Multiplying two fractions using a unit square

The idea

A fraction times a fraction follows the same steps: with a unit square as the whole, divide the multiplicand into as many equal parts as the multiplier's denominator, then multiply that part by the multiplier's numerator.

Half an hour of a slow walk

The tortoise walks 1/4 km in one hour. How far does it walk in half an hour? We need 1/2 × 1/4.

How far does it walk in half an hour?

• 1/8 km
• 1/6 km
• 1/2 km

Divide the fraction into equal parts

The unit square is the whole. The blue rows are the multiplicand. Cut it into columns, then take columns.

The same two steps as before

To find 1/2 × 1/4, divide the 1/4 into 2 equal parts. The whole is now 4 rows × 2 columns = 8 equal parts. One of them is gold: 1/8.

And 6/20 = 3/10 in lowest form. The whole is cut into 5 × 4 = 20 parts, and 2 of them lie in one column.

A fraction times a fraction follows the same steps. On the unit square, divide the multiplicand into as many equal parts as the denominator of the multiplier, then multiply that part by the numerator of the multiplier.

• Product | Step 1: divide | Step 2: multiply
• 1/2 × 1/4 | 1/4 ÷ 2 = 1/8 | 1 × 1/8 = 1/8
• 3/4 × 2/5 | 2/5 ÷ 4 = 2/20 | 3 × 2/20 = 6/20

Notes

A fraction times a fraction follows the same steps. On the unit square, divide the multiplicand into as many equal parts as the denominator of the multiplier, then multiply that part by the numerator of the multiplier.

Check yourself

Shade 1/5 of a unit square, divide it into 3 equal parts and take 2 of them. That is 2/3 × 1/5. What fraction of the whole is it? Type a fraction like 3/10.

Answer: 2/15

5 rows × 3 columns = 15 equal parts. The gold part is 1 row × 2 columns = 2 parts. So 2/3 × 1/5 = 2/15.

To find 3/4 × 2/5 on a unit square, Rhea cuts the whole into 4 + 5 = 9 parts. What is right?

A snail crawls 3/4 metre in one minute. How far does it crawl in 2/3 minute? Type a fraction like 3/10.

Answer: 1/2

The whole has 4 × 3 = 12 parts. The gold part is 3 × 2 = 6 parts: 6/12 = 1/2 metre.

Which steps find 3/4 × 2/5?

• 9 is right: the rows and the columns are added.. The number of cells in a grid is rows times columns, not rows plus columns.
• Only 4 parts, because the multiplier is divided into 4 parts.. Dividing into 4 columns cuts every one of the 5 rows into 4 cells.
• The whole has 5 rows and 4 columns: 5 × 4 = 20 equal parts. — correct. Yes! Rows and columns make a grid, and a grid has rows × columns cells.
• Divide 2/5 into 3 equal parts, then take 4 of them. The steps are swapped. The denominator of the multiplier (4) is for dividing.
• Add 3/4 and 2/5. This is a multiplication story: divide into parts, then take parts.
• Divide 2/5 into 4 equal parts, then take 3 of them — correct. Yes! The denominator 4 makes the parts, the numerator 3 says how many to take.