© 2013 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Tennessee Department of Education High School Mathematics Algebra

© 2013 UNIVERSITY OF PITTSBURGH

Supporting Rigorous Mathematics Teaching and Learning

Tennessee Department of Education

High School Mathematics

Algebra 1

Deepening our Understanding of the CCSS Via

a Constructed Response Assessment


Forms of Assessment

Assessment for LearningAssessment of Learning

Assessment as Learning


Session Goals

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.


Overview of Activities

Participants will:

• analyze Constructed Response (CRAs) in order to determine the way they are assessing the CCSS for Mathematics;

• analyze and discuss the CCSS for Mathematical Content and CCSS for 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.


The Common Core State Standards

The standards consist of:

The CCSS for Mathematical Content

The CCSS for Mathematical Practice


Analyzing a

Constructed Response Assessment


Tennessee Focus Clusters Algebra 1

Create equations that describe numbers or relationships.

Solve equations and inequalities in one variable. Represent and solve equations and inequalities

graphically. Interpret functions that arise in applications in terms

of the context.


Analyzing Assessment Items(Private Think Time)

Four assessment items have been provided:

Buddy Bags

Disc Jockey Decisions

What’s the point?

Paulie’s Pen

For each assessment item:• solve the assessment item; and

• make connections between the standard(s) and the assessment item.


1. Buddy BagsFor a student council fundraiser, Anna and Bobby have spent a total of $55.00 on supplies to create Buddy Bags. They plan to charge $2.00 per Buddy Bag sold.Anna created the graph below from an equation to represent the profit from the number of Buddy Bags sold.

0 10 20 30 40 50 60

-60

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

70

Number of Buddy Bags SoldP

rofi

t in

Do

llar

s

a. Determine the equation Anna used to create the graph if x represents the number of Buddy Bags sold and y represents the profit in dollars. Use mathematical reasoning to explain your equation.

b. Bobby claims that Anna’s graph is incorrect because it does not show that they plan to charge $2.00 per Buddy Bag. Do you agree or disagree with Bobby? Use mathematical reasoning to support your decision.

c. Anna says, “I connected the points to represent the equation, but by connecting the points I am not representing the context of the problem.” Use mathematical reasoning to explain why she is correct.


2. Disc Jockey Decisions

The student council has asked Dion to be the disc jockey for the Fall Banquet. He has been asked to play instrumental music during the first hour while the students are eating dinner. During the last 15 minutes of the banquet the school choir will sing. For the remaining time, Dion will choose popular songs to play.

a. Write an equation to determine the number of popular songs, p, that Dion can choose if the songs Dion chooses have an average run time of 3.5 minutes and the total time for the Fall Banquet is t minutes. Use mathematical reasoning to justify that your equation is correct.

b. Use your equation from Part a to determine the number of popular songs that Dion can choose if the banquet will be held from 6:00 – 10:00pm.

c. Dion decides to organize the music another way. He decides to play 50 popular songs. Write and solve an algebraic equation to determine the average run time, r, of the 50 popular songs Dion can choose if the average run time is represented in minutes by r. Use mathematical reasoning to justify that your equation is correct.


3. What’s the point?

Mr. Williams asks his Algebra 1 class to find the solutions to an equation in two variables with a domain in the set of real numbers.

Colton correctly creates the table below using values from the domain of the equation. He then uses his table to create a graph.

Colton’s Tablex y0 -42 -1.5-2 -6.55 2.25-8 -1410 8.5

a. Determine the equation Colton used to create the table. Use mathematical reasoning to justify that the equation is correct.

b. Destiny sees Colton’s work and argues that any table contains just a subset of the solutions to Mr. Williams’ equation. Do you agree or disagree with Destiny? Explain why or why not.

-10 -5 0 5 10

-15

-10

-5

0

5

10

Colton's Graph


4. Paulie’s Pen Scarlet has to build a rectangular pen for her peacock, Paulie. Scarlet has P feet of fencing.

a. Scarlet decides the length of the pen will be 12 feet. Write an equation describing the relationship between P and the width of the pen, w. Explain your thinking in the context of the problem.

b. Solve the equation for w. Show your work.

c. Scarlet changes her mind and decides that the width of Paulie’s pen will be half of the length of his pen. Write an equation describing the relationship between P and the width of the pen, w. Explain your thinking in the context of the problem.

d. Solve the equation for w. Show your work.


Discussing Content Standards (Small Group Time)

For each assessment item:

With your small group, discuss the connections

between the content standard(s) and the assessment

item.


Deepening Understanding of the Content Standards via the Assessment Items(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?

The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra

Common Core State Standards, 2010

Creating Equations★

(A –CED)Create equations that describe numbers or relationshipsA-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.

A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

★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.



Reasoning with Equations and Inequalities (A-REI)Solve equations and inequalities in one variableA.-REI.B.3 Solve linear equations and inequalities in one variable,

including equations with coefficients represented by letters.

A.-REI.B.4 Solve quadratic equations in one variable.

A.-REI.B.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.

A.-REI.B.4b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.



Reasoning with Equations and Inequalities (A-REI)Represent and solve equations and inequalities graphicallyA.-REI.D.10 Understand that the graph of an equation in two variables is the set of

all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A.-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.★

A.-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

★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.

The CCSS for Mathematical ContentCCSS Conceptual Category – Functions


Interpreting Functions (F-IF)

Interpret functions that arise in applications in terms of the contextF-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.★.

F-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.★

F-IF.B.6 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.★

★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and



Determining the Standards for

Mathematical Practice Associated with

the Constructed Response

Assessment


Getting Familiar with the CCSS for Mathematical Practice(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.

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.

21

Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO


Discussing Practice Standards(Small Group Time)

Each person has 2 minutes to share important

information about his/her assigned Mathematical

Practice.

22


Discussing Practice Standards(Small Group Time)

For each assessment item:

With your small group, discuss the connections

between the practice standards and the assessment

item.


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.



Deepening Understanding of the Practice Standards via the Assessment Items(Whole Group)

Which standards for mathematical practice do you

better understand?

What are you still wondering about?


Assessing Conceptual Understanding


Rationale

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.


Assessing for Conceptual Understanding

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?


Conceptual Understanding

• 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?

Developing Conceptual Understanding

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

The CCSS on Conceptual Understanding

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

Assessing Concept Image

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


Developing and Assessing Understanding

Why is it important, when assessing a student’s

conceptual understanding, to vary items in these

ways?


Using the Assessment to Think About Instruction

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?


Step Back

• 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 the

Standards for Mathematical Content and Standards

for Mathematical Practice?

Functions and Modeling

In modeling situations, knowledge of the context and statistics are sometimes used together to find algebraic expressions that best fit an observed relationship between quantities. Then the algebraic expressions can be used to interpolate (i.e., approximate or predict function values between and among the collected data values) and to extrapolate (i.e., to approximate or predict function values beyond the collected data values). One must always ask whether such approximations are reasonable in the context.

http://commoncoretools.wordpress.com/2012/04/ccss_progression_functions_2012_12 04.bis.pdf, pg. 4

http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_sp_hs_2012_11_121.pdf

1. Buddy Bags


Creating Equations★

(A-CED)Create equations that describe numbers or relationshipsA-CED.A.2 Create equations in two or more variables to represent relationships

between quantities; graph equations on coordinate axes with labels and scales.

Interpreting Functions (F-IF)Interpret functions that arise in applications in terms of the contextF-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.★

F-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. ★

★ 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.

2. Disc Jockey Decisions


Creating Equations★ (A-CED)

Create equations that describe numbers or relationships

A.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.

A.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Reasoning with Equations and Inequalities (A-REI)

Solve equations and inequalities in one variable

A.-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.


3. What’s the point?

Common Core State Standards, NGA Center/CCSSO, 2010

Creating Equations★ (A-CED)Create equations that describe numbers or relationshipsA.CED.A.2 Create equations in two or more variables to represent

relationships between quantities; graph equations on coordinate axes with labels and scales.

Reasoning with Equations and Inequalities (A-REI)

Represent and solve equations and inequalities graphically

A.-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and


4. Paulie’s Pen


Creating Equations★ (A-CED)

Create equations that describe numbers or relationships

A-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.

A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.


Documents

© 2013 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Tennessee Department of Education High School Mathematics Algebra