Dr. Rena Gibbs Amy Edmundson. What does it mean to “invert and multiply”?

FRACTIONS “THE WHOLE STORY”

Dr. Rena Gibbs Amy Edmundson

What does it mean to “invert and multiply”?

Let’s say we want to divide a whole number by a fraction. For example…

This is a visual representation of the first part of our equation “seven wholes”.

This is a visual representation of the number 7 divided by .

So this division problem is really asking us “How many ‘1/3rds’ can we ‘fit’ into the seven wholes?”

Three “thirds” fit in one.

Six “thirds” fit in two.

Nine “thirds” fit in three.

Twelve “thirds” fit in four.

Fifteen “thirds” fit in five.

Eighteen “thirds” fit in six.

Twenty-one “thirds” fit in seven.

So because twenty-one

“1/3rds” fit into seven wholes.

÷ = 21

We can verify this result in many ways, but the most common way taught in schools is to “invert and multiply”: ÷ is equivalent to

21Because fits in each “whole” three times.

We have seven “wholes” that we multiply by the three that fit into each whole and 7 times 3 is 21, therefore

x = 21

MEMORIZATION VS. CONCEPTUALIZATION

÷ = ● “Copy, dot, flip” OR “there are three

rds in each whole”.

So how do we build this conceptual understanding? By starting at the beginning…

USING A 100’S CHART TO COMBINE UNLIKE UNITS

(FINDING COMMON DENOMINATORS)

Let’s say we want to do this math problem

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

4 8 12 16 20 24 28 32 36 40 44 48

5 10 15 20 25 30 35 40 45 50 55 60

6 12 18 24 30 36 42 48 54 60 66 727 14 21 28 35 42 49 56 63 70 77 848 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12011 22 33 44 55 66 77 88 99 110 121 13212 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

4 8 12 16 20 24 28 32 36 40 44 48

5 10 15 20 25 30 35 40 45 50 55 60

6 12 18 24 30 36 42 48 54 60 66 727 14 21 28 35 42 49 56 63 70 77 848 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12011 22 33 44 55 66 77 88 99 110 121 13212 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

4 8 12 16 20 24 28 32 36 40 44 48

5 10 15 20 25 30 35 40 45 50 55 60

6 12 18 24 30 36 42 48 54 60 66 727 14 21 28 35 42 49 56 63 70 77 848 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12011 22 33 44 55 66 77 88 99 110 121 13212 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

5 10 15 20 25 30 35 40 45 50 55 60

6 12 18 24 30 36 42 48 54 60 66 727 14 21 28 35 42 49 56 63 70 77 848 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12011 22 33 44 55 66 77 88 99 110 121 13212 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

5 10 15 20 25 30 35 40 45 50 55 60

6 12 18 24 30 36 42 48 54 60 66 727 14 21 28 35 42 49 56 63 70 77 848 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12011 22 33 44 55 66 77 88 99 110 121 13212 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

5 10 15 20 25 30 35 40 45 50 55 60

6 12 18 24 30 36 42 48 54 60 66 727 14 21 28 35 42 49 56 63 70 77 848 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12011 22 33 44 55 66 77 88 99 110 121 13212 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

5 10 15 20 25 30 35 40 45 50 55 60

6 12 18 24 30 36 42 48 54 60 66 727 14 21 28 35 42 49 56 63 70 77 848 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12011 22 33 44 55 66 77 88 99 110 121 13212 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

5 10 15 20 25 30 35 40 45 50 55 60

6 12 18 24 30 36 42 48 54 60 66 727 14 21 28 35 42 49 56 63 70 77 848 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12011 22 33 44 55 66 77 88 99 110 121 13212 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

4 8 12 16 20 24 28 32 36 40 44 48

5 10 15 20 25 30 35 40 45 50 55 60

6 12 18 24 30 36 42 48 54 60 66 727 14 21 28 35 42 49 56 63 70 77 848 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12011 22 33 44 55 66 77 88 99 110 121 13212 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

4 8 12 16 20 24 28 32 36 40 44 48

5 10 15 20 25 30 35 40 45 50 55 60

6 12 18 24 30 36 42 48 54 60 66 727 14 21 28 35 42 49 56 63 70 77 848 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12011 22 33 44 55 66 77 88 99 110 121 13212 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

4 8 12 16 20 24 28 32 36 40 44 48

5 10 15 20 25 30 35 40 45 50 55 60

6 12 18 24 30 36 42 48 54 60 66 727 14 21 28 35 42 49 56 63 70 77 848 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12011 22 33 44 55 66 77 88 99 110 121 13212 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

4 8 12 16 20 24 28 32 36 40 44 48

5 10 15 20 25 30 35 40 45 50 55 60

5 10 15 20 25 30 35 40 45 50 55 607 14 21 28 35 42 49 56 63 70 77 848 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12011 22 33 44 55 66 77 88 99 110 121 13212 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

4 8 12 16 20 24 28 32 36 40 44 48

5 10 15 20 25 30 35 40 45 50 55 60

5 10 15 20 25 30 35 40 45 50 55 6011 22 33 44 55 66 77 88 99 110 121 1328 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12012 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

4 8 12 16 20 24 28 32 36 40 44 48

5 10 15 20 25 30 35 40 45 50 55 60

5 10 15 20 25 30 35 40 45 50 55 6011 22 33 44 55 66 77 88 99 110 121 1328 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12012 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

4 8 12 16 20 24 28 32 36 40 44 48

5 10 15 20 25 30 35 40 45 50 55 60

5 10 15 20 25 30 35 40 45 50 55 6011 22 33 44 55 66 77 88 99 110 121 1328 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12012 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

4 8 12 16 20 24 28 32 36 40 44 48

5 10 15 20 25 30 35 40 45 50 55 60

5 10 15 20 25 30 35 40 45 50 55 6011 22 33 44 55 66 77 88 99 110 121 1328 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12012 24 36 48 60 72 84 96 108 120 132 144

1 2 3 4 5 6 7 8 9 10 11 122 4 6 8 10 12 14 16 18 20 22 24

3 6 9 12 15 18 21 24 27 30 36 39

4 8 12 16 20 24 28 32 36 40 44 48

5 10 15 20 25 30 35 40 45 50 55 60

5 10 15 20 25 30 35 40 45 50 55 6011 22 33 44 55 66 77 88 99 110 121 1328 16 24 32 40 48 56 64 72 80 88 969 18 27 36 45 54 63 72 81 90 99 108

10 20 30 40 50 60 70 80 90 100 110 12012 24 36 48 60 72 84 96 108 120 132 144

3 6 9 12 15 18 21 24 27 30 33 36

5 10 15 20 25 30 35 40 45 50 55 60

11 22 33 44 55 66 77 88 99 110 121 132

3 6 9 12 15 18 21 24 27 30 33 36

5 10 15 20 25 30 35 40 45 50 55 60

11 22 33 44 55 66 77 88 99 110 121 132

3 6 9 12 15 18 21 24 27 30 33 36

5 10 15 20 25 30 35 40 45 50 55 60

11 22 33 44 55 66 77 88 99 110 121 132

3 6 9 12 15 18 21 24 27 30 33 36

5 10 15 20 25 30 35 40 45 50 55 60

11 22 33 44 55 66 77 88 99 110 121 132

+ + = or 1

3 6 9 12 15 18 21 24 27 30 33 36

5 10 15 20 25 30 35 40 45 50 55 60

11 22 33 44 55 66 77 88 99 110 121 132

This same data can be used to determine that >

Dr. Rena Gibbs Amy Edmundson. What does it mean to “invert and multiply”?

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