Investigating Ratios As Instructional Tasks MTL Meeting April 15 and 27, 2010 Facilitators Melissa HedgesKevin McLeod Beth SchefelkerMary Mooney DeAnn HuinkerConnie Laughlin The Milwaukee Mathematics Partnership (MMP), an initiative of the Milwaukee Partnership Academy (MPA), is supported with funding by the National Science Foundation.

WALT We are learning to explore ratios (part to part, part to whole) We will be successful when we analyze ratios in instructional tasks.

Ahhh Grasshopper… A grasshopper can jump further than a person. Do you agree or disagree? What justification do you have for your answer. Turn and talk with a person at your table.

Two Types of Thinking Absolute thinking - thinking additively Relative thinking - thinking multiplicatively Which type of thinking were you using? If you used relative thinking what comparisons did you use to justify your reasoning?

What is a ratio? An ordered pair of numbers that express A multiplicative (relative) comparison of two quantities or measures. Types of ratios Part-to-Part:number of girls to number of boys 2:3 Part-to-Whole:number of girls to number of children in the family 2:5

Studying ratios Proportional thinking is developed through activities involving comparing and determining the equivalence of ratios and solving proportions in a wide variety of problem based contexts and situations without recourse to rules or formulas To the student beginning to develop an understanding of ratio, different settings or contexts may seem like different ideas even though they are essentially the same from a mathematical viewpoint. Van de Walle,J. (2009). Elementary and middle school teaching developmentally. Boston, MA: Pearson Education.

Interpreting Ratios in Instructional Tasks If you are told the ratio of girls to boys in a class is 3:4, what can you tell about the class?

Orange Juice To Water You have a 30% concentration of orange juice in water. If you take a cup of the mixture, what percent will be orange juice? What is the ratio in this situation? How is this situation similar to the previous task? How is it different?

Interpreting information in ratios situations In order to understand the different nuances that ratios bring to a contextual situation, it is important to discuss all of the issues and understandings related to that situation. Explicit information Implicit information Lamon,S Teaching Fractions and Ratios for Understanding. Lawrence Erlbaum Associates.

Auditorium problem There are 100 seats in the theatre with 30 in the balcony and 70 on the main floor. Eighty tickets were sold for the matinee performance, including all of the seats on the main floor. What fraction of the seats were sold? What is the ratio of balcony seats to empty seats? What is the ratio of empty seats to occupied seats? What is the ratio of empty seats to occupied seats in the balcony?

Does the ratio remain the same? John is 25 years old and his son is 5 years old. Does this ratio remain constant as John and his son age? Is this relationship multiplicative or additive?

Fathers and Sons The ratio of a father’s age to his son’s age is 5:1 What are some possible ages that the father and son could be?

Big Idea A key developmental milestone is the ability of a student to begin to think of a ratio as a distinct entity, different from the two measures that made it up. Ratios and proportions involve multiplicative rather than additive comparisons. Equal ratios result from multiplication or division not from addition or subtraction. Van de Walle,J. (2009). Elementary and middle school teaching developmentally. Boston, MA: Pearson Education.