Explain Equivalence Using Multiplication of Numerator and Denominator

Students learn that multiplying both the numerator and denominator of a fraction by the same number n creates an equivalent fraction (n×a)/(n×b). They use visual models to justify why this works and practice generating equivalent fractions.

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Grade 4 · Equivalent Fractions · question 1 of 3

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20 rungs. A star at the top.

A sign in Emi's toy shop is cut into 10 equal parts, and she paints 9 of them. She cuts every part into smaller equal pieces, making 100 pieces in all. How many small pieces are painted?

How it is taught

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Equivalent fractions name the same amount with different parts.

Multiply the top and the bottom by the same number.

Read it: Same amount, different name

Two friends share a chocolate bar. Leo breaks his half into two pieces and says he has more, because two is more than one. Look at the chocolate, not the pieces.

Maya has one piece out of two: 1/2.
Leo cut his half into two, so he has two pieces out of four: 2/4.
Slide Leo's two pieces together. They are exactly the size of Maya's one.

Cutting a piece into more pieces changes how many pieces there are. It does not change how much chocolate there is.

A fraction is a name for an amount. The same amount can carry more than one name — 1/2, 2/4, 3/6, 4/8 all name Maya's half.

When the top and the bottom are both multiplied by the same number, the name changes and the amount stays.

Multiply the top and the bottom by the same number: the fraction has a new name and the same value.

Reading 2/4 as more than 1/2 because 2 is more than 1.

The bottom number says how small the pieces are. Fourths are smaller than halves, so two of them are only worth one half.

Multiply the top and the bottom of 3/5 by 2. 3/5 = ___/10 — 6. 3 × 2 = 6 and 5 × 2 = 10, so 3/5 = 6/10: the same amount, in smaller pieces.

1/2 = 3/6

Today we'll learn a cool trick. Two fractions can look different but be exactly the same amount.

Here is one-half. The bar is split into two equal parts, and one part is shaded.

Can we write one-half a different way, using more pieces?

Watch this. We'll cut each piece into three smaller pieces.

Now the shaded part is three small pieces out of six total. That's three-sixths!

Look — the same amount is shaded. So one-half equals three-sixths.

Here's the rule. To find an equivalent fraction, multiply the top and bottom by the same number.

Multiply the numerator: one times three equals three.

Multiply the denominator: two times three equals six.

Grade 4 · Multiply top and bottom by the same number

Why does this work? Multiplying by three over three is the same as multiplying by one — and one doesn't change the value.

One-half equals three-sixths. Same amount, just more pieces. Nice work!

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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