Moving from Additive to Multiplicative Thinking: The Road to Proportional Reasoning

1. Moving from Additive to Multiplicative Thinking:The Road to Proportional Reasoning MTL Meeting March 16 and 18, 2010 Facilitators Melissa Hedges Kevin McLeod Beth Schefelker Mary Mooney DeAnn Huinker Connie Laughlin The Milwaukee Mathematics Partnership (MMP), an initiative of the Milwaukee Partnership Academy (MPA), is supported with funding by the National Science Foundation.

2. Root Beer or Cola? During dinner at a local restaurant, the five people sitting at Table A and the ten people sitting at Table B ordered the drinks shown below. Later, the waitress was heard referring to one of the groups as the “root beer drinkers.” To which table was she referring? Table A Table B

3. Sharing Your Thinking • Share your answer and thinking with a neighbor. • How are your thoughts alike and how are they different?

4. Two Perspectives on Thinking Absolute Thinking (additive) Comparing the actual number of root beer bottles fromTableA toTable B. • How might an additive thinker answer which is the root beer table? • How might they justify their reasoning? Relative (multiplicative) Comparing amount of root beers to the total amount of beverages for each table. • How might a relative thinker respond to this task? • How might they justify their reasoning? Table A Table B

5. Learning Intention and Success Criteria • We are learning to… develop an awareness of proportional situations in every day life. • By the end of the session you will be able to…recognize the difference between additive thinking (absolute) and multiplicative thinking (relative) in student work.

6. Which family has more girls? The Jones Family (GBGBB) The King Family (GBBG)

7. Thinking about “more” from an absolute and relative perspective After you’ve read turn and talk: • How would an additive thinker interpret “more” in this context? • How would a relative thinker interpret “more” in this context? • In what way will questioning strategies surfacing relative thinking?

8. Surfacing relative (multiplicative) thinking… Keeping the relative amount of boys to girls the same, what would happen if… The Jones Family grew to 50? The King family grew to 40?

9. What happens when… Keeping the ratios of boys to girls the same…. • The Jones Family grew to 100? • The King family grew to 100?

10. M&M’s were featured in the weekly advertisement from two different stores. Greenwall’s Drug: 2 – 16 oz packages of M & M’s $ 3.00. Drekmeier Pharmacy: 3 – 16 oz packages of M & M’s $ 4.00. Which store offered a better deal? Which is a better deal?

11. Proportional Reasoning Proportional reasoning has been referred to as the capstone of the elementary curriculum and the cornerstone of algebra and beyond. It begins with the ability to understand multiplicative relationships, distinguishing them from relationships that are additive. Van de Walle,J.(2009). Elementary and middle school teaching developmentally.Boston, MA:Pearson Education.

12. Proportional Reasoning vs Proportions Proportional reasoning goes well beyond the notion of setting up a proportion to solve a problem—it is a way of reasoning about multiplicative situations. In fact, proportional reasoning, like equivalence is considered a unifying theme in mathematics.

13. Ratio and ProportionWhat’s the difference? Ratio and proportion do not develop in isolation. They are part of an individual’s multiplicative conceptual field, which includes other concepts such as multiplication, division, and rational numbers. Lo, J., & Watanabe, T. (1997). Developing ratio and proportional schemes: A story of a fifth grader. Journal for Research in Mathematics Education, 28, 216-236.

14. What is a ratio? An ordered pair of numbers that express a multiplicative (relative) comparison. Types of ratios Part-to-Part: number of girls to number of boys Part-to-Whole: number of girls to number of children in the family

15. What is a proportion? • A proportion is a statement of equality between two ratios. • Jones Family 2:5 = 20:50 = 40:100 • King Family 2:4 = 20:40 = 50:100 • What do these proportions represent?

16. Big Ideas of Fractions as Ratios • A ratio is a multiplicative comparison of quantities. • Different types of comparisons can be represented as ratios. • Ratios give the relative sizes of the quantities being compared, not necessarily the actual sizes.

17. MPS StudentsBenchmark 3 – Grade 7 CR Item From a shipment of 500 batteries, a sample of 25 was selected at random and tested. If 2 batteries in the sample were found to be defective, how many defective batteries would be expected in the entire shipment?

19. Which Tree Grew More? Before tree A was 8’ tall and tree B was 10’ tall. Now, tree A is 14’ tall and tree B is 16’ tall. Which tree grew more? 8’ 10’ 14’ 16’ A B A B

20. How Do Student’s Make Sense of the Tree Growth? • Discuss the student responses to the tree question. • Which response is correct? • How are they different? • Order the responses according to the sophistication of reasoning? Lamon, S. J. (2005). Teaching fractions and ratios for understanding. New York: Routledge

21. Student Responses

22. Benchmark #3 Grade 8 Selected Response Item #54 Michelle’s softball team had a 15:2 win-to-loss ratio this season. If the team lost 4 games, how many games did they win? A. 15 B. 30 C. 45 D. 60