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Investigating Ratios As Instructional Tasks

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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.

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

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.

- 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?

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

- 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.

If you are told the ratio of girls to boys in a

class is 3:4, what can you tell about the

class?

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?

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. 2005. Teaching Fractions and Ratios for Understanding. Lawrence Erlbaum Associates.

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?

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?

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?

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.