# Compare Without Guessing > **Which is bigger, 3/8 or 4/7?** 4/7 is bigger. Half of 8 is 4, so 3/8 is less than one half. Half of 7 is three and a half, so 4/7 is more than one half. A fraction above one half is always bigger than a fraction below it. - Big idea: Compare by position, not digits. 3/8 is less than one half and 4/7 is more than one half, so 4/7 is bigger. - Common confusion: When the tops and bottoms are both different, fractions can only be compared by guessing or by the digits: 3/8 must be bigger than 4/7, because 8 is bigger than 7. - It often sounds like this: “The tops and bottoms are all different, so I just guessed.” - Family: Same Value, Different Forms (slice 15 of 20) - Try it yourself (about 4 minutes, free): https://slices.org/slices/compare-without-guessing ## In plain words 3/8 and 4/7 have different top numbers and different bottom numbers, so we can’t just count pieces or compare piece sizes. Instead, ask where each fraction sits compared with one half. Half of 8 is 4, so 3/8 is less than one half. Half of 7 is three and a half, so 4/7 is more than one half. Any fraction less than one half is smaller than any fraction more than one half. So 4/7 > 3/8, and no guessing was needed. On a number line from 0 to 1, 3/8 lands to the left of the halfway point and 4/7 lands to the right of it. The number further to the right is the bigger number. One half is called a benchmark. 0 and 1 are useful benchmarks too. When both fractions are on the same side of one half, use another strategy, such as rewriting both with a common denominator. ## A worked example Ana and Ben have same-size pizzas. Ana eats 3 of her 8 slices; half would have been 4 slices, so she ate less than half. Ben eats 4 of his 7 slices; half would have been three and a half slices, so he ate more than half. Ben ate more: 4/7 > 3/8. ## Why this confusion is common When both numbers are different, the shortcuts for same tops or same bottoms no longer apply, so children often fall back on comparing digits (“8 is bigger than 7”) or on guessing. Comparing each fraction with one half gives them a reliable way to reason about size, and it builds the idea that every fraction is a number with its own place on the number line. ## Try this away from the screen Mark 0 and 1 at the ends of a long paper strip and fold it in half to find one half. Call out fractions such as 2/5, 5/8, 3/7 and 6/10. For each one, decide together whether it belongs to the left or the right of the fold. Ask: how can you tell without measuring? ## Helpful ways to talk about it - Instead of “Just cross-multiply.”, try: “Is each fraction more or less than one half? How do you know?” - Instead of “Look at which numbers are bigger.”, try: “What would half of 8 be? So is 3/8 more or less than one half?” ## Practice questions 1. Which is bigger: 5/8 or 3/7? Use one half to decide. **Answer:** 5/8. 5/8 is more than one half (half of 8 is 4) and 3/7 is less than one half (half of 7 is three and a half). So 5/8 > 3/7. 2. Two same-size pizzas. Which is more: 2/5 of one, or 5/9 of the other? **Answer:** 5/9 of a pizza. Half of 5 is two and a half, so 2/5 is less than half a pizza. Half of 9 is four and a half, so 5/9 is more than half. So 5/9 > 2/5. 3. Which is bigger: 7/12 or 2/5? **Answer:** 7/12. 7/12 is more than one half, because 7 is more than half of 12. 2/5 is less than one half, because 2 is less than half of 5. So 7/12 > 2/5. 4. Which is the largest: 4/9, 6/11 or 5/12? **Answer:** 6/11. 4/9 and 5/12 are both less than one half. 6/11 is more than one half, because 6 is more than half of 11. So 6/11 is the largest. ## Questions people ask ### How do you compare fractions with different numerators and denominators? One reliable way is to compare each fraction with one half. If one is less than one half and the other is more, the one above one half is bigger. You can also rewrite both fractions with a common denominator and compare the top numbers. ### How can you tell if a fraction is more than one half? Double the top number and compare it with the bottom number. If double the top is more than the bottom, the fraction is more than one half: 4/7 is more than one half because 8 is more than 7. If it is less, the fraction is less than one half. ### What if both fractions are more than one half? Then one half can’t decide it, so try another benchmark. For 3/4 and 5/6, think about how far each is from 1: 3/4 is one fourth short and 5/6 is one sixth short, a smaller gap, so 5/6 is bigger. A common denominator works too: 9/12 and 10/12. ### What is a benchmark fraction? A familiar fraction, such as 1/2, used as a reference point. 0, 1/2 and 1 are the most useful benchmarks for comparing fractions between 0 and 1. ## Curriculum connections (US Common Core) - 4.NF.A.2: Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. ## Related slices - [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. - [Same Bottom, Compare the Top](https://slices.org/slices/same-bottom-compare-the-top): Why 5/7 is bigger than 2/7: the pieces are the same size, so the one with more pieces is bigger. - [Same Top, Compare the Pieces](https://slices.org/slices/same-top-compare-the-pieces): Why 3/5 is bigger than 3/8: the same number of pieces, but fifths are bigger pieces than eighths. - [The Bigger Numerator Trap](https://slices.org/slices/bigger-numerator-trap): Why 2/3 is bigger than 3/8, even though 3 is bigger than 2.