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1. Welcome Welcome to content professional development sessions for Grades 6-8. The focus is Proportional Reasoning. Proportional reasoning includes fractions as ratios, rates, ratios, and proportions. It extend understanding about division and part-whole relationships. The goal is to help you understand this mathematics better to support your implementation of the Mathematics Standards. Proportional Reasoning: Grades 6-8: slide 1

2. Introductions of Facilitators INSERT the names and affiliations of the facilitators Proportional Reasoning: Grades 6-8: slide 2

3. Introduction of Participants In a minute or two: 1. Introduce yourself. 2. Describe an important moment in your life that contributed to your becoming a mathematics educator. 3. Describe a moment in which you hit a “mathematical wall” and had to struggle with learning. Proportional Reasoning: Grades 6-8: slide 3

4. Overview Some of the problems may be appropriate for students to complete, but other problems are intended ONLY for you as teachers. As you work the problems, think about how you might adapt them for the students you teach. Also, think about what Performance Expectations these problems might exemplify. Proportional Reasoning: Grades 6-8: slide 4

5. Role of Understanding of Fractions • A deep understanding of fractions is the foundation of proportional thinking. • Proportional thinking is the foundation of linearity. • Think of this “mathematical story”: integers --> fractions and ratios --> proportions --> direct variation-- > linear relationships Proportional Reasoning: Grades 6-8: slide 5

6. Usefulness of Understanding Fractions Being fluid with fractions AND having a flexible understanding of fractions allow students to have access to multiple ways of thinking about and representing proportional relationships. Proportional Reasoning: Grades 6-8: slide 6

7. Problem Set 1 The focus of Problem Set 1 is fractions. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. Proportional Reasoning: Grades 6-8: slide 7

8. Problem 1.1 Look at the picture below. Can you see 3/5? What is the unit? Can you see 2/3? What is the unit? Can you see 5/3? What is the unit? Can you see 2/3 of 3/5? What is the unit? Proportional Reasoning: Grades 6-8: slide 8

9. Problem 1.1: Additional Questions Look at the picture below. Can you see 3/2 of 2/3? What is the unit? Can you see 3/5 of 5/3? What is the unit? Can you see 5/3 of 3/5? What is the unit? Proportional Reasoning: Grades 6-8: slide 9

10. Problem 1.2 Find three fractions between 4/7 and 5/7. Find three fractions between 5/7 and 5/6. How are your solution strategies alike? How are they different? Proportional Reasoning: Grades 6-8: slide 10

11. Problem 1.2: More Questions • Find three fractions equally spaced between 4/7 and 5/7. • Can you generalize this for all pairs of fractions a/b and (a+1)/b? • 4.5 is half way between 4 and 5. So is 4.5/7 half way between 4/7 and 5/7? Explain. Proportional Reasoning: Grades 6-8: slide 11

12. Problem 1.2: Even More Questions • Find three fractions equally spaced between 5/7 and 5/6. • Can you generalize this for all pairs of fractions a/(b+1) and a/b? • 6.5 is half way between 6 and 7, so is 5/6.5 halfway between 5/7 and 5/6? Explain. Proportional Reasoning: Grades 6-8: slide 12

13. Problem 1.3 Is it correct to think of 3/7 as 3 parts out of 7? Is it correct to think of 7/3 as 7 parts out of 3? Why doesn’t the same mental image work for both 3/7 and 7/3? What visual model or mental image might help students conceptualize both 3/7 and 7/3? Proportional Reasoning: Grades 6-8: slide 13

14. Problem 1.4 Can every fraction equivalent to 8/12 be found by multiplying the numerator and denominator by some counting number, n? Can every fraction equivalent to 2/3 be found by multiplying the numerator and denominator by some counting number, n? How are your answers alike and different for these two problems? Proportional Reasoning: Grades 6-8: slide 14

15. Problem 1.5 Think of all the fractions equivalent to 8/12. What percentage of them can be found by multiplying the numerator and denominator by some counting number, n? Explain your answer. Proportional Reasoning: Grades 6-8: slide 15

16. Reflection What did you learn (or re-learn) about fractions by working on these problems? How might your understanding help you understand students’ thinking? Proportional Reasoning: Grades 6-8: slide 16

17. Problem Set 2 The focus of Problem Set 2 is multiplicative reasoning. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. Proportional Reasoning: Grades 6-8: slide 17

18. Problem 2.1 Write answers to these problems using mental math ONLY. a. What is 50% of 40? b. What is 200% of 40? c. What is 150% of 40? d. What is 10% of 40? e. What is 60% of 40? f. What is 260% of 40? g. What is 5% of 40? h. What is 15% of 40? i. What is 55% of 40? j. What is 35% of 40? Proportional Reasoning: Grades 6-8: slide 18

19. Problem 2.2 What is the volume of this box? Explain your answer or explain why you cannot find the volume. Proportional Reasoning: Grades 6-8: slide 19

20. Problem 2.3 We can think of 5 x 4 as “add 4 five times” or as “5 fours.” The latter lets us make sense of 2 2/3 x 4 1/5; that is, think of 2 2/3 copies of 4 1/5. Draw a picture of 2 2/3 copies of 4 1/5. Proportional Reasoning: Grades 6-8: slide 20

21. Problem 2.4 What is the area of a rectangle that is 5 inches long and 3 centimeters wide? Proportional Reasoning: Grades 6-8: slide 21

22. Problem 2.5 Solve each problem. a. Find the quotient, 6 ÷ 2/3. b. Draw a picture to show what 6 ÷ 2/3 means. c. 6 is 2/3 of what number? Proportional Reasoning: Grades 6-8: slide 22

23. Reflection How might a deep understanding of multiplication and division help students better understand fractions and ratios? Proportional Reasoning: Grades 6-8: slide 23

24. Problem Set 3 The focus of Problem Set 3 is introductory proportional reasoning. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. Proportional Reasoning: Grades 6-8: slide 24

25. Problem 3.1 In 1980 the populations of Towns A and B were 5000 and 6000, respectively. In 1990 the populations of Towns A and B were 8000 and 9000, respectively. Brian claims that from 1980 to 1990 the two town’s populations grew by the same amount. Use mathematics to explain how Brian might have justified his answer. Darlene claims that from 1980 to 1990 the population of Town A had grown more. Use mathematics to explain how Darlene might have justified her answer. Proportional Reasoning: Grades 6-8: slide 25

26. Problem 3.2 Melissa bought 0.43 of a pound of wheat flour for which she paid \$0.86. How many pounds of flour could she buy for one dollar? Proportional Reasoning: Grades 6-8: slide 26

27. Problem 3.3 Melissa bought 0.46 of a pound of wheat flour for which she paid \$0.83. How many pounds of flour could she buy for one dollar? Proportional Reasoning: Grades 6-8: slide 27

28. Problem 3.4 A school system reported that they had a student-teacher ratio of exactly 30:1. How many more teachers would they need to hire to reduce the ratio to exactly 25:1. Proportional Reasoning: Grades 6-8: slide 28

29. Reflection What are the main characteristics of proportional reasoning? How is proportional reasoning (like Darlene’s) different from additive reasoning (like Brian’s)? Proportional Reasoning: Grades 6-8: slide 29

30. Problem Set 4 The focus of Problem Set 4 is standard proportional reasoning. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. Proportional Reasoning: Grades 6-8: slide 30

31. Problem 4.1 Proportional Reasoning: Grades 6-8: slide 31

32. Problem 4.2 Proportional Reasoning: Grades 6-8: slide 32

33. Problem 4.2 Proportional Reasoning: Grades 6-8: slide 33

34. Problem 4.3 Solve this problem in at least three different ways: To make one glass of lemonade, use 3 tablespoons of lemonade mix and 6 oz. of water. How much lemonade mix do you need to make 2 quarts of lemonade? Proportional Reasoning: Grades 6-8: slide 34

35. Problem 4.4 Can you enlarge a photo whose size is 3 ½ inches by 5 inches so that it is 8 ½ inches by 11 inches? Explain. Proportional Reasoning: Grades 6-8: slide 35

36. Reflection • How might solving these “standard proportional reasoning problems” help students learn to reason proportionally? • How has your thinking about proportional reasoning changed as a result of working on these problems? • How might that shift affect your instructional practice? Proportional Reasoning: Grades 6-8: slide 36

37. Problem Set 5 The focus of Problem Set 5 is more complicated proportional reasoning. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. Proportional Reasoning: Grades 6-8: slide 37

38. Problem 5.1 Gertrude has an “interest only” mortgage on her house. Each month, she pays only the required interest payment. She pays down the principal whenever she has money to spare. As she pays down the principal, the monthly payment decreases. Currently her mortgage is \$200,000 and her monthly payment is \$1,000. a. What is her annual interest rate? b. If she pays down \$35,000 of the principal, what will her new monthly payment be? Proportional Reasoning: Grades 6-8: slide 38

39. Problem 5.2 A man was stranded on a desert island with enough water to last him 27 days. After 3 days, he saved a woman on a small life raft. If they can keep their water supply from evaporating, they figure that they can share their water equally for 18 days. What portion of the man’s original daily ration was allotted to the woman? Proportional Reasoning: Grades 6-8: slide 39

40. Problem 5.3 In an adult condominium complex, 2/3 of the men are married to 3/5 of the women. What part of the residents are married? Proportional Reasoning: Grades 6-8: slide 40

41. Problem 5.4 For any linear measurement, let Y = number of yards for that measurement, and let F = number of feet for that measurement. Write an equation showing the relationship of these two variables. Proportional Reasoning: Grades 6-8: slide 41

42. Problem 5.5 a. How are the two graphs below alike? How are they different? Proportional Reasoning: Grades 6-8: slide 42

43. Problem 5.5 b. Joe walks down a straight path and then turns around a walks back to the starting point. The graph below displays how far away he was from the starting point. Sketch the graph of his walking speed(s). Proportional Reasoning: Grades 6-8: slide 43

44. Reflection Which one of these problems was most difficult for you? Which one was least difficult? Are there any of these problems that you think most of your students could solve? Are there any of your students (from last year) that you think could solve all of these problems? Proportional Reasoning: Grades 6-8: slide 44

45. Problem Set 6 The focus of Problem Set 6 is understanding π as a ratio. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. Proportional Reasoning: Grades 6-8: slide 45

46. Problem 6.1 Complete the Pi Ruler activity. The “multipliers” for some common trees are given below: 2.5 white elm, tulip, chestnut 3 black walnut 3.5 black oak, plum 4 birch, sweet gum, sycamore, oak, red oak, apple 5 ash, white ash, pine, pear 6 beech, sour gum, sugar maple 7 fir, hemlock 8 shagbark, hickory, larch Proportional Reasoning: Grades 6-8: slide 46

47. Reflection How might completing the Pi Ruler activity help students understand what π is? What objects (other than trees) could the Pi Ruler be used to measure? Proportional Reasoning: Grades 6-8: slide 47

48. Problem Set 7 The focus of Problem Set 7 is reflection on thinking. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others. Proportional Reasoning: Grades 6-8: slide 48

49. Words Associated with “Fraction” At your table, make a list of mathematical terms or vocabulary words that are associated with the word “fraction.” Proportional Reasoning: Grades 6-8: slide 49

50. Definitions For EACH the terms: fraction, ratio, proportion 1. Write a “teacher” definition. 2. Write a “student” definition, if you think it should be different. 3. Give an example and a non-example. Proportional Reasoning: Grades 6-8: slide 50