When dividing by a fraction, I find it helpful to think to myself a question like, How many one-halves are there in 3 wholes?
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- Lizbeth Daniels
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1 Whole Number Divided by a Unit Fraction (1 of 2) Example: 3 ½ When dividing by a fraction, I find it helpful to think to myself a question like, How many one-halves are there in 3 wholes? 1/
2 Whole Number Divided by a Unit Fraction (2 of 2) Example: 3 ½ /2 1/2 1/2 1/2 1/2 1/2 There are 6 one-halves in 3 wholes so, 3 ½ = 6.
3 Whole Number Divided by a Unit Fraction (1 of 2) Example: 3 1/3 How many one-thirds are there in 3 wholes? 1/
4 Whole Number Divided by a Unit Fraction (2 of 2) Example: 3 1/3 1/3 1 1/3 1/ /3 1/3 1/3 1/3 1/3 1/3 There are 9 one-thirds in 3 wholes so, 3 1/3 = 9.
5 Whole Number Divided by a Non-Unit Fraction (1 of 6) Example: 3 2/3 My measurement division question stays the same, How many two-thirds are there in 3 wholes? 2/
6 Whole Number Divided by a Non-Unit Fraction (2 of 6) Example: 3 2/ /3 2/3 2/3 2/3 Hmm, if I stop at 4 two-thirds there is some extra space.
7 Whole Number Divided by a Non-Unit Fraction (3 of 6) Example: 3 2/ /3 2/3 2/3 2/3 2/3 But if I use 5 two-thirds, then I go past 3 wholes.
8 Whole Number Divided by a Non-Unit Fraction (4 of 6) Example: 3 2/ /3 2/3 2/3 2/3? That means there are more than 4 two-thirds in 3 wholes, but less than 5 twothirds. That means 4 < (3 2/3) < 5.
9 Whole Number Divided by a Non-Unit Fraction (5 of 6) Example: 3 2/ /3 2/3 2/3 2/3 1/3 If I think about it, in order to complete the third whole I need a one-third piece (2/3 + 1/3 = 1). The question I have to answer is, what fraction of a two-third piece am I using?
10 Whole Number Divided by a Non-Unit Fraction (6 of 6) Example: 3 2/ /3 2/3 2/3 2/3 1/3 1/3 2/3 One-third is half of two-thirds, so I need half of a two-thirds piece. So altogether there are 4 two-thirds pieces and ½ of a two-thirds piece in 3 wholes. That means 3 2/3 = 4½. This is really the trickiest part of dividing by a fraction, understanding that you need a fraction of a piece which can result in a fraction you might not be expecting to see in the answer. In this case I divided by 2/3, but ended up with ½ in my answer because I needed half of a 2/3-size piece.
11 Whole Number Divided by a Non-Unit Fraction (1 of 6) Example: 2 ¾ My measurement division question stays the same, How many three-fourths are there in 2 wholes? 3/4 1 2
12 Whole Number Divided by a Non-Unit Fraction (2 of 6) Example: 2 ¾ 1 2 3/4 3/4 Hmm, if I stop at 2 three-fourths there is some extra space.
13 Whole Number Divided by a Non-Unit Fraction (3 of 6) Example: 2 ¾ 1 2 3/4 3/4 3/4 But if I use 3 three-fourths, then I go past 2 wholes.
14 Whole Number Divided by a Non-Unit Fraction (4 of 6) Example: 2 ¾ 1 2 3/4 3/4? That means there are more than 2 three-fourths in 2 wholes, but less than 3 three-fourths. That means 2 < (2 ¾) < 3.
15 Whole Number Divided by a Non-Unit Fraction (5 of 6) Example: 2 ¾ 1 2 3/4 3/4 2/4 If I think about it, so far I have used 2 three-fourths, which is equal to 1 2/4 (¾ + ¾ = 1 2/4). In order to complete the second whole I need a two-fourths piece (1 2/4 + 2/4 = 2). The question I have to answer is, what fraction of a three-fourths piece am I using?
16 Whole Number Divided by a Non-Unit Fraction (6 of 6) Example: 2 ¾ 1 2 3/4 3/4 2/4 1/4 Two-fourths is 2/3 of three-fourths, so I need 2/3 of a three-fourths piece. So altogether there are 2 three-fourths pieces and 2/3 of a three-fourths piece in 2 wholes. That means 2 3/4 = 2 2/3. Again, you needed a fraction of a piece which results in a fraction you might not be expecting to see in the answer. In this case I divided by 3/4, but ended up with 2/3 in my answer because I needed 2/3 of a 3/4-size piece. 3/4
17 Unit Fraction Divided by a Whole Number (1 of 3) Example: ½ 3 When dividing by a whole number, I tend to think a bit differently, How large will each part be if I divide this fraction into 3 equal parts? 1/2 1/2??? When I divide ½ into three equal parts, I need to consider the whole to help me understand what is happening.
18 Unit Fraction Divided by a Whole Number (2 of 3) Example: ½ 3 1/2 1/2?????? It would take 6 of these equal parts to make a whole, so what I ve done is divide the whole into sixths.
19 Unit Fraction Divided by a Whole Number (3 of 3) Example: ½ 3 1/2 1/2 1/6 1/6 1/6 That means the size of each piece is 1/6, so ½ 3 = 1/6.
20 Unit Fraction Divided by a Whole Number (1 of 3) Example: 1/4 2 How large will each part be if I divide this fraction into 2 equal parts? 1/4 1/4 1/4 1/4?? When I divide ¼ into two equal parts, I need to consider the whole to help me understand what is happening.
21 Unit Fraction Divided by a Whole Number (2 of 3) Example: 1/4 2 How large will each part be if I divide this fraction into 2 equal parts? 1/4 1/4 1/4 1/4???????? It would take 8 of these equal parts to make a whole, so what I ve done is divide the whole into eighths.
22 Unit Fraction Divided by a Whole Number (3 of 3) Example: 1/4 2 How large will each part be if I divide this fraction into 2 equal parts? 1/4 1/4 1/4 1/4 1/8 1/8 That means the size of each piece is 1/8, so ¼ 2 = 1/8.
23 Unit Fraction Divided by a Whole Number (1 of 2) Example: 1/4 2 Another way to think of it is How many 2-whole size pieces can I get out of a piece that is ¼ of a whole? 1/4 2 The question is somewhat silly because clearly I can t even get a 2-whole size piece one time without going over. I m going to get some fraction of a 2-whole size. That means (1/4 2) < 1.
24 Unit Fraction Divided by a Whole Number (2 of 2) Example: 1/4 2 1/4 1/4 1/4 1/4 1/4 1/4 1/4 1/4 2 One-fourth is 1/8 of 2 wholes, so I have 1/8 of a 2-whole size piece. That means 1/4 2 = 1/8.
25 Non-Unit Fraction Divided by a Whole Number (1 of 2) Example: 3/4 2 How many 2-whole size pieces can I get out of a piece that is 3/4 of a whole? 3/4 2 If I try to measure out 2 wholes, it immediately goes too far. I can t even do it one time. That means (3/4 2) < 1.
26 Non-Unit Fraction Divided by a Whole Number (2 of 2) Example: 3/4 2 1/4 1/4 1/4 1/4 1/4 1/4 1/4 1/4 3/4 2 I know one-fourth is 1/8 of 2 wholes, so I have 3/8 of a 2-whole size piece. That means 3/4 2 = 3/8.
27 Fraction Divided by a Fraction (1 of 2) Example: ¼ ½ How many ½-size pieces can I get out of a piece that is ¼ of a whole? 1/4 1/2 If I try to measure out 1/2, it immediately goes too far. I can t even do it one time. That means (1/4 1/2) < 1.
28 Fraction Divided by a Fraction (1 of 2) Example: ¼ ½ How many ½-size pieces can I get out of a piece that is ¼ of a whole? 1/4 1/4 1/2 One-fourth is 1/2 of one half, so I have 1/2 of a ½-size piece. That means 1/4 1/2 = 1/2.
29 Fraction Divided by a Fraction (1 of 2) Example: 1/2 ¼ How many 1/4-size pieces can I get out of a piece that is 1/2 of a whole? 1/4 1/2
30 Fraction Divided by a Fraction (2 of 2) Example: 1/2 ¼ 1/2 1/4 1/4 There are 2 one-fourths in 1/2 so, 1/2 1/4 = 2.
31 Fraction Divided by a Fraction (1 of 3) Example: 1/2 2/3 How many 2/3-size pieces can I get out of a piece that is 1/2 of a whole? 1/2 2/3 If I try to measure out 2/3, it immediately goes too far. I can t even do it one time. That means (1/2 2/3) < 1.
32 Fraction Divided by a Fraction (2 of 3) Example: 1/2 2/3 I don t know off the top of my head what fraction of 2/3 one half is so I m going to use common denominators to help me. I m going to change the denominators to 6, so I end up with 3/6 4/6: 1/6 1/6 3/6 1/6 1/64/61/6 1/6
33 Fraction Divided by a Fraction (3 of 3) Example: 1/2 2/3 3/6 1/6 4/6 Three-sixths is 3/4 of four-sixths, so I need 3/4 of a four-sixths piece. That means 3/6 4/6 = 3/4. In other words, ½ 2/3 = 3/4
34 Fraction Divided by a Fraction (1 of 6) Example: 2/3 ½ How many 1/2-size pieces can I get out of a piece that is 2/3 of a whole? 1/2 2/3 Hmm, if I stop at 1 one-half there is some extra space.
35 Fraction Divided by a Fraction (2 of 6) Example: 2/3 ½ 2/3 1/2 1/2 But if I use 2 one-halves, then I go past 2/3.
36 Fraction Divided by a Fraction (3 of 6) Example: 2/3 ½ 2/3 1/2? That means there is more than 1 one-half in 2/3, but less than 2 one-halves. That means 1 < (2/3 1/2) < 2.
37 Fraction Divided by a Fraction (4 of 6) Example: 2/3 ½ 2/3 1/2? I know ½ +? = 2/3. For the sake of consistency right now, I m going to use common denominators again to help me find what fraction is missing.
38 Fraction Divided by a Fraction (5 of 6) Example: 2/3 ½ I ll use 6 again, so this time I ll end up with 4/6 3/6. 4/6 3/6 1/6 Now it s much easier to see that the missing piece is 1/6 because 3/6 + 1/6 = 4/6.
39 Fraction Divided by a Fraction (6 of 6) Example: 2/3 ½ I ll use 6 again, so this time I ll end up with 4/6 3/6. 4/6 3/6 1/6 1/6 1/6 3/6 One-sixth is 1/3 of three-sixths, so I need 1/3 of a three-sixths piece. So altogether there is 1 three-sixths piece and 1/3 of a three-sixths piece in 4/6. That means 4/6 3/6 = 1 1/3. In other words, 2/3 ½ = 1 1/3
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