Multiplying fractions

What is this skill?

Multiplying with fractions has two forms in fifth grade: a whole number times a fraction, and a fraction times a fraction. Both rest on one idea. Multiplying by a fraction means taking that part of something. 4 × 2/3 is four groups of two-thirds, which is 8/3, or 2 2/3. 2/3 × 3/4 means two-thirds of three-fourths, which comes to 6/12, or 1/2 in simplest form.

The procedure is short: multiply the numerators, multiply the denominators, and simplify. What matters more is the picture behind it. A rectangle split into 4 columns with 3 shaded, then cut into 3 rows with 2 of them marked, shows 6 pieces out of 12 shaded. Finding a part of a part makes the result smaller than either factor when both fractions are less than 1. In the practice here, the fractions are all proper (less than 1), and answers need to be in simplest form.

Why it matters

Taking a fraction of a quantity is something people do all the time without writing it down: half of a recipe, three-quarters of a tank, two-fifths of a class. Fifth grade is where that everyday reasoning is connected to a written operation.

This skill also sets up the next one, dividing with fractions, and it feeds into area with fractional side lengths and eventually into proportions and rates. A child who thinks of multiplication as “a part of” is much better placed than one who has memorized a rule.

Common mistakes

• Expecting multiplication to always make a number bigger. Multiplying by a fraction less than 1 makes it smaller, and that is correct, not a mistake.
• Looking for a common denominator first, as they would to add. For multiplication it is not needed, and it only adds work.
• Multiplying the whole number by both the top and the bottom, so 4 × 2/3 becomes 8/12. The whole number multiplies the numerator only.
• Forgetting to simplify the product, or simplifying incorrectly by dividing only the top or only the bottom.
• Mixing up what the whole is when drawing the model. “Two-thirds of three-fourths” means the three-fourths is the new whole.

How to help at home

• Fold a sheet of paper in half, then in thirds the other way, and shade one part to show 1/2 × 1/3 = 1/6.
• Use measuring cups. Half of 2/3 cup is a real question when a recipe is being cut down.
• Ask for a prediction first: will the answer be larger or smaller than the starting fraction?
• Rewrite a whole number as a fraction over 1, like 4/1. It makes the rule the same for every case.
• Before simplifying, look for common factors between any numerator and any denominator. Spotting them early saves work.

Worked examples

These are real questions from the Tutorsaurus multiplying fractions practice, with the answers worked out.

1. Example 1

   8 × 1/3 = ? (simplest form)

   Answer: 2 2/3. 8 × 1 = 8, so the answer is 8/3, which is 2 2/3 in simplest form.

2. Example 2

   4/6 × 2/3 = ? (simplest form)

   Answer: 4/9. Multiply tops: 4 × 2 = 8. Multiply bottoms: 6 × 3 = 18. That gives 8/18, which simplifies to 4/9.

3. Example 3

   3/5 × 3/4 = ? (simplest form)

   Answer: 9/20. Multiply tops: 3 × 3 = 9. Multiply bottoms: 5 × 4 = 20. That gives 9/20.