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#### Presentation Transcript

**1. **Mathematical Models Tools for Re-Teaching Logan Toone
Davis School District Curriculum Department

**2. **Re-Teaching What do we do when a kid doesn’t understand …

**3. **Re-Teaching

**4. **Re-Teaching Effective re-teaching is not merely “teaching again.”

**5. **One Way to Teach Differently
Mathematical Models

**6. **What is a Mathematical Model? Could mean one of 2 things.
A model for a real-life situation using mathematical language
Example: population growth modeled by exponential functions

**7. **What is a Mathematical Model?
Not all real-life situations can be modeled mathematically…

**8. **What is a Mathematical Model? The other definition of mathematical model
A concrete representation used to give meaning and clarity to a mathematical concept.
We’ll look at two examples of models.
A model for multiplying integers
The bar model for problem solving

**9. **Multiplying Integers: A Model for Teaching Developed by
Logan Toone
and
Patty Norman

**10. **Common Models for Multiplication Algebraic Model - Multiplication as repeated addition of sets:
Example

**11. **Common Models for Multiplication Geometric Model - Multiplication as the area of a rectangle:
Example:

**12. **A Comprehensive Model ? These two models for multiplication work primarily with whole numbers, which is great …
Is there a model that includes negative integers as well?

**13. **A Model for All Integers We can map the possible combinations of positive and negative integers on a quadrant grid.

**14. **A Model for All Integers We continue the model of multiplication as repeated addition of sets.

**15. **A Model for All Integers

**16. **In Other Words …

**17. **Convention for this Model The first number tells us how many sets.
The second number tells us how many elements in each set.
The commutative property would also allow us to work with five sets of three, but we chose this convention for the model.

**18. **Quadrant 1 Positive x Positive

**19. **Quadrant 2 Negative x Positive

**20. **Quadrant 3 Negative x Negative

**21. **Quadrant 4 Positive x Negative

**22. ** Summary

**23. **Bar Models: A Model for Teaching Logan Toone
Davis School District Curriculum Department
Examples adapted from:
Bisk, R. (2008). Problem Solving with Model Drawing. wwwfac.wrocester.edu/smip

**24. **What is a Bar Model? In a bar model, the bars are representative of numbers (known or unknown) with the length of the bar corresponding to the magnitude of the number.

**25. **What is a Bar Model? The bars can be divided into units which may be known or unknown.

**26. **Simple Application A worker earned three times as much this week as she did last week. If she earned a total of $200 in the two weeks, how much more did she earn this week than last week?

**27. **Simple Application The sum of two numbers is 30. The smaller number is two-thirds of the larger number. What are the two numbers?

**28. **Simple Application One of your stock certificates lost 2/5 of its total value in the last two weeks. If it originally was worth $60.00, how much is it worth now?

**29. **More Challenging Application A box of Cheez-Its was divided among Paul, Rita, and Anna in the ratio of 3:5:7. If Rita got 40 Cheez-Its, how many more Cheez-Its did Anna get than Paul?

**30. **More Challenging Application In a stadium, one fourth of the seats are in the top tier. One third of the seats are in the center tier, and there are 11,250 seats on the floor. How many total seats are there in the stadium

**31. **More Challenging Application You spent 60% of your paycheck during the first week of the month. You spent 1/3 of the remainder during the second week. If you spent $220.00 in the two weeks, how much of the paycheck was left.

**32. **Reminder As with all other models, bar models are not intended to be a comprehensive strategy for problem solving. (They don’t work smoothly for all problems).
Rather, they are a way of helping students visualize what is happening so they can begin to move from concrete ideas to abstract thought and mathematical representation.