Understanding Division with Fractions: Invert and Multiply Method Explained
Discover the concept of dividing a whole number by a fraction using the invert and multiply method. Explore visual representations and step-by-step explanations to gain a solid grasp on this fundamental math operation.
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Presentation Transcript
FRACTIONS THE WHOLE STORY Dr. Rena Gibbs Amy Edmundson
What does it mean to invert and multiply ?
Lets say we want to divide a whole number by a fraction. For example 7 1 1 3
7 1 1 3 This is a visual representation of the first part of our equation seven wholes .
7 1 1 3 This is a visual representation of the number 7 divided by 1 3 .
7 1 1 3 So this division problem is really asking us How many 1/3rds can we fit into the seven wholes?
7 1 1 3 = 21 So because twenty-one 1/3rds fit into seven wholes.
We can verify this result in many ways, but the most common way taught in schools is to invert and multiply : 7 1 1 3 is equivalent to 7 1 x 3 1 = 21 Because1 3fits 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 7 1 x 3 1 = 21
MEMORIZATION VS. CONCEPTUALIZATION 7 1 OR there are three 1 1 3 = 7 3 1 1 Copy, dot, flip OR each whole . 3rds in So how do we build this conceptual understanding? By starting at the beginning
USING A 100S CHART TO COMBINE UNLIKE UNITS (FINDING COMMON DENOMINATORS) Let s say we want to do this math problem 3 5 + 5 11
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 4 6 8 9 12 12 16 15 20 18 24 21 28 24 32 27 36 30 40 36 44 39 48 5 10 15 20 25 30 35 40 45 50 55 60 6 7 8 9 10 11 12 12 14 16 18 20 22 24 18 21 24 27 30 33 36 24 28 32 36 40 44 48 30 35 40 45 50 55 60 36 42 48 54 60 66 72 42 49 56 63 70 77 84 48 56 64 72 80 88 96 54 63 72 81 90 99 108 120 132 144 60 70 80 90 100 110 120 110 121 132 66 77 88 99 72 84 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 4 6 8 9 12 12 16 15 20 18 24 21 28 24 32 27 36 30 40 36 44 39 48 5 10 15 20 25 30 35 40 45 50 55 60 6 7 8 9 10 11 12 12 14 16 18 20 22 24 18 21 24 27 30 33 36 24 28 32 36 40 44 48 30 35 40 45 50 55 60 36 42 48 54 60 66 72 42 49 56 63 70 77 84 48 56 64 72 80 88 96 54 63 72 81 90 99 108 120 132 144 60 70 80 90 100 110 120 110 121 132 66 77 88 99 72 84 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 4 6 8 9 12 12 16 15 20 18 24 21 28 24 32 27 36 30 40 36 44 39 48 5 10 15 20 25 30 35 40 45 50 55 60 6 7 8 9 10 11 12 12 14 16 18 20 22 24 18 21 24 27 30 33 36 24 28 32 36 40 44 48 30 35 40 45 50 55 60 36 42 48 54 60 66 72 42 49 56 63 70 77 84 48 56 64 72 80 88 96 54 63 72 81 90 99 108 120 132 144 60 70 80 90 100 110 120 110 121 132 66 77 88 99 72 84 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 5 6 10 9 15 12 20 15 25 18 30 21 35 24 40 27 45 30 50 36 55 39 60 5 10 15 20 25 30 35 40 45 50 55 60 6 7 8 9 10 11 12 12 14 16 18 20 22 24 18 21 24 27 30 33 36 24 28 32 36 40 44 48 30 35 40 45 50 55 60 36 42 48 54 60 66 72 42 49 56 63 70 77 84 48 56 64 72 80 88 96 54 63 72 81 90 99 108 120 132 144 60 70 80 90 100 110 120 110 121 132 66 77 88 99 72 84 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 5 6 10 9 15 12 20 15 25 18 30 21 35 24 40 27 45 30 50 36 55 39 60 5 10 15 20 25 30 35 40 45 50 55 60 6 7 8 9 10 11 12 12 14 16 18 20 22 24 18 21 24 27 30 33 36 24 28 32 36 40 44 48 30 35 40 45 50 55 60 36 42 48 54 60 66 72 42 49 56 63 70 77 84 48 56 64 72 80 88 96 54 63 72 81 90 99 108 120 132 144 60 70 80 90 100 110 120 110 121 132 66 77 88 99 72 84 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 5 6 10 9 15 12 20 15 25 18 30 21 35 24 40 27 45 30 50 36 55 39 60 5 10 15 20 25 30 35 40 45 50 55 60 6 7 8 9 10 11 12 12 14 16 18 20 22 24 18 21 24 27 30 33 36 24 28 32 36 40 44 48 30 35 40 45 50 55 60 36 42 48 54 60 66 72 42 49 56 63 70 77 84 48 56 64 72 80 88 96 54 63 72 81 90 99 108 120 132 144 60 70 80 90 100 110 120 110 121 132 66 77 88 99 72 84 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 5 6 10 9 15 12 20 15 25 18 30 21 35 24 40 27 45 30 50 36 55 39 60 5 10 15 20 25 30 35 40 45 50 55 60 6 7 8 9 10 11 12 12 14 16 18 20 22 24 18 21 24 27 30 33 36 24 28 32 36 40 44 48 30 35 40 45 50 55 60 36 42 48 54 60 66 72 42 49 56 63 70 77 84 48 56 64 72 80 88 96 54 63 72 81 90 99 108 120 132 144 60 70 80 90 100 110 120 110 121 132 66 77 88 99 72 84 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 5 6 10 9 15 12 20 15 25 18 30 21 35 24 40 27 45 30 50 36 55 39 60 5 10 15 20 25 30 35 40 45 50 55 60 6 7 8 9 10 11 12 12 14 16 18 20 22 24 18 21 24 27 30 33 36 24 28 32 36 40 44 48 30 35 40 45 50 55 60 36 42 48 54 60 66 72 42 49 56 63 70 77 84 48 56 64 72 80 88 96 54 63 72 81 90 99 108 120 132 144 60 70 80 90 100 110 120 110 121 132 66 77 88 99 72 84 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 4 6 8 9 12 12 16 15 20 18 24 21 28 24 32 27 36 30 40 36 44 39 48 5 10 15 20 25 30 35 40 45 50 55 60 6 7 8 9 10 11 12 12 14 16 18 20 22 24 18 21 24 27 30 33 36 24 28 32 36 40 44 48 30 35 40 45 50 55 60 36 42 48 54 60 66 72 42 49 56 63 70 77 84 48 56 64 72 80 88 96 54 63 72 81 90 99 108 120 132 144 60 70 80 90 100 110 120 110 121 132 66 77 88 99 72 84 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 4 6 8 9 12 12 16 15 20 18 24 21 28 24 32 27 36 30 40 36 44 39 48 5 10 15 20 25 30 35 40 45 50 55 60 6 7 8 9 10 11 12 12 14 16 18 20 22 24 18 21 24 27 30 33 36 24 28 32 36 40 44 48 30 35 40 45 50 55 60 36 42 48 54 60 66 72 42 49 56 63 70 77 84 48 56 64 72 80 88 96 54 63 72 81 90 99 108 120 132 144 60 70 80 90 100 110 120 110 121 132 66 77 88 99 72 84 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 4 6 8 9 12 12 16 15 20 18 24 21 28 24 32 27 36 30 40 36 44 39 48 5 10 15 20 25 30 35 40 45 50 55 60 6 7 8 9 10 11 12 12 14 16 18 20 22 24 18 21 24 27 30 33 36 24 28 32 36 40 44 48 30 35 40 45 50 55 60 36 42 48 54 60 66 72 42 49 56 63 70 77 84 48 56 64 72 80 88 96 54 63 72 81 90 99 108 120 132 144 60 70 80 90 100 110 120 110 121 132 66 77 88 99 72 84 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 4 6 8 9 12 12 16 15 20 18 24 21 28 24 32 27 36 30 40 36 44 39 48 5 10 15 20 25 30 35 40 45 50 55 60 5 7 8 9 10 11 12 10 14 16 18 20 22 24 15 21 24 27 30 33 36 20 28 32 36 40 44 48 25 35 40 45 50 55 60 30 42 48 54 60 66 72 35 49 56 63 70 77 84 40 56 64 72 80 88 96 45 63 72 81 90 99 108 120 132 144 50 70 80 90 100 110 120 110 121 132 55 77 88 99 60 84 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 4 6 8 9 12 12 16 15 20 18 24 21 28 24 32 27 36 30 40 36 44 39 48 5 10 15 20 25 30 35 40 45 50 55 60 5 11 8 9 10 12 10 22 16 18 20 24 15 33 24 27 30 36 20 44 32 36 40 48 25 55 40 45 50 60 30 66 48 54 60 72 35 77 56 63 70 84 40 88 64 72 80 96 45 99 72 81 90 108 120 132 144 50 110 121 132 80 88 90 99 100 110 120 55 60 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 4 6 8 9 12 12 16 15 20 18 24 21 28 24 32 27 36 30 40 36 44 39 48 5 10 15 20 25 30 35 40 45 50 55 60 5 11 8 9 10 12 10 22 16 18 20 24 15 33 24 27 30 36 20 44 32 36 40 48 25 55 40 45 50 60 30 66 48 54 60 72 35 77 56 63 70 84 40 88 64 72 80 96 45 99 72 81 90 108 120 132 144 50 110 121 132 80 88 90 99 100 110 120 55 60 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 4 6 8 9 12 12 16 15 20 18 24 21 28 24 32 27 36 30 40 36 44 39 48 5 10 15 20 25 30 35 40 45 50 55 60 5 11 8 9 10 12 10 22 16 18 20 24 15 33 24 27 30 36 20 44 32 36 40 48 25 55 40 45 50 60 30 66 48 54 60 72 35 77 56 63 70 84 40 88 64 72 80 96 45 99 72 81 90 108 120 132 144 50 110 121 132 80 88 90 99 100 110 120 55 60 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 4 6 8 9 12 12 16 15 20 18 24 21 28 24 32 27 36 30 40 36 44 39 48 5 10 15 20 25 30 35 40 45 50 55 60 5 11 8 9 10 12 10 22 16 18 20 24 15 33 24 27 30 36 20 44 32 36 40 48 25 55 40 45 50 60 30 66 48 54 60 72 35 77 56 63 70 84 40 88 64 72 80 96 45 99 72 81 90 108 120 132 144 50 110 121 132 80 88 90 99 100 110 120 55 60 96 108
1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20 11 22 12 24 3 4 6 8 9 12 12 16 15 20 18 24 21 28 24 32 27 36 30 40 36 44 39 48 5 10 15 20 25 30 35 40 45 50 55 60 5 11 8 9 10 12 10 22 16 18 20 24 15 33 24 27 30 36 20 44 32 36 40 48 25 55 40 45 50 60 30 66 48 54 60 72 35 77 56 63 70 84 40 88 64 72 80 96 45 99 72 81 90 108 120 132 144 50 110 121 132 80 88 90 99 100 110 120 55 60 96 108
3 6 9 12 15 18 21 24 27 30 33 36 5 10 15 20 25 30 35 40 45 50 55 60 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 5 10 15 20 25 30 35 40 45 50 55 60 11 22 33 44 55 66 77 88 99 110 121 132
33 3 6 9 12 15 18 21 24 27 30 36 55 5 10 15 20 25 30 35 40 45 50 60 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 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 5 + 5 = 58 3 55 33 55 + 25 55or 1 11 55
33 3 6 9 12 15 18 21 24 27 30 36 55 5 10 15 20 25 30 35 40 45 50 60 20 25 5 10 15 30 35 40 45 50 55 60 44 55 11 22 33 66 77 88 99 110 121 132 3 5 > 5 This same data can be used to determine that 11
