the national council of supervisors of mathematics

The Common Core State Standards Illustrating the Standards for

Mathematical Practice:

Model with Mathematics

www.mathedleadership.org

The National Council of Supervisors of Mathematics

National Council of Supervisors of MathematicsIllustrating the Standards for Mathematical Practice: Modeling with Mathematics

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Common Core State Standards

Mathematics

• Standards for Content

• Standards for Practice


Today’s Goals

• To explore the mathematical standards for Content and Practice

• To consider how the Common Core State Standards (CCSS) are likely to impact your mathematics program and plan next steps

In particular, participants will• Examine opportunities to develop skill in

modeling with mathematics


Standards for Mathematical Practice

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.


Standards for Mathematical Practice

“The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important ‘processes and proficiencies’ with longstanding importance in mathematics education.” (CCSS, 2010)

National Council of Supervisors of Mathematics

Illustrating the Standards for Mathematical Practice: Modeling with Mathematics

Today’s Session

• Part 1—An example of modeling with mathematics: Lunch Money

• Part 2—What does modeling with mathematics mean? What content standards lend themselves to work on modeling?

• Part 3—Investigating a relationship between two variables: The Penny Jar

• Part 4—The beginnings of modeling: Connection between MP#4 and MP#2


Part 1: An example of modeling with mathematics: Lunch Money

Lunches in our school cost $2 each. How much do 2 lunches cost? 3 lunches, 4 lunches . . . 10 lunches? More lunches?

Create at least 2 models of this situation. You can choose a physical model, a table, a graph, and/or an equation. Your model should show number of lunches and cost of the lunches. You should be able to use your model to find the cost of a certain number of lunches.


Lunch Money: student work

One lunch costs $2.

Number of lunches Number of dollars

1 lunch 2 dollars

2 lunches 4 dollars

3 lunches

4 lunches

5 lunches


Lunch Money: student work

Individually, read through the transcript beginning with line 66, then consider the following questions:

•How does each student model show the number of lunches and the cost of that number of lunches?

•If you only had time to introduce two of these representations to the class for the whole-group discussion, which two would you choose and why?


Part 2: What does modeling with mathematics mean? What content standards lend themselves

to work on modeling?

Read the description of modeling on the handout. Discuss with a few others: •What are the essential elements of modeling at grades K-5? •How did you see modeling occur in the work of the first and second graders in the Lunch Money problem?•What are your questions about this?


Elements of K-5 Modeling with Mathematics

• Express a situation using mathematical representations such as physical objects, diagrams, graphs, tables, number lines, or symbols

• Operate within the mathematical context to solve the problem

• Use the solution to answer the original question; interpret this result in the context of the situation

• Improve the model if needed


What content standards lend themselves to work on modeling?

• Foundations of multiplication: Equal groups (grade 2)

• Understanding multiplication; sorting out multiplicative from additive situations (grades 3-4)

• Using equations to represent situations, including use of a letter to stand for the unknown (grades 3-4)

• Generate and analyze patterns that have additive and multiplicative elements (grade 4)

• Graphing on a coordinate plane, including comparing graphs of related situations (grade 5)


Part 3: Investigating a relationship between two variables—the Penny Jar

• Contexts in which students think through the relationship between two varying quantities are good contexts for modeling

• A relationship between two variables lends itself to modeling with physical objects, tables, graphs, numbers and equations

• Modeling contexts that relate two variables can start simply and be extended to provide more challenge


The Basic Penny Jar Situation

There is some number of pennies in a jar.

The same number of pennies is added to the jar each day (or “each round”).

Example: There are 8 pennies in the penny jar. Anna adds 4 pennies to the jar each day.


Your Task

Situation: There are 8 pennies in the penny jar. Anna adds 2 pennies to the jar each day. How many pennies are in the jar after 1 day, 2 days, 3 days, . . . up to10 days?

Model this situation in the following ways:•With a table•With a graph•With an equation

Write a statement in words about the relationship between the number of days and the number of pennies in the jar.


Discussion

• What can you see in the table about the relationship between the number of days and total number of pennies?

• What can you see in the graph?

• How are the table and graph related to each other?


Discussion

• What general statements did you write to describe the relationship between the number of days and total number of pennies?

• Is there an equation that describes this relationship?


Video 1: Penny Jar - Grade 4

Start with 3 pennies in the Penny Jar.Add 4 each round.

Take notes as you watch: 1. How are they using a representation to

model the Penny Jar situation? 2. What are the different ways they use

equations to model the Penny Jar situation?


Video 1 – Part 1Penny Jar, Grade 4


Video 1 – Part 2Penny Jar, Grade 4


Extending the Penny Jar Situation: Comparing two penny jars

Penny Jar A:Start with 8 penniesAdd 2 pennies each round

Penny Jar B:Start with 0 penniesAdd 4 pennies each round

Compare the two jars. Which jar starts with more pennies? Does the other jar ever catch up or not? Why? What will happen if you continue to extend the table or graph? What does the table make visible? What does the graph make visible?


Grade 4 Discussion: Comparing Penny Jars

• What aspects of the situation do students notice by modeling with a table?

• What aspects of the situation do students notice by modeling with a graph?

• How do aspects of their model help them understand the situation they are modeling?

• What are the connections to the content standards listed earlier?


A grade 1 Discussion: Comparing a Penny Jar situation with a Staircase Tower situation

Penny Jar: Start with 1 penny in the jar. Add 3 pennies each day.

Staircase Tower: Start with a tower of 1. Add 3 cubes for each new tower.


Elements of Modeling with Mathematics at K-5

• Express a situation using mathematical representations such as physical objects, diagrams, graphs, tables, number lines, or symbols

• Operate within the mathematical context to solve the problem

• Use the solution to answer the original question; interpret this result in the context of the situation

• Improve the model if needed


Part 4: The beginnings of modeling — Connection between MP #4 and MP #2

• SMP #2: Reason abstractly and quantitatively:

Abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own

• SMP #4: Model with mathematics:

Apply the mathematics they know to solve problems . . . in early grades, this might be as simple as writing an addition equation to describe a situation


Video 2Grade 1: How many wheels on 3 cars?

• Take notes on how these students are abstracting aspects of the situation using numbers and expressions—what are the different ways they do this?


Video 2 – Part 1How many wheels on 3 cars?


Video 2 – Part 2How many wheels on 3 cars?


Video 3 - Writing an Equation


Reflecting on the session

• Explore the Standards for Content and Practice.

• To consider how the Common Core State Standards (CCSS) are likely to impact your mathematics program and plan next steps

In particular, participants will• Examine opportunities to develop skill in

modeling with mathematics


What content standards lend themselves to work on modeling?

• Foundations of multiplication: Equal groups (grade 2)

• Understanding multiplication; sorting out multiplicative from additive situations (grades 3-4)

• Using equations to represent situations, including use of a letter to stand for the unknown (grades 3-4)

• Generate and analyze patterns that have additive and multiplicative elements (grade 4)

• Graphing on a coordinate plane, including comparing graphs of related situations (grade 5)


Modeling with Mathematics

1. Individually review the Standard for Mathematical Practice #4, Modeling with mathematics

2. Choose a partner at your table and discuss a new insight you had into this practice, then discuss the following question:

What implications might this standard of mathematical practice have for your classroom?



End of Day Reflections

1. Are there any aspects of your own thinking and/or practice that our work today has caused you to consider or reconsider? Explain.

2. Are there any aspects of your students’ mathematical learning that our work today has caused you to consider or reconsider? Explain.



About This Module

• Developed by Deborah Schifter and Susan Jo Russell, based on work from the projects Developing Mathematical Ideas (DMI) and Investigations in Number, Data, and Space. Video clips, student work, and student dialogue are used with permission.

• Further cases of modeling can be found in the professional development module, Patterns, Functions, and Change (DMI series, Pearson, 2008).

• This work was funded in part by the National Science Foundation through grants ESI-9254393, ESI-9731064, and ESI-0242609 to EDC and ESI-0095450 to TERC. Any opinions, findings, conclusions, or recommendations expressed here are those of the authors and do not necessarily reflect the views of the National Science Foundation.


http://www.noycefdn.org/National Council of Supervisors of MathematicsIllustrating the Standards for Mathematical Practice: Modeling with Mathematics

Join us in thanking the

Noyce Foundation

for their generous grant to NCSM that made this series possible!

Project Contributors

• Geraldine Devine, Oakland Schools, Waterford, MI• Aimee L. Evans, Arch Ford ESC, Plumerville, AR• David Foster, Silicon Valley Mathematics Initiative, San José

State University, San José, California• Dana L. Gosen, Ph.D., Oakland Schools, Waterford, MI• Linda K. Griffith, Ph.D., University of Central Arkansas• Cynthia A. Miller, Ph.D., Arkansas State University• Valerie L. Mills, Oakland Schools, Waterford, MI• Susan Jo Russell, Ed.D., TERC, Cambridge, MA• Deborah Schifter, Ph.D., Education Development Center,

Waltham, MA• Nanette Seago, WestEd, San Francisco, California
