Fractions (II): Same denominator

1Summary 重點摘要

15 + 25 = 35. Denominator stays, add numerators.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Part 1: Same denominators add directlyFractions with the same denominator can be added directly. For 15 + 25: both fractions cut the whole into 5 parts of the same size, so add the numerators, 1 + 2 = 3, and the answer is 35. It is like 1 cake plus 2 cakes making 3 cakes, every cake the same size; just count them. A common mistake is to add the denominators too, writing 15 + 25 = 310. The denominator shows the size of each part; since the size has not changed, the denominator stays. First check that the two denominators are the same; only then may the numerators be added directly.
2. Part 2: Why the denominator staysWhy does the denominator stay? 15 + 25 = 35, not 310. Think of 15 as a block: one block plus two blocks makes three blocks, and each block is still the same size from start to finish, so the denominator stays 5 and does not become 10. If the denominator became 10, every block would be cut half as small, and three small blocks would not equal the original three large ones. A common mistake comes from whole-number habits, adding every number in sight. Keep the roles clear: the numerator is a count, and counts are added; the denominator is the size of each part, and an unchanged size is left alone.
3. Part 3: Subtraction works the sameSubtraction works the same way. For 47 − 17: keep the denominator 7 and subtract the numerators, 4 − 1 = 3, so the answer is 37. It is like taking 1 block from 4 blocks; the blocks never change size. Another example: 59 − 29 = 39. A common mistake is to reverse the subtraction, writing 17 − 47 with the smaller on top; subtraction takes the second numerator from the first, and the order may not be swapped. Another is subtracting the denominators as well, which would give 7 − 7 = 0 and is completely wrong; the denominator stays 7 throughout.
4. Part 4: When numerators pass the denominatorCan the numerator pass the denominator? Yes. For example, 38 + 78 = 108, which is more than 1: one whole plus 28 extra. Since 88 is already the whole, 108 means the whole plus 28, and this way of writing it is correct. Another example: 56 + 46 = 96, which is 1 and 36. A common mistake is to think a numerator larger than the denominator must be wrong and to shrink it, or to forget to say how many wholes the result contains. When the numerator reaches or passes the denominator, write the result as it is, then say how it contains a whole: 96 = 1 + 36. Both forms are correct, and the meaning must be stated.
5. Part 5: Subtracting from a wholeTo subtract from a whole, first turn 1 into a fraction. A whole cake is 88: after eating 38, how much is left? 88 − 38 = 58. As long as you think of 1 as 88, the subtraction can go ahead; whatever the denominator is, 1 becomes that many parts: if the whole is cut into 5 parts, 1 is 55. A common mistake is to write 1 − 38 and not know what numerator to use, or to turn 1 into some fraction other than eighths. Remember the rule: to turn a whole into a fraction, copy the denominator and make the numerator the same: 1 = 88 = 55 = 66. Then subtract as same-denominator fractions.
6. Part 6: Check with a pictureCheck with a picture. Draw a bar divided into 6 parts: shade 1 part for 16 and 2 parts for 26, making 3 shaded parts, which is 36. This matches the calculation 16 + 26 = 36, so the picture and the sum confirm each other. Subtraction can be drawn too: fill 5 of the 6 parts for 56, wipe out 2, and 4 remain, that is 56 − 26 = 46. A common mistake is a picture whose number of parts disagrees with the denominator, drawing 8 parts for sixths. Before shading, count that the parts equal the denominator; only then does the picture come out right and the check mean anything.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Same-denominator sums: keep the bottom, add or subtract the top
• 15 + 25 = 35, never 310
• Subtraction too: 47 − 17 = 37
• Turn 1 into matching parts first: 1 = 88
• Check answers by shading a picture