1. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Ratio and Proportional Relationships April 30, 2013 Common Core Leadership in Mathematics2 (CCLM) 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.

2. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 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. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Strategies to Reason About Proportions

4. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 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. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Building Understanding of Proportional Relationships Press Students Conceptually Take Students Ideas Seriously Encourage Multiple Strategies/Models Focus on the Structure of the Mathematics Address Misconceptions

6. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Learning Intentions Analyze student reasoning in proportional situations Examine a progression of student strategies Summarize the CCSS related to reasoning about Proportions

7. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Anticipating Student Strategies What 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.

8. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 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?

9. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 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 pro portional relationships? How would you sort them?

10. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013

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

15. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Composed Unit View How many blue for 9 orange? +2 +6 +2 +6 +2 +6 +1 +3

16. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Composed Unit View How many blue for 9 orange? 6 x 4.5 = 27 1 12 2 3 3 4 4 4.5

17. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Student Work – Composed Unit Which student work shows a Composed Unit point of view? Student A, Student B, Student C, Student D

18. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Multiplicative Comparison (within relationship) How many orange for 9 blue? 2 9 6 27 x3

19. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Student Work Multiplicative Comparison Which student work shows Multiplicative Comparison? Student F and Student I

20. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 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.

21. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Graphs and Equations Where do you see… Unit rate? Composed Unit Thinking? Multiplicative Comparison Thinking ? 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?

22. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Learning Intentions Analyze student reasoning in proportional situations Examine a progression of student strategies Summarize the CCSS related to reasoning about Proportions

23. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Additive Scaling Up Scaling Up and Down Multiplicative Comparison Linear Relationships -------- Composed Unit View of Rate -------- ------ 3rd through 5 th grade 6 th grade 7 th grade 8 th grade 6th Grade “Understand ratio concepts and use ratio r easoning to solve problems” Tools: Ratio table Unit Rates Coordinate plane Bar model or number line Equations Progression in the CCSS

24. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Additive Scaling Up Scaling Up and Down Multiplicative Comparison Linear Relationships -------- Composed Unit View of Rate -------- ------ 3rd through 5 th grade 6 th grade 7 th grade 8 th 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 Progression in the CCSS

25. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Additive Scaling Up Scaling Up and Down Multiplicative Comparison Linear Relationships -------- Composed Unit View of Rate -------- ------ 3rd through 5 th grade 6 th grade 7 th grade 8 th grade 8th Grade ““Understand the connection between proportional relationships, lines and linear equations” Tools: Tables Coordinate plane Equations/Functions Progression in the CCSS

26. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 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