Start with folding, not numbers
Give every child three identical paper strips. Fold the first in half and shade one part. Fold the second in half and then in half again, and shade two of the four parts. Fold the third into eighths and shade four. Line them up: the shaded amount is exactly the same each time. That is what equivalent means, and children have made it with their hands before anyone writes a fraction.
Only now label the strips one half, two quarters and four eighths. The notation is describing something they can see.
Use a fraction wall to find families
A fraction wall (a whole split into halves, thirds, quarters, fifths, sixths, eighths, tenths and twelfths, stacked) lets children find equivalent fractions by looking straight down. Everything that lines up with one third is a member of the same family: two sixths, four twelfths. Everything that lines up with three quarters is another family: six eighths, nine twelfths.
Ask children to write every fraction on the wall that equals one half. They will find two quarters, three sixths, four eighths, five tenths and six twelfths, and the pattern in the numbers (the top is always half the bottom) tends to jump out without being told.
Then introduce the rule, with the reason
Multiplying the numerator and the denominator by the same number gives an equivalent fraction. The reason: you are cutting every piece into the same number of smaller pieces. Cutting each quarter into three gives twelfths, and the three shaded quarters become nine shaded twelfths. Nothing has been added or taken away, so the amount is unchanged.
Worked example: two thirds. Multiply both parts by 2 to get four sixths. By 3 to get six ninths. By 5 to get ten fifteenths. All equal two thirds. On a fraction wall, four sixths sits exactly under two thirds.
Dividing both parts by the same number works the same way in reverse, and that is simplifying. Twelve sixteenths: divide both by 4 to get three quarters. Children who can find equivalent fractions in both directions are ready to compare and add fractions with different denominators.
A teaching sequence that works
- Fold and shade strips: halves, quarters, eighths. Then thirds, sixths, twelfths.
- Fraction wall: find every fraction equal to a given one. Write the families down.
- Spot the pattern: what happens to the numerator and denominator across a family?
- The rule: multiply or divide both parts by the same number. Always ask what the multiplier was.
- Missing-number problems: three fifths equals something over 20. What was 5 multiplied by? Do the same to 3.
- Simplify: divide out common factors until none are left. Link it to the factors work children have already done.
- Apply: compare two fractions by rewriting them with the same denominator.
Worked examples with answers
Which is bigger, three quarters or five eighths? Rewrite three quarters as six eighths. Six eighths is bigger than five eighths, so three quarters is bigger.
Fill the gap: two sevenths equals something over 21. 7 times 3 is 21, so 2 times 3 is 6. Six twenty-firsts.
Simplify eighteen twenty-fourths. Both share a factor of 6: 18 divided by 6 is 3, and 24 divided by 6 is 4. Three quarters.
Is four tenths equivalent to two fifths? Divide both parts of four tenths by 2: two fifths. Yes.
Common mistakes and how to fix them
- Adding instead of multiplying: turning two thirds into three quarters by adding one to each part. The strips show three quarters is bigger, so it cannot be equivalent. Go back to cutting pieces.
- Multiplying only the numerator or only the denominator. Ask: did you cut every piece the same way?
- Believing a fraction with bigger numbers is a bigger amount. Eight sixteenths looks bigger than one half but is the same. The fraction wall settles it.
- Stopping simplifying too early: leaving six eighths as six eighths when three quarters is simpler. Teach children to check for a common factor one more time.