Eleventh grade Lesson Unit Assessment: Polynomial Functions and Expressions

Objective

Unit Assessment: Polynomial Functions and Expressions

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Objective

Students will demonstrate facility with polynomial arithmetic and classification and mastery of all aspects of quadratic functions, expressions and equations.

Big Idea

Show that you are able to rewrite and classify polynomial expressions, use quadratic equations to solve problems, and express quadratic functions in various forms.

Lesson Author

Colleen Werner

Ipswich, MA

Grade Level

Eleventh grade

Subjects

Math

quadratic factorization

quadratic formula

quadratic models

quadratic functions

maximum points

minimum points

Standards

HSA-SSE.A.1a

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

HSA-SSE.B.3

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

HSA-SSE.B.3a

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

HSA-SSE.B.3b

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

HSA-SSE.B.4

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.★

HSA-APR.A.1

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.

HSA-CED.A.1

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.

HSA-CED.A.2

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

HSA-REI.B.4

Solve quadratic equations in one variable.

HSA-REI.D.11

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.*

HSF-IF.B.4

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.*

HSF-IF.B.5

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.*

HSF-IF.C.7a

Graph linear and quadratic functions and show intercepts, maxima, and minima.

HSF-IF.C.8a

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.

HSF-IF.C.9

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.

MP1

Make sense of problems and persevere in solving them.

MP2

Reason abstractly and quantitatively.

MP3

Construct viable arguments and critique the reasoning of others.

MP4

Model with mathematics.

MP5

Use appropriate tools strategically.

MP6

Attend to precision.

MP7

Look for and make use of structure.

Unit Assessment: Polynomial Functions and Expressions

90 minutes

How I Create Unit Assessments

Students will use this period to complete Test Polynomial Functions and Expressions. This assessment has a "calculator page" and a "non-calculator page" because I want to make sure that students know how to factor, graph and solve without the relying on the CAS (computer algebra system) functionality of our calculators. As a math department, we have decided on a list of skills that students should be able to do without a graphing calculator so I am careful to assess these skills throughout the year.

I give students both pages of the test at the beginning of the period and instruct them not to put their calculators under their chairs. They may not use a calculator on the exam until they have delivered the non-calculator page to my desk. This allows them to budget their time, as they can see the entire test at once. Because the non-calculator page has a highly visible symbol at the top, it is easy for me to scan the room to make sure no student is using a calculator inappropriately.

This test is designed to assess the Standards for Mathematical Practice as well as content standards. I discuss this in the video, Assessing Math Practice Standards with Quadratics.

The test assesses the math practice standards in the following way:

MP1 - students must look for starting points, monitor their own progress and keep track of time remaining. They are checking their solutions and considering whether their answers are reasonable.

MP2 - the application problems require that students decontextualize in order to represent the scenario with a function and use that function to develop a solution

MP3 - the initial two questions are writing prompts that require students to explain their conceptual understanding of this unit

MP4 - the application problems demand that students model financial and physics scenarios with quadratic functions

MP5 - students decide how much of the test to complete before surrendering the non-calculator page. Forcing them to make this decision help students consider the best use of the graphing calculator

MP6 - The stems of the question contain content-specific vocabulary that students must make sense of and the conceptual questions require precise use of vocabulary

MP7 - Much of this exam assesses students ability to make use of the structure of an expression in order to rewrite an expression (factoring, completing the square), solve an equation, or graph a function.

MP8 - This assessment does not really address this standard because I feel that MP8 is not best addressed with tight time contraints.

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