Different Numbers, Same Amount

In plain words

Is 1/2 the same as 2/4?

Yes. 1/2 and 2/4 are the same amount. Split each half into 2 equal pieces and the whole has 4 equal pieces, with 2 of them shaded. The pieces are half the size and there are twice as many, so the amount does not change. Both fractions sit at the same point on the number line.

The big idea

Different numbers can name the same amount. Split every piece into the same number of equal pieces and the amount stays the same: 1/2 = 2/4 = 4/8.

The common confusion

Fractions written with different numbers must be different amounts, so 1/2 and 2/4 can’t be equal. Often 2/4 seems bigger, because its numbers are bigger. It often sounds like this: “2/4(two fourths) is bigger than 1/2(one half). The numbers are bigger.”

Why it works

Shade one of two equal parts of a bar. That is 1/2.

Now split every part into 2 equal pieces. The bar has 4 equal parts, and the shaded region is made of 2 of them, so it is 2/4. Nothing was added or taken away; we only drew more lines.

The pieces are half the size, and there are twice as many of them. Those two changes balance out, so the amount stays the same: 1/2 = 2/4. Split again and you get 4/8, still the same amount.

On a number line, 1/2, 2/4 and 4/8 all sit at exactly the same point. Fractions that name the same number are called equivalent fractions.

Multiplying the top and bottom by the same number is the same move written in symbols. Splitting every third into 2 turns 2/3 into 4/6: twice as many parts in the whole, twice as many shaded, each half the size.

A worked example

You cut a sandwich in half and eat one half. Your friend cuts an identical sandwich into 4 equal pieces and eats 2 of them. You both ate exactly the same amount: 1/2 = 2/4. Your friend just ate it in smaller pieces.

Why this confusion is so common

With whole numbers, different numbers always mean different amounts, so it is natural to expect 2/4 to be more than 1/2. In a fraction, the two numbers work together: the bottom number sets the size of the pieces and the top number counts them. Doubling both makes the pieces half as big and twice as many, so the amount stays the same. Seeing one amount with several names takes time and plenty of pictures.

Try this away from the screen

Fold a paper strip in half, unfold it and color one half. Fold it in half twice, unfold it and count: the color covers 2 of 4 equal parts. Fold it in half three times and count again: 4 of 8. Ask: did the colored amount change, or only the number of parts?

Helpful things to say

Practice questions

For grown-ups helping out: the answers are tucked away so learners can try first.

1. Two same-size pizzas, one cut into 3 equal slices and one into 6. Which is more pizza: 2/3 or 4/6?

   Show the answer

   Answer: They are the same amount. Cut each third of a pizza in two and you get 2 sixths, so 2 thirds is 4 sixths: 2/3 = 4/6. The slices are smaller, but there are twice as many.

2. Which point is in the same place as 3/4?

   Show the answer

   Answer: 6/8. Each fourth splits into 2 eighths, so 3 fourths is 6 eighths. 3/4 and 6/8 are two names for the same point on the number line.

3. This bar is cut into 6 equal parts. Shade the same amount as 1/3.

   Show the answer

   Answer: shade parts of the bar so that 1/3 of it is shaded. Each third of the bar is 2 sixths, so 1/3 = 2/6. Shading 2 of the 6 parts covers exactly one third.

4. Which fraction is the same amount as 2/5?

   Show the answer

   Answer: 4/10. Doubling the top and bottom gives twice as many pieces, each half the size, so 2/5 = 4/10. Changing only one number, or adding 1 to both, changes the amount.

Questions people ask

Words to know

Equivalent fractions

Fractions that name the same amount, such as 1/2, 2/4 and 4/8. They share one point on the number line.

Numerator

The top number of a fraction. It counts how many of the equal parts we have.

Denominator

The bottom number of a fraction. It tells how many equal parts make one whole.

Number line

A line where every point is a number. Fractions have their own places between the whole numbers.

Curriculum connections

We don’t assume any particular curriculum. For teachers in the United States, here’s how this slice lines up with the Common Core State Standards:

• 3.NF.A.3.a

  Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

• 3.NF.A.3.b

  Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.

• 4.NF.A.1

  Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

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