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Explain Why Multiplying by a Fraction Greater Than 1 Enlarges and Less Than 1 Shrinks

Students explain and justify using models or reasoning why multiplying a number by a fraction greater than 1 yields a product larger than the original number, and multiplying by a fraction less than 1 yields a product smaller than the original number. Students use number lines and area models to support their explanations.

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Explain Why Multiplying by a Fraction Greater Than 1 Enlarges and Less Than 1 Shrinks

Grade 5 · Multiplication as Scaling · question 1 of 3

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20 rungs. A star at the top.

Which of these is the least: 21, 22, 4/2 × 21 or 5/2 × 21?

How it is taught

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Multiplying by a fraction greater than 1 makes a number grow.

Multiplying by a fraction less than 1 makes a number shrink.

Multiplying by 1 leaves the number the same.

Read it: Scaling by a fraction

8 × 3/4 is less than 8, because 3/4 is less than 1. 8 × 5/4 is more than 8, because 5/4 is more than 1. Hold the fraction against 1, and you know without multiplying.

  1. Look at the fraction you multiply by.
  2. Hold it against 1: are there fewer parts shaded than make 1, or more?
  3. Less than 1 makes the product less than the number; more than 1 makes it greater.

3/4 of 8 is only part of 8, so it is less than 8: 8 × 3/4 = 6.

5/4 of 8 is all of 8 and 1/4 of 8 more, so it is more than 8: 8 × 5/4 = 10.

4/4 is 1, and 8 × 4/4 = 8: times 1 leaves a number as it is.

Times a fraction less than 1, the product is less than the number; times a fraction more than 1, it is greater.

Thinking a product is always greater than the number: 8 × 3/4 > 8.

3/4 is less than 1, so 8 × 3/4 is only part of 8: 8 × 3/4 < 8.

14 × 9/7 ___ 14 — >. 9/7 is more than 1, so 14 × 9/7 > 14.

6 × 3/2 = 9 vs 6 × 1/2 = 3

When we multiply a number by a fraction, the answer can come out bigger OR smaller. Let's see why.

Here's our starting number, six. We'll mark it on the number line and watch what fractions do to it.

First, six times three-halves. Three-halves is a fraction GREATER than one.

Three-halves means one and a half groups of six — that's nine.

Nine lands to the RIGHT of six. The number got bigger!

Now six times one-half. One-half is a fraction LESS than one.

One-half of six is just half of six — that's three.

Three lands to the LEFT of six. The number got smaller!

Here's the rule. Multiplying by more than one whole stretches the number. Less than one whole shrinks it.

So before you even compute, you can predict the answer's size by looking at the fraction. Greater than one grows. Less than one shrinks.

Does multiplying always make numbers bigger?Not when fractions are involved…0126our starting number6 × 32 = ?(32 is greater than 1)6 × 32 = 99BIGGER!6 × 12 = ?(12 is less than 1)6 × 12 = 33SMALLER!The Rulefraction > 1 → product is BIGGERfraction < 1 → product is SMALLERfraction = 1 → product stays the SAME39REMEMBER>1 grows<1 shrinks
Grade 5 · Fractions greater than 1 stretch, less than 1 shrink

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
Climbing now
Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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