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Problem Solving and Modeling. Day 5: 6-8 Algebra MSTA Region 11 Teacher Center. Problem Solving & Modeling. Goals Teachers helping students develop generalization abilities – fundamental to algebraic thinking – and mathematical connections through representation and translation.

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problem solving and modeling

Problem Solving and Modeling

Day 5: 6-8 Algebra

MSTA Region 11 Teacher Center

problem solving modeling

Problem Solving & Modeling


Teachers helping students develop generalization abilities – fundamental to algebraic thinking – and mathematical connections through representation and translation.

Teachers learn how to unpack student solutions to problem solving problems by orchestrating meaningful discussions.

Teachers helping students elicit their mathematical thinking through modeling.

lesh translation model
Lesh Translation Model

Lesh & Doerr (2003)

gear ratios

Gear Ratios

Why are gears useful? Where are they used? For what purpose?

What is the gear ratio between the yellow and red gears?

Why might it be useful to string several gears together?

gear ratio investigations
Gear Ratio Investigations

String all 4 gears together.

  • What is the gear ratio between the first gear and last gear on the train?
  • If the yellow gear rotates at 6 revolutions per minute, how many revolutions per minute does the red gear rotate?
engineering challenge modeling
Engineering Challenge - Modeling

Engineers working on wind turbines must decide what gears to use in their wind turbine. They know that the blades turn at one speed and the generator needs to turn at a different speed. What they don’t know is how to figure out how fast gears rotate when linked based on their size.

engineering challenge
Engineering challenge

Please provide a general procedure for the engineers to use that will help them with their problem. Use the question below as a prompt.

  • If you have any two gears and the primary gear is rotating at x rpm, how fast does the secondary gear rotate?

Be prepared to present your solutions to the engineers.

representations in the gears
Representations in the Gears
  • What representations did you use?
  • What other representations can you imagine students using in this problem?
  • What can you do to promote representations and translations in this problem?
extensions gears on a bike
Extensions – gears on a bike
  • On a multiple gear bike, how do you order the gears from easiest to pedal to hardest?
  • How does this extend the concept of gear ratio?
  • What representations do you expect to see from students if you asked a question like this? How can you encourage translations?
lunch and planning time
Lunch and Planning Time
  • Eat lunch with the teachers from your school.
    • We will use this time to plan for next year on how to implement all of this year’s academy training into your math program.
    • Use your curricular materials in this planning time.
    • You have until ___(time)____ to begin planning for next year and eat lunch!
five practices for facilitating mathematical discussions
Five Practices for Facilitating Mathematical Discussions






(Stein, et al., 2008)

caterpillar problem anticipating
Caterpillar Problem - Anticipating

A sixth-grade class needs 5 leaves each day to feed its 2 caterpillars. How many leaves would they need each day for 12 caterpillars? (NAEP, 1996)


Use drawings, words, or numbers to show how you got your answer.

caterpillar problem monitoring
Caterpillar Problem – Monitoring

You’ve decided to implement the problem

  • Look at the range of solutions in your class.
    • What strategies did each student use to solve the problem?
    • What are the connections between those strategies?
  • How might you be able to assess this information while the students are working on this problem? (individual/group)
caterpillar problem selecting and sequencing
Caterpillar Problem – Selecting and Sequencing
  • Choose the solutions that you want students to present from your class and the order for presentation.
  • Create a strategy for how you will implement the student presentations?
    • Why did you choose those students to present?
    • Why did you choose that order?
    • Why did you choose that strategy for implementation?
caterpillar problem connecting
Caterpillar Problem – Connecting
  • Compare and contrast Martin’s picture to Jamal’s table.
  • Compare and contrast Jamal and Janine’s responses.
  • What other connections should we make and for what purpose?
complex modeling activities
Complex Modeling Activities
  • We are going to participate in a modeling activity called “Historic Hotels”
  • There are multiple solution paths to this problem, and the answer will not be easily apparent at first. You’ll have to work at it.
    • Teachers: Solve this problem as if you have the mathematical background of your students.
historic hotels
Historic Hotels


  • Read the newspaper article and answer the readiness questions.

In your teams:

  • Read the problem statement and answer the team questions together.
  • Create a procedure for Mr. Graham
  • Be prepared to share your solutions in a 2 minute presentation.
representations revisited
Representations Revisited
  • In your groups, jot down your ideas about what representations were elicited in this problem.
  • How can you as a teacher foster multiple representations on this problem? Translations between representations?
  • What are the “big” mathematical ideas that are elicited in this problem?

More model-eliciting activities can be found at:

plan for discussion
Plan for discussion
  • Read over the teacher materials that come with this problem.
  • Make an implementation plan with your team. This plan should include all parts of the LESA model.
    • Launch, Explore, Share/Summarize, Application
  • Include ideas for using the 5 practices for facilitating discussions.
  • Include ideas for extensions to this problem.
review of the year
Review of the Year
  • Day 0 – Equality
  • Day 1 – Ratio and Proportion
  • Day 2 – Pattern Generalization
  • Day 3 – Equations and Inequalities
  • Day 4 – Function and Representation
  • Day 5 – Problem Solving and Modeling
  • Whew!!!