Lesson Guide
Explanation
Start with the easy case. When two fractions have the same bottom number, the pieces are all the same size, so you just count them. Two fifths against four fifths: four pieces of the same size beats two, so four fifths is bigger. The bottom number tells you how big each piece is; the top number tells you how many you have. Same bottom means same size pieces, so the top number decides.
Your Turn
Which is bigger, 3/8 or 5/8? And 7/10 or 3/10? Check: 5/8 and 7/10.
Watch Out
This shortcut only works when the bottom numbers MATCH. It is the whole reason the rest of this lesson exists.
Explanation
Here is the second easy case, and it feels backwards at first. When the top numbers match, the fraction with the BIGGER bottom number is the SMALLER fraction. One third is bigger than one sixth, which is bigger than one tenth. Think about the cake: cutting it into ten pieces instead of three does not give you more cake, it gives you smaller slices. A bigger bottom number means more cuts, and more cuts means less on each plate.
Your Turn
Put these in order, biggest first: 1/4, 1/8, 1/2. Check: 1/2, then 1/4, then 1/8.
Watch Out
This is where the biggest number instinct betrays you. 1/10 has the biggest digit in it and is the smallest fraction here.
Explanation
Now the real case: 2/3 against 3/4. The bottom numbers are different, so the pieces are different sizes and you cannot just count them. The fix is to re-cut both into pieces of the same size. Thirds and quarters both fit neatly into twelfths: 2/3 becomes 8/12, and 3/4 becomes 9/12. Now the pieces match, so you count — nine beats eight, and 3/4 wins. Renaming does not change how much you have, only what you call it.
Your Turn
Rename 1/2 and 2/5 into tenths, then compare. Check: 5/10 and 4/10, so 1/2 is bigger.
Watch Out
When you re-cut, multiply the TOP and the BOTTOM by the same number. Changing only one changes the amount.
Explanation
How do you know what size to cut into? Count up in both bottom numbers until you hit the same one. For quarters and sixths: fours go 4, 8, 12, 16 and sixes go 6, 12, 18 — twelve appears in both, so twelfths it is. Then 3/4 becomes 9/12 and 5/6 becomes 10/12, and ten beats nine. This is just your times tables doing a job. If you cannot spot a shared number quickly, multiplying the two bottom numbers together always works, even if it gives more pieces than you strictly need.
Your Turn
What size would you cut thirds and fifths into? Check: threes go 3, 6, 9, 12, 15 and fives go 5, 10, 15 — so fifteenths.
Watch Out
You need a number in BOTH lists. Picking a number from only one list leaves you comparing different-sized pieces again.
Explanation
One last trick that saves a lot of work. Compare each fraction to a half instead of to each other. For 3/8: half of 8 is 4, and the top is only 3, so it is under a half. For 4/6: half of 6 is 3, and the top is 4, so it is over a half. Something under a half can never beat something over a half, so 4/6 wins without any re-cutting at all. This works whenever the two fractions land on opposite sides of a half — which happens more often than you would think.
Your Turn
Use the half check on 2/5 and 5/9. Check: half of 5 is 2.5 and 2 is less, so under; half of 9 is 4.5 and 5 is more, so over. 5/9 is bigger.
Watch Out
If both fractions are on the SAME side of a half, this check cannot decide it. Go back to re-cutting.
📌 Key Points — this lesson in a nutshell
- Same bottom number → just compare the tops.
- Same top number → the bigger bottom is the smaller fraction.
- Otherwise, re-cut both into pieces of the same size and then count.
- Find that size by counting up in both bottom numbers until they meet.
- Quick check: is each fraction over or under a half?
Step 1: Introduction
Step 1 of 5
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