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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 2 Introductions of Facilitators INSERT the names and affiliations of the facilitators

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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 doesnt 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 19 Problem 2.2 What is the volume of this box? Explain your answer or explain why you cannot find the volume.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 21 Problem 2.4 What is the area of a rectangle that is 5 inches long and 3 centimeters wide?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 23 Reflection How might a deep understanding of multiplication and division help students better understand fractions and ratios?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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 towns 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 29 Reflection What are the main characteristics of proportional reasoning? How is proportional reasoning (like Darlenes) different from additive reasoning (like Brians)?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 31 Problem 4.1

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2008 May 29Proportional Reasoning: Grades 6-8: slide 32 Problem 4.2

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2008 May 29Proportional Reasoning: Grades 6-8: slide 33 Problem 4.2

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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 mans original daily ration was allotted to the woman?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 42 Problem 5.5 a. How are the two graphs below alike? How are they different?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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).

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 46 Problem 6.1 Complete the Pi Ruler activity. The multipliers for some common trees are given below: 2.5white elm, tulip, chestnut 3black walnut 3.5black oak, plum 4birch, sweet gum, sycamore, oak, red oak, apple 5ash, white ash, pine, pear 6beech, sour gum, sugar maple 7fir, hemlock 8shagbark, hickory, larch

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 49 Words Associated with Fraction At your table, make a list of mathematical terms or vocabulary words that are associated with the word fraction.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 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.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 51 Problem 7.1 What is a fraction? Is a fraction a number? Is it two numbers? Is it a symbol?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 52 Problem 7.2 What is a ratio? Is a ratio a number? Is it two numbers? Is it a symbol?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 53 Problem 7.3 How do these problem illustrate the Mathematics Standards for Grades 6-8?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 54 Problem 7.4 Look at Core Content 6.1, 6.3, 7.1, 7.2, 8.1, 8.5. Identify tasks from the Problem Sets that are examples for specific Performance Expectations. Identify tasks from the Problem Sets that cut across multiple Performance Expectations. Be ready to share some of your examples.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 55 Problem Set 8 The focus of Problem Set 8 is extensions of proportional reasoning problems. You may work alone or with colleagues to solve these problems. When you are done, share your solutions with others.

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2008 May 29Proportional Reasoning: Grades 6-8: slide 56 Problem 8.1 For each graph below, create a table of values that might generate the graph. Graph 1

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2008 May 29Proportional Reasoning: Grades 6-8: slide 57 Problem 8.1 For each graph below, create a table of values that might generate the graph. Graph 2

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2008 May 29Proportional Reasoning: Grades 6-8: slide 58 Problem 8.1 For each graph below, create a table of values that might generate the graph. Graph 3

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2008 May 29Proportional Reasoning: Grades 6-8: slide 59 Problem 8.2 A boy can bike a mile in 5 minutes and walk a mile in 20 minutes. How much time does he save if he bikes to his dads office, 8 miles away, rather than walking?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 60 Problem 8.3 A biker rides at a speed of 10 mph for about half an hour and then turns around and walks home on the same route, at a speed of 4 miles per hour. What is his average speed for the entire trip? Does it matter if about half an hour means 28 minutes or 32 minutes?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 61 Problem 8.4 Complete these problems. a. Change the speed, 1 foot per second, to miles per hour. b. If you walk 50 feet in 20 seconds, how fast is that in miles per hour? c. How can you use the solution to problem a to help solve problem b?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 62 Problem 8.5 a. There is a dripping faucet in the kitchen. Every 53 seconds, 31 milliliters of water drips out. How much water will drip out in one minute? b. The bathtub faucet is also dripping. Every 97 seconds, 18 milliliters of water drips out. How much water will drip out in one hour?

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2008 May 29Proportional Reasoning: Grades 6-8: slide 63 Closing Comments Implementing the K-8 Mathematics Standards will require a deeper focus of mathematics ideas at each grade. Personal understanding of these ideas will make the implementation process easier.

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