Number & Operations — Fractions

Understanding Fractions: Number Lines and Equivalence

Grade 3

✓By the end of this lesson students will be able to understand a fraction as a number representing parts of a whole.
✓By the end of this lesson students will be able to locate and represent fractions on a number line diagram.
✓By the end of this lesson students will be able to recognize and generate simple equivalent fractions.
✓By the end of this lesson students will be able to explain why fractions are equivalent using visual models and number lines.

Key concepts

What is a Fraction?

A fraction is a number that represents a part of a whole. It is made up of two parts: a numerator and a denominator. The denominator tells you how many equal parts the whole is divided into. The numerator tells you how many of those equal parts you are considering. For example, in the fraction 3/4, the '4' (denominator) means the whole is divided into 4 equal parts, and the '3' (numerator) means we are looking at 3 of those parts.

Fractions on a Number Line

A fraction can be shown as a point on a number line. To place a fraction like a/b on a number line, you first look at the denominator (b). This tells you to divide the space between 0 and 1 (which represents one whole) into 'b' equal parts. Then, you count 'a' of those equal parts starting from 0 to find the location of the fraction a/b. Each mark represents a unit fraction, like 1/b, 2/b, and so on.

Equivalent Fractions

Equivalent fractions are fractions that name the same amount or represent the same point on a number line, even though they have different numerators and denominators. For example, 1/2 and 2/4 are equivalent fractions because they both represent half of a whole. You can often find equivalent fractions by dividing or multiplying both the numerator and the denominator by the same non-zero number, or by using visual models like fraction bars or number lines to show they cover the same amount.

Key facts to remember

1A fraction represents a part of a whole.
2The denominator tells how many equal parts the whole is divided into.
3The numerator tells how many of those equal parts are being considered.
4On a number line, the space between 0 and 1 represents one whole.
5To locate a fraction a/b on a number line, divide the whole into 'b' equal parts and count 'a' parts from 0.
6Equivalent fractions name the same amount or are at the same point on a number line.
7You can use visual models like fraction bars or number lines to show equivalent fractions.

Worked examples

Example 1

Locate the fraction 3/4 on a number line.

IDraw a number line from 0 to 1.

IILook at the denominator, which is 4. This means we need to divide the whole (the space between 0 and 1) into 4 equal parts.

IIIMark three equally spaced points between 0 and 1. These points will represent 1/4, 2/4, and 3/4.

IVCount 3 parts from 0 and mark the point. This point is 3/4.

Answer

The point representing 3/4 is located at the third mark from 0 on a number line divided into 4 equal parts.

Remember that the first mark after 0 is 1/4, the second is 2/4, and the third is 3/4.

Example 2

Show that 1/2 is equivalent to 2/4 using a visual model.

IDraw two identical rectangles or circles to represent two wholes.

IIFor the first whole, divide it into 2 equal parts and shade 1 of those parts to represent 1/2.

IIIFor the second whole, divide it into 4 equal parts and shade 2 of those parts to represent 2/4.

IVCompare the shaded areas of both models. You will see that the shaded amount for 1/2 is exactly the same as the shaded amount for 2/4.

Answer

Both visual models show the same amount shaded, demonstrating that 1/2 and 2/4 are equivalent fractions.

Make sure your two whole shapes are exactly the same size before dividing them.

Example 3

Find a fraction equivalent to 1/3 using a number line.

IDraw a number line from 0 to 1. Divide it into 3 equal parts and mark the point for 1/3.

IIDraw a second identical number line directly below the first one, also from 0 to 1.

IIIFor the second number line, choose a denominator that is a multiple of 3, such as 6. Divide this number line into 6 equal parts.

IVObserve where the point for 1/3 on the first number line aligns with a point on the second number line. The point for 1/3 aligns with the second mark on the number line divided into 6 parts, which is 2/6.

VTherefore, 1/3 is equivalent to 2/6.

Answer

1/3 is equivalent to 2/6.

You can also think of multiplying the numerator and denominator of 1/3 by 2 to get 2/6.

Common mistakes

✗Not dividing the whole into *equal* parts when drawing fraction models or number lines.
✗Counting the tick marks instead of the intervals (spaces) when locating fractions on a number line.
✗Confusing the numerator and the denominator, leading to incorrect fraction representation.
✗Thinking that fractions with different numbers in the numerator and denominator can never be equivalent.

Exam tips

★Always draw clear diagrams, such as number lines or fraction shapes, to help you visualize fractions and solve problems.
★When drawing models, use a ruler or grid paper to ensure that all parts are truly equal in size.
★For number line problems, label the whole (0 and 1) and all the unit fractions (like 1/4, 2/4, 3/4) to avoid errors.
★When checking for equivalent fractions, ask yourself if they cover the exact same amount of a whole.

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