1. Relative and Absolute Thinking

2. I love chocolate, so I’m going to get a slice of the 6-layer cake! I want the one that has more chocolate flavor, so I’m getting the 3-layer cake! The Chocolatey Cake Debate If you wanted to buy a slice of the cake that had the most chocolate flavor, which slice of cake would you buy?

3. Chocolatey Cake? Work on your own to decide which cake is more chocolatey Put your paper aside when you are done so that we know that you have finished Then work with your group members to decide on an answer Are you tempted to change your answer that you wrote down?

4. Mr. Short and Mr. Tall When Mr. Short is measured in paper clips, he is 6 paper clips tall. When he is measured in buttons, he is 4 buttons tall. Mr. Short has a friend named Mr. Tall. When Mr. Tall is measured in buttons, he is 6 buttons tall. How many paper clips tall is Mr. Tall?

5. How many paper clips? Work on your own to decide the height of Mr. Tall in paper clips Put your paper aside when you are done so that we know that you have finished Then work with your group members to decide on an answer Are you tempted to change your answer that you wrote down?

6. Student Work

7. Proportional Reasoning Proportional Thinkers understand that (Van de Walle): –There is a clear difference between proportional relationships and non- proportional relationships, especially in the real world –There are a variety of strategies for solving proportions or comparing ratios (that are not prescribed algorithms) –There are relationships where 2 quantities vary together (covariation)

8. Proportional Reasoning It is important to develop proportional reasoning both in ourselves and our students Develop slowly over middle school years, not just a couple weeks in 6 th grade Common Core addresses this

9. California Common Core

10. 6.RP3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

11. California Common Core

12. 7.RP2 and 7.RP3 Recognize and represent proportional relationships between quantities. –Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. Use proportional relationships to solve multistep ratio and percent problems.

13. California Common Core

14. 8.EE5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

15. Proportional Reasoning Activities for developing proportional reasoning (Van de Walle): –Identifying multiplicative situations –Equivalent-Ratio –Comparing Ratios –Scaling with Ratio Tables –Construction and Measurement What type of activities are “Chocolatey Cake” and “Mr. Short and Mr. Tall”?

16. Relative and Absolute Thinking

20. What are you doing or can you do to distinguish the difference?