What's Really Behind "Multiply and It Gets Smaller" — Fraction Multiplication and Division
"Multiply, and the answer gets smaller." "Divide, and it gets bigger." Once grade 6 gets into fraction multiplication and division, the sense that worked reliably with whole numbers keeps flipping on its head.
Why does the sense that worked with whole numbers fall apart here? Because the meaning of fraction multiplication and division itself has been quietly rebuilt, a little at a time, grade after grade.
Why the answer moves the wrong way
Cause 1: "multiplying means bigger, dividing means smaller" stops holding
Multiply by 2/3, and the result is smaller than the original number. Divide by 4/5, and it's bigger. The sense that held true throughout the world of whole numbers — "multiplying makes it bigger, dividing makes it smaller" — stops working the moment the other number is smaller than 1. This isn't a phenomenon unique to fractions; it had already started happening with Multiplying and Dividing Decimals. On the meaning of fraction multiplication and division, the commentary states: "Teach so that children can apply the meaning of multiplication and division as it was extended from multiplication and division of whole numbers to multiplication and division of decimals." (Grade 6, Numbers and Calculation) The sense doesn't suddenly break down with fractions — the experience of it breaking down has been building for a while already.
Cause 2: multiplying the whole-number and fraction parts of a mixed number separately
When multiplying 1 1/2 by 2/3, some answers never convert the mixed number to an improper fraction at all. What stands out is an answer that multiplies the whole-number part (1) and the fraction part (1/2) by 2/3 separately. Fraction multiplication is a procedure of multiplying denominators together and numerators together, so the procedure itself doesn't work while a number stays a mixed number. If the answer comes out much smaller than it should, suspect this pattern. Looking into it, I found the commentary states: "For multiplication and division of fractions, expressing a number as an improper fraction rather than a mixed number makes the calculation easier to carry out." I didn't know this until I read that sentence. diglabo's worksheets also give the multiplication unit a dedicated step — "mixed number × whole number, whole number × mixed number" — so a child can practise this before moving on to fraction × fraction.
Cause 3: mixing up which number gets flipped in division
Fraction division works by flipping the divisor into its reciprocal and turning the calculation into multiplication. A common mistake here is flipping the dividend instead, which turns the answer upside down. In division with three fractions, some answers flip only the first divisor and multiply the rest as they are.
How to teach it at home
Step 1: Show what multiplying and dividing by a number less than 1 looks like
Take 2/3 of a strip of tape, and it ends up shorter than the original. Draw this out, and it breaks down the assumption that "multiplying always makes it bigger." Do the same with division — confirm, using a concrete quantity, that dividing by 4/5 makes the answer bigger.
Step 2: Convert mixed numbers to improper fractions before you start calculating
After your child writes out a multiplication or division expression, check first whether the mixed number has been converted to an improper fraction. Build the habit of checking the same order every time: convert first, then calculate — not calculate, then convert.
Step 3: Point to the number that gets flipped
In a division expression, have your child press a finger on the divisor (the fraction after the ÷) before flipping it. Adding this step — confirming with both eyes and hand which number gets flipped — cuts down on the mistake of flipping the dividend instead.
What to avoid
On fraction calculation, the commentary states: "rather than teaching needlessly complicated calculations, [the aim is] for children to be able to apply fraction calculation to daily life and future learning." At home, too, there's no need to have your child work through fractions with more digits or more complicated reductions just for extra practice. Figuring out exactly which of the patterns above your child is stuck on is the faster route.
How to practise
- Multiplication of Fractions starts with fraction × whole number and whole number × fraction, goes through a dedicated mixed-number stage — mixed number × whole number, whole number × mixed number — and then moves on to fraction × fraction. Treat the mixed-number stage lightly, and the trouble surfaces later at fraction × fraction
- Division of Fractions starts with fraction ÷ whole number, whole number ÷ fraction, and mixed number ÷ whole number, then moves on to proper fraction ÷ proper fraction, forms that include mixed numbers, and forms that mix fractions and decimals
Mixed number ÷ whole number is made its own step for the same reason.
- Both branches go on to cover area, amount per unit quantity, word problems asking for a compared amount, and multiplication and division with three fractions. The commentary also stresses "not teaching needlessly complex calculations, but making fraction calculation something children can put to use in daily life and in later learning." There's no need to force your child through fractions with awkwardly large numbers
Frequently asked questions
Q. Why does division involve multiplying by the reciprocal?
A. The commentary states: "teach that division where the divisor is a fraction can be converted to multiplication by using the reciprocal, and that multiplication of whole numbers, decimals, and fractions can always be carried out by expressing them as fractions." Converting division into multiplication is a way of making calculations look the same. It's also meant to let children handle everything using just the multiplication procedure they've already learned. Telling your child that nothing about the underlying idea is changing — it's just being made consistent — tends to make more sense to them.
Q. When should reducing happen?
A. It's fine to multiply denominators and numerators first and then reduce the result all at once. But when the numbers get big and hard to work with, reducing by a common factor before multiplying makes it more manageable. Either order gives the same answer.
Multiply, and it gets smaller. Divide, and it gets bigger. This mismatch isn't some special wall unique to fractions. It's the sense your child built up back in the whole-number days getting rearranged, just this once. When the answer doesn't match intuition, there's no need to panic.
This article is written by a parent who builds diglabo. I am not a teacher or an education specialist. The general guidance here is based on how the topic is taught in Japanese primary schools, checked against Japan's national curriculum guidelines.
diglabo's worksheets follow the Japanese curriculum. What that means if you are outside Japan