Mathematics Standards Clarification for Algebra Conceptual ...

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

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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 Standards for Mathematical Practice

Instructional Strategies

Prerequisite Skills

Connections Within and Beyond High School

Instructional Examples/Lessons/Tasks

Exemplars

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.

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.

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. 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.

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 +8 + 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

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

Mathematical Practice

expression

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

Instructional Strategies

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

Recognize special patterns such as difference of squares and

Prerequisite Skills

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.

Structure of an Expression

Equivalent Expressions

Instructional

(Illustrative Mathematics)

Examples/Lessons/Tasks

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 Standards for Mathematical Practice Instructional Strategies

Prerequisite Skills

Connections Within and Beyond High School

Instructional Examples/Lessons/Tasks

Exemplars

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.

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. 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.

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.

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 () are shown. Select the form that reveals the zeros of () without changing the form of the equation. () = -22 + 24 - 54 () = -2( - 3)( - 9) () = -2( - 6)2 + 18 Part B: Select all values of for which () = 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

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