Understand a Fraction as a Multiple of a Unit Fraction

Students understand that a/b equals a × (1/b). For example, 5/3 = 5 × (1/3). They use repeated addition and visual models to connect counting unit fractions to multiplication notation.

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Grade 4 · Multiplying Fractions by Whole Numbers · question 1 of 3

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20 rungs. A star at the top.

4/5 = × 1/5.

How it is taught

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A fraction counts copies of a unit fraction.

3/4 means 3 copies of 1/4.

Read it: Copies make a longer run

4/5 is 4 pieces of a bar cut into 5. Take 2 copies of it and the run is 8 pieces, each still 1/5: 2 × 4/5 = 8/5, which is more than 1 bar.

Shade 1 copy: 4/5 is 4 pieces of 1/5.
Take it 2 times: 2 copies of 4 pieces make 8 pieces.
Keep the bottom number, because the pieces are still fifths: 2 × 4/5 = 8/5.

Multiplying by a whole number takes copies of the fraction, and each copy adds pieces of the same size.

Only the top number counts pieces, so 2 × 4/5 multiplies the top: 2 × 4 = 8 pieces of 1/5.

8/5 is more than 1 whole: 5 of the pieces fill 1 bar, and the other 3 start a second.

To multiply a fraction by a whole number, multiply the top number and keep the bottom number.

Multiplying the bottom number by 2 as well, and writing 8/10.

8/10 is 8 pieces of a bar cut into 10, smaller pieces than these. The copies are still fifths, so the answer is 8/5.

14 × 1/5 = ___/5 — 14. 14 copies of 1/5 make 14 pieces of 1/5: 14/5.

5/3 = 5 × (1/3)

What is five-thirds? Let's see it as five copies of one-third.

First, our building block. One-third is one piece when something is split into three equal parts.

Here's a number line from zero to two, split into thirds.

Start at zero. Hop one-third. We land on one-third.

Hop another one-third. Now we're at two-thirds.

Another hop of one-third lands us on three-thirds, which is one whole.

Keep going. Hop one-third more. We're at four-thirds.

One last hop of one-third lands us on five-thirds!

Grade 4 · Five copies of one-third

We added one-third five times: one-third plus one-third plus one-third plus one-third plus one-third equals five-thirds.

Adding the same thing five times is the same as multiplying. So five-thirds equals five times one-third.

Any fraction is just a count of unit fractions! Five-thirds is five copies of one-third.

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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