Equivalent fractions

What is this skill?

Equivalent fractions are different names for the same amount. 1/2, 2/4, 3/6 and 4/8 all describe the same share of a whole. In fourth grade children learn why: if you cut every part of a fraction into the same number of smaller equal parts, the amount does not change, only the number of pieces does.

In symbols, multiplying the top and bottom of a fraction by the same number gives an equivalent fraction: 2/5 becomes 6/15 when both numbers are multiplied by 3. Dividing the top and bottom by a shared factor works the other way and simplifies the fraction. Children see this with fraction bars and number lines first, where the equal shares line up, and then meet the rule as a shortcut for what the pictures showed.

Why it matters

Equivalent fractions are the key that opens the rest of fraction work. To compare 5/8 and 2/3, or to add 1/4 and 1/6, a child has to rewrite them with the same denominator, and that means finding equivalent fractions. Without this skill, fractions stay a set of unrelated rules.

It also teaches a wider idea in math: the same quantity can be written in many ways, and choosing a convenient way is part of problem solving. That idea comes back with decimals, ratios and algebra.

Common mistakes

• Adding the same number to the top and bottom instead of multiplying. 1/2 plus 2 on each part gives 3/4, which is a different amount.
• Multiplying only the denominator, or only the numerator, so the fraction changes size.
• Believing that a bigger denominator means a bigger fraction. 1/8 is smaller than 1/4, because the pieces are smaller.
• Not seeing that the fraction is the same when the picture is cut differently. Draw both and shade them to compare.
• Treating 2/4 and 1/2 as unrelated numbers because they look different. Shading both on a bar shows the same amount.

How to help at home

• Fold a strip of paper in half, then half again, then again, and write the fraction for the same shaded part each time.
• Cut a sandwich or pizza in different ways and ask if the portion size has changed. A half is still a half.
• Ask “what did we multiply the bottom by?” and then “what must we do to the top?” Keep the question the same each time.
• Use a ruler marked in inches. Halves, quarters and eighths line up and show equivalents at a glance.
• Make a fraction wall from strips of paper in different colors, with each row cut into a different number of equal parts.

Worked examples

These are real questions from the Tutorsaurus equivalent fractions practice, with the answers worked out.

1. Example 1

   Which fraction is equivalent to 2/4?

   Choices: 6/13, 5/7, 6/12, 7/12

   Answer: 6/12. 6/12 = 2/4 because both parts were multiplied by 3.

2. Example 2

   Which fraction is equivalent to 1/3?

   Choices: 3/9, 4/9, 4/6, 3/10

   Answer: 3/9. 3/9 = 1/3 because both parts were multiplied by 3.

3. Example 3

   Which fraction is equivalent to 2/5?

   Choices: 4/10, 5/10, 4/11, 4/7

   Answer: 4/10. 4/10 = 2/5 because both parts were multiplied by 2.