Making Sense of Math: Thinking Rationally

1. Making Sense of Math:Thinking Rationally Amy Lewis Math Specialist IU1 Center for STEM Education

2. Goals for the course • Use a variety of tools to deepen their understanding of rational numbers and explore proportional relationships to connect fractional meanings and representations. • Participate collaboratively in solving problems in other base systems to strengthen reasoning skills. • Connect new understandings of ratios and fractions to classroom practice.

3. Day 2: • Use physical models to represent and manipulate fractions in order to make sense of the operations. • Use and compare fractions and ratios. • Examine student work to analyze uses of fractions and ratios.

4. Homework • Examine the student responses. • Which artifact demonstrates the strongest understanding of fractional representations? Why? • Which artifact demonstrates the weakest understanding of fractional representations? Why? • How would you address one of the misconceptions demonstrated by one of these responses?

5. Question • Think carefully about the following question. Write a complete answer. You may use drawings, words, and numbers to explain your answer. Be sure to show all of your work. • José ate ½ of a pizza. • Ella ate ½ of another pizza. • José said that he ate more pizza than Ella, but Ella said they both ate the same amount. Use words and pictures to show that José could be right.

6. Student #1 Minimal

7. Student #2 Satisfactory

8. Student #3 Partial

9. Student #4 Extended

10. Student #5 Satisfactory

11. Performance Data • 1992 Mathematics Assessment • 4th Grade • ECR – Extended Constructed Response • “Hard” • Minimal – 18% • Partial – 2% • Satisfactory – 8% • Extended – 16% • Omitted – 57%

12. Operations with Fractions • Addition • Subtraction • Multiplication • Division

13. Comparing Candy

14. Comparing Candy Students on the 7th grade field trip committee at STEM Middle School are planning a fund raising event. They can’t decide what candy to sell -- Blocko Choco or Choca Latta. Both candies cost the same so it is a matter of which candy will sell more. They use a lunchtime survey to determine the preference of their customers, the students in their school. When the results were tabulated each member of the committee designed an ad to present their results.

15. Comparing Candy • 3 out of 5 students prefer Blocko Choco to Choca Latta. • The students have spoken, students who preferred Blocko Choco outnumbered those who preferred Choca Latta by a ratio of 3 to 2. • The students have spoken, students who preferred Blocko Choco outnumbered those who preferred Choca Latta by a ratio of 792 to 528. • 264 more students preferred Blocko Choco to Choca Latta. • 60% prefer Blocko Choco to Choca Latta. • 3/5 of the students prefer Blocko Choco. • 40% of the students prefer Choca Latta to Blocko Choco. • The number of students who prefer Blocko Choco is 1.5 times the number who prefer Choca Latta.

16. Comparing Candy • Examine the 8 statements. • Discuss the questions with your partner or group.

17. Eight Ways to Compare • What do each of the 8 statements mean? How does each show a comparison? What is being compared and how is it being compared? • Could all of the 8 statements be based on the same survey data?

18. Eight Ways to Compare • What information is lost in each form of comparison? • Which is the most accurate way to compare the data? Why?

19. Eight Ways to Compare • Which would be the most effective way to advertise for Blocko Choco? • Which would be the most effective way to advertise for Choca Latta?

20. Eight Ways to Compare • Which statements are misleading? • If one statement was going to be used for a newspaper ad, as a consumer, which statement would influence you the most?

21. Connecting to the Mathematics • How are ratios like fractions? How are they different? • What is being compared with decimals and with percents? • What does a scaling comparison tell you?

22. Subtraction as Comparison Many important mathematical applications involve comparing quantities. In instances where we need to know which quantity is greater or how much greater, we subtract to find a difference. Since addition and subtraction come first in a students’ experience with mathematics, this way of thinking becomes pervasive in any situation requiring comparison.

23. Ways to Compare beyond Subtraction Beyond subtraction we can compare quantities using ratios, fractions, decimals, percents, unit rates, and scaling. Students must learn different ways to reason proportionally and to recognize when such reasoning is appropriate.

24. Making Punch

25. Making Punch Each year, your class presents its mathematics portfolio to parents and community members. This year, your homeroom is in charge of the refreshments for the reception that follows the presentations.

26. Making Punch Four students in class give their recipes for lemon-lime punch. The class decides to analyze the recipes to determine which one will make the Fruitiest-tasting punch. The recipes are shown below. Adam’s RecipeBobbi’s Recipe 4 cups lemon-lime concentrate 3 cups lemon-lime concentrate 8 cups club soda 5 cups club soda Carlos’ RecipeZeb’s Recipe2 cups lemon-lime concentrate 1 cups lemon-lime concentrate3 cups club soda 4 cups club soda

27. Making Punch In your group, determine: • Which recipe has the strongest taste of lemon-lime? Explain your reasoning. • Which recipe has the weakest taste of lemon-lime? Explain your reasoning. Record your solutions on poster paper and display it near your table.

28. Making Punch Participate in a gallery walk and contrast the methods used by the groups. Whole group discussion: • What methods of comparison were used by the different groups? • What are the similarities and differences across all the methods? • Did everyone get the same answers?

29. Examining Student Work In your group examine each team’s solution. • What comparison model was used? • What misconceptions if any are evident? • Which solutions are mathematically equivalent?

30. Examining Student Work

31. Examining Student Work

32. Examining Student Work

33. Examining Student Work

34. Examining Student Work

35. Examining Student Work

36. Examining Student Work

37. Examining Student Work

38. Making Punch Every participant will receive ½ cup of punch. For each recipe, how much concentrate and how much club soda are needed to make punch for 240 people? Explain your answer.

39. Making Punch Summarize the work that you have done so far in the table below.

40. Making Punch • What patterns do you notice in the table?

41. Making Connections Each person in the group should take one of the recipes and make a table.

42. Making Connections Next, each person in the group should make a graph of the cups of concentrate as a function of cups of punch. • What type of relationship is shown by each graph? • Write an equation for each graph. • How can the equations or graphs be used to solve the original problem? • In the graph, what does the steepness of the data represent?

43. Candy Jar

44. The Candy Jar (right) contains Jawbreakers (the circles) and Jolly Ranchers (the rectangles). Use this Candy Jar to solve the following problems. Candy Jar

45. Candy Jar • Suppose you had a new treat tin with the same ratio of Jawbreakers to Jolly Ranchers as shown above, but it contained 10 Jolly Ranchers. How many Jawbreakers would you have? • Suppose you had a treat tin with the same ratio of Jolly Ranchers to Jawbreakers as shown above, but it contained 720 candies. How many of each kind of candy would you have? • Suppose that you are making treats to hand out to trick-or-treaters on Halloween. Each treat is a small bag that contains 5 Jolly Ranchers and 13 Jawbreakers. You have 50 Jolly Ranchers and 125 Jawbreakers. How many small bags could you make up?

46. Candy Jar • Record your solutions on poster paper. • Reflect on your strategies: • Did you use the same or different strategy for each problem? Why? • How do your strategies for these problems compare to those you used for the Making Punch problem?

47. Extension Kandies-R-Us sells a super-sized 500-piece Tub-O-Treats. It contains chocolate kisses in addition to Jawbreakers and Jolly Ranchers. If the ratio of Jolly Ranchers to Jawbreakers to kisses is 7:8:10, how many of each candy are in the Tub-O-Treats?

48. Fractions and Ratios

49. Treat Tins The treat tin containsJawbreakers (circles)and Jolly Ranchers (rectangles). • Write as many different ratios as you can to describe the contents of the tin. • Which, if any, of your ratios are fractions? Why? • Which, if any, of your ratios are NOT fractions. Why?

50. Fractions and Ratios • A ratio is a comparison of any two quantities. • Part-to-Whole Ratios • Part-to-Part Ratios • Rates as Ratios • A rate is a comparison of two different things or quantities, e.g., mile per hour. • A fraction is a comparison of parts-to-wholes.