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Fraction Region Models, Set Models, and Linear Models

This lesson explains the three main ways fractions are pictured in elementary math: region (area) models, set models, and linear (number line) models. Students learn to identify the whole, count equal parts, and match a shaded picture or a point on a line to a fraction like three fourths.

Introduction

A fraction like \( \frac{3}{4} \) can be pictured in more than one way. Sometimes it is a shaded piece of a shape, sometimes it is a small group inside a larger collection, and sometimes it is a single point sitting on a line. These three pictures are called the region model, the set model, and the linear model. Learning to move between them helps you see that a fraction is really just a comparison between a part and a whole, no matter what that whole looks like.

Before working through these models, it helps to be comfortable with the basic idea of a fraction from the Introduction to Fractions lesson, since every model below is built on the same numerator-over-denominator idea.

Region (Area) Models

A region model, also called an area model, starts with one whole shape, usually a rectangle or a circle. The shape is split into equal-sized parts, and some of those parts are shaded. The denominator tells you how many equal parts the whole was cut into, and the numerator tells you how many of those parts are shaded.

The rectangle is cut into 4 equal columns, and 3 are shaded, so the shaded region shows \( \frac{3}{4} \).

The most important rule in a region model is that every part must be the same size. If the rectangle above were cut into four pieces of different widths, shading three of them would not represent \( \frac{3}{4} \), because a fraction only makes sense when the whole is split into equal shares. For more shapes and shading practice, see Fractions - Pictorial Representations.

Set Models

A set model uses a group of separate objects instead of one connected shape. The whole is the entire group, and the fraction compares the number of objects with a certain property (shaded, circled, chosen) to the total number of objects in the group.

Out of a set of 4 circles, 3 are shaded, so \( \frac{3}{4} \) of the set is shaded.

Set models are useful for everyday counting situations, such as 3 red marbles out of a bag of 4 marbles, or 3 students out of a group of 4 who chose pizza. Unlike a region model, the objects in a set do not need to touch or line up, but they do need to be counted as equal, individual items.

Linear Models (Number Lines)

A linear model places a fraction as a point along a number line. The segment from 0 to 1 represents one whole, and it is split into equal-length pieces that match the denominator. Counting pieces from 0 tells you which fraction each mark represents.

Dividing the segment from 0 to 1 into 4 equal parts locates \( \frac{3}{4} \) three steps from 0.

Linear models make it easy to compare fraction sizes and to see equivalent fractions, since a fraction like \( \frac{2}{4} \) lands on the exact same point as \( \frac{1}{2} \). A closer look at these ideas is available in Fractions - Number Lines.

Comparing the Three Models

The three models all describe the same fraction, but they highlight different situations:

A region model works well when the whole is one continuous object, like a pizza, a garden, or a sheet of paper.
A set model works well when the whole is a collection of separate, countable objects.
A linear model works well when you need to compare fraction sizes, measure a distance, or line fractions up in order.

Worked Example

Suppose a class has 8 students, and 6 of them ride the bus to school. This is naturally a set model, so the fraction of students who ride the bus is \( \frac{6}{8} \). If you wanted to show this on a region model, you could shade 6 out of 8 equal parts of one rectangle. On a number line, you would divide the segment from 0 to 1 into 8 equal parts and mark the point 6 steps from 0. All three pictures represent the exact same fraction, \( \frac{6}{8} \), which is equivalent to \( \frac{3}{4} \).

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