to both set A and set B. It a. Describe the sample space for the experiment.
is denoted by 𝐴 ∩ 𝐵 and is b. Describe the subset of outcomes for the union of two events.
read “A intersection B”
 The union of two sets A For sets A and B:

and B is the set of 𝐴∪𝐵
elements, which are in A
or in B, or in both. It is
denoted by𝐴 ∪ 𝐵, and is
read “A union B”
 The complement of the set For sets A and B:

𝐴 ∪ 𝐵 is the set of (𝐴 ∪ 𝐵)’
elements that are members
of the universal set 𝑈 but
are not in 𝐴 ∪ 𝐵. It is
denoted by (𝐴 ∪ 𝐵)’
NC.M2.S-CP.3a
Develop and understand independence and conditional probability.
a. Use a 2-way table to develop understanding of the conditional probability of A given B (written P(A|B)) as the likelihood that A will occur
given that B has occurred. That is, P(A|B) is the fraction of event B’s outcomes that also belong to event A.

 Understand patterns of association from two-way tables in bivariate categorical Generally, all SMPs can be applied in every standard. The following SMPs can be
data (8.SP.4) highlighted for this standard.
 Represent data on two categorical by constructing two-way frequency tables of As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
data and use the table to determine independence (NC.M2.S-CP.4) in all oral and written communication.
New vocabulary – independence, conditional probability
 Recognize and explain the concepts of conditional probability and
independence (NC.M2.S-CP.5)
 Find conditional probabilities and interpret in context (NC.M2.S-CP.6)
Students created two-way tables of categorical Students can use two-way tables to find conditional probabilities.
data and used them to examine patterns of Example: Each student in the Junior class was asked if they had to complete Curfew
association in MS. They also displayed chores at home and if they had a curfew. The table represents the data.
frequencies (counts) and relative frequencies Yes No Total
a. What is the probability that a student who has chores also has a curfew?
(percentages) in two-way tables. This standard
b. What is the probability that a student who has a curfew also has chores? Yes 51 24 75
Chores
uses two-way tables to establish an
understanding for conditional probability, that c. Are the two events have chores and have a curfew independent?
is given the occurrence of one event the Explain. No 30 12 42
probability of another event occurs.
Total 81 36 117
Students understand conditional probability as the probability of A occurring given B has occurred.
Example: What is the probability that the sum of two rolled number cubes is 6 given that you rolled doubles?
Example: There are two identical bottles. A bottle is selected at random

and a single ball is drawn. Use the tree diagram at the right to determine
each of the following:
a. 𝑃 (𝑟𝑒𝑑|𝑏𝑜𝑡𝑡𝑙𝑒 1)
b. 𝑃 (𝑟𝑒𝑑|𝑏𝑜𝑡𝑡𝑙𝑒 2)
The rows/columns determine the condition.

Using the example above, the probability that
you select a left-handed person, given that it is
a girl is the number of left-handed girls divided
by the total number of girls 
10
𝑃(𝐿𝑒𝑓𝑡 − ℎ𝑎𝑛𝑑𝑒𝑑|𝐺𝑖𝑟𝑙) = ≈ .43. The
23
condition in this problem is a girl therefore, the
number of girls represents the total of the
conditional probability.
NC.M2.S-CP.3b
Develop and understand independence and conditional probability.
b. Understand that event A is independent from event B if the probability of event A does not change in response to the occurrence of event B.
That is P(A|B) = P(A).

 Understand patterns of association from two-way tables in bivariate Generally, all SMPs can be applied in every standard. The following SMPs can be
categorical data (8.SP.4) highlighted for this standard.
 Connections Disciplinary Literacy

 Represent data on two categorical by constructing two-way frequency As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
tables of data and use the table to determine independence (NC.M2.S-CP.4) all oral and written communication.
New vocabulary – independence, conditional probability
 Apply the general Multiplication Rule, including when A and B are
independent, and interpret in context (NC.M2.S-CP.8)

Students can use two-way tables to find conditional probabilities. Curfew
Example: Each student in the Junior class was asked if they had to
complete chores at home and if they had a curfew. The table represents the Yes No Total
data. Are the two events have chores and have a curfew independent?
51 24 75
Chores
Explain Yes
No 30 12 42
Total 81 36 117
Conditional Probabilities 1 NEW
NC.M2.S-CP.4
Represent data on two categorical variables by constructing a two-way frequency table of data. Interpret the two-way table as a sample space to
calculate conditional, joint and marginal probabilities. Use the table to decide if events are independent.

 Understand patterns of association from two-way tables in bivariate Generally, all SMPs can be applied in every standard. The following SMPs can be
categorical data (8.SP.4) highlighted for this standard.
 Develop and understand independence and conditional probability As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
(NC.M2.S-CP.3a, NC.M2.S-CP.3b) in all oral and written communication.
New vocabulary – joint probabilities, marginal probabilities

This standard builds upon the study of bivariate Students can create a two-way frequency table for data and calculate probabilities from the table.
categorical data from MS. This standard Example: Collect data from a random sample of students in your school on their favorite subject among math,
supports data analysis from the statistical science, history, and English. Estimate the probability that a randomly selected student from your school will favor
process. science given that the student is in tenth grade. Do the same for other subjects and compare the results.
The statistical process includes four essential steps:
Students can use a two-way table to evaluate independence of two variables.
1. Formulate a question that can be answered with data.
2. Design and use a plan to collect data.
Example: The Venn diagram to the right shows the data collected at a
3. Analyze the data with appropriate methods.
sandwich shop for the last six months with respect to the type of bread
4. Interpret results and draw valid conclusions. people ordered (sourdough or wheat) and whether or not they got cheese on
their sandwich. Use the diagram to construct a two-way frequency table and
Students created two-way tables of categorical then answer the following questions.
data and used them to examine patterns of a. P (sourdough)
association in 8th grade. They also displayed b. P (cheese | wheat)
frequencies (counts) and relative frequencies c. P (without cheese or sourdough)
(percentages) in two-way tables. Additionally, d. Are the events “sourdough” and “with cheese” independent events? Justify your reasoning.
students have determined the sample space of
simple and compound events in 7th grade. This
standard expands on both of the 7th and 8th
grade concepts to using the table to determine
independence of two events. Example: Complete the two-way frequency table at the
right and develop three conditional statements regarding
the data. Determine if there are any set of events that
independent. Justify your conclusion.
NC.M2.S-CP.5
Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

 Generally, all SMPs can be applied in every standard. The following SMPs can be
 Develop and understand independence and conditional probability As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
(NC.M2.S-CP.3a, NC.M2.S-CP.3b) in all oral and written communication.
 Find conditional probabilities and interpret in context (NC.M2.S-CP.6)

This standard is about helping students make Students can use everyday language to determine if two events are dependent.
meaning of data and statistical questions. It is about Example: Felix is a good chess player and a good math student. Do you think that the events “being good at
communicating in their own language what the playing chess” and “being a good math student” are independent or dependent? Justify your answer.
data/graphs/information is “saying.”
The statistical process includes four essential steps: Example: Juanita flipped a coin 10 times and got the following results: T, H, T, T, H, H, H, H, H, H. Her math
1. Formulate a question that can be answered with data. partner Harold thinks that the next flip is going to result in tails because there have been so many heads in a row.
2. Design and use a plan to collect data. Do you agree? Explain why or why not.
3. Analyze the data with appropriate methods.
4. Interpret results and draw valid conclusions.
Students can explain conditional probability using everyday language.
This standard supports the idea of helping students Example: A family that is known to have two children is selected at random from amongst all families with two
to process the information around them presented in 1
different formats or combination of formats children. Josh said that the probability of having two boys is . Do you agree with Josh? Why or why not?
3
(graphs, tables, narratives with percentages, etc.) Explain how you arrived at your answer?
NC.M2.S-CP.6
Use the rules of probability to compute probabilities of compound events in a uniform probability model.
Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in context.

 Develop and understand independence and conditional probability (NC.M2.S- Generally, all SMPs can be applied in every standard. The following SMPs can be
CP.3a, NC.M2.S-CP.3b) highlighted for this standard.
 Recognize and explain the concepts of conditional probability and independence As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
(NC.M2.S-CP.5) in all oral and written communication.
 Apply the general Multiplication Rule, including when A and B are independent,
and interpret in context (NC.M2.S-CP.8)

This standard should build on conditional Students can find the conditional probability of compound events.
probability and lead to the introduction of the Example: If a balanced tetrahedron with faces 1, 2, 3, 4 is rolled twice.
addition and general multiplication rules of (A): Sum is prime
probability. Venn diagrams and/or tables of
(B): A 3 is rolled on at least one of the rolls.
outcomes should serve as visual aids to build to
the rules for computing probabilities of a. Create a table showing all possible outcomes (sample space) for rolling the two tetrahedron.
compound events. b. What is the probability that the sum is prime (A) of those that show a 3 on at least one roll (B)?
c. Use the table to support the answer to part (b).
The sample space of an experiment can be
modeled with a Venn diagram such as: Example: Peter has a bag of marbles. In the bag are 4 white marbles, 2 blue marbles, and 6 green marbles. Peter
randomly draws one marble, sets it aside, and then randomly draws another marble. What is the probability of Peter
drawing out two green marbles? Note: Students must recognize that this a conditional probability P(green | green).
𝑃(𝐴 𝑎𝑛𝑑 𝐵) Example: A teacher gave her class two quizzes. 30% of the class passed both quizzes and 60% of the class passed
So, the 𝑃(𝐴|𝐵) =
𝑃(𝐵)
the first quiz. What percent of those who passed the first quiz also passed the second quiz?
NC.M2.S-CP.7
Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in context.

 Describe events as subsets of the outcomes in a sample space based on Generally, all SMPs can be applied in every standard. The following SMPs can be
characteristics of the outcomes or as unions, intersections or complements highlighted for this standard.
of other events (NC.M2.S-CP.1) 2 – Reason abstractly and quantitatively
 Apply the general Multiplication Rule, including when A and B are As stated in SMP 6, the precise use of mathematical vocabulary is the expectation in
independent, and interpret in context (NC.M2.S-CP.8) all oral and written communication.

Students should apply the addition rule for computing probabilities of Students can apply the general addition rule for calculating conditional probabilities.
compound events and interpret them in context. Students should Example: Given the situation of drawing a card from a standard deck of cards, calculate the
understand 𝑃 (𝐴 𝑜𝑟 𝐵) OR 𝑃(𝐴 ∪ 𝐵) to mean all elements of A and all probability of the following:
elements of B excluding all elements shared by A and B.
a. Drawing a red card or a king
The Venn diagram shows that when you include everything in both b. Drawing a ten or a spade
sets the middle region is included twice, therefore you must subtract c. Drawing a four or a queen
the intersection region out once. The probability for calculating joint
events is… Example: In a math class of 32 students, 18 boys and 14 are girls. On a unit test, 5 boys and
𝑃 (𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃 (𝐴 𝑎𝑛𝑑 𝐵) 7 girls made an A grade. If a student is chosen at random from the class, what is the
probability of choosing a girl or an A student?
Students may recognize that if two events A and B are mutually

exclusive, also called disjoint, the rule can be simplified to
𝑃 (𝐴 𝑜𝑟 𝐵) = 𝑃 (𝐴) + 𝑃 (𝐵) since for mutually exclusive events
𝑃 (𝐴 𝑎𝑛𝑑 𝐵) = 0.
NC.M2.S-CP.8
Apply the general Multiplication Rule P (A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in context. Include the case where A and
B are independent: P (A and B) = P(A) P(B).

 Describe events as subsets of the outcomes in a sample space based on Generally, all SMPs can be applied in every standard. The following SMPs can be
characteristics of the outcomes or as unions, intersections or complements highlighted for this standard.
of other events (NC.M2.S-CP.1) 2 – Reason abstractly and quantitatively
 Apply the Addition Rule and interpret in context (NC.M2.S-CP.7) As stated in SMP 6, the precise use of mathematical vocabulary is the expectation
in all oral and written communication.

Students should understand 𝑃 (𝐴 𝑎𝑛𝑑 𝐵) OR 𝑃(𝐴 ∩ 𝐵) to mean all Students can apply the general multiplication rule for computing conditional probabilities.
elements of A that are also elements of B excluding all elements shared Example: You have a box with 3 blue marbles, 2 red marbles, and 4 yellow marbles. You
by A and B. Two events must be independent to apply the general are going to pull out one marble, record its color, put it back in the box and draw another
multiplication rule
marble. What is the probability of pulling out a red marble followed by a blue marble?
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴)𝑃(𝐵|𝐴) = 𝑃(𝐵)𝑃(𝐴|𝐵)
The general rule can be explained based on the definitions of Example: Consider the same box of marbles as in the previous example. However, in this
independence and dependence. Events are either independent or case, we are going to pull out the first marble, leave it out, and then pull out another marble.
dependent. What is the probability of pulling out a red marble followed by a blue marble?
 Two events are said to be independent if the occurrence of one
event does not affect the probability of the occurrence of the Example: Suppose you are going to draw two cards from a standard deck. What is
other event. the probability that the first card is an ace and the second card is a jack (just one of several
ways to get “blackjack” or 21)?
 Two events are dependent if the occurrence of one event does, in
fact, affect the probability of the occurrence of the other event. Students can use the general multiplication rule to determine whether two events are independent.
Sampling with and without replacement are opportunities to model

independent and dependent events.

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The Math Resource for Instruction for

Concepts and Skills The Standards for Mathematical Practices

Mastering the Standard

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Concepts and Skills The Standards for Mathematical Practices

Mastering the Standard

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Concepts and Skills The Standards for Mathematical Practices

Connections Disciplinary Literacy

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FAL: Evaluating Statements About Rational and Irrational Numbers (Mathematics

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Concepts and Skills The Standards for Mathematical Practices

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Students should connect what they have learned

Students should be able to express solutions to a

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Functions represented as graphs, tables or verbal descriptions in context

A Progression of Learning of Functions through Algebraic Reasoning

Concepts and Skills The Standards for Mathematical Practices

Mastering the Standard

The Physics Professor (Illustrative Mathematics)

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Concepts and Skills The Standards for Mathematical Practices

Mastering the Standard

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Concepts and Skills The Standards for Mathematical Practices

Mastering the Standard

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Concepts and Skills The Standards for Mathematical Practices

Connections Disciplinary Literacy

Mastering the Standard

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Concepts and Skills The Standards for Mathematical Practices

Connections Disciplinary Literacy

Mastering the Standard

Students should be able to create equations using right triangle trigonometry.

Throwing a Ball (Illustrative Mathematics)

Concepts and Skills The Standards for Mathematical Practices

Connections Disciplinary Literacy

Mastering the Standard

Marbleslides: Parabolas (Desmos.com) NEW

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Concepts and Skills The Standards for Mathematical Practices

Connections Disciplinary Literacy

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Algebra – Reasoning with Equations and Inequalities

Concepts and Skills The Standards for Mathematical Practices

Mastering the Standard

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Concepts and Skills The Standards for Mathematical Practices

Mastering the Standard

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Concepts and Skills The Standards for Mathematical Practices

Mastering the Standard

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Concepts and Skills The Standards for Mathematical Practices