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MATH · GRADE 7Number

Multiplying Fractions

Multiplying by a proper fraction shrinks the number; that's not a mistake.

Grade 7
04812× 3/412?longer, or shorter…
In this lesson

How to multiply two fractions

The rule is short: top × top, bottom × bottom.

ab×cd=a×cb×d\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

So 23×34=2×33×4=612=12\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}. Multiply across, then reduce.

The picture is a rectangle. Shade a rows for the first fraction, c columns for the second; the cells shaded both ways are the product.

Try it

Drag the two handles. The shaded area is the product.

011140.25120.5120.5
12×12=14·0.25

Shaded area: 0.25 of the unit square

Drag the bottom handle along the x-axis and the side handle along the y-axis. The shaded rectangle's area IS the product.

Drag the handles. The shaded area, as a fraction of the whole, is the product.

That same idea points to a broader rule: the product of two fractions is equivalent to the product of any equivalent forms. Replace 2/3 with 4/6 or 6/9; the product still simplifies to the same value. Try it: (2/3) × (3/4) = 6/12 = 1/2, and (4/6) × (3/4) = 12/24 = 1/2. Same answer, different forms of the factor.

When a factor is bigger than 1

A proper fraction has a numerator smaller than its denominator (2/3, 3/4), a value between 0 and 1. An improper fraction is the other case (5/4, 7/3), a value of 1 or more.

A mixed number like 1½ is an improper fraction written in two pieces. To multiply with one, convert it first: multiply the whole part by the denominator, add the numerator, keep the same denominator.

112=2×1+12=321\tfrac{1}{2} = \frac{2 \times 1 + 1}{2} = \frac{3}{2}

Then multiply across the same way as before:

112×12=32×12=341\tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{3}{2} \times \tfrac{1}{2} = \tfrac{3}{4}

The diagram below uses a 2×2 grid so factors greater than 1 have room to draw.

Try it

Drag the two handles. The shaded area is the product — note that the rectangle can extend past the first unit square.

01212340.75112=32=1.512=0.5
112×12=32×12=34·0.75

Shaded area: 0.75 unit squares

Try 1 1/2 × 1/2. Try 1 1/2 × 1 1/2. When a factor is bigger than 1, the rectangle reaches into the next unit square.

Try 1½ × 1½. Both factors are bigger than 1, so the rectangle covers more than one unit square. The product, 9/4 = 2¼, is bigger than each factor.

Where it shows up in real life

Scaling a recipe is fraction multiplication that you can taste. A small-batch Saskatoon-berry jam — the kind you'd find at a u-pick near Smoky Lake or a kitchen in Bonnyville — uses 4 cups of berries to 3 cups of sugar.

Picked half the berries you planned? Multiply the whole recipe by ½. Sugar drops from 3 to 12×3=32=112\tfrac{1}{2} \times 3 = \tfrac{3}{2} = 1\tfrac{1}{2} cups. Berries drop from 4 to 2. The ratio holds; only the size of the batch changes.

Now flip it. Found extra berries on the bush and want 1½ times the recipe? Sugar climbs to 32×3=92=412\tfrac{3}{2} \times 3 = \tfrac{9}{2} = 4\tfrac{1}{2} cups — because the factor is bigger than 1, the product grew. Same area model, same rule.

Worksheet

Try these to put the idea into practice. The goal is fluency with the rule and a feel for when the product is smaller than the factors.

Practice · Not graded

MA.7.NUM.4

Practice the idea

01 / 05

What is 1/2 × 1/3?

Multiple choice: what is one-half times one-third? Four answer cards: one-sixth, two-fifths, one-fifth, two-sixths.

Area model

top × top, bottom × bottom

Show common mistakes

Student says

“1/2 × 1/3 = 2/5. I added the tops and the bottoms.”

What it reveals

Treating multiplication like addition. The student combined numerators and denominators column-by-column, which is the addition rule (and even there, only with a common denominator).

Targeted response

Top × top, bottom × bottom: multiply, not add. 1/2 × 1/3 = (1×1)/(2×3) = 1/6. The area model shows it: a 1/2 by 1/3 strip covers 1/6 of the square.

Student says

“3/4 × 2 must be smaller than 3/4. I'm multiplying.”

What it reveals

Carrying the whole-number bias 'multiplication makes bigger' the wrong way: assuming any product is smaller than each factor.

Targeted response

The size rule: factors bigger than 1 grow the result, factors smaller than 1 shrink it. 3/4 × 2 = 6/4 = 1½, bigger than 3/4 because the factor 2 is bigger than 1.

Student says

“1½ × 1/2 = 1/2. I just used the half.”

What it reveals

Skipped converting the mixed number to an improper fraction. The whole-number part dropped out.

Targeted response

Convert first: 1½ = 3/2. Then 3/2 × 1/2 = 3/4. The whole-part 1 is part of the factor; it doesn't get to sit out.

Going further

The same area-model picture comes back in algebra. Multiplying two binomials (x + 1)(x + 2) is geometrically the same idea: the area of a rectangle whose sides each split into two terms.

Decimal multiplication is the same idea using base-10 fractions: 0.4 × 0.3 = 0.12 is just 410×310=12100\tfrac{4}{10} \times \tfrac{3}{10} = \tfrac{12}{100}.