# Different Numbers, Same Amount > **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. - 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. - 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 is bigger than 1/2. The numbers are bigger.” - Family: Same Value, Different Forms (slice 11 of 20) - Try it yourself (about 3 minutes, free): https://slices.org/slices/different-numbers-same-amount ## In plain words 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 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 ways to talk about it - Instead of “Just multiply the top and bottom by 2.”, try: “What happens to the shaded part if we cut every piece in half?” - Instead of “2/4 is just 1/2 with bigger numbers.”, try: “How many fourths fit in one half? How could we check?” ## Practice questions 1. Two same-size pizzas, one cut into 3 equal slices and one into 6. Which is more pizza: 2/3 or 4/6? **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? **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. **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? **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 ### Why are 1/2 and 2/4 equal when the numbers are different? Because both numbers change together. In 2/4 the pieces are half the size of halves, and there are twice as many of them. Smaller pieces, more of them, same amount. ### What are equivalent fractions? Fractions that name the same amount, such as 1/2, 2/4 and 4/8. They are written with different numbers but sit at exactly the same point on the number line. ### How do you make an equivalent fraction? Multiply the top and bottom by the same whole number: 2/3 = 4/6 = 6/9. In a picture, that means splitting every part into the same number of equal pieces. Adding the same number to the top and bottom does not work: 2/3 and 3/4 are different amounts. ### How can I help my child see that 2/4 = 1/2? Show it rather than state it. Color half of a paper strip, fold it into fourths and count the colored parts. Then ask whether the colored amount changed. Let the model do the convincing. ## Curriculum connections (US Common Core) - 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. ## Related slices - [Simplifying Doesn’t Shrink](https://slices.org/slices/simplifying-doesnt-shrink): Why 6/8 and 3/4 are the same amount, and why simplifying a fraction never makes it smaller. - [The Bigger Denominator Trap](https://slices.org/slices/bigger-denominator-trap): Why 1/8 is smaller than 1/6, even though 8 is bigger than 6. - [One Number, Not Two](https://slices.org/slices/one-number-not-two): Why 2/5 is one number between 0 and 1, not a 2 and a 5. - [Adding Different Pieces](https://slices.org/slices/adding-different-pieces): Why 1/2 + 1/4 is 3/4, not 2/6: make the pieces the same size before you count them.