Fractions (unlike denominator)

1Summary 重點摘要

Find common denominator first.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Part 1: Why direct adding failsIn 12 + 14, a half and a quarter are bricks of different sizes, so the tops cannot simply be added. Why not? Because adding the tops assumes every piece is the same size, just as 3 apples plus 2 oranges make 5 fruit but never 5 apples. Suppose we add tops with tops and bottoms with bottoms to get 36, then check with a picture: 12 is half a sheet and 14 is a quarter of a sheet, and together they clearly make three quarters of a sheet, not a half, so 36 is wrong. A common mistake is exactly this: add the tops and add the bottoms; fractions with unlike denominators must be converted to a common denominator first, and only then added as like fractions.
2. Part 2: Finding a common denominatorTo match 12 with 14, find a denominator they can both use: since 4 is a multiple of 2, the number 4 works as the common denominator. Why is the smallest common denominator best? Because small numbers keep the arithmetic simple and the work least likely to go wrong: for 12 and 14 the number 4 is enough, with no need for 8 or 12. Likewise for 13 and 16, since 6 is a multiple of 3, choose 6; for 14 and 18 choose 8. If neither denominator is a multiple of the other, as with 13 and 14, take their least common multiple 12. A common mistake is to pick a wrong multiple, thinking 5 is a multiple of 2; the test is to divide the larger denominator by the smaller — if it divides exactly, the larger one is the common denominator, otherwise find the least common multiple.
3. Part 3: How to convertThe method of converting: whatever the denominator is multiplied by, the numerator is multiplied by the same number. To change 12 into a fraction with denominator 4: the bottom 2 × 2 = 4, so the top must also be multiplied by 2, and 1 × 2 = 2, giving 24. Why does multiplying the top and the bottom by the same number keep the value unchanged? Because it means every piece is further divided into two equal parts; the number of pieces grows, but the number taken grows just the same, so the amount never changes — like cutting half a sheet into two quarters, which is still half a sheet. A picture confirms it: the shaded regions of 12 and 24 are exactly the same, so the fraction before and after conversion is equal. A common mistake is to multiply only the denominator, wrongly converting 12 into 14; top and bottom must be multiplied by the same number.
4. Part 4: Then add as usualAfter converting, 12 + 14 becomes 24 + 14. The two denominators now match and every piece is the same size, so the tops can be added directly: 2 + 1 = 3, the denominator stays 4, and the answer is 34. Why does the denominator stay the same? Because the denominator tells the size of each piece; after converting, the pieces already match, and what is added is only the count of pieces, which cannot change the size of each piece. A check: 34 as a decimal is 0.75, and 12 + 14 by decimals is 0.5 + 0.25 = 0.75, so the two agree and the calculation is right. A common mistake is to add the denominators too, writing 24 + 14 = 38; when denominators match, add only the tops and keep the bottom — this is the core rule of fraction addition.
5. Part 5: Subtraction works the sameSubtracting fractions with unlike denominators follows exactly the same steps as adding: convert first, subtract the tops next, and finally check whether the answer simplifies. For example, 12 − 14: change 12 into 24 first, then work out 24 − 14 = 14. Another: 56 − 13, where 6 is a multiple of 3; convert 13 into 26, so 56 − 26 = 36, which simplifies to 12. Why the same steps? Because subtraction likewise requires every piece to be the same size before taking away — the reasoning is identical, only the operation changes from combining to removing. A common mistake is to forget to simplify, leaving 36 as the final answer; another is to subtract tops and bottoms separately, computing 5 − 1 = 4 and 6 − 3 = 3 to get 43, which is larger than 1 and clearly wrong — the check 'the difference is smaller than the number subtracted from' exposes it at once.
6. Part 6: Checking with a pictureDrawing is a fine way to check calculations with unlike denominators: draw two strips of equal length, divide the first into 2 parts and shade 1 for 12, and divide the second into 4 parts and shade 1 for 14. Re-divide the first strip into 4 parts, and the old 1 part becomes exactly 2, showing 24 + 14 = 34 at a glance — the picture confirms the sum. Why does drawing work? Because a fraction describes the relation between a part and the whole, and a picture turns abstract numbers into visible area, while the converting process stands out clearly: changing the same strip from halves into quarters makes each piece exactly half as big. A common mistake is to draw the two strips of different lengths, which ruins the comparison; the first thing to confirm in a picture check is equal total length and even parts, and only then count the shaded pieces.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• First step for unlike denominators: convert
• Pick the smallest common denominator
• Multiply top and bottom together to keep the value
• After converting, add or subtract as usual
• Simplify the answer at the end