Mathematics

Standards Clarification for

Algebra Conceptual Category

High School

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Contents Seeing Structure in Expressions (SSE) ......................................................................................................................................... 4

Arithmetic with Polynomial & Rational Expressions (APR) ............................................................................................... 12

Creating Equations (CED) .............................................................................................................................................................. 20

Reasoning with Equations & Inequalities (REI) ...................................................................................................................... 28

Acknowledgements .......................................................................................................................................................................... 44

References ........................................................................................................................................................................................... 45

4

Seeing Structure in Expressions

Cluster: Interpret the structure of expressions. NVACS HSA.SSE.A.1.a and HSA.SSE.A.1.b Interpret expressions that represent a quantity in terms of its context.*

Element Exemplars

Standards for Mathematical Practice

● MP 7: Look for and make use of structure to connect real world examples algebraically.

● MP 8: Look for and express regularity in repeated reasoning in an algebra context.

Instructional Strategies

● Modeling Standard ● Students need to understand and explain terms like factor,

coefficient, term and like terms in the context of expressions. ● Students should be able to identify these terms in expressions. ● Provide students with representations of expressions so they can

compare terms in context. ● Provide opportunities for students to use pictures, manipulatives,

and symbols to make sense of equivalent expressions.

Prerequisite Skills

● Know the meaning of terms, factors, and coefficients so they can begin to interpret expressions with those elements.

● For linear and constant terms in functions, interpret the rate of change and the initial value.

● Interpreting parts of expressions in context.

Connections Within and Beyond High School

● These standards can be reinforced continually as new expressions, equations, and formulas are introduced.

● Interpreting changes in the parameters of a linear and exponential function in context.

● Interpret one variable rational equations. ● Interpret statements written in piecewise function notation. ● Understand the effects on transformations on functions. ● Use completing the square to write equivalent form of quadratic

expressions to reveal extrema.

Instructional Examples/Lessons/Tasks

● Interpreting Expressions Delivery Trucks Seeing Dots The Physics Professor (Illustrative Mathematics)

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Assessment Examples

● Students should recognize that in the expression 2x + 1, “2” is the coefficient, “2” and “x” are factors, and “1” is a constant, as well as “2x” and “1” being terms of the binomial expression. Also, a student recognizes that in the expression 4(3)x, 4 is the coefficient, 3 is the factor, and x is the exponent. Development and proper use of mathematical language is an important building block for future content. Using real-world context examples, the nature of algebraic expressions can be explored.

● Example: The expression −4.9t2 + 17t + 0.6 describes the height in meters of a basketball t seconds after it has been thrown vertically into the air. Interpret the terms and coefficients of the expression in the context of this situation. (The Math Resource for Instruction for NC Math 1)

● Example: The expression 35000(0.87)t describes the cost of a new car t years after it has been purchased. Interpret the terms and coefficients of the expression in the context of this situation. (The Math Resource for Instruction for NC Math 1)

● Example: The area of a rectangle can be represented by the expression x2 +8𝑥𝑥 + 12. What do the factors of this expression represent in the context of this problem? (The Math Resource for Instruction for NC Math 2)

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Seeing Structure in Expressions

Cluster:

Interpret the structure of expressions.

NVACS HSA.SSE.A.2

Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). Element Exemplars Standards for Mathematical Practice

● MP 7: Look for and make use of the structure of an algebraic expression

Instructional Strategies

● Students should be able to explain how specific structures are seen in different expression

● Students should be able to rewrite expressions to identify important components of the expression

● Provide students with problems that allow students to discover special patterns that occur with the structure of the expression

● Use problems like compound interest where students see the structure of the expression and how exponents help rewrite the expression

Prerequisite Skills

● Recognize special patterns such as difference of squares and greatest common factors so they can use them in new situations. Students should understand the area model for difference of squares.

Connections Within and Beyond High School

● In Algebra 1, the focus of this standard would be linear, exponential, and quadratic expressions. Algebra 2 may focus more on polynomial and rational expressions.

Instructional Examples/Lessons/Tasks

● Structure of an Expression Equivalent Expressions (Illustrative Mathematics)

● Complex Numbers Computing with Complex Numbers (Illustrative Mathematics)

Assessment Examples ● PARCC Practice Test PARCC Practice Test Item p 21

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Seeing Structure in Expressions

Cluster:

Write expressions in equivalent forms to solve problems.

NVACS HSA.SSE.B.3.a

Factor a quadratic expression to reveal the zeros of the function it defines.

Element Exemplars

Standards for Mathematical Practice

● MP6: Attend to precision by finding exact value of the zeros. ● MP7: Look for and make use of structure by representing equations

in equivalent forms.

Instructional Strategies

● Students need exposure to standard form, factored form, and vertex form. Rather than just memorizing the names, students need to understand what information they can gather from each form. This standard focuses on factored form.

● Teachers do not have to put emphasis on simplest form.

Prerequisite Skills

● Understand slope intercept form and how it relates to a graph. ● Factoring and expanding linear expressions with rational

coefficients . ● Understand that rewriting expressions into equivalent forms can

reveal other relationships between quantities.

Connections Within and Beyond High School

● Graphing linear equations and higher order polynomials. ● Understanding the relationships between factors, solutions, and

zeros. ● Interpreting the factors in context. ● Solving quadratic equations. ● Rewriting quadratic functions into different forms to show key

features of different functions.

Instructional Examples/Lessons/Tasks

● Equivalent Forms Graphs of Quadratic Functions Profit of a Company (Illustrative Mathematics)

● Interpret Expressions Seeing Dots (Illustrative Mathematics)

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Assessment Examples

● Khan Academy (249 Questions and 10 Skills) Scroll down to HSA.SSE.B.3.a

● Students should understand that the reasoning behind rewriting quadratic expressions into factored form is to reveal different key features of a quadratic function, namely the zeros/x-intercepts.

● Example: The expression –4x2 + 8x + 12 represents the height of a coconut thrown from a person in a tree to a basket on the ground where x is the number of seconds.

○ a) Rewrite the expression to reveal the linear factors. ○ b) Identify the zeros and intercepts of the expression and

interpret what they mean in regard to the context. ○ c) How long is the ball in the air? ○ (The Math Resource for Instruction for NC Math 1)

● Example: Part A: Three equivalent equations for (𝑥𝑥) are shown. Select the form that reveals the zeros of (𝑥𝑥) without changing the form of the equation.

○ 𝑓𝑓(𝑥𝑥) = −22 + 24𝑥𝑥 − 54 ○ 𝑓𝑓(𝑥𝑥) = −2(𝑥𝑥 − 3)(𝑥𝑥 − 9) ○ 𝑓𝑓(𝑥𝑥) = −2(𝑥𝑥 − 6)2 + 18

● Part B: Select all values of 𝑥𝑥 for which (𝑥𝑥) = 0. ○ −54, −18, −9, −6, −3, 0, 3, 6, 9, 18, 54

(from the Smarter Balanced Assessment Consortium) ● PARCC Practice Test Page 27

PARCC Practice Test Item

9

Seeing Structure in Expressions

Cluster:

Write expressions in equivalent forms to solve problems.

NVACS HSA.SSE.B.3.b

Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Element Exemplars

Standards for Mathematical Practice

● MP4: Model with mathematics by connecting the area of a square to quadratics.

● MP7: Look for and make use of structure to connect quadratic squares to minimum/maximum values.

Instructional Strategies

● Students need exposure to standard form, factored form, and vertex form. Rather than just memorizing the names, students need to understand what information they can gather from each form. This standard focuses on vertex form.

● Teachers do not have to put emphasis on simplest form. ● Algebra tiles

Prerequisite Skills ● Rearranging literals and formula ● Factoring trinomials

Connections Within and Beyond High School

● Simplifying quadratic expressions so they become solvable with square roots.

● Deriving the quadratic formula. ● Rewriting equations of conics in standard form.

Instructional Examples/Lessons/Tasks

● Completing the Square Rewriting a Quadratic Expression (Illustrative Mathematics)

Assessment Examples

● Reveal the vertex of a quadratic expression using the process of completing the square.

● Example: Write each expression in vertex form and identify the minimum or maximum value of the function.

○ a) x2−4x+5 ○ b) x2+5x+8 ○ c) 2x2+12x−18 ○ d) 3x2−12x−1 ○ e) 2x2−15x+3

(The Math Resource for Instruction for NC Math 2)

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Seeing Structure in Expressions

Cluster:

Write expressions in equivalent forms to solve problems.

NVACS HSA.SSE.B.3.c

Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

Element Exemplars

Standards for Mathematical Practice

● MP7: Students use structure to extend their use of the laws of exponents to transform expressions.

● MP8: Students use repeated reasoning to extend their use of the law of exponents to transform expressions.

Instructional Strategies ● Real World Scenarios with compound interest

Prerequisite Skills ● Use the properties of exponents to rewrite expressions with rational exponents.

Connections Within and Beyond High School

● Use the structure of an expression to identify ways to write equivalent expressions.

● Analyze and compare functions for key features. ● Building functions from graphs, descriptions and ordered pairs.

Instructional Examples/Lessons/Tasks

● Equivalent Expressions Forms of Exponential Expressions (Illustrative Mathematics)

● TI Calculator Exponential Functions (Texas Instruments)

Assessment Examples

● Exponential Functions Algebra Mod 3 Topic D Lesson 23 Algebra 2 Mod 3 Topic D Lesson 26 (Engage NY)

● PARCC Practice Items, Page 31 PARCC Practice Test Item

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Seeing Structure in Expressions Cluster:

Write expressions in equivalent forms to solve problems.

NVACS HSA.SSE.B.4 (Major Supporting Work)

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.*

Element Exemplars Standards for Mathematical Practice

● MP8: Students express regularity in repeated reasoning to derive the formula for the sum of a finite geometric series.

Instructional Strategies ● Financial Literacy

Prerequisite Skills

● Vocabulary: compounded interest, measures of time ie bi-annual, annual, quarterly, etc.

● Interest Formula ● Series = Sum of sequence terms

Connections Within and Beyond High School

● Summation Notation ● Geometric Sequences

Instructional Examples/Lessons/Tasks

● Virtual Manipulatives (CPALMS.org)

● Geometric Series (Engage NY)

● TI Calculator Activity Summing Up Geometric Series Geometric Series (Texas Instruments)

Assessment Examples ● Math Baseball (Shmoop)

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Arithmetic with Polynomial & Rational Expressions

Cluster:

Perform arithmetic operations on polynomials.

NVACS HSA.APR.A.1 (Major Supporting Work)

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Element Exemplars Standards for Mathematical Practice

● MP7: Students see how the structure of arithmetic is also used when working with expressions.

Instructional Strategies

● Combine like terms: ○ Evaluate expressions using grouping methods (e.g.: color

coding, shapes, etc.) ● Distributive property:

○ Area model diagram (box method, Punnett square) ○ Algebra Tiles

Prerequisite Skills

● Add, subtract, factor and expand linear expressions. ● Understand that rewriting expressions into equivalent forms can

reveal other relationships between quantities. ● Operations with polynomials ● Identify like terms, coefficients, and degrees. ● Distributive property ● Properties of exponents (multiplication) ● Rewrite expressions with radicals and rational exponents using the

properties of exponents.

Connections Within and Beyond High School

● Multiplying and factoring trinomials are inverse operations ● Quadratic formula ● Finding roots ● Building functions to model a relationship ● Solving systems of linear and quadratic equations

Instructional Examples/Lessons/Tasks

● Arithmetic with Polynomials and Rational Expressions (Khan Academy)

● Funding the Future (Illustrative Mathematics)

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Assessment Examples

● Students should be able to rewrite polynomial expressions using the properties of operations.

● Write at least two equivalent expressions for the area of the circle with a radius of 5𝑥𝑥 − 2 kilometers (The Math Resource for Instruction for NC Math 1).

• Page #32 ● Simplify each of the following: (The Math Resource for Instruction

for NC Math 1) • (4𝑥𝑥 + 3) − (2𝑥𝑥 + 1) • (𝑥𝑥2 + 5𝑥𝑥 − 9) + 2𝑥𝑥(4𝑥𝑥 − 3)

● The area of a trapezoid is found using the formula 𝐴𝐴 = 12 ℎ(𝑏𝑏1 + 𝑏𝑏2),

where 𝐴𝐴 is the area, ℎ is the height, and 𝑏𝑏1 and 𝑏𝑏2 are the lengths of the bases (The Math Resource for Instruction for NC Math 1).

What is the area of the above trapezoid? • 𝐴𝐴 = 4𝑥𝑥 + 2 • 𝐴𝐴 = 4𝑥𝑥 + 8 • 𝐴𝐴 = 2x2 + 4𝑥𝑥 − 21 • 𝐴𝐴 = 2𝑥𝑥2+ 8𝑥𝑥 − 42

● Students should be able to rewrite polynomials into equivalent

forms through addition, subtraction and multiplication. ● Simplify and explain the properties of operations apply (The Math

Resource for Instruction for NC Math 1). • (𝑥𝑥3 + 3𝑥𝑥2 − 2𝑥𝑥 + 5)(𝑥𝑥 − 7) • 4𝑏𝑏(𝑐𝑐𝑏𝑏 − 𝑧𝑧𝑑𝑑) • (4𝑥𝑥2 − 3𝑦𝑦2 + 5𝑥𝑥𝑦𝑦) − (8𝑥𝑥𝑦𝑦 + 3𝑦𝑦3) • (4𝑥𝑥2 − 3𝑦𝑦2 + 5𝑥𝑥𝑦𝑦) + (8𝑥𝑥𝑦𝑦 + 3𝑦𝑦2) • (𝑥𝑥 + 4)(𝑥𝑥 − 2)(3𝑥𝑥 + 5)

● PARCC Sample Items PARCC Practice Test Items (NMSU)

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Arithmetic with Polynomial & Rational Expressions

Cluster:

Understand the relationship between zeros and factors of polynomials.

NVACS HSA.APR.B.2 (Major Supporting Work)

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

Element Exemplars

Standards for Mathematical Practice

● MP1: Students make sense of the problem by using their knowledge of division of polynomials to make sense of the Remainder Theorem.

Instructional Strategies

● Factoring: ○ GCF, quadratic trinomial when leading coefficient is 1 and

greater than 1, perfect square trinomials, differences of square

○ Consider leading coefficient in quadratic trinomial ○ Methods: Completing the Square, Quadratic Formula,

Graphing ● Dividing Polynomials

○ Synthetic Division ○ Long Division

Prerequisite Skills

● Find Greatest Common Factor. ● Understand the roots of a polynomial can be written as factors. ● Understand that if x = a is a root, then x – a is a factor. ● A factor evenly divides a polynomial.

Connections Within and Beyond High School

● Minimizing surface area ● Multiplicity ● Graphing and solving rational functions ● Polynomial identities ● Synthetic division ● Analyze and compare functions for key features ● Building functions from graphs, descriptions and ordered pairs

Instructional Examples/Lessons/Tasks

● Polynomial Factoring Bingo (RPDP)

● The Remainder Theorem (Illustrative Mathematics)

Assessment Examples ● Polynomials and Factoring Practice Test (RPDP)

15

Arithmetic with Polynomial & Rational Expressions

Cluster:

Understand the relationship between zeros and factors of polynomials.

NVACS HSA.APR.B.3 (Major Supporting Work)

Identify zeros of polynomials when suitable factorizations are available and use the zeros to construct a rough graph of the function defined by the polynomial.

Element Exemplars

Standards for Mathematical Practice

● MP1: Students make sense of the problem by using their knowledge of division of polynomials, the Remainder Theorem, and substitution to make graphs of functions.

● MP2: Students reason about the shapes of graphs and what a zero means on a graph.

● MP7: Students use the structure of polynomial functions, their knowledge of zeros. and plotting points to make a sketch of a function.

Instructional Strategies

● Checking your work • Verify zeros by substitution • Real world applications • Projectile motion • Maximizing profit/height

Prerequisite Skills ● Factoring ● Solving quadratic equations ● Determine the general shape of a linear & quadratic

Connections Within and Beyond High School

● State end behavior; characteristics of a function ● Making connections to functions such as (but not limited to) cubic,

quartic, quintic, inverse, etc. ● Fundamental Theorem of Algebra

Instructional Examples/Lessons/Tasks

● Football and Parabolas (RPDP) ● Illuminations Egg Launch Contest (RPDP) ● Analyzing Polynomial Functions (RPDP) ● Will it Hit the Hoop? (Desmos)

Assessment Examples

● Polynomial Graphing Class Activity (RPDP) ● Rational Root Theorem and Finding Zeros (RPDP) ● Modeling with Polynomials (RPDP) ● PARCC Practice Test Item (NMSU)

• Pages 34–35

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Arithmetic with Polynomial & Rational Expressions

Cluster:

Use polynomial identities to solve problems.

NVACS HSA.APR.C.4 (Major Supporting Work)

Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.

Element Exemplars Standards for Mathematical Practice

● MP8: Students look for repeated reasoning by exploring several cases of an identity.

Instructional Strategies

● Derive difference of squares ● Generate Pythagorean Triple by proving

(𝑎𝑎2 + 𝑏𝑏2)2 = (𝑎𝑎2 + 𝑏𝑏2)+(2𝑎𝑎𝑏𝑏)2 ● Define restrictions on 2 chosen integers

Prerequisite Skills ● Pythagorean Theorem ● Distributive Property ● Polynomial Arithmetic

Connections Within and Beyond High School

● Geometry Proofs ● Binomial Expansion

Instructional Examples/Lessons/Tasks

● Identities Polynomial Identities (Part 1) Polynomial Identities (Part 2) (Illustrative Mathematics)

● Pythagorean Theorem Trina’s Triangle (Illustrative Mathematics)

Assessment Examples ● Identities

Polynomial Identities (Khan Academy)

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Arithmetic with Polynomial & Rational Expressions

Cluster:

Use polynomial identities to solve problems.

NVACS HSA.APR.C.5 (Major Supporting Work)

(+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.1

Element Exemplars

Standards for Mathematical Practice

● MP5 and MP7: Students will look for and make use of structure when building Pascal’s Triangle and comparing to the coefficients of various binomial expansions.

Instructional Strategies ● Relate to combinations because they are easier to visualize and

explain ● Blank Pascal's Triangle (Cabrillo College)

Prerequisite Skills ● Multiplying polynomials ● Combining like terms

Connections Within and Beyond High School

● Binomial Probabilities ● Combinations

Instructional Examples/Lessons/Tasks

● Binomial Theorem Powers of 11 (Illustrative Math) Binomial Expansion with Pascal’s Triangle (Illustrative Math) Precalculus and Advanced Topics Module 3, Topic A, Lesson 4 (Engage NY)

● Patterns in the Coefficients Precalculus and Advanced Topics Module 3, Topic A, Lesson 5 (Engage NY)

● CCSS Examples Sample Assignments (shmoop.com)

Assessment Examples ● Polynomial arithmetic, HSA.APR.C.5, 16 Questions and 1 Skill

Algebra: Arithmetic with Polynomials and Rational Expressions (Khan Academy)

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Arithmetic with Polynomial & Rational Expressions

Cluster:

Rewrite rational expressions.

NVACS HSA.APR.D.6 (Major Supporting Work)

Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Element Exemplars Standards for Mathematical Practice

● MP7: Students see how the structure of arithmetic is also used when working with rational expressions.

Instructional Strategies

● Start with rational numbers, review how to rewrite improper fractions as mixed numbers. Then translate this into rewriting rational expressions into the form q(x) + 𝑟𝑟(𝑥𝑥)

𝑏𝑏(𝑥𝑥) . ● Students will be given a rational expression and need to decide if

they can just look at the problem and be able to rewrite it as q(x) + 𝑟𝑟(𝑥𝑥)𝑏𝑏(𝑥𝑥) , or if they will need to use long division to rewrite. They

should also be able to work backward to go from q(x) + 𝑟𝑟(𝑥𝑥)𝑏𝑏(𝑥𝑥) ,

to 𝑎𝑎(𝑥𝑥)𝑏𝑏(𝑥𝑥), checking their work.

Prerequisite Skills

● Ratios and proportions ● Operations with fractions ● Operations with polynomials ● Synthetic/long division of polynomials ● Remainder Theorem

Connections Within and Beyond High School ● Partial fraction decomposition

Instructional Examples/Lessons/Tasks

● Rewrite Ration Expressions Combined Fuel Efficiency (Illustrative Math) Egyptian Fractions II (Illustrative Math)

Assessment Examples ● Rational Expressions

Rewrite Simple Rational Expressions in Different Forms (ACT Academy)

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Arithmetic with Polynomial & Rational Expressions

Cluster:

Rewrite rational expressions.

NVACS HSA.APR.D.7 (Major Supporting Work)

(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Element Exemplars Standards for Mathematical Practice

● MP7: Students see how the structure of arithmetic is also used when working with rational expressions.

Instructional Strategies ● Start with add, subtract, multiply, and divide fractions. Go from the known to the unknown.

Prerequisite Skills ● Order of operations ● Properties of addition and multiplication

Connections Within and Beyond High School ● Rational functions

Instructional Examples/Lessons/Tasks

● Tutorials and Assessments Related Resources (CPalms)

● Multiply and Divide Rational Expressions Algebra II Module 1, Topic C, Lesson 24 (Engage NY) Dividing Rational Expressions (Open Middle) Rational Expression Multiplication (Texas Instruments)

● Adding and Subtracting Rational Expressions Algebra II Module 1, Topic C, Lesson 25 (Engage NY)

Assessment Examples

● Rational Expressions Adding and Subtracting Rational Expressions (RPDP) Multiplying and Dividing Rational Expressions (RPDP)

20

Creating Equations

Cluster:

Create equations that describe numbers or relationships.

NVACS HSA.CED.A.1 (Major Supporting Work)

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Element Exemplars

Standards for Mathematical Practice

● MP1 and MP4: Students use equations and inequalities to model different situations and to make sense of problems.

● MP3 and MP6: When students consider the viability of their own and of others’ solutions, they construct arguments and justifications. This also means students are using proper vocabulary and attending to the precision of their symbols and work.

Instructional Strategies

● Students look for patterns in one variable in data, contextual situations, and other numeric patterns and create equations or inequalities for them. Then, the equations or inequalities are used to solve contextual problems.

● Present situation in one variable so that students may create equations or inequalities that model them. For example, four people may be seated at a single rectangular table. If two tables are placed together, 6 people may be seated at the table. How many people can be seated when 10 tables are placed together? How many tables are needed for 100 people?

● Provide a variety of different types of relationships for students to model, including linear functions, exponential functions, quadratic functions, and simple rational functions.

● Choose from among linear functions, exponential functions, quadratic functions, and simple rational functions in order to model a situation.

● Have students explain their reasoning and steps when creating an equation or inequality.

Prerequisite Skills

● Write numeric and algebraic expressions. ● Write and solve numeric and algebraic equations and inequalities

from real-world and mathematical problems. ● Apply the order of operations. ● Use properties (i.e. distributive, associative, etc.) to write equations

in different forms.

Connections Within and Beyond High School

● Solve inverse variation, square root and quadratic equations. ● Use trig ratios to solve problems. ● Solve systems of equations. ● Write a system of equations as an equation or write an equation as a

system of equations to solve.

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Instructional Examples/Lessons/Tasks

● Creating a linear equation task example Paying the rent (Illustrative Math)

● Creating a simple rational equation task example Buying a Car (Illustrative Math)

● Creating a quadratic equation task example Squares to Stairs (Illustrative Math)

● Creating an exponential inequality task example Paper Folding (Illustrative Math)

Assessment Examples

● Sample Assessments Shmoop (Shmoop)

● SBAC Sample Item Sample Items (Item 3360) (SBAC)

● PARCC Sample Items PARCC Practice Test Item p 38, 45 (PARCC)

● A student earns $7.50 per hour at her part-time job. She wants to earn at least $200. Enter an inequality that represents all of the possible numbers of hours (h) the student could work to meet her goal. Enter your response in the first response box. Enter the least whole number of hours the student needs to work in order to earn at least $200. Enter your response in the second response box.

22

Creating Equations

Cluster:

Create equations that describe numbers or relationships.

NVACS HSA.CED.A.2 (Major Supporting Work)

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Element Exemplars

Standards for Mathematical Practice

● MP4: Students will collect data or look at data sets to determine a model that best fits a scatterplot of the data.

● MP6: Students will graph equations on coordinate axes with appropriate labels and scales.

● MP7: Students use properties of operations to transform equations. Students use structure to interpret equations that represent real-world situations.

Instructional Strategies

● Have students look for patterns in bivariate (two-variable matched) data, contextual situations, and other numeric patterns and create equations or inequalities for them. Equations, inequalities, or systems of equations or inequalities are used to solve contextual problems.

● Present situations in two variables so that students may create equations or inequalities that model them. Students can find a line of fit for a set of data.

● Provide a variety of different types of relationships for students to model, including linear and exponential contexts.

● Require students to explain their reasoning when they have created an equation, inequality, or systems.

● Students are hesitant to move to the use of two variables for a situation and may try to create equations with only one variable. Using real data situations can help clear this up, as there are two very distinct quantities. (Rate and distance problems are one such context.)

Prerequisite Skills

● Construct a linear function that models the relationship between two quantities.

● Graph linear equations. ● The graph of a function is the set of ordered pairs consisting of

input and a corresponding output. ● Understand that the graph of a two-variable equation represents the

set of all solutions to the equation. ● Interpret parts of an expression in context. ● Use the structure of an expression to identify ways to write

equivalent expressions.

23

Connections Within and Beyond High School

● Interpret parts of an expression in context. ● Creating linear equations for a system. ● Solving for a variable of interest in a formula. ● The graph of a function f is the graph of the equation y = f(x). ● Interpret a function’s domain and range in context. ● Identify key features of linear, exponential, and quadratic functions. ● Building a function through patterns or by combining other

functions. ● Analyze and compare functions. ● Understand the relationship between the factors of a polynomial,

solutions and zeros. ● Use logarithms to express solutions to exponential equations.

Instructional Examples/Lessons/Tasks

● Desmos ○ Match My Line ○ Build a Bigger Field ○ Marbleslides Parabolas

Assessment Examples

● Shmoop ● The floor of a rectangular cage has a length 4 feet greater than its

width, w. James will increase both dimensions of the floor by 2 feet. Which equation represents the new area, 𝑁𝑁, of the floor of the cage? (North Carolina, EOC released item)

○ a) N = w2 + 4w ○ b) N = w2 + 6w ○ c) N = w2 + 6w + 8 ○ d) N = w2 + 6w + 12

● 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. (The Math Resource for Instruction for NC Math 2)

24

Creating Equations

Cluster:

Create equations that describe numbers or relationships.

NVACS HSA.CED.A.3 (Major Supporting Work)

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Element Exemplars

Standards for Mathematical Practice

● MP2: Students use contexts to determine constraints for equations or inequalities and by systems of equations or inequalities, analyze their solutions to determine if it is viable for the given context.

● MP4: Students identify important quantities in a practical situation involving systems and map their relationships using graphs and equations. They can analyze those relationships mathematically to draw conclusions. They interpret their mathematical results in the context of the situation and reflect on whether the results make sense.

● MP5: Students can analyze graphs of functions and solutions generated using a graphing calculator or software to determine the solutions of a system.

Instructional Strategies

● Provide contextual examples of equations or inequalities and by systems of equations and/or inequalities that require constraints.

● Require students to explain the reasoning behind their use of constraints in a task.

● Consistently expect to check solutions to determine if they make sense in a context and to explain any decisions they make about potential solutions.

● Ensure students have opportunities to contextually, analytically, and graphically check a solution set of inequalities to determine the viability of a solution.

Prerequisite Skills

● Understanding a system of equations. ● Creating linear equations in two variables. ● Create and graph two variable equations. ● Create equations for a system of equations in context. ● Interpret parts of an expression in context. ● Create equations in two variables.

25

Connections Within and Beyond High School

● Interpret parts of an expression in context. ● Use tables, graphs, and algebraic methods to solve systems of linear

equations. ● Represent the solution to a system of linear inequalities as a region

of the plane. ● Solve systems of equations. ● Write a system of equations as an equation or write an equation as a

system of equations to solve. ● Use function notation to evaluate piecewise functions.

Instructional Examples/Lessons/Tasks

● Illustrative Mathematics: Dimes and Quarters ● Desmos: Solutions to Systems of Linear Equations

Assessment Examples

● PARCC Sample Items PARCC Practice Test Item p 74 – 75 (PARCC)

● A club is selling hats and jackets as a fundraiser. Their budget is $1500 and they want to order at least 250 items. They must buy at least as many hats as they buy jackets. Each hat costs $5 and each jacket costs $8. (The Math Resource for Instruction for NC Math 1)

○ a) Write a system of inequalities to represent the situation. ○ b) Graph the inequalities. ○ c) If the club buys 150 hats and 100 jackets, will the

conditions be satisfied? ○ d) What is the maximum number of jackets they can buy

and still meet the conditions? ● 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 (𝑛𝑛) = −0.05𝑛𝑛 2 + 20𝑛𝑛 and (𝑛𝑛) = 5𝑛𝑛 + 250. Determine values for which (𝑛𝑛) = (𝑛𝑛) and explain what the solution(s) reveal about the prospects of the pizza sale fundraiser. (The Math Resource for Instruction for NC Math 2)

26

Creating Equations

Cluster:

Create equations that describe numbers or relationships.

NVACS HSA.CED.A.4 (Major Supporting Work)

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

Element Exemplars

Standards for Mathematical Practice

● MP3: Students understand and use previously established results in constructing arguments to justify their reasoning for how they transform an equation to another form.

● MP7: Students use properties of operations to transform equations.

Instructional Strategies

● Provide contextual situations that use literal equations (e.g., rewrite the perimeter formula to solve for a missing length).

● Make connections between solving equations and rearranging formulas.

● Require students to explain their solution process for rearranging a formula.

Prerequisite Skills

● Solve linear equations in one variable. ● Use square root and cube root symbols to represent solutions to

equations of the form x2 = p and x3 = p where p is a positive rational numbers.

● Justify a solution method and each step in the solving process. Connections Within and Beyond High School

● Create an equation in two variables that represent a relationship between quantities.

Instructional Examples/Lessons/Tasks

● Rearranging Formulas Rewriting Equations (Illustrative Mathematics)

27

Assessment Examples

● Shmoop: Click Samples Assignments for drills with solutions and explanations

● Energy and mass are related by the formula E = mc2 ■ 𝑚𝑚 is the mass of the object ■ 𝑐𝑐 is the speed of light

○ Which equation finds 𝑚𝑚, given 𝐸𝐸 and 𝑐𝑐? (The Math Resource for Instruction for NC Math 1)

○ A) m = E – c2 ○ B) m = Ec2 ○ C) 𝑚𝑚 = 𝑐𝑐2

𝐸𝐸

○ D) 𝑚𝑚 = 𝐸𝐸𝑐𝑐2

● The equation for an object that is launched from the ground is given

by h(t) = –16t2 + vt where ℎ is the height, 𝑡𝑡 is the time, and v is the initial velocity. What is the initial velocity of an object that is one-hundred feet off the ground four seconds after it is launched? (The Math Resource for Instruction for NC Math 1)

28

Reasoning with Equations & Inequalities

Cluster:

Understand solving equations as a process of reasoning and explain the reasoning.

NVACS HSA.REI.A.1 (Major Supporting Work)

Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Element Exemplars

Standards for Mathematical Practice

● MP 1: Students will make sense of problems and persevere in solving problems by rewriting expressions.

● MP 2: Students will reason abstractly and quantitatively by making sense of the quantities and relationships within the context of the problem.

● MP 3: Students will construct viable arguments using givens, properties and definitions. Students will also critique the reasoning of others.

Instructional Strategies

● Start with linear equations in one variable move to more complex equations.

● Make sure the students understand the goal of solving an equation is to isolate a certain variable (coefficient and exponent of 1) on one side of the equal sign.

Prerequisite Skills

● Order of operations ● Inverse operations ● Properties of equality ● Properties on inequality ● Properties of real numbers ● Operations with fractions, reciprocals, ratios, and proportions

Connections Within and Beyond High School

● Solving equations in one, two, and three variables ● Solving inequalities in one and two variables ● Proofs in geometry strand

29

Instructional Examples/Lessons/Tasks

● Properties and Roots Products & Reciprocals Complex Square Roots Zero Product Property 1 Reasoning with linear inequalities Zero Product Property 2 Zero Product Property 3 Zero Product Property 4 (Illustrative Math)

● Linear Equations in one variable Method to My Mathness (CPalms)

Assessment Examples ● Summative Exit Ticket (CPalms)

30

Reasoning with Equations & Inequalities

Cluster:

Understand solving equations as a process of reasoning and explain the reasoning.

NVACS HSA.REI.A.2 (Major Supporting Work)

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Element Exemplars

Standards for Mathematical Practice

● MP 6: Attend to precision by checking for extraneous solutions. ● MP 7: Look for and make use of the structure of radical and rational

equations to identify ways to solve it.

Instructional Strategies

● Review inverse operations. ● Review solving absolute value equations and checking their

solutions. Remind them how sometimes solutions are not valid, so they have to check the solutions they obtain from solving.

Prerequisite Skills

● Properties of exponents ● Understand proportions ● Combine, manipulate, and simplify radicals ● Inverse operations ● Solving equations ● Evaluating functions (checking their solutions to equations)

Connections Within and Beyond High School ● Absolute value equations

Instructional Examples/Lessons/Tasks

● Sample Items Shmoop (Shmoop)

● Sample Lessons Khan Academy (Khan Academy)

Assessment Examples

● Solve Equations Illustrative Mathematics (Illustrative Mathematics)

● Sample Assessments RPDP: Math, High School, Algebra 2, Unit 5 Create, Graph, and Solve Notes, Pages 22–23 RPDP: Math, High School, Algebra 2, Unit 6 Rational Functions Notes, Pages 20–22 (RPDP)

31

Reasoning with Equations & Inequalities

Cluster:

Solve equations and inequalities in one variable.

NVACS HSA.REI.B.3 (Major Supporting Work)

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Element Exemplars Standards for Mathematical Practice

● MP 5: Students will model mathematics using real world applications to create equations and inequalities in one variable.

Instructional Strategies

● To solve linear equations/inequalities: ○ Know inverse operations. ○ Know properties of equality. ○ Know that a solution is a value that, when substituted for a

variable, makes an equation or inequality true.

Prerequisite Skills ● Define variable. ● Order of Operations, simplifying expressions ● Translating verbal statements into mathematical statements.

Connections Within and Beyond High School

● Solving literal equations. ● Solving linear and quadratic inequalities in two variables ● Graphing linear and quadratic inequalities in two variables

Instructional Examples/Lessons/Tasks

● Equations and Their Solutions (Illustrative Mathematics)

Assessment Examples

● PARCC Practice Test PARCC Practice Test Item p 33

● Equations and Inequalities Unit 2 Equations Notes (pdf) Unit 3 Linear Inequalities & Absolute Equations/ Inequalities Notes (RPDP)

32

Reasoning with Equations & Inequalities

Cluster:

Solve equations and inequalities in one variable.

NVACS HSA.REI.B.4 (Major Supporting Work)

Solve quadratic equations in one variable.

Element Exemplars

Standards for Mathematical Practice

● MP 4: Students model quadratic equations through real world applications.

● MP 7: Students look and make use of the structure of quadratic equations in order to determine the best method of factoring and solving.

Instructional Strategies ● Twelve Days of Solving Quadratics

Prerequisite Skills ● Solving equations in one variable. ● Multiplication facts

Connections Within and Beyond High School

● Solving quadratics: ○ By taking square roots ○ Completing the square ○ Quadratic Formula ○ Factoring

● Graphing Quadratics ○ Vertex, maximum and minimum values, domain and range,

x and y intercepts

Instructional Examples/Lessons/Tasks

● Class Notes Solve by Factoring Class Notes (pdf) Solve by Completing the Square Class Notes (pdf) Solve by the Quadratic Formula Class Notes (pdf) Solving Quadratic Inequalities Class Notes Angry Bird Quadratic Modeling Activity (RPDP)

● Quadratic Equation Building a General Quadratic Function Braking Distance Completing the Square Visualizing Completing the Square Vertex of the Parabola with Complex Roots Quadratic Sequence 1 Quadratic Sequence 2 Quadratic Sequence 4 Zero Product Property 4 (Illustrative Math)

Assessment Examples ● Sample Items

Khan Academy: Scroll down to HSA.REI.B.4 (Khan Academy)

33

Reasoning with Equations & Inequalities

Cluster:

Solve equations and inequalities in one variable.

NVACS HSA.REI.B.4A (Major Supporting Work)

Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form.

Element Exemplars

Standards for Mathematical Practice

● MP 2: Students will reason abstractly as they derive the quadratic formula by completing the square using the standard form of a quadratic.

● MP 4: Students can use models to demonstrate understanding of completing the square.

Instructional Strategies ● Algebra tiles ● Area model ● Algebraically

Prerequisite Skills

● Knowledge of perfect trinomial squares ● Knowledge of square roots and perfect squares ● Solving equations for x ● Knowledge of quadratic functions and what the x values are ● Foundational understanding of quadratics ● What is the meaning of maximum and minimum ● Inverse operations ● What the “a” value in a quadratic means in terms of the direction a

parabola opens ● Solving equations using square roots. ● Vertex form of a parabola

Connections Within and Beyond High School

● Quadratic graphs ● Factoring ● Quadratic formula ● Complex numbers

Instructional Examples/Lessons/Tasks

● Math Resources RPDP Worksheets (RPDP)

● Completing the Square ● Visualizing Completing the Square

(Illustrative Mathematics) Hip to be (completing the) Square (CPalms) Completing the Square Completing the Square Algebraically (Texas Instruments)

34

Assessment Examples

● Exit Ticket Completing the square exit ticket (Engage NY)

● PARCC Sample Items PARCC Practice Test Item p 20 (PARCC)

35

Reasoning with Equations & Inequalities

Cluster:

Solve equations and inequalities in one variable.

NVACS HSA.REI.B.4B (Major Supporting Work)

Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Element Exemplars

Standards for Mathematical Practice

● MP 1: Students should be able to make sense of a problem and determine a procedure for solving a quadratic equation that is convenient based on the form of the equation.

● MP 2: Students can quantitatively reason and make sense of their answers using the discriminant.

Instructional Strategies ● Algebra tiles ● Algebraically

Prerequisite Skills

● Knowledge of perfect trinomial squares ● Knowledge of square roots and perfect squares ● Solving equations for x ● Knowledge of quadratic functions and what the x values are ● Foundational understanding of quadratics ● What is the meaning of maximum and minimum ● Inverse operations ● What the “a” value in a quadratic means in terms of the direction a

parabola opens ● Solving equations using square roots

Connections Within and Beyond High School

● Quadratic graphs ● Factoring ● Quadratic formula ● Complex numbers

Instructional Examples/Lessons/Tasks

● Illustrative Math ● TI Nspire Activities ● Four Ways to Solve Quadratics ● Where Would The Angry Birds Have Landed?

Assessment Examples

● Sort Quadratic Equations The Quadratic Quandary (CPalms)

● Mini-Assessment Quadratic Equations Mini Quiz (Achieve the Core)

● PARCC Practice Items PARCC Practice Test Item p 22, 70 (PARCC)

36

Reasoning with Equations & Inequalities

Cluster:

Solve systems of equations.

NVACS HSA.REI.C.5 (Major Supporting Work)

Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Element Exemplars

Standards for Mathematical Practice

● MP5: The process of adding multiples of two different equations to eliminate one variable to solve a system can be done with little understanding, but students can connect this to graphing different sets of equations to see why elimination works.

Instructional Strategies ● To solve systems of equations algebraically using substitution and

elimination strategies. ● Explanation of real-life problems to identify the linear system.

Prerequisite Skills ● Combining like terms ● Solving equations for a single variable ● Graphing linear equations and inequalities

Connections Within and Beyond High School

● Linear Programming ● Linear equations

Instructional Examples/Lessons/Tasks

● Solving Two Equations in Two Unknowns ● TI Nspire Activities ● NCTM Illustrations: Movie Rental, Candy Problem ● Solving Systems of Equations and Inequalities Notes and Examples ● Exploring Systems with Piggies, Pizzas and Phones ● 13 activities to solve system of equations ● Grade 9 solving systems of equations (exit slip included)

Assessment Examples ● Systems of Equations

9th grade lesson assessment of system of equations (Better Lesson)

37

Reasoning with Equations & Inequalities Cluster:

Solve systems of equations.

NVACS HSA.REI.C.6 (Major Supporting Work)

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Element Exemplars

Standards for Mathematical Practice

● MP2: Students reason abstractly and quantitatively by solving systems of equations with and without models.

● MP5: Students are using tools when graphing, either by hand or with technology, and when they are using algebraic methods, also by hand or with CAS.

Instructional Strategies ● Graphing Calculator ● Graph paper ● Line graphing

Prerequisite Skills ● Graph linear equations by hand or with technology ● Make a table of (x,y) values for equations

Connections Within and Beyond High School

● All system of equations and what the solution means. ● Parallel and perpendicular lines on the coordinate plane. ● Independent, dependent, inconsistent solutions ● Graphing systems with different shapes including parabolas, circles,

etc.

Instructional Examples/Lessons/Tasks

● TI Nspire Linear Equations ● Illuminations: Road Rage ● Desmos Linear Systems ● Illustrative Math ● 11 graphing activities for solving systems of linear equations ● Grade 9 solving systems (exit slip included)

Assessment Examples ● System of Equations

9th grade lesson assessment of system of equations (Better Lesson)

38

Reasoning with Equations & Inequalities

Cluster:

Solve systems of equations.

NVACS HSA.REI.C.7 (Major Supporting Work)

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3.

Element Exemplars

Standards for Mathematical Practice

● MP1: Students should check the algebraic or graphical solutions in the system to make sense of the solution.

● MP4: Students will show graphically the solutions to non-linear systems.

Instructional Strategies ● Graphing Calculator

Prerequisite Skills ● Graphing system of equations with 2 lines. ● Solving system of equations with 2 lines

Connections Within and Beyond High School

● Solving systems of equations with 2 lines ● Circle formula and graph ● Parabola formulas and graphs ● Solving systems by graphing

Instructional Examples/Lessons/Tasks

● Eureka math lesson including exit slip ● Illustrative Math ● Non Linear Systems of Equations ● Solving systems with circles and lines ● Solving systems with parabolas or circles and lines

Assessment Examples ● System of Equations

Ninth grade assessment (at the bottom of the page) (Better Lesson)

39

Reasoning with Equations & Inequalities

Cluster:

Solve systems of equations.

NVACS HSA.REI.C.8 (Major Supporting Work)

(+) Represent a system of linear equations as a single matrix equation in a vector variable.

Element Exemplars

Standards for Mathematical Practice

● MP2: Students will transform a system of linear equations into a single matrix equation. Separating the variables and the coefficients and understanding the importance of where each piece is placed can be abstract for most students.

Instructional Strategies ● Graphing Calculator

Prerequisite Skills ● Matrix operations ● Discriminant ● Meanings of the solutions to Systems of Equations

Connections Within and Beyond High School

● Solving systems using substitution, elimination and graphing ● use of graphing calculator

Instructional Examples/Lessons/Tasks

● Illuminations: Pick’s Theorem ● TI Graphing Calculators: Algebra 2 Solving Systems with Row

Operations 2, All Systems Go!, Reduce It!, Cramer’s Rule ● RPDP Notes Systems & Matrices ● EngageNY Systems of Equations with Matrices

Assessment Examples ● Systems of Equations

EngageNY Systems of Equations with Matrices (EngageNY)

40

Reasoning with Equations & Inequalities

Cluster:

Solve systems of equations.

NVACS HSA.REI.C.9 (Major Supporting Work)

(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for atrices of dimension 3 × 3 or greater).

Element Exemplars

Standards for Mathematical Practice

● MP5: Students will use matrices to solve systems of linear equations. When the matrix dimensions are 3 X 3 and greater they will use technology to solve the systems.

Instructional Strategies ● Graphing Calculator

Prerequisite Skills

● Matrix operations ● Discriminant ● Meanings of the solutions to Systems of Equations ● Inverses of Matrices

Connections Within and Beyond High School

● Cramer’s Rule ● Reduced Row Echelon Form

Instructional Examples/Lessons/Tasks

● TI Graphing Calculators Solving Systems w/Row Operations 2 All Systems Go Reduce It Cramer's Rule

● RPDP Systems of Equations & Matrices

● EngageNY Topic C Lesson 27 Matrices to Solve Systems

Assessment Examples ● Solving Systems of Equations

Overview Topic C Matrices to Solve Systems (Engage NY)

41

Reasoning with Equations & Inequalities

Cluster:

Represent and solve equations and inequalities graphically.

NVACS HSA.REI.D.10 (Major Supporting Work)

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Element Exemplars

Standards for Mathematical Practice

● MP1: Make sense of the problem by analyzing the relationships and goals. Students can also create different representations of an equation to help them make sense of the question.

● MP3: Students should be able to clearly communicate their justifications and conclusions.

● MP5: Students should use appropriate tools to help them make sense. A student could use a pencil and graph paper or technology to create a graph to analyze the equation.

Instructional Strategies ● Have the students graph different types of functions. Give them

ordered pairs to test that are and are not on the curve. This will help to illustrate each of these graphs have more than one solution

Prerequisite Skills ● Graphing different types of equations ● Evaluating expressions

Connections Within and Beyond High School

● Solving equations and inequalities in one variable ● Solving different types of equations in two variables ● Solving different types of inequalities in two variables

Instructional Examples/Lessons/Tasks

● Illustrative Mathematics ● Khan Academy ● Open Ed

Assessment Examples

● Represent Equations Illustrative Mathematics (Illustrative Mathematics)

● PARCC Practice Items PARCC Practice Test Item p 47–48 (PARCC)

42

Reasoning with Equations & Inequalities

Cluster:

Represent and solve equations and inequalities graphically.

NVACS HSA.REI.D.11 (Major Supporting Work)

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.*

Element Exemplars Standards for Mathematical Practice

● MP2: Students reason abstractly and know that there are different representations to find the solution(s) between two functions.

Instructional Strategies ● Students verify solutions using more than one way (algebraically, graphically, creating table of values, guess & check)

Prerequisite Skills ● Solving Systems of Equations ● Polynomial Arithmetic

Connections Within and Beyond High School

● Maximum and minimum values ● Domain and range ● Linear Programming

Instructional Examples/Lessons/Tasks

● Maximizing Your Efforts, TI Activity; Topics: write and solve a system of linear inequalities relating to real world application of maximizing profit

● Illustrative Mathematics

Assessment Examples ● Graph Linear Inequalities

Illustrative Mathematics (Illustrative Mathematics)

43

Reasoning with Equations & Inequalities

Cluster:

Represent and solve equations and inequalities graphically.

NVACS HSA.REI.D.12 (Major Supporting Work)

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Element Exemplars

Standards for Mathematical Practice

● MP 4: Students will use graphs of inequalities and systems of inequalities to show the solutions set for each situation.

● MP 6: Students will attend to precision in their graphing to correctly model the solution set for inequalities and systems of inequalities.

Instructional Strategies

● Lecture with examples ● Link to inequalities in one-variable ● Link to graphing different forms of a linear equation ● Use test points to help determine which part of the coordinate plane

to shade ● Make sure the students understand where the solutions are ● Centers/ stations ● Sorting activities ● Scavenger hunts

Prerequisite Skills

● Understand the meaning of inequality symbols ● Know how to graph a linear equation ● Rearrange literals and formulas ● Evaluating functions

Connections Within and Beyond High School

● Systems of linear inequalities ● Linear programming ● Graphing quadratics inequalities

Instructional Examples/Lessons/Tasks ● Illustrative Mathematics

Assessment Examples ● RPDP: High School, Math, Algebra 1, Unit 5, Unit 7

44

Acknowledgements

Debra Brown Clark County School District

Vicki Collaro

Washoe County School District

Becky Curtright Washoe County School District

Stefanie Falk

Douglas County School District

Teresa Holyoak Clark County School District

Linda Koyen

Washoe County School District

Kathy Lawrence Washoe County School District

Sarah Lobsinger

Carson City School District

Dawn Lockett Clark County School District

Jennifer Loescher

Southern RPDP

Jennifer Lopez Clark County School District

Sarah Mascarenas

Washoe County School District

Candice Meiries Southern RPDP

Amy Mitchell

Clark County School District

Amy Nelson Clark County School District

Barbara Perez

Clark County School District

Douglas Speck Southern RPDP

Carly Strauss

Douglas County School District

Sara Swanson Clark County School District

Trent Tietje

Douglas County School District

Lynn Trell Clark County School District

Lauren Wachter

Clark County School District

Wendy Weatherwax Clark County School District

45

References

Mills, R. H. & Williams, L. A. (2016). The common core mathematics companion: The standards decoded, grades 6-8: What they say, what they mean, how to teach them. Thousand Oaks, CA: Corwin.

Gojak, L. M. & Miles, R. A. (2015). The common core mathematics companion: The standards decoded, grades 3-5: What they say, what they mean, how to teach them. Thousand Oaks, CA: Corwin.

46

Jhone Ebert

Superintendent of Public Instruction

Jonathan Moore, Ed.D

Deputy Superintendent for Student Achievement

Dave Brancamp

Director of Standards and Instructional Support

Tracy Gruber

K-12 Mathematics Education Program Professional