Helping whole numbers × fractions make sense
This multiplication is best introduced as repeated equal lots of the same fraction. Build the lots first. Only then compress the picture into multiplication.
Start with repeated lots
Before writing a multiplication sign, make several identical fractional amounts.
Make one group containing 2 third-pieces. Copy that group 4 times. Ask: “How many third-pieces are there altogether?”
Only after this is clear, write 4 × 2/3 = 8/3.
Use different kinds of models
As in earlier lessons, vary what counts as the whole. A learner may understand one representation and not another.
Use circles, rectangles or strips. Three lots of 1/4 means placing three quarter-pieces together.
If 8 counters make one whole collection, then 3 lots of 2/8 means three identical groups of two counters.
Three lots of 1/4 of £1 means 25p + 25p + 25p = 75p. The same with $1 gives 25¢ + 25¢ + 25¢ = 75¢. Many children find this easier to grasp than abstract fraction notation.
Keep the piece size fixed
The whole number tells us how many lots we have. The numerator tells us how many fraction-pieces are in each lot. The denominator still tells us the size of each piece.
How many lots?
How many pieces in each lot?
What size are the pieces?
Useful phrase: “We are making more lots, not changing the size of the pieces.”
Connect to repeated addition
This is a useful place to connect fraction multiplication to multiplication the learner already knows.
For example:
If ordinary multiplication is shaky, keep a times-table grid nearby. Do not turn a fraction lesson into a memory test for multiplication facts.
When the answer is bigger than one
Repeated lots can easily produce an improper fraction. This is a good opportunity to connect back to Lesson 5.
Four lots of 2/3 make 8/3.
Now regroup the 8 thirds into wholes: 8/3 = 2 2/3. The multiplication gives the total number of thirds; the mixed number helps us picture the amount.
Common misconceptions to watch for
If the learner changes thirds into twelfths in 4 × 2/3, return to the four physical groups. The pieces are still thirds; there are simply more of them.
Keep saying “4 lots of…” rather than immediately presenting 4 × 2/3 as a symbolic object.
Those are related but conceptually different starting stories. In this lesson, make the repeated-lots interpretation explicit first.
Questions that reveal understanding
Cross-topic links worth noticing
4 lots of 1/4 of £1 make £1; 4 lots of 1/4 of $1 make $1. Three lots make 75p or 75¢.
Three lots of a quarter-hour make 45 minutes; four lots make one hour.
Three lengths of 1/2 metre make 1 1/2 metres.
This is the same structural idea as 4 × 3 meaning 3 + 3 + 3 + 3.
When to move on
The learner is ready when they can:
- build a whole-number number of identical fractional lots;
- translate the picture into repeated addition;
- explain why the denominator stays unchanged;
- multiply the whole number by the numerator as shorthand for counting all the pieces;
- regroup an improper result into a mixed number when useful.
Use the learner lesson alongside this guide
The learner page builds repeated lots visually, then compresses them into multiplication and reconnects answers greater than one to mixed numbers.
Open Lesson 6: Whole numbers × fractions →