Cut finer and you get more pieces but the same amount — equivalence, lowest terms, and common denominators.
Hand a chocolate bar to one friend and snap it in half, or score it into four pieces and give two, or into six and give three — your friend gets exactly the same amount of chocolate each way. The cut changed, the amount did not. That is the whole idea of this lesson: a single amount can be written with many different fractions, and they are all equal. To move between these names you do one honest thing — multiply or divide the top and the bottom by the same number. That lets you reduce a fraction to its simplest, smallest-number form, and it lets you give two fractions a shared denominator so you can finally line them up, compare them, and (next lesson) add them. By the end you will read 68 and 34 as two faces of one number.
Take one half and fold it again, straight down the middle. Each half splits into two, so the whole bar is now in four equal pieces — and the shaded region that used to be one piece is now two pieces. Nothing was added or taken away; only the knife came down more often. So one half and two fourths cover the very same length: 12 = 24. Fold once more and you get three sixths. Fractions that name the same amount like this are called equivalent fractions.
Cutting every piece into k smaller pieces multiplies the bottom by k and the top by k. That is why multiplying top and bottom by the same number gives an equivalent fraction: ab = a × kb × k. The picture proves it: more pieces, same shaded amount.
The same thing must happen to both numbers. Adding 1 to the top and 1 to the bottom does not work: 12 turned into 23 is a bigger amount, not an equal one. You may only multiply (or divide) top and bottom by the same number.
Equivalence runs both ways. If multiplying top and bottom by the same number gives a bigger-looking equal fraction, then dividing top and bottom by a common factor gives a smaller-looking equal fraction. Reducing means doing exactly that until you cannot go any further — until the only number that divides both top and bottom is 1. That smallest-number form is called lowest terms.
Take 68. Both 6 and 8 are even, so divide each by 2: 68 = 34. Now 3 and 4 share no factor but 1, so we stop. The fastest route is to divide by the greatest common divisor (gcd) of top and bottom in one move; you can also chip away by smaller factors and arrive at the same place.
Reduce 1218. Both are even → divide by 2 → 69. Both are multiples of 3 → divide by 3 → 23. The gcd of 12 and 18 is 6, and dividing by 6 in one step lands on the same 23.
Two fractions with different bottoms are like two grids cut differently — their pieces are different sizes, so you can’t line them up yet. To match them, you need a number of pieces that both denominators can reach: a common multiple of the two bottoms. The smallest such number is the least common multiple (lcm), and when it is the new shared denominator we call it the least common denominator (lcd).
Halves come in counts 2, 4, 6, 8, … ; thirds come in counts 3, 6, 9, 12, …. The first count they share is 6, so sixths is the size into which both a half and a third recut cleanly. That makes 6 the least common denominator of 12 and 13.
The common denominator is a piece-size both fractions can be recut into. The least common denominator is the lcm of the two bottoms — the smallest such size, which keeps the numbers small.
Once you know the shared denominator, recut each fraction into it using the equivalence rule from §3.2.1 — multiply top and bottom by whatever turns the old bottom into the new one. For 12 and 13 over sixths: a half needs ×3 (2 → 6) so it becomes 36; a third needs ×2 (3 → 6) so it becomes 26. Now both are measured in the same-size pieces and can be compared or combined.
Recut 34 and 56. The lcd of 4 and 6 is 12. Then 34 = 912 (×3) and 56 = 1012 (×2). Same-size pieces at last.
Now you can decide which of two fractions is larger. Three cases:
For example, which is bigger, 23 or 35? Recut both to fifteenths: 23 = 1015 and 35 = 915. Now 10 > 9, so 23 is larger.
Don’t judge 13 against 12 by saying “3 > 2, so thirds are bigger.” A bigger bottom means the whole was cut into more pieces, so each piece is smaller. With the same top, the smaller bottom is the larger fraction.
| To do this… | …do this | Example |
|---|---|---|
| Make an equivalent fraction | × top & bottom by the same number | 12 = 36 (×3) |
| Reduce to lowest terms | ÷ top & bottom until gcd = 1 | 68 = 34 |
| Find a common denominator | use the lcm of the two bottoms | lcd of 2, 3 is 6 |
| Compare two fractions | common denominator, then compare tops | 23 = 1015 > 915 = 35 |
| Never do this | add the same number to top & bottom | 12 ≠ 23 |
Fill in the missing number: 34 = ?12.
The bottom went from 4 to 12, that is ×3, so do the same to the top: 3 × 3 = 9. So 34 = 912.
Reduce 1015 to lowest terms.
10 and 15 are both multiples of 5; the gcd is 5. Divide top and bottom by 5: 1015 = 23. Now 2 and 3 share no factor but 1, so it is fully reduced.
What is the least common denominator of 14 and 16? Recut both.
Multiples of 4: 4, 8, 12, …; of 6: 6, 12, …. Their first shared count is 12, so the lcd is 12. Then 14 = 312 (×3) and 16 = 212 (×2).
Which is larger, 34 or 58?
Give them the common denominator 8: 34 = 68. Now compare tops over the same bottom: 6 > 5, so 34 > 58.
A classmate writes “25 = 36 because I added 1 to the top and 1 to the bottom.” What went wrong?
Adding the same number is not allowed — only multiplying or dividing keeps the amount equal. Check: 25 = 0.4 but 36 = 12 = 0.5, a bigger amount. To make a true equivalent, multiply: 25 = 410 (×2).
Put in order from smallest to largest: 12, 23, 35.
A common denominator of 2, 3, 5 is 30: 12 = 1530, 23 = 2030, 35 = 1830. Compare the tops: 15 < 18 < 20, so the order is 12 < 35 < 23.
Six questions to lock it in. Tap the answer you think is right.
This lesson builds the equivalence and comparison strand of the Common Core fraction standards: 3.NF.A.3 (recognize and generate equivalent fractions and compare them with a visual model), 4.NF.A.1 (generate equivalent fractions by multiplying top and bottom by the same number), and 4.NF.A.2 (compare two fractions by creating common denominators or numerators, recording the result with >, <, or =). Every shaded bar, reduced form, common denominator, and comparison in the page is computed from the same fraction helpers, so the figures and the words always agree. A good follow-up question at home: “Can you write three different fractions that all equal one half?”