Lessons · Teaching · equivalent fractions and simplest form

Equivalent fractions: the same number, cut more finely

Multiplying the numerator and the denominator by the same number does not change the value, because it cuts every piece into the same number of smaller pieces and takes proportionately more of them. One half and two quarters are the same point on the number line.

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What it is for

This single fact is what makes adding fractions possible at all, and it is also what makes simplifying possible. A pupil who can see that 6/8 and 3/4 are one number, written twice, has the tool for the next three lessons. One who thinks simplifying makes a fraction smaller has a misconception that will cost them.

How to think about it

To simplify, find the largest number that divides both top and bottom, and divide both by it. To build up, multiply both by the same number. Check by asking whether the value moved: it must not have.

Worked example

1/2 = 2/4 = 3/6 = 50/100, all the same point on the line

One number with infinitely many names.

Simplify 18/24: the largest common factor is 6, so divide both by 6 to get 3/4

Finding the largest factor in one go beats simplifying three times.

Build 3/4 into twelfths: multiply both by 3 to get 9/12

Building up is how two fractions are made ready to add.

Simplifying does not make the fraction smaller, only the digits

18/24 and 3/4 are equal. Only the writing got shorter.

Your turn

Name what must be done to the numerator and denominator to keep a fraction's value.

Multiply or divide both by the  number

The trap

Adding the same number to top and bottom instead of multiplying. 1/2 with 1 added to each is 2/3, which is a different number. The rule is multiply or divide, never add or subtract.

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