## Description

- Overview:
- This lesson unit is intended to help you assess how well students are able to manipulate and calculate with polynomials. In particular, it aims to identify and help students who have difficulties in: switching between visual and algebraic representations of polynomial expressions; and performing arithmetic operations on algebraic representations of polynomials, factorizing and expanding appropriately when it helps to make the operations easier.

- Level:
- Lower Primary, Upper Primary, Middle School, High School
- Grades:
- Kindergarten, Grade 1, Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7, Grade 8, Grade 9, Grade 10, Grade 11, Grade 12
- Material Type:
- Assessment, Lesson Plan
- Provider:
- Shell Center for Mathematical Education, U.C. Berkeley
- Provider Set:
- Mathematics Assessment Project (MAP)
- Date Added:
- 04/26/2013

- License:
- Creative Commons Attribution Non-Commercial No Derivatives
- Media Format:
- Downloadable docs, Text/HTML

## Standards

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 – (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 – y^2)(x^2 + y^2).

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: 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.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Arithmetic with Polynomials and Rational FunctionsCluster: Perform arithmetic operations on polynomials

Standard: 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.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x –1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x^2 + x + 1), and (x – 1)(x^3 + x^2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Arithmetic with Polynomials and Rational FunctionsCluster: Rewrite rational expressions

Standard: 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.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Arithmetic with Polynomials and Rational FunctionsCluster: Rewrite rational expressions

Standard: (+) 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.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Arithmetic with Polynomials and Rational FunctionsCluster: Understand the relationship between zeros and factors of polynomial

Standard: 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.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Interpret parts of an expression, such as terms, factors, and coefficients.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)^n as the product of P and a factor not depending on P.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Interpret expressions that represent a quantity in terms of its context.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Use the properties 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).

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Arithmetic with Polynomials and Rational FunctionsCluster: Use polynomial identities to solve problems

Standard: (+) Know and apply that the Binomial Theorem gives 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. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Arithmetic with Polynomials and Rational FunctionsCluster: Use polynomial identities to solve problems

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

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Arithmetic with Polynomials and Rational FunctionsCluster: Understand the relationship between zeros and factors of polynomial

Standard: 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).

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

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

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

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

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Interpret expressions that represent a quantity in terms of its context.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Use the properties 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).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

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

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

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

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Arithmetic with Polynomials and Rational Functions

Standard: Perform arithmetic operations on polynomials

Indicator: 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Arithmetic with Polynomials and Rational Functions

Standard: Rewrite rational expressions

Indicator: (+) 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Interpret parts of an expression, such as terms, factors, and coefficients.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)^n as the product of P and a factor not depending on P.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x -1) = 3. Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x^2 + x + 1), and (x - 1)(x^3 + x^2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Arithmetic with Polynomials and Rational Functions

Standard: Use polynomial identities to solve problems

Indicator: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: 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.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Arithmetic with Polynomials and Rational Functions

Standard: Use polynomial identities to solve problems

Indicator: (+) Know and apply that the Binomial Theorem gives 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. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Arithmetic with Polynomials and Rational Functions

Standard: Rewrite rational expressions

Indicator: 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Arithmetic with Polynomials and Rational Functions

Standard: Understand the relationship between zeros and factors of polynomial

Indicator: 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Arithmetic with Polynomials and Rational Functions

Standard: Understand the relationship between zeros and factors of polynomial

Indicator: 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).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize"Óto abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents"Óand the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Degree of Alignment: Not Rated (0 users)

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# Tags (8)

- Mathematics
- Algebraic Expressions
- Arithmetic Operations
- CCSS
- Common Core Math
- Common Core PD
- ODE Learning
- Polynomials

## Comments