Multiplying fractions with an area model

What the area model shows

The area model shows multiplying fractions as overlapping shading on one whole square. The first fraction, such as 2/3, shades columns in blue. The second, such as 3/4, shades rows in yellow. Where they overlap, the square turns purple: that is 3/4 of 2/3. The square is now cut into small rectangles, 3 columns by 4 rows, and 2 × 3 = 6 of the 12 are purple, so the product is 6/12. The readout gives the product, simplified when it can be, and a toggle hides it so students can count first.

Why the picture matters

"Multiply the tops and multiply the bottoms" is an easy rule to follow and a hard one to believe. The area model shows where each number comes from: the denominators decide how many columns and rows cut up the whole, and the numerators decide how many of them are shaded. It also explains a result that surprises many students, that multiplying two fractions less than 1 gives a smaller answer. Taking 3/4 of something that is already less than a whole can only make it smaller.

Common mix-ups

• Expecting multiplication to make things bigger. 1/2 × 1/2 = 1/4, because you are taking half of a half.
• Finding a common denominator first. That step belongs to adding and subtracting. To multiply, the denominators simply multiply.
• Counting all the shading instead of the overlap. The product is only the purple part, where both shadings meet.
• Forgetting to simplify. 2/3 × 3/4 is 6/12, which is the same amount as 1/2.

A 10-minute activity

• Minutes 1 to 2: set 1/2 × 1/2. Ask what fraction of the square is purple and why.
• Minutes 3 to 5: hide the product and set 2/3 × 3/4. The child counts the purple rectangles and the total, writes the fraction and simplifies it, then shows the product to check.
• Minutes 6 to 8: ask the child to predict, then check, 3/5 × 1/3, 3/5 × 2/3 and 3/5 × 3/3.
• Minutes 9 to 10: press Challenge and find a pair that makes the target. Then look for a second pair.

Where it fits by grade

• Grade 5: multiplying a fraction by a fraction (Common Core 5.NF.B.4), finding the area of a rectangle with fractional side lengths (5.NF.B.4b), and multiplication as scaling, including why multiplying by a fraction less than 1 gives a smaller product (5.NF.B.5).
• Grade 6: multiplying fractions inside fraction division, ratio and percent problems.
• England: Year 6 pupils multiply simple pairs of proper fractions and write the answer in its simplest form, such as 1/4 × 1/2 = 1/8.

Learning more

Multiplying fractions is a skill on Eurekami's skill map, right after the equivalent fractions and simplifying skills it relies on. Lessons draw area models from the numbers in each question, and Daily Practice brings the skill back when it starts to fade.