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Proportional Reasoning Strategies: Analyzing Student Work

Explore and analyze student strategies in solving ratio problems, focusing on proportional relationships. Study Common Core Standards and differentiating student reasoning depth. Implement visually represented solutions and anticipate student strategies.

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Proportional Reasoning Strategies: Analyzing Student Work

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  1. Ratio and Proportional Relationships This material was developed for use by participants in the Common Core Leadership in Mathematics (CCLM^2) project through the University of Wisconsin-Milwaukee. Use by school district personnel to support learning of its teachers and staff is permitted provided appropriate acknowledgement of its source. Use by others is prohibited except by prior written permission. April 30, 2013 Common Core Leadership in Mathematics2 (CCLM)

  2. 6.RP.1 Understand ratio concepts and use ratio reasoning to solve problems. 1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.”“For every vote candidate A received, candidate C received nearly three votes.”

  3. Strategies to Reason About Proportions

  4. Learning Intentions We Are Learning To…. • Analyze student thinking strategies to reason proportionally • Examine a progression of those strategies • Study and summarize the CCSS related to Proportional Reasoning

  5. Learning Intentions Analyze student reasoning in proportional situations Examine a progression of student strategies Summarize the CCSS related to reasoning about Proportions

  6. Anticipating Student StrategiesWhat we did last class Solve the four problems as though you are a 7th grader who does not yet know the cross multiplication algorithm. Represent at least one of your solutions visually.

  7. Ellie estimates that it takes her 5 hours to walk 8 miles. How many hours would she walk if she walked 48 miles? Jane estimates that she takes 8 hours to go 12 miles. How many miles would she walk in 42 hours? Quinten is an extreme trail runner and estimates that he takes 3 hours to run 9 miles. How many hours would it take for him to run 24 miles? Sierra is also a trail runner. She estimates that she runs 8 miles in 3 hours. If she runs for 2 miles, how long has she run?

  8. Examining Student Work What is similar or different about the way the students solved the problems? What in the context or number choice of the problem might have led students to use a particular strategy or model? From the student work, what can you say about each student’s  depth  of  understanding  about  proportional   relationships? How would you sort them?

  9. Learning Intentions Analyze student reasoning in proportional situations Examine a progression of student strategies Summarize the CCSS related to reasoning about Proportions

  10. Composed Unit View +2 +2 +2 +1 +6 +6 +6 +3 How many blue for 9 orange?

  11. Composed Unit View 4.5 1 2 4 3 1 2 3 4.5 4 How many blue for 9 orange? 6 x 4.5 = 27

  12. Student Work – Composed Unit Which student work shows a Composed Unit point of view? Student A, Student B, Student C, Student D

  13. Multiplicative Comparison(within relationship) 9 2 x3 6 27 How many orange for 9 blue?

  14. Student WorkMultiplicative Comparison Which student work shows Multiplicative Comparison? Student F and Student I

  15. Shift 3 – From Composed-unit Strategies to Multiplicative Comparisons Read Shift 3 pp. 69-71 in Developing Essential Understanding of Ratios, Proportions and Proportional Reasoning. Share 2 big ideas from this reading that relate to the gear problems or the exploration of student work.

  16. Graphs and Equations Jane estimates that it takes 8 hours to go 12 miles. d = distance (miles) t = time (hours) d = 1.5t t = What is the relationship b-between -Miles to hours? -Hours to miles? How do ratio tables emphasize both composed unit and multiplicative thinking? Where do you see… Unit rate? Composed Unit Thinking? Multiplicative Comparison Thinking?

  17. Learning Intentions Analyze student reasoning in proportional situations Examine a progression of student strategies Summarize the CCSS related to reasoning about Proportions

  18. Progression in the CCSS -------- Composed Unit View of Rate -------------- 3rd through 5th grade 6th grade 6th Grade “Understand  ratio  concepts  and  use   ratio  reasoning  to  solve  problems” Tools:Ratio table Unit Rates Coordinate planeBar model or number line Equations 7th grade 8th grade

  19. Progression in the CCSS -------- Composed Unit View of Rate -------------- 3rd through 5th grade 6th grade 7th Grade “Analyze  proportional   relationships   and use them to solve real-world and mathematical  problems.” Tools: Ratio table Unit rate Coordinate plane Equations Diagrams 7th grade 8th grade

  20. Progression in the CCSS -------- Composed Unit View of Rate -------------- 3rd through 5th grade 6th grade 8th Grade ““Understand  the  connection   between proportional relationships, lines  and  linear  equations” Tools:Tables Coordinate plane Equations/Functions 7th grade 8th grade

  21. CCLM Success Criteria We will know we are successful when we can • Use various strategies to solve ratio and proportion problems. • Justify our thinking when solving problems involving ratio and proportion . • Clearly explain and provide examples for specific CCSS standards

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