Helping a fraction of a fraction make sense
Keep the word of visible. Start with one fraction of the whole, then take a fraction of that selected part. The multiplication rule should emerge from repeated subdivision, not arrive as a trick.
Start with “of”, not “multiply”
Say one third of one fifth before writing 1/3 × 1/5. First show one fifth of the whole. Then split that selected fifth into 3 equal pieces and take one of them.
The whole has now been divided into 5 groups, and each of those groups has been divided into 3 smaller groups.
Preserve the parent pieces
When you subdivide, keep the original grouping visible. If one fifth is split into thirds, the three new pieces should still look as though they belong to that original fifth.
This helps the learner see where the multiplication of denominators comes from: there are 5 parent groups, each split into 3 child pieces.
Use coloured strips, paper rectangles, LEGO rows or MathLink cubes arranged in parent groups. Different pale colours for the parent groups can make the structure easier to follow.
Move from one selected piece to several
Once 1/3 of 1/5 is understood, try 2/3 of 1/7. Start with one seventh, split it into 3, and select 2 of those 3 smaller pieces.
The whole now has 7 × 3 = 21 equal small pieces, and 2 are selected, giving 2/21.
“How many parent pieces did the whole have?” “How many smaller pieces did each parent become?” “How many small pieces are selected?”
Why the numerators and denominators multiply
Only introduce the general rule after the learner can describe the picture.
Count how finely the whole has been split. If thirds are split into fifths, there are 3 × 5 equal small pieces.
Count how many of those smaller pieces are selected. If 2 of 3 parent parts and 4 of 5 child parts are selected, there are 2 × 4 selected small pieces.
The calculation is a compressed record of the subdivision the learner has already seen.
Money gives a very friendly “of” example
Use money because children often understand the amount before they understand the notation.
One half of one half of £1 is 25p. First take half of £1 = 50p, then half of 50p = 25p.
One half of one half of $1 is 25¢. The same two-step reasoning gives a quarter of the original dollar.
This makes 1/2 of 1/2 = 1/4 feel unsurprising before the symbolic multiplication is shown.
Common misconceptions to watch for
Return to the two-stage picture: first choose one fraction, then take a fraction of that selected amount.
Keep the whole visible while subdividing. The selected part is still part of the same original whole.
Ask the learner to reconstruct one multiplication with a grid or strip whenever the rule becomes detached from the picture.
Questions that reveal understanding
When to move on
The learner is ready when they can build a fraction of a fraction visually, explain the two stages, connect “of” with multiplication, and explain why both numerator and denominator products appear. Simplifying can wait until Lesson 10.
Use the learner lesson alongside this guide
The learner page begins with one third of one fifth, preserves the parent-group structure, then generalises to multiplying numerators and denominators.
Open Lesson 7: A fraction of a fraction →