1. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Ratio and Proportional Relationships April 23, 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 Agenda Welcome Unpacking 6th grade CCSSM Standards RP Connections to other standards

3. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Learning Intention & Success Criteria We are learning to… Deepen conceptual understanding of ratio and rates Unpack the 6 th grade CCSSM standards about ratios.

4. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Learning Intention & Success Criteria We will be successful when we can…  Use various strategies to solve ratio and proportion problems.  Justify our thinking when solving problems involving ratios and proportions.  Clearly explain and provide examples for specific CCSSM standards

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

6. Why a focus on ratio and proportion? Proportional reasoning …has been called the backbone, the cornerstone, the gateway to higher levels of mathematics success, and is considered as a “capstone” of primary school mathematics Kilpatrick, Swafford & Findell, 2001; Lamon, 1999; Lesh, Post & Behr, 1988 …is one of the best indicators that students have attained an understanding of rational numbers. Lamon (1999)

7. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Launch Consider this phrase Two pounds of potatoes for $1.00 What does this mean?

8. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Making Sense Read the statements on the handout. Which ones make sense? Why? What distinguishes those that make sense from those that don’t make sense?

9. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 From Additive to Multiplicative Comparisons Students need to distinguish between an additive and multiplicative situation. Students need to make a transition from making additive comparisons to forming a ratio between two quantities.

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

11. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 What Have You Discovered? If the small gear turns clockwise, which direction does the big gear turn? Why? If you turn the small gear a certain number of times, does the big gear turn more revolutions, fewer, or the same amount? How can you tell?

12. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Gear Up! Part I Suppose you have turned the gears until they have returned to their original positions. –How many revolutions does the small gear make? –How many revolutions does the big gear make? (Find a way to keep track of the number of revolutions both gears make until they return to their original positions.)

13. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Shift from One Quantity to Two Quantities Students need to make a transition from focusing on only one quantity to realizing that two quantities are important.

14. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Gear Up! Part II Someone found out that when the small gear turns 5 times, the medium gear turns 3 times. What are some other rotation pairs for the gears?

15. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Gear Up! Part III Graph the ordered pairs. At your table surface some important features of the graph.

16. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Gear Up! Part IV If your gears have different number of teeth how would this affect the relationships you have found?

17. 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

18. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Oh no! What’s the answer?