1. Proportional Reasoning: Looking At Student Work Learning About Student Thinking Identifying Next Steps MTL Meeting May 18 and 20, 2010 Facilitators Melissa HedgesKevin McLeod Beth SchefelkerMary 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. Session Goals We Are Learning To…build Math Knowledge of Teaching (MKT) about proportional reasoning. We will be successful when we can look at student work and identify student strengths, challenges, and misconceptions.

3. Thinking Back… As an adult learner of mathematics what connections have you made as we’ve explored proportional reasoning over the past 3 months?

4. A Definition of Proportionality When two quantities are related proportionally, the ratio of one quantity to the other is invariant, or the numerical values of both quantities change by the same factor. Developing Essential Understandings of Ratios, Proportions & Proportional Reasoning, Grades 6-8. National Council of Teachers of Mathematics, 2010, pg. 11.

5. Essential Understandings A ratio is a multiplicative comparison of two quantities, or it is a joining of two quantities in a composed unit. Possible ages of fathers and sons with a 5:1 ratio. Miles to kilometers (canal problem)

6. Essential Understanding A ratio is a multiplicative comparison of two quantities. Absolute Thinking (Additive) Relative Thinking (Multiplicative)

7. Essential Understanding A number of mathematical connections link ratios and fractions. Ratios are often expressed in fraction notation, although ratios and fractions do not have identical meaning. Ratios are often used to make “part-part” comparisons, but fractions are not. 30% concentration of orange juice in water 3:7 concentrate to water

8. Essential Understanding A proportion is a relationship of equality between two ratios. 3 girls to 4 boys is the same ratio as 6 girls to 8 boys

9. What should student work look like? Turn and talk: In order to know if students understand proportions, what traits would you want to see demonstrated on their work? Share out ideas with the whole group.

10. Bonita’s Faucet Bonita had a leaky faucet, which dripped 6 ounces of water in 8 minutes. She wanted to find out how much water dripped in 4 minutes. Show two different ways that Bonita could have used to find the solution. Show how Bonita could find how much water dripped in 40 minutes.

11. Looking At Student Work Look at the 5 pieces of student work. What relationships did students find? How did the relationships support proportional thinking? What happened when students were asked for a second strategy?

12. ABonita’s Faucet

13. BBonita’s Faucet

14. CBonita’s Faucet

15. DBonita’s Faucet

16. EBonita’s Faucet

17. What did we find in our student work collection? Many students had no entry point in solving the problem. More middle school students approached the problem through reasoning and making sense of the numbers. We didn’t find as much cross multiplying as expected. We didn’t find kids stating a proportion and supporting it with reasoning.

18. Essential Understanding A rate is a set of infinitely many equivalent ratios.

19. Cassandra’s Faucet Cassandra has a leaky faucet in her bathtub. She put a bucket underneath the faucet in the morning and collected data throughout the day to see how much water was in the bucket. Use the data Cassandra collected to determine how fast the faucet was leaking. TimeAmount of Water 7:00 a.m.2 ounces 8:30 a.m.14 ounces 9:30 a.m.22 ounces 11:00 a.m.34 ounces 2:00 p.m.58 ounces 5:30 p.m.86 ounces 9:30 p.m.118 ounces

20. ACassandra’s Faucet

21. B Cassandra’s Faucet

22. C Cassandra’s Faucet

23. DCassandra’s Faucet

24. E Cassandra’s Faucet

25. Cassandra’s Faucet Student Work How did student’s make sense of the table? What were the various entry points? What conclusions can you make about how students are thinking as they engaged in purposeful struggle to understand rate?

26. In Conclusion.. Proportional reasoning has been referred to as the capstone of the elementary curriculum and the cornerstone of algebra and beyond. Van de Walle,J. (2009). Elementary and middle school teaching developmentally. Boston, MA: Pearson Education.

27. Think About… Proportional reasoning may at first seem straightforward, developing an understanding of it is a complex process for students.