Understanding Fractions: Meanings, Models, and Applications

5 Meanings of Fractions

Part-Whole

It goes well beyond shading a region. The circle model is particularly effective in illustrating the part-whole relationship. Focus: How many parts.

Measurement

It involves identifying a length and then using that length as a measurement piece to determine the length of an object. Focus: How much rather than how many parts.

Division

This is often not connected to fractions, which is unfortunate. Consider the idea of sharing $10 with 4 people; each person will receive 1/4 of the money or $2.50.

Operator

Fractions can be used to indicate an operation, as in 4/5 of 20 squares. This situation indicates a fraction of a whole number, and students might be able to use mental math to solve it. Knowing how to represent fractions doesn't mean students will know how to operate with them.

Ratio

This can be part-part or part-whole, so students need to pay attention to the context of the problem. For example, the ratio 3/4 could be the ratio of those wearing jackets (part) to those not wearing jackets (part), or it could be the part-whole, meaning those wearing jackets (part) to those in the class (whole).

3 Types of Models for Fractions

Area

Circular fraction piece models are the most used area model. One advantage of the circular model is that it emphasizes the part-whole concept of fractions and the meaning of the relative size of a part to the whole. Commercial versions of area models are available in a wide variety, including circular and rectangular pieces.

Length Models

With length models, lengths or measurements are compared instead of areas. Either lines are drawn and subdivided or physical materials are compared on the basis of length. Length models are very important in developing student understanding of fractions. Number lines help students understand a fraction as a number and help develop other fraction concepts.

Set Models

The whole is understood to be a set of objects, and subsets of the whole make up fractional parts. For example, 3 objects are 1/4 of a set of 12 objects. The set of 12, in this example, represents the whole.

Grouping and Regrouping

Grouping

Counting by groups. Example: 53 - "5 tens and 3 ones."

Regrouping

Flexible thinking about place values should be practiced prior to decimals. Students should be thinking about regrouping 2,451 into 24 hundreds, 245 tens, or 2,451 ones.

Additive vs. Multiplicative Comparison

Additive Comparison:

• 8 & 12
• +3
• 11 & 15

Multiplicative Comparison:

• 3 over 8, 12
• 38% - 3/8
• 25% - 3/12

Real-World Application

3 widgets for $2.40. How much would 10 cost at that price?

Widgets: 1 --- 3 --- 9 --- 10

Money: $0.80 --- $2.40 --- $7.20 --- $8.00