1. Division of Fractions with Models
Dr. Sarah Ledford
In this powerpoint I have taken some good Illustrative Mathematics problems & their solutions and made slides that would be easy for a teacher to use in his/her classroom. Therefore, some of the ideas in this ppt are not mine alone.

2. Fact Families
There are 3 numbers in every fact family. It is SUPER helpful for students to know their fact families. If the numbers are 2, 3, & 5, then we have: = = 5 (commutative property) 5 – 3 = 2 (subtraction is inverse of addition) 5 – 2 = 3

3. Fact Families
If the numbers are 3, 4, & 12, then we have: 3 x 4 = 12 4 x 3 = 12 (commutative property) 12 ÷ 3 = 4 (division is inverse of multiplication) 12 ÷ 4 = 3

4. Multiplication Models
Array Model 2 x 3 2 groups of 3
Area Model 2 x 3 2 rows of 3

5. Now on to fractions…

6. Multiplication of Fractions Making Sense
We want to model multiplication of fractions with an area model.
We can also try to make it make sense.
We will need to start with one whole.
We will need to think about what 2/3 of the whole looks like and what ¾ of the whole looks like.
Draw three wholes – one will be the whole, then shade 2/3 of another whole, & shade ¾ of the last whole.

7. Multiplication of Fractions Making Sense
2/3 of the whole
3/4 of the whole
One whole

8. Multiplication of Fractions Making Sense
3/4 of the whole
This means “what is 2/3 of ¾?”
In looking at the ¾ of the whole, can you see 2/3 of it?
2 of the 3 shaded parts?
What part of the whole is that?
1/2

9. Multiplication of Fractions Making Sense
3/4 of the whole /3 of ¾ = ½ of the whole of the whole (blue) (blue)

10. Multiplication of Fractions Making Sense
Commutative Property of Multiplication
2/3 of the whole
We could also look at “what is 3/4 of 2/3?”
In looking at the 2/3 of the whole, can you see 3/4 of it?
3 of the 4 shaded parts?
We don’t have 4 shaded parts.
How can we make 4 shaded parts?

11. Multiplication of Fractions Making Sense
Commutative Property of Multiplication
2/3 of the whole
Now that we have 4 parts representing the 2/3 of the whole…. Can you see 3/4 of it? 3 of the 4 shaded parts? What part of the whole is that? 3 of the 6 pieces in the whole? 1/2

12. Multiplication of Fractions Making Sense
2/3 of the whole
¾ of 2/3
= ½ of the whole

13. Multiplication of Fractions Area Model
We want to find 2/3 of ¾ so we will model ¾ first. We break the whole into fourths & shade 3 of them. (It doesn’t matter if you break it vertically or horizontally.) I will do this vertically. Then we break the whole into thirds & shade 2 of them. I will do this horizontally. The overlap in shading is the solution as shaded parts to whole number of parts.

14. Multiplication of Fractions Area Model
¾ of the whole
2/3 of the whole
2/3 of ¾ the overlap is the answer as the purple shaded parts to the number of parts in the whole
6 out of 12 = 6/12 = 1/2

15. Now on to division …

16. How Many _____ are in ….?
standards/6/NS/A/1/tasks/692 Solve each problem using pictures and using a number sentence involving division.

17. How many twos are in 6 ?

18. There are 3 sets of two in 6. 6 ÷ 2 = 3
How many twos are in 6 ?
There are 3 sets of two in 6. 6 ÷ 2 = 3

19. How many fives are in 15 ?

20. There are 3 sets of five in 15. 15 ÷ 5 = 3
How many fives are in 15 ?
There are 3 sets of five in ÷ 5 = 3

21. How many threes are in 12 ? Your turn… There are 4 sets of 3 in 12
12 divided by 3 = 4

22. How many halves are in 1 ?

23. There are 2 half-sized pieces in 1 whole. 1 ÷ ½ = 2
How many halves are in 1 ?
There are 2 half-sized pieces in 1 whole. 1 ÷ ½ = 2
This is a 5th grade standard – divide whole numbers by unit fractions.

24. How many halves are in 2 ?

25. There are 4 half-sized pieces in 2 wholes. 2 ÷ ½ = 4
How many halves are in 2 ?
There are 4 half-sized pieces in 2 wholes. 2 ÷ ½ = 4

26. How many halves are in 3?

27. There are 6 half-sized pieces in 3 wholes. 3 ÷ ½ = 6
How many halves are in 3?
There are 6 half-sized pieces in 3 wholes. 3 ÷ ½ = 6

28. Observations?
How many halves in each whole? In the last example, we had three wholes broken into two halves each. What other mathematical statement could we write to show this relationship? 3 x 2 = 6 Hmmmmm… 3 ÷ ½ = 3 x 2 = 6

29. How many sixths are in 4?

30. There are 24 sixth-sized pieces in 4. 4 ÷ 1/6 = 24 4 x 6 = 24
How many sixths are in 4?
There are 24 sixth-sized pieces in 4. 4 ÷ 1/6 = 24 4 x 6 = 24

31. How many thirds are in 2?
Your turn…

32. How many two-thirds are in 2?

33. How many two-thirds are in 2?
Each whole yields a two-thirds and one half of another two-thirds, therefore 3 sets of two-thirds can be made.

34. How many two-thirds are in 2?
Each whole yields a two-thirds and one half of another two-thirds, therefore 3 sets of two-thirds can be made.

35. How many two-thirds are in 2?
We can see that there are 2 wholes with 3 thirds in each whole. There are 2 x 3 thirds in 2. Because we want to know how many two-thirds there are, we have to make groups of 2 thirds. We have to divide the number of thirds we have by 2. So, there are (2 x 3) ÷ 2 = 6 ÷ 2 = 3 two-thirds in 2.

36. How many three-fourths are in 3?

37. How many three-fourths are in 3?
Your turn…

38. How many three-fourths are in 2?
2 complete sets of three-fourths can be made and 2 of the 3 pieces need to make another ¾ are left over, so we have another 2/3 of a three-fourths.

39. How many three-fourths are in 2?
2 complete sets of three-fourths can be made and 2 of the 3 pieces need to make another ¾ are left over, so we have another 2/3 of a three-fourths.

40. How many ¾ are in 2?
We can see that there are 2 wholes with 4 fourths in each whole. There are 2 x 4 fourths in 2. Because we want to know how many three-fourths there are, we have to make groups of 3 fourths. We have to divide the number of fourths we have by 3. So, there are (2 x 4) ÷ 3 = 8 ÷ 3 = 2 2/3 three-fourths in 2.

41. Observations? Hope to see the division of fractions algorithm!!
4 x 6 = 24  24 divided by 1 = 24
2 x 3 = 6  6 divided by 2 = 3
2 x 4 = 8  8 divided by 3 = 8/3

42. How many 1/6’s are in 1/3?
It takes two 1/6 to make a 1/3. .

43. How many 1/6’s are in 1/3?
It takes two 1/6 to make a 1/3. There are 6 sixths in a whole. 1/3 of those 6 pieces are shaded. 1/3 of 6 = 1/3 x 6 = 2

44. It takes four 1/6 to make a 2/3. .
How many 1/6’s are in 2/3?
It takes four 1/6 to make a 2/3. .

45. How many 1/6’s are in 2/3?
It takes four 1/6 to make a 2/3. There are 6 sixths in a whole. 2/3 of those 6 pieces are shaded. 2/3 of 6 = 2/3 x 6 = 4

46. How many 1/4’s are in 2/3?

47. How many 1/4’s are in 2/3?
The picture shows the ones broken into twelfths. Why twelfths? 4 twelfths represent 1/3. 3 twelfths represent ¼.

48. This picture shows 8 twelfths OR 2/3 shaded.
How many 1/4’s are in 2/3?
This picture shows 8 twelfths OR 2/3 shaded.

49. How many 1/4’s are in 2/3?
This picture shows 8 twelfths OR 2/3 shaded. Why does this picture have dark purple and light purple pieces shaded?

50. This picture shows the light purple pieces moved to the top right. .
How many 1/4’s are in 2/3?
This picture shows the light purple pieces moved to the top right. .

51. How many 1/4’s are in 2/3?
We can now relate this picture to the picture of ¼ in the original pictures. We can see that we have two whole ¼’s and 2/3 of another ¼. Therefore, there are 2 2/3 one-fourths in 2/3.

52. How many 1/4’s are in 2/3?

53. How many 1/4’s are in 2/3?
Another explanation: 2/3 = 8/12 and ¼ = 3/12 We want to know how many groups of 3 are in 8. 8 ÷ 3 = 8/3 OR 2 2/3

54. How many 5/12’s are in ½?
Your turn… Just try it!!

55. How many 5/12’s are in ½?
One complete set of 5/12 fits into ½. 1/12 of the whole is left over. That 1/12 of the whole is 1/5 of 5/12. Therefore, there are 1 and 1/5 five- twelfths in ½.

56. How many 5/12’s are in ½? We know
So, we really want to know how many groups of 5 are in 6 OR 6 ÷ 5 = 1 1/5.

57. Observations??
Hopefully they “see” the standard division of fractions algorithm. I am MORE hopeful that they see a different algorithm of getting a common denominator.

58. Dan’s Division Strategy
Dan observes that
He says, “I think that if we are dividing a fraction by a fraction with the same denominator, then we can just divide the numerators.”
Is Dan’s conjecture true for all fractions?
Explain how you know.

59. 6 ÷ 2 = 3
I have a group of 6 circles. I want to put them in groups with 2 circles in each group. How many groups would I have?

60. 6 circles ÷ 2 circles = 3 groups There are 3 groups of two in 6.
6 ÷ 2 = 3
6 circles ÷ 2 circles = 3 groups There are 3 groups of two in 6.

61. Division
In the previous problem, we had 6 circles broken into 3 groups of 2 circles. A “circle” is just a thing. It could have been flowers, cakes, bunnies, etc. They are all “things.” 1/10 is also a “thing.” So, 6/10 is 6 things and 2/10 is 2 of those SAME things. Instead of 6 “circles” broken into 3 groups of 2 “circles,” we have 6 “tenths” broken into 3 groups of 2 “tenths.”

62. Division
Would this always work? Get a common denominator for these division problems & divide the numerators.

63. Division
Would this always work? Get a common denominator for these division problems & divide the numerators.

64. Division of Fractions
Another algorithm for dividing fractions is… You can get a common denominator & divide the numerators!! Neat!!