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Hands-on Division with Fractions for Student Understanding

Jim Olsen, Ph.D., retired

Western Illinois University

NCTM Annual Conference 2024 ~ Chicago, IL

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Description

The goal of this approach is to enhance student understanding (in grades 4-8) of division, fractions, & division involving fractions.

Shared will be:

Progression of problems: 8 objectives, from 4.OA.3 through 6.NS.1
Two ways to model division: with physical manipulatives & with a online tool (Polypad).
Handouts with problems: for each of the 8 objectives in the Progression.

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Outline

Introduction - The two meanings of division and the challenges of crossing over to fractions.
First Example - using Connecting Fraction Tiles
Rationale
Fractions on Polypad (from https://mathigon.org/) - demo/notes
Second Example - using Polypad.
The Division Progression - 8 objectives
More examples.
Files on the website.
Conclusion and your feedback.

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Introduction

There are two primary meanings of division:

Group Size Unknown and
Number of Groups Unknown.

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Introduction

See the website for more comments on the two primary meanings of division.

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Introduction - Examples of Basic Division Scenarios

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Introduction - Examples of Basic Division Scenarios

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Introduction - the problem of moving to fractions.

Fractional Pieces

In group size unknown there may be some pieces that need to be broken into fractional parts (in order to “fairly divide”)

Fractional Groups

The number of groups might need to be a fraction.

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Introduction - the problem of moving to fractions.

To get Fractional Pieces we like

Connecting Fraction tiles (physical)
Polypad (from https://mathigon.org/) (virtual)

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First Example

A biologist is adding liquid protein to cell samples to study how the cells will grow. She reported in her study that he divided ¾ of an ounce of liquid protein evenly among 6 cell samples. How much liquid protein did she give to each cell sample?

We’ll use Connecting Fraction Tiles

Objective 5 - HW #2.

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Rationale

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Rationale

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Rationale

We want to build connections in the brain (understanding) by connecting:

Real-world problem (context)
Meaning (of division)
Diagram (representation)
Explanation
Number sentence (which includes the answer).

As you probably have gathered, we are trying to build conceptual understanding. Procedural skills are also important, and should be taught, but not the focus here.

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Comments and Disclaimers

Comment 1: We do think students should be able to draw things and explain things.

Comment 2: We’re not going to round (up, down, or off). Mathematically, you can have 4⅓ busses.

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Comments and Disclaimers

Disclaimer 1: We are showing ways to represent and explain division. There is flexibility. Students may find other ways to represent and explain their process.

Disclaimer 2: For each objective, we’ve attempted to write problems for both meanings of division. If you find a question unreasonable, don’t use it.

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Fractions on Polypad - https://mathigon.org/

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Fractions on Polypad - ⋮More tools

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Second Example

Some students are sharing four bags of candy. If each student gets ⅓ of the bag, how many students get candy?

We’ll use Polypad

Objective 4 - HW #2.

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The Division Progression

The Division Progression is a listing of 8 objectives involving division when fractions are involved (as part of a problem statement, answer, or both). Each objective is a problem type.

Keyed to CCSSM.
Grades 4 through 6 are listed…However, considering the challenging nature of these problems, these problem types should probably be used frequently through the rest of middle school and into high school.

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The Division Progression-Go to the website.

Document: The Division Progression and Teacher Notes

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More examples (#3)

We have ½ cup of flour remaining. The recipe calls for 4 cups of flour. What fraction of the recipe can we make?

We’ll use tiles.

Objective 3 - HW #2.

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More examples (#4)

A landscape architect bought 2 tons of soil. Each flower bed in the garden needs ⅗ of a ton of soil. How many flower beds can she fill?

We’ll use Polypad.

Objective 6 - Example #1.

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More examples (#5) The fractional group

We have 8 ¾ cups of scrambled eggs. We need 3 ½ servings (for three adults and one child). What is the serving size?

We’ll use Polypad.

Objective 8 - Example #2.

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More examples

Try these: Think-Pair-Share

You have 4 ⅔ cups of hot chocolate powder. Each serving requires ⅔ cup of hot chocolate powder. How many servings can you make? Objective 7 - HW#2.

You have 3 ½ feet of ribbon. To make a bow, you use ¾ feet of ribbon. How many bows can you make? Objective 8 - Assessment.

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Concluding Remarks:

Understanding division is tough.
Understanding fractions is tough (and A and B are related)
Understanding division with fractions is even tougher.
…

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Concluding Remarks:

Our goals and principles should still be:

Strive for multiple representations,
Understanding through multiple connections,
Conceptual and procedural understanding,
Problem solving,
Real-world Contexts.

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Questions/Comments

What Questions/Comments do you have?

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Thank You.

Jim Olsen

JR-Olsen@wiu.edu

Western Illinois University, retired.

Files (and slides) on the website.

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