1. Dr. Rena Gibbs Amy Edmundson

2. What does it mean to “invert and multiply”?

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

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

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

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

7. Three “thirds” fit in one.

8. Six “thirds” fit in two.

9. Nine “thirds” fit in three.

10. Twelve “thirds” fit in four.

11. Fifteen “thirds” fit in five.

12. Eighteen “thirds” fit in six.

13. Twenty-one “thirds” fit in seven.

14. So because twenty-one “1/3rds” fit into seven wholes.

15. We can verify this result in many ways, but the most common way taught in schools is to “invert and multiply”: 21 We have seven “wholes” that we multiply by the three that fit into each whole and 7 times 3 is 21, therefore

17.  Let’s say we want to do this math problem

18. 123456789101112 24681012141618202224 369121518212427303639 4812162024283236404448 51015202530354045505560 61218243036424854606672 71421283542495663707784 81624324048566472808896 918273645546372819099108 102030405060708090100110120 112233445566778899110121132 1224364860728496108120132144

19. 123456789101112 24681012141618202224 369121518212427303639 4812162024283236404448 51015202530354045505560 61218243036424854606672 71421283542495663707784 81624324048566472808896 918273645546372819099108 102030405060708090100110120 112233445566778899110121132 1224364860728496108120132144

20. 123456789101112 24681012141618202224 369121518212427303639 4812162024283236404448 51015202530354045505560 61218243036424854606672 71421283542495663707784 81624324048566472808896 918273645546372819099108 102030405060708090100110120 112233445566778899110121132 1224364860728496108120132144

21. 123456789101112 24681012141618202224 369121518212427303639 51015202530354045505560 51015202530354045505560 61218243036424854606672 71421283542495663707784 81624324048566472808896 918273645546372819099108 102030405060708090100110120 112233445566778899110121132 1224364860728496108120132144

22. 123456789101112 24681012141618202224 369121518212427303639 51015202530354045505560 51015202530354045505560 61218243036424854606672 71421283542495663707784 81624324048566472808896 918273645546372819099108 102030405060708090100110120 112233445566778899110121132 1224364860728496108120132144

23. 123456789101112 24681012141618202224 369121518212427303639 51015202530354045505560 51015202530354045505560 61218243036424854606672 71421283542495663707784 81624324048566472808896 918273645546372819099108 102030405060708090100110120 112233445566778899110121132 1224364860728496108120132144

24. 123456789101112 24681012141618202224 369121518212427303639 51015202530354045505560 51015202530354045505560 61218243036424854606672 71421283542495663707784 81624324048566472808896 918273645546372819099108 102030405060708090100110120 112233445566778899110121132 1224364860728496108120132144

25. 123456789101112 24681012141618202224 369121518212427303639 51015202530354045505560 51015202530354045505560 61218243036424854606672 71421283542495663707784 81624324048566472808896 918273645546372819099108 102030405060708090100110120 112233445566778899110121132 1224364860728496108120132144

26. 123456789101112 24681012141618202224 369121518212427303639 4812162024283236404448 51015202530354045505560 61218243036424854606672 71421283542495663707784 81624324048566472808896 918273645546372819099108 102030405060708090100110120 112233445566778899110121132 1224364860728496108120132144

27. 123456789101112 24681012141618202224 369121518212427303639 4812162024283236404448 51015202530354045505560 61218243036424854606672 71421283542495663707784 81624324048566472808896 918273645546372819099108 102030405060708090100110120 112233445566778899110121132 1224364860728496108120132144

28. 123456789101112 24681012141618202224 369121518212427303639 4812162024283236404448 51015202530354045505560 61218243036424854606672 71421283542495663707784 81624324048566472808896 918273645546372819099108 102030405060708090100110120 112233445566778899110121132 1224364860728496108120132144

29. 123456789101112 24681012141618202224 369121518212427303639 4812162024283236404448 51015202530354045505560 51015202530354045505560 71421283542495663707784 81624324048566472808896 918273645546372819099108 102030405060708090100110120 112233445566778899110121132 1224364860728496108120132144

30. 123456789101112 24681012141618202224 369121518212427303639 4812162024283236404448 51015202530354045505560 51015202530354045505560 112233445566778899110121132 81624324048566472808896 918273645546372819099108 102030405060708090100110120 1224364860728496108120132144

31. 123456789101112 24681012141618202224 369121518212427303639 4812162024283236404448 51015202530354045505560 51015202530354045505560 112233445566778899110121132 81624324048566472808896 918273645546372819099108 102030405060708090100110120 1224364860728496108120132144

32. 123456789101112 24681012141618202224 369121518212427303639 4812162024283236404448 51015202530354045505560 51015202530354045505560 112233445566778899110121132 81624324048566472808896 918273645546372819099108 102030405060708090100110120 1224364860728496108120132144

33. 123456789101112 24681012141618202224 369121518212427303639 4812162024283236404448 51015202530354045505560 51015202530354045505560 112233445566778899110121132 81624324048566472808896 918273645546372819099108 102030405060708090100110120 1224364860728496108120132144

34. 123456789101112 24681012141618202224 369121518212427303639 4812162024283236404448 51015202530354045505560 51015202530354045505560 112233445566778899110121132 81624324048566472808896 918273645546372819099108 102030405060708090100110120 1224364860728496108120132144

35. 369121518212427303336 51015202530354045505560 51015202530354045505560 112233445566778899110121132

36. 369121518212427303336 51015202530354045505560 51015202530354045505560 112233445566778899110121132

37. 36912151821242730 33 36 5101520253035404550 55 60 51015202530354045505560 112233445566778899110121132

38. 369121518212427303336 51015202530354045505560 51015202530354045505560 112233445566778899110121132

39. 36912151821242730 33 36 5101520253035404550 55 60 5101520 25 30354045505560 11223344 55 66778899110121132 This same data can be used to determine that