## Description

- Overview:
- This lesson unit is intended to help teachers assess how well students are able to translate between graphs and algebraic representations of polynomials. In particular, this unit aims to help you identify and assist students who have difficulties in: recognizing the connection between the zeros of polynomials when suitable factorizations are available, and graphs of the functions defined by polynomials; and recognizing the connection between transformations of the graphs and transformations of the functions obtained by replacing f(x) by f(x + k), f(x) + k, -f(x), f(-x).

- Subject:
- Mathematics
- 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
- 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

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).

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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: 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: Functions: Interpreting Functions

Standard: Understand the concept of a function and use function notation.

Indicator: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Understand the concept of a function and use function notation.

Indicator: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ‰ä´ 1 (n is greater than or equal to 1).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: (+) Verify by composition that one function is the inverse of another.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Understand the concept of a function and use function notation.

Indicator: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*

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: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: Write a function that describes a relationship between two quantities.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph linear and quadratic functions and show intercepts, maxima, and minima.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.*

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: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2(x^3) or f(x) = (x+1)/(x-1) for x ‰äĘ 1 (x not equal to 1).

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: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: Find inverse functions.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: (+) Produce an invertible function from a non-invertible function by restricting the domain.

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: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Interpret functions that arise in applications in terms of the context

Indicator: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Interpret functions that arise in applications in terms of the context

Indicator: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Interpret functions that arise in applications in terms of the context

Indicator: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^(12t), y = (1.2)^(t/10), and classify them as representing exponential growth and decay.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: Determine an explicit expression, a recursive process, or steps for calculation from a context.

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: 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.

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