Multiplying and dividing fractions with area models

What is it about?

Multiplying with an area model: a square is one whole. The first fraction shades vertical strips (blue), the second horizontal strips (orange). Where both overlap (purple) is the product. For $\frac{2}{3} \cdot \frac{3}{4}$ the square is cut into $3 \cdot 4 = 12$ equal cells, and $2 \cdot 3 = 6$ of them overlap:

In general: numerator times numerator, denominator times denominator, so $\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$.

• “Of” means “times”: $\frac{2}{3}$ of $\frac{3}{4}$ of a pizza is $\frac{2}{3} \cdot \frac{3}{4} = \frac{1}{2}$ of a pizza.
• Multiplying does not always make bigger: multiplying by a number less than $1$ makes the result smaller: $\frac{2}{3} \cdot \frac{3}{4} < \frac{3}{4}$. Multiplying by a number greater than $1$ makes it bigger.
• Cancel before multiplying: you may cancel any numerator with any denominator, also diagonally: $\frac{4}{9} \cdot \frac{3}{8} = \frac{1 \cdot 1}{3 \cdot 2} = \frac{1}{6}$ ($4$ and $8$ by $4$, $3$ and $9$ by $3$). The numbers stay small.
• Whole and mixed numbers are written as fractions first: $3 = \frac{3}{1}$, $1\frac{1}{2} = \frac{3}{2}$.

Dividing means measuring: $\frac{3}{4} \div \frac{2}{3}$ asks: how often does $\frac{2}{3}$ fit into $\frac{3}{4}$? With a common denominator the pieces have the same size, and you only count pieces:

So the strip $\frac{2}{3}$ fits in once completely, plus $\frac{1}{8}$ of it.

The reciprocal rule: $\frac{2}{3}$ fits into $1$ exactly $\frac{3}{2}$ times – this is the reciprocal of $\frac{2}{3}$. Into $\frac{3}{4}$ the strip fits $\frac{3}{4}$ times as often:

In general: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$ – you multiply by the reciprocal of the divisor. Dividing by a number less than $1$ makes the result bigger.

Try it

1. Load 2/3 · 3/4 and press Build up. Count the cells of one whole and the purple cells.
2. Drag the blue handle beyond $1$, e.g. to 3/2 · 4/5. When is the product larger than the second factor?
3. Is “multiplying makes bigger” true? Test 1/2 · 1/3.
4. Press Swap factors. What changes in the picture, what in the result?
5. Turn on cancelling: 4/9 · 3/8. Which numbers are cancelled?
6. Switch to dividing: 3/4 ÷ 1/4. Press Measure and count along.
7. For 3/4 ÷ 2/3 something is left over. Which part of the strip still fits in?
8. For 1/4 ÷ 3/4 the strip does not fit in completely even once. What is the result of the division?
9. How often does $\frac{1}{12}$ fit into $3$? Check

Tasks

Task 1: Area model

Work out and check with the area model: a) $\frac{3}{5} \cdot \frac{2}{3}$ b) $\frac{3}{4}$ of $\frac{4}{5}$ Load task a

Task 2: Cancel first, then multiply

a) $\frac{5}{12} \cdot \frac{9}{10}$ b) $\frac{8}{9} \cdot \frac{3}{4}$ Load task a

Task 3: Leftover pizza

$\frac{3}{4}$ of a pizza is left. Tim eats $\frac{2}{3}$ of it. What part of the whole pizza does he eat?

Task 4: Portions

a) $2$ pizzas are cut into portions of $\frac{2}{3}$ of a pizza each. How many portions are there? b) How often does $\frac{3}{8}$ fit into $\frac{3}{4}$? Load task a

Task 5: Two ways

Work out $\frac{5}{6} \div \frac{2}{3}$ once with a common denominator and once with the reciprocal. Load task

Task 6: How often does it fit?

Work out a) $\frac{3}{4} \div \frac{1}{8}$ b) $\frac{2}{3} \div \frac{4}{5}$ c) $3 \div \frac{1}{4}$. Is the result greater or less than $1$? Load task a

Task 7: Bigger by dividing? (Puzzle)

Explain why $\frac{3}{4} \div \frac{1}{2}$ is greater than $\frac{3}{4}$. For which divisors is the result less than the dividend?

Ideas for the classroom

• Use: About grade 6, multiplying and dividing fractions: solid mental models “part of” (area model) and “sharing out/measuring” (division). For introducing the topic in class and for practice; dividing uses the idea of expanding to a common denominator from “Equivalent fractions and comparing”.
• Model: factors up to $3$ (so up to $3 \times 3$ wholes in the picture) and denominators up to $12$. Numerators are at least $1$, so division by $0$ cannot occur. The handles at the edge snap in steps of $\frac{1}{b}$ or $\frac{1}{d}$. With “Build up” the picture appears step by step: first one factor, then the other, then the overlap with numbered cells.
• Division: the simulation uses the idea of measuring (quotition): “How often does … fit?”. If the divisor is a whole number, the idea of sharing (partition) is more natural: $\frac{3}{4} \div 3$ means “sharing $\frac{3}{4}$ among $3$ people”, so $\frac{1}{4}$ each. Both ideas lead to $\frac{3}{4} \cdot \frac{1}{3}$.
• Common stumbling blocks:
  • “Multiplying makes bigger, dividing makes smaller.” – This is only true for factors or divisors greater than $1$.
  • Finding a common denominator before multiplying: not wrong, but unnecessary, and it makes the numbers big.
  • Taking the reciprocal of the wrong fraction: $\frac{3}{4} \div \frac{2}{3} \neq \frac{4}{3} \cdot \frac{2}{3}$. Only the divisor is turned over.
  • Cancelling diagonally in additions: you must not cancel diagonally in $\frac{2}{3} + \frac{3}{4}$ – that only works when multiplying.