Tennessee Department of Education
High School Mathematics
Algebra 2
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Assessment as Learning
Assessment of Learning Assessment for Learning
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Participants will:
• deepen understanding of the Common Core State
Standards (CCSS) for Mathematical Practice and
Mathematical Content;
• understand how Constructed Response
Assessments (CRAs) assess the CCSS for both
Mathematical Content and Practice; and
• understand the ways in which CRAs assess students’ conceptual understanding.
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Participants will:
• analyze Constructed Response Assessments
(CRAs) in order to determine the way they are assessing the CCSSM;
• analyze and discuss the CCSS for Mathematical
Content and Mathematical Practice;
• discuss what it means to develop and assess conceptual understanding; and
• discuss the CCSS related to the tasks and the implications for instruction and learning.
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The standards consist of:
 The CCSS for Mathematical Content
 The CCSS for Mathematical Practice
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Algebra 2
 Extend the properties of exponents to rational exponents.
 Write expressions in equivalent forms to solve problems.
 Understand the relationship between zeros and factors of polynomials.
 Build a function that models a relationship between two quantities.
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7

(Private Think Time)
Four assessment items have been provided:

Car Depreciation

Writing a Polynomial

Patterns in Patterns

One Rocket, Three Equations
For each assessment item:
• solve the assessment item; and
• make connections between the standard(s) and the assessment item.
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After you purchase a new car, it begins to lose value. As the years pass, more and more value is lost. This process is known as depreciation. a.
Carmen buys a new car for $24,500. Carmen’s new car loses 14% of its value each year through depreciation.
Write a function that can be used to model the value of the car at the end of each year that Carmen owns the car.
Justify your equation mathematically.
b. After Carmen buys the car, he adds an audio and speaker system worth $500. The audio system loses 15% of its value the minute it is installed into the car, and 7% of the remaining value each year through depreciation. Write a function that can be used to model the value of the car with the audio system and speakers at the end of each year that Carmen owns the car.
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Recall that polynomial functions with only real number zeros can be written in factored form as follows: y
 a

 x
 z
1
  x
 z
2
 p
2

...

 x
 z n
 p n where each z n represents some real root of the function, and each is a whole number exponent greater than or equal to 1.
p n
Consider the graph of the polynomial function below.
a. Lisa claims that, since the point (0, 6) is on the graph, ( x – 6) is a factor of this polynomial. Explain why you agree or disagree with Lisa’s claim.
Identify all the zeroes of the function and use that information in your explanation.
1 b. Suppose a =
12
. Write a function in factored form to represent this graph.
Justify your equation mathematically.
4
3
6
5
2
1
0
-5 -4 -3 -2 -1 -1 0
-2
14
13
12
11
10
9
8
7
1 2 3 4
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5

Laura creates a design of circles embedded in each other for a poster. The largest circle has a diameter of
28 inches, and each successive circle has a diameter of
3 the previous circle.
4 a. Write a function that can be used to determine the diameter of any circle drawn in the poster in this way. Explain the meaning of each term in your expression in the context of the problem. b. Laura eventually draws 10 circles. Write and use a formula for the sum of a series to find the sum of the circumferences of the 10 circles, accurate to two decimal places. Show your work.
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28 inches

For his science project, Grady designs and launches a model rocket from the rooftop of an empty building. He places a sensor on the rocket to provide him with data about the height of the rocket (from ground level in feet) over time (in seconds) during the flight.
From the data returned by the sensor, Grady was able to write and graph a function modeling the height of the rocket over time.
𝐻 𝑡 = −16𝑡 2 + 96𝑡 + 112 a. Rewrite the function in vertex form. Then describe what this form of the function reveals in the context of the situation. b. Rewrite the function by factoring completely. Then describe what this form of the function reveals in the context of the situation.
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(Small Group Time)
For each assessment item:
With your small group, discuss the connections between the content standard(s) and the assessment item.
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(Whole Group)
As a result of looking at the assessment items, what do you better understand about the specifics of the content standards?
What are you still wondering about?
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CCSS Conceptual Category – Number and Quantity
The Real Number System (N-RN)
Extend the properties of exponents to rational exponents.
N-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3
= 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal 5.
N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Common Core State Standards, 2010

CCSS Conceptual Category – Algebra
Seeing Structure in Expressions (A-SSE)
Write expressions in equivalent forms to solve problems
A-SSE.B.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by an expression.
★
A-SSE.B.3c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15
t can be rewritten as (1.15
1/12 ) 12t ≈
1.012
12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
A-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.
★
★
Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (
★
). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010

CCSS Conceptual Category – Algebra
Arithmetic with Polynomials and Rational Expressions (A-APR)
Understand the relationship between zeros and factors of polynomials
A-APR.B.2
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 ).
A-APR.B.3
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.
Common Core State Standards, 2010

CCSS Conceptual Category – Functions
Building Functions (F-BF)
Build a function that models a relationship between two quantities
F-BF.A.1
Write a function that describes a relationship between two quantities.
★
F-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context.
F-BF.A.1b 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.
F-BF.A.2
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
★
★
Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (
★
). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010

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(Private Think Time)
Count off by 8. Each person reads one of the CCSS for Mathematical Practice.
Read your assigned Mathematical Practice. Be prepared to share the “gist” of the Mathematical
Practice.
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20

1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO
21

(Small Group Time)
Each person has 2 minutes to share important information about his/her assigned Mathematical
Practice.
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22

(Small Group Time)
For each assessment item:
With your small group, discuss the connections between the practice standards and the assessment item.
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1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Common Core State Standards, 2010

(Whole Group)
Which Standards for Mathematical Practice do you better understand?
What are you still wondering about?
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We have now examined assessment items and discussed their connection to the CCSS for
Mathematical Content and Practice. A question that needs considering, however, is if and how these assessments will give us a good means of measuring the conceptual understandings our students have acquired.
In this activity, you will have an opportunity to consider what it means to develop conceptual understanding, as described in the CCSS for
Mathematics, and what it takes to assess for it.
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The set of CRA items are designed to assess student understanding of expressions and equations.
Look across the set of related items. What might a teacher learn about a student’s understanding by looking at the student’s performance across the set of items as a whole?
What is varying from one item to the next?
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•
What do the authors mean by conceptual understanding?
•
How might analyzing student performance on this set of assessments help us determine if students have a deep understanding of the assessed standards?
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Knowledge that has been learned with understanding provides the basis of generating new knowledge and for solving new and unfamiliar problems. When students have acquired conceptual understanding in an area of mathematics, they see connections among concepts and procedures and can give arguments to explain why some facts are consequences of others.
They gain confidence, which then provides a base from which they can move to another level of understanding.
Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics.
Washington, DC: National Academy Press

In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical
Content and the Standards for Mathematical Practice.
These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.
Common Core State Standards for Mathematics, 2010, p. 8, NGA Center/CCSSO

Tall (1992) differentiates between the mathematical definition of a concept and the concept image, which is the entire cognitive structure that a person has formed related to the concept. This concept image is made up of pictures, examples and non-examples, processes, and properties.
A strong concept image is a rich, integrated, mental representation that allows the student to flexibly move between multiple formulations and representations of an idea. A student who has connected mathematical ideas in this way can create and use a model to analyze a situation, uncover patterns and synthesize them to form an integrated picture. They can also use symbols meaningfully to describe generalizations which then provides a base from which they can move to another level of understanding.
Brown, Seidelmann, & Zimmermann. In the trenches: Three teachers’ perspectives on moving beyond the math wars.
http://mathematicallysane.com/analysis/trenches.asp

Why is it important, when assessing a student’s conceptual understanding, to vary items in these ways?
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In order for students to perform well on the CRA, what are the implications for instruction?
• What kinds of instructional tasks will need to be used in the classroom?
• What will teaching and learning look like and sound like in the classroom?
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• What have you learned about the CCSS for
Mathematical Content that surprised you?
• What is the difference between the CCSS for
Mathematical Content and the CCSS for
Mathematical Practice?
• Why do we say that students must work on both
Standards for Mathematical Content and the
Standards for Mathematical Practice?
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A container of ice cream is taken from the freezer and sits in a room for t minutes. Its temperature in degrees Fahrenheit is 𝑎 − 𝑏 2 −𝑡 + 𝑏 , where 𝑎 and 𝑏 are positive constants. Write this expression in a form that shows that the temperature is always
1. Less than 𝑎 + 𝑏
2. Greater than 𝑎
A.SSE.3
http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf
, pg. 5

Susan is told to take 250 mg doses of an antibiotic for an ear infection regularly every 12 hours for 10 days. Susan does some research online and finds out that at the end of 12 hours, about 4% of the drug is still in the body.
1. What quantity of the drug is in the body right after the first dose, the second dose, the third dose?
2. When will the total amount of the antibiotic in Susan’s body be the highest? What is that amount?
A.SSE.B.4
http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf
, pg. 5

1. What quantity of the drug is in the body right after the first dose, the second dose, the third dose?
Let 𝑄
1 be the amount of the drug in the body after the first dose
𝑄
1
= 250
To find 𝑄
2
, the amount of the drug in the body after the second dose, we have to find out how much of the first dose is still present and add the new dose. Since after 12 hours about 4% of the drug is still present in the body we get:
𝑄
2
= 0.04𝑄
1
+ 250 = 0.04 250 + 250 = 260
We use the same reasoning to find 𝑄
3
𝑄
3
= 0.04𝑄
2
:
+ 250 = 0.04 0.04 250 + 250 + 250 = 260 .4
Or 𝑄
3
= 0.04
2 𝑄
1
+ 0.04𝑄
1
+ 𝑄
1 http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf
, pg. 5

2. When will the total amount of the antibiotic in Susan’s body be the highest? What is that amount?
Since the amount of antibiotic in Susan’s body increases every time she takes a dose, we need to find the amount in her body after the last (20 th ) dose:
𝑄
20
= 0.04
19
𝑄
1
+ 0.04
18
𝑄
1
+ ⋯ + 0.04𝑄
1
+ 𝑄
1
𝑄
20
= 𝑄
1
0.04
19 + 0.04
18 + ⋯ + 0.04 + 1
𝑄
20
=
𝑄
1
1−0.04
20
1−0.04
or 𝑄
20
=
𝑄
1
−𝑄
1
0.04
1−0.04
20
Therefore, the amount of antibiotic in Susan’s system after 20 does of
250 mg is:
𝑄
20
=
250−250 0.04
1−0.04
20
= 260.4167 mg ≈ 260.4 mg
, pg. 5

Building Functions (F-BF)
Build a function that models a relationship between two quantities
F-BF.A.1
Write a function that describes a relationship between two quantities.
★
F-BF.A.1b 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.
★
Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (
★
). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010

Arithmetic with Polynomials and Rational Expressions (A-APR)
Understand the relationship between zeros and factors of polynomials
A-APR.B.3 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.
Building Functions (F-BF)
Build a function that models a relationship between two quantities
F-BF.A.1
Write a function that describes a relationship between two quantities.
★
★
Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (
★
). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010

Building Functions (F-BF)
Build a function that models a relationship between two quantities
F-BF.A.2
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
Seeing Structure in Expressions A-SSE
Write expressions in equivalent forms to solve problems
A.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.
★
★
Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (
★
). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010

Seeing Structure in Expressions A-SSE
Write expressions in equivalent forms to solve problems
A.SSE.B.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by an expression.
★
★
Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (
★
). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.
Common Core State Standards, 2010