Investigating Ratios As Instructional Tasks

MTL Meeting

April 15 and 27, 2010 Facilitators

Melissa Hedges Kevin McLeod

Beth Schefelker Mary Mooney

DeAnn Huinker Connie 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

2:3Part-to-Whole: number of girls to number

of children in the family2:5

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 themixture, what percent will be orangejuice? 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. 2005. Teaching Fractions and Ratios for Understanding. Lawrence Erlbaum Associates.

Auditorium problemThere are 100 seats in the theatre with 30 in the balconyand 70 on the main floor. Eighty tickets were sold forthe matinee performance, including all of the seats onthe 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.