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Ratio and Proportion

Ratio and Proportion. Today’s Goals. Get acquainted; Review our learning related to the concept of equality; Learn strategies to help students think proportionally and multiplicatively; Differentiate proportional from non-proportional situations;

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Ratio and Proportion

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  1. Ratio and Proportion

  2. Today’s Goals Get acquainted; Review our learning related to the concept of equality; Learn strategies to help students think proportionally and multiplicatively; Differentiate proportional from non-proportional situations; Understand the PLC tasks related to ratio and proportion strategies.

  3. By calculating 5 x 84 = 420 10 x 42 = 420 So, 5 x 84 = 10 x 42 By thinking relationally 5 x 84 = 5 x (2 x 42) = (5 x 2) x 42 = 10 x 42 …or? T or F: 5 x 84 = 10 x 42

  4. TASK • Work as a group • Solve your equation in two ways: • by calculating • by thinking relationally • Show your thinking • Be ready to explain your work to the larger group

  5. Relational Thinking… Focuses on relations rather than only on calculating answers (red) Uses fundamental properties of arithmetic to relate or transform expressions (green) Looks at expressions and equations in their entirety rather than as procedures to be carried out step by step (orange) Flexibly uses operations and relations (yellow) from Tom Carpenter’s slide show on September 23rd

  6. Walk-about and Discuss • Take a ‘wall walk’ with your group • Look for examples of these relational thinking characteristics in these problems. • Use post-its to identify examples when you see them

  7. So…. • What is the connection between relational thinking and solving equations?

  8. TASK • Select a PLC task of your choice: • Baseline Assessment • Teaching a lesson • Interviewing students • Post Assessment • Find 2-3 others who share your interest • Find the related discussion Q’s • Discuss the Q’s with your small group • Discuss the Follow-up Q’s • Be ready to report out

  9. Summary Questions What do we want our kids to know about equality? What can we do to improve their understanding of equality as an equivalence relationship? How will this work impact students understanding of algebra?

  10. Quantitative Comparison

  11. TASK: • Work as a group • Examine the student work and be able to explain the strategy used • Compare the samples of student work • What strategies are similar? • What strategies are different?

  12. Student Responses (from Glenda Lappan)

  13. Student Responses (from Glenda Lappan)

  14. Student Responses (from Glenda Lappan)

  15. Student Responses (from Glenda Lappan)

  16. Student Responses (from Glenda Lappan)

  17. Wes Problem #1 Two baseballs cost $5. How much will seven baseballs cost?

  18. Two baseballs cost $5. How much will seven baseballs cost? The unit rate approach is characterized when students find the constant factor across or between the two measure spaces. x 2.5 x 2.5 17.50

  19. Two baseballs cost $5. How much will seven baseballs cost? The scale factor approach is characterized when students find the multiplicative relationship among elements within the measurement space. X 3.5 X 3.5 17.50

  20. Two baseballs cost $5. How much will seven baseballs cost? The table strategy is a strategy that is a mix of unit rate and factor of change. 7 17.50

  21. Two baseballs cost $5. How much will seven baseballs cost? The graphing strategy has an interesting equation. y = 2.5x 20 17.50 15 Dollars ($) 10 5 0 7 17.50 7 0 2 4 6 8 baseballs

  22. Two baseballs cost $5. How much will seven baseballs cost? The fraction strategy is devoid of the problem context. x 3.5 = 17.50 x 3.5

  23. Two baseballs cost $5. How much will seven baseballs cost? Cross-multiplication is also a strategy but is hard for students to explain why it works and they tend not to use it when solving problems on their own. = baseball-dollar

  24. Wes Problem #2 I can buy 6 baseballs for $20. How much will 18 baseballs cost?

  25. Wes Problem #3 I can buy 6 baseballs for $18. How much will 11 baseballs cost?

  26. Wes Problem #4 I can buy 7 baseballs for $12. How much will 11 baseballs cost?

  27. The Marissa Problem Marissa bought 0.46 of a pound of wheat flour for which she paid $0.83. How many pounds could she buy for one dollar?

  28. The Track Problem Wendy and Brooke run at the same speed around a track. Wendy was running lap 3 when Brooke was running lap 7. What lap will Brooke be on when Wendy is running lap 9?

  29. Proportional Situation or Not? Taxi cabs in Minneapolis charge $2.50 for the first one-fifth mile and each succeeding fifth will cost $.38. How much will the fare be for a 2 mile trip?

  30. Proportional Situation or Not? The action figure of a famous wrestler has a scale of 1/12. If the doll is 7 inches tall, how tall is the actual wrestler?

  31. Proportional Situation or Not? Dara swims four laps across the pool in the same time as Michael swims three laps. They begin swimming at the same time. How far has Michael swum when Dara completes lap 10?

  32. Proportional Situation or Not? Jake typically exchanges 200 text messages a day. The plan he signed up for charges a $30 flat fee per month and does not charge for the first 1000 text messages exchanged. Each message exchanged after the first 1000 costs $0.05. What is the amount for Jake’s bill for the month of October?

  33. Proportional Situation or Not? Lance is riding his bicycle in France. He rides 18 km in half an hour. If he rides at the same rate, how long will it take him to ride 162 km?

  34. Proportional Situation or Not? Two U.S. dollars can be exchanged for three Euros. How many U.S. dollars can be exchanged for 21 Euros?

  35. Proportional Situation or Not?

  36. Proportional Situation or Not?

  37. Proportional Situation or Not?

  38. Proportional Situation or Not?

  39. Proportional Situation or Not?

  40. Proportional Situation or Not?

  41. What is proportional thinking? (partial list adapted from Lamon) the ability to use proportionality as a mathematical model to organize information in appropriate contexts; the ability to distinguish situations in which proportionality is not an appropriate mathematical model from situations in which it is useful; knowing that the graph of a direct proportional situation is a straight line through the origin; the ability to explain the difference between functions of the form y = mx and functions of the form y = mx + b. In the latter function, y is not proportional to x, but rather the change in y is proportional to the change in x.

  42. Minnesota State Algebra Standards (7th grade) Understand the concept of proportionality in real-world and mathematical situations, and distinguish between proportional and other relationships. Recognize proportional relationships in real-world and mathematical situations; represent these and other relationships with tables, verbal descriptions, symbols and graphs; solve problems involving proportional relationships and explain results in the original context. Apply understanding of order of operations and algebraic properties to generate equivalent numerical and algebraic expressions containing positive and negative rational numbers and grouping symbols; evaluate such expressions. Represent real-world and mathematical situations using equations with variables. Solve equations symbolically, using the properties of equality. Also solve equations graphically and numerically. Interpret solutions in the original context.

  43. Upcoming PLC Tasks • Administer a Baseline Assessment on Ratio and Proportion to at least one class • Teach a Lesson • Interview at least 3 students • Administer a Post Assessment • Meet in your PLCs to discuss your observations/insights

  44. Be a Student… • Work the Baseline Assessment • Predict how your students will respond to the questions • Ask Q’s about anything that is unclear

  45. Baseline Assessment #1 2 baseballs cost $5. How much will 7 baseballs cost? Answer: Use drawings, words, or numbers to show how you got your answer.

  46. Baseline Assessment #2 A sixth-grade class needs 5 leaves each day to feed its 2 caterpillars. How many leaves would they need each day for 12 caterpillars? (NAEP, 1996) Answer: Use drawings, words, or numbers to show how you got your answer.

  47. Baseline Assessment #3 The length of Mr. Short is 4 large buttons. The length of Mr. Tall is 6 large buttons. When paper clips are used to measure Mr. Short and Mr. Tall: The length of Mr. Short is 6 paper clips. What is the length of Mr. Tall in paper clips?

  48. Baseline Assessment #4 Wendy and Brooke run at the same speed around a track. Wendy was running lap 3 when Brooke was running lap 7. What lap will Brooke be on when Wendy is running lap 9?

  49. Baseline Assessment #5 Doughman Instant Pancake mix requires 3 cups of water for every 4 cups of mix. Xiong needs to make a large number of pancakes. She has 23 cups of mix and wants to use it all. How much water does Xiong need to add to make her pancakes?

  50. TASK: Prepare to teach a lesson • Turn to page 7 in the handout. • Work with a partner to create sample problems for each of the strategies.

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