In plain words
Why does multiplying by a fraction make a number smaller?
Because multiplying by a fraction less than one takes only part of an amount. 1/2 × 1/2 means one half of one half, which is 1/4, and that’s smaller than the 1/2 we started with.
- The big idea
- Multiplying by a fraction less than one takes only part of an amount, so the answer comes out smaller: 1/2 × 1/2 is 1/4.
- The common confusion
- Multiplying always makes a number bigger, so 1/2 × 1/2 can’t be smaller than 1/2. It often sounds like this: “Times always makes it bigger, so (one half) × (one half) has to be more than (one half).”
Why it works
With whole numbers, multiplying by 2, 3 or 10 makes a number bigger, so it is natural to expect every multiplication to do the same.
Multiplying by a fraction means taking that fraction of an amount. 1/2 × 1/2 means one half of one half.
On an area model, shade half of a rectangle one way, then half of it the other way. The part shaded both ways is half of the first half, which is one of four equal parts. So 1/2 × 1/2 = 1/4, which is smaller than the 1/2 we started with.
The same happens whenever a positive amount is multiplied by a number between 0 and 1, because we keep only part of it. 1/3 × 6 = 2, which is less than 6.
Multiplying by exactly 1 leaves an amount unchanged, and multiplying by a number greater than 1, such as 3/2, makes it bigger. The size of the multiplier tells us what to expect.
A worked example
A vegetable patch covers 3/4 of a garden, and tomatoes grow in 1/3 of the patch. The tomatoes cover 1/3 of 3/4 of the garden, and 1/3 × 3/4 = 3/12, which is 1/4 of the garden. That is less than the patch, because the tomatoes use only part of it.
Why this confusion is so common
Most of us first meet multiplication as repeated addition of whole numbers, where (apart from multiplying by 0 or 1) the answer is always bigger than either number. That experience becomes a belief that multiplication makes bigger. Researchers have documented it for decades, in learners of every age, and it affects which operation people choose in word problems. Reading × as “of” and drawing an area model let the size of the answer make sense.
Try this away from the screen
Fold a sheet of paper in half one way and color one half. Unfold it, fold it in half the other way and draw stripes on one half. The part that is both colored and striped is half of a half, which is a quarter of the sheet. Then try folding the second way into thirds and striping one third, to find 1/3 of 1/2.
Helpful things to say
| Instead of | Try |
|---|---|
| “Multiplying makes numbers bigger.” | “What does it mean to take 1/2 of something? Will we end up with more or less?” |
| “Just multiply the tops and multiply the bottoms.” | “Read × as ‘of’. What is half of a half?” |
Practice questions
These questions check the same idea with new numbers and new pictures. The best way to use them is one at a time, with pictures you can change and a reply to every answer. If you’d rather read them here, each answer stays hidden until you open it, so whoever is practicing can try first.
Try these questions one at a timeQuestion 1
Use the area model to show (two thirds) of (three fourths).
Show the answer
To answer it, shade (three fourths) of the rectangle one way and (two thirds) the other way; the overlap is the answer. Shade 3 of 4 columns to show (three fourths), then shade 2 of 3 rows to take (two thirds) of it. The overlap is 6 of the 12 equal parts, so (two thirds) of (three fourths) is (six twelfths), or (one half), which is less than (three fourths).
Question 2
What is (three fourths) × 8? You can use the 8 counters to help.
The choices are 2, 6 and 24.
Show the answer
The answer is 6. (three fourths) × 8 means (three fourths) of 8. If you share the 8 counters into 4 equal groups of 2 and take 3 of the groups, you get 6. That’s less than 8, because (three fourths) is less than one.
Question 3
What is (one half) × (three fourths)?
The choices are (one fourth), (three eighths) and (six fourths).
Show the answer
The answer is (three eighths). (one half) × (three fourths) means half of (three fourths). On the line, (three fourths) is six eighths from 0, and half of six eighths is three eighths. So (one half) × (three fourths) = (three eighths), which is less than (three fourths).
Question 4
Half a pizza is left, and you eat (one fourth) of that half. What fraction of the whole pizza is that?
Show the answer
The answer is (one eighth). The half is 4 of the 8 slices, and (one fourth) of those 4 slices is 1 slice. So (one fourth) of (one half) is (one eighth) of the whole pizza.
Question 5
Kai says (three fifths) × 10 = 30. Can that be right?
The choices are yes and no.
Show the answer
The answer is no. (three fifths) × 10 means (three fifths) of 10. One fifth of 10 is 2, so three fifths is 6. Multiplying 10 by a fraction less than one gives less than 10, so 30 can’t be right.
Questions people ask
Why does multiplying fractions make the answer smaller?
Multiplying by a fraction less than one means taking part of an amount. Half of something is less than all of it, so 1/2 × 1/2 = 1/4, which is less than 1/2. Multiplying a positive amount by a number greater than one still makes it bigger.
Does multiplying by a fraction always give a smaller answer?
No, only when the fraction is less than one and the amount is positive. 3/2 × 4 = 6, which is more than 4, because 3/2 is more than one. Multiplying by a fraction equal to one, such as 4/4, changes nothing.
Why does × mean “of” with fractions?
3 × 4 can mean 3 groups of 4. In the same way, 1/2 × 4 means half of a group of 4, which is 2. Reading × as “of” keeps one meaning for whole numbers and fractions.
How do you multiply two fractions?
Multiply the numerators and multiply the denominators: 1/2 × 1/2 = 1/4. The area model shows why, because halving one way and halving the other way cuts the whole into 2 × 2 = 4 equal parts, and 1 × 1 = 1 of them is shaded both ways.
Words to know
- Area model
- An area model is a rectangle split one way to show one fraction and the other way to show a second, and the overlap shows their product.
- Fraction of a set
- A fraction of a set is a fraction of a group of objects, and you find it by sharing the group out into equal groups.
- Whole
- The whole is the complete thing that a fraction is part of. It might be a pizza, a length on a number line or a group of objects.
Curriculum connections
We don’t assume any particular curriculum, but for teachers in the United States, here’s how this lesson lines up with the Common Core State Standards:
- 5.NF.B.4.a
Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = (ac)/(bd).)
- 5.NF.B.4.b
Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
- 5.NF.B.5.b
Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.
Related lessons
Division Can Make Bigger
Why does dividing by a fraction give a bigger answer?
Dividing 1/2 by 1/4 asks how many fourths fit into a half, and two of them do, so the answer is 2. · About 3 minutes
Fractions of Collections
What is (one fourth) of 12?
A group of things can be a whole too, and you find 1/4 of 12 by sharing the 12 into 4 equal groups. · About 3 minutes
What’s the Whole?
Is half of a large pizza the same amount as half of a small pizza?
Half of a big pizza is more than half of a small one, because a fraction is always a fraction of something. · About 3 minutes
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