FRACTIONS:

From Part-Whole to Quotients

Melissa Hedges

DeAnn Huinker

Connie Laughlin

Dan Lotesto

Kevin McLeod

Mary Mooney

Beth Schefelker

Content Goals for the

2009-2010 School Year

  • Develop a conceptual understanding of rational numbers and proportional reasoning.
  • Develop an understanding of student misconceptions of rational numbers and identify teacher moves needed to remediate/address the errors.
  • Support MTLs in linking content ideas to the curricular resources in their buildings.
  • MTLs recognize the 4 levels of cognitive demand and transfer to classroom practice.

Learning Intention

To explore fractions as quotients

We Are Learning To...

  • understand how fractions describe division.

Success Criteria

By the end of the session you will be able to:

Use the idea of fair sharing to develop fraction concepts

Relate the action of fair sharing to the interpretation of fractions as quotients.

What is a fraction?

Consider the following 3rd grade CABS:

Mariah’s cat had 24 kittens. Mariah kept of the kittens. How many kittens does Mariah have?

Solve this problem individually.

Explain your thinking to your table partners.

Is “Part-Whole” sufficient?

This Mystery Box contains something that four people are going to share.

If they share the contents equally, how much will each person get?

Why does it make sense that each person will get of the contents of the box?

Open the Box!

The box contained 3 apples. Now what do we do?

Individually

  • Devise a strategy to divide the 3 apples fairly amongst 4 people.
  • Think
  • Visualize
  • Record using pictures and symbols
  • Place face down in middle of table when done

As a table group:

  • Examine how others at your table approached this task.

Keep this situation in mind as you talk:

If 4 people share 3 apples,

How much is one share?

How many apples is one share?

Is “Part-Whole” Sufficient? (Continued)

Did your strategies for sharing 3 apples amongst 4 people have a straightforward “part-whole” interpretation?

“Unfortunately, fraction instruction has traditionally focused on only one interpretation of rational numbers, that of part-whole comparisons, after which the algorithms for symbolic operation are introduced.”

Lamon, S. (2005). Teaching fractions and ratios for understanding.

What Constitutes A

Fair Share?

  • Total amount of what is being shared is used up.
  • The shares are not overlapping.
  • Each share contains the same amount.

“Fair sharing has been a part of children’s everyday experiences since they were toddlers. They have had the need for fractions when sharing did not work out nicely and they had to break something up.”

Lamon, S. (2005). Teaching fractions and ratios for understanding.

Big Ideas of Fair Sharing

  • Fair sharing means sectioning a whole into equal amounts.
  • To quantify a share it is necessary to give both a number and a unit.

From Fair Shares to Quotients

What is a quotient?

“It is the answer to a division problem.”

What does that mean?

Remember that division is the inverse operation to multiplication.

Division as Inverse to Multiplication

Why is 6 2 = 3 ?

Because 2 x 3 = 6

Why is 3 4 = ?

Because 4 x = 3

A quotient is a missing factor!

Missing Factors

What goes in each box?

5 x = 7

14 x = 175

12 x = 60

3 x =

From Part-Whole to Fair Shares to Quotients

1. Tell the story that matches the picture on the card.

For example:

___ people share ___ items

2. Use the following questions to guide your thinking:

  • How much is one share?
  • How many ______is one share?

3. How would you convince a student that the answer to the fair share problem is a quotient (i.e. a missing factor)?

Quotients on the

Number Line

Draw a number line.

Locate on your number line.

Use your number line to show why

4 x = 3 .

Jeopardy

The answer is . What is the question?

Try to give questions that are appropriate for each interpretation of fractions we have discussed:

  • Part-whole interpretation
  • Fair share interpretation
  • Quotient interpretation

  • Big Ideas
  • Instruction on fractions and rational numbers should not be restricted to the "part-whole" interpretation
  • The "fair share" interpretation of fractions leads naturally to fractions as quotients
  • Middle school students should come to understand that division is the inverse operation to multiplication, so that a qoutient is really a missing factor in a multiplication problem.