Decimals are Fractions, Too! Helping Students Connect Fraction and Decimal Understanding (Grades 4-7) by Kimberly Rimbey at the MEAD Conference, Tucson, AZ, 1-20-18 Have you ever wanted to help more students connect fractions and decimals? Do your students struggle to see the connections once you begin adding and subtracting them? Come and learn engaging student-active ways to help them make sense of the connections and truly "see" their interrelatedness. Making sense of decimal concepts and operations requires strong connections with fractions and place value. Join us as we explore fraction and decimal multiplication using concrete and visual models, connected to written work, and grounded in problem solving. Formative assessment strategies will be included.

1. Decimals are Fractions, Too!
Helping Students Connect Fraction
and Decimal Understanding
Kimberly Rimbey, Ph.D.
National Board Certified Teacher – EA Mathematics
kpkimrimbey@gmail.com
MEAD – January 20, 2018

2. How do you feel about teaching
fractions and decimals?

7. Together, we can do it!

8. Today’s Session
Mathematical Goal: Understand the relationship
between fraction and decimal multiplication.
Explorations:
– Decimal-Fraction Connection
– Multiplicative Thinking
– Multiple Representations
– Contextual Situations

9. DECIMAL FRACTIONS

10. About Decimal Fractions
Decimal fractions exhibit attributes of the base-ten system and
of fractions. To fully understand operations with decimal
fractions, one must attend to both systems.
Base-Ten Fractions
Decimals

11. About Fractions

12. The Chocolate Experiment

13. About Fractions
1. Whole numbers tell how many…fractions tell how much.
2. You don’t know how much the fraction is until you know the
whole.

14. About Decimal Fractions – “The Whole”
In the examples presented here, we will define the unit as one
large blue square.
1 whole Tenths Hundredths

15. About Decimal Fractions – Place Value
p. 6

16. MULTIPLICATION

17. About Multiplication
It’s common for children to think that both numbers in a
multiplication problem represent quantities. For example, with 3
x 4, they may think there is 3 of something and 4 of something.
3 x 4

18. About Multiplication
However, when multiplying, only one number represents the
original quantity. The other is the multiplier, or the number of
times one increases (or decreases) the original quantity.
3 x 4

19. About Multiplication
This helps with understanding what’s happening when the multiplier is…
> 1
< 1
= 1

20. About Multiplication
> 1
2 x 4 …. 2 groups of 4 = 8
< 1
½ x 4 … ½ of a group of 4 = 2
½ x ¼ … ½ of ¼ of a group = 1/8 of the group

21. Multiplying Fractions
x

22. 2/3 x 3/4
After his party, Chris has 3/4 of a cake left.
His dog eats 2/3 of what is left.
How much of the whole cake did his dog eat?

23. 2/3 x 3/4
10 1/2 3/41/4

24. 2/3 x 3/4
0 1/3 2/3 1

25. 2/3 x 3/4
10 1/2 3/41/4

26. 2/3 x 3/4 - Put it all together
10 1/2 3/41/4
0 1/3 2/3 1

27. You Try It!
1/2 x 2/3
3/4 x 4/5
3/4 x 1/2

28. Array Model 2/3 x 3/4

29. Array Model 2/3 x 3/4

30. Array Model 2/3 x 3/4

31. Array Model 2/3 x 3/4

32. Array Model 2/3 x 3/4

33. Array Model 2/3 x 3/4

34. Array Model 2/3 x 3/4

35. Array Model 2/3 x 3/4

36. Array Model 2/3 x 3/4

37. Array Model 2/3 x 3/4

38. You Try It!
3/4 x 1/3
1/3 x 4/5
1/4 x 5/8

39. 3 X 0.4
Multiplying Decimal Fractions

40. 3 x 0.4
Tommy had 3 juice boxes. Each box has 0.4 L of
juice.
He dumps all the juice into a pitcher for the
chugging contest.
How many liters of juice are in the pitcher?

41. Multiplying Decimal Fractions 3 x 0.4
Define the unit (the whole).

42. Multiplying Decimal Fractions 3 x 0.4
The whole is composed of 10 tenths.

43. Multiplying Decimal Fractions 3 x 0.4
Show 3 groups of 4 tenths.

44. Multiplying Decimal Fractions 3 x 0.4
Combine the tenths to show the total.
0.4
4/10+
0.4
4/10+
+ +
4/10
0.4

45. Multiplying Decimal Fractions 3 x 0.4
Combine the tenths to show the total.

46. Multiplying Decimal Fractions 3 x 0.4
Combine the tenths to show the total.

47. Multiplying Decimal Fractions 3 x 0.4
Combine the tenths to show the total.

48. Multiplying Decimal Fractions 3 x 0.4
Combine the tenths to show the total.

49. Multiplying Decimal Fractions 3 x 0.4
3 groups of 4 tenths is the same as 1 whole and 2 tenths.

50. Multiplying Decimal Fractions 3 x 0.4
3 groups of 4 tenths is the
same as 1 whole and 2 tenths.
0.4 + 0.4 + 0.4 = 1.2
4/10 + 4/10 + 4/10 = 12/10 =
1 2/10
3 x 0.4 = 1.2

51. You Try It!
2 x 0.4
6 x 0.3
Design a context problem to match.

52. 0.3 X 0.4
Multiplying Decimal Fractions – KP Ten-Frame Tiles

53. 0.3 x 0.4
After the party, Jeannie has 0.4 kg of popcorn.
She spills 0.3 of the popcorn in the fish pond.
If they eat it all, how much of a kg of popcorn do
the fish eat?

54. Multiplying Decimal Fractions 0.3 x 0.4
Define the whole.

55. Multiplying Decimal Fractions 0.3 x 0.4
The whole is
composed of 10
tenths.

56. Multiplying Decimal Fractions 0.3 x 0.4
Show 4 tenths of
the whole.
Think, “I want to
show 3 tenths of
the 4 tenths; that
is, I want to show 3
out of each of the 4
tenths.”

57. Multiplying Decimal Fractions 0.3 x 0.4
“Open” the 4
tenths to show
that each tenth is
composed of 10
pieces.

58. Multiplying Decimal Fractions 0.3 x 0.4
Show 3 tenths (3
out of 10) of each
of the 4 tenths.

59. Multiplying Decimal Fractions 0.3 x 0.4
Combine the pieces
to show the total.

60. Multiplying Decimal Fractions 0.3 x 0.4
Combine the pieces
to show the total.

61. Multiplying Decimal Fractions 0.3 x 0.4
Combine the pieces
to show the total.

62. Multiplying Decimal Fractions 0.3 x 0.4
Combine the pieces
to show the total.

63. Multiplying Decimal Fractions 0.3 x 0.4
Combine the pieces
to show the total.

64. Multiplying Decimal Fractions 0.3 x 0.4
Combine the pieces
to show the total.

65. Multiplying Decimal Fractions 0.3 x 0.4
Combine the pieces
to show the total.

66. Multiplying Decimal Fractions 0.3 x 0.4
Think, “3 tenths of the 4 tenths is
the same as 12 hundredths of the
whole.”
0.3 x 0.4 = 0.12
0.3 of 0.4 = 0.12
3/10 of 4/10 = 12/100

67. You Try It!
Create a context problem for 0.9 x 0.5.
In your group, show 0.9 x 0.5 using physical models.
Extra Practice:
– 0.8 x 0.4
– 0.7 x 0.3

68. 0.3 X 0.4
Multiplying Decimal Fractions – Array Model

69. Array Model 0.3 x 0.4
Define the
whole.

70. Array Model 0.3 x 0.4
The whole is
composed of ten
rows. Each row is
1 tenth of the
whole.
Each row is
composed of 10
hundredths. The
whole contains
100 hundredths.

71. Array Model 0.3 x 0.4
Show 4 tenths of
the whole; that
is, show four
rows of tenths.

72. Array Model 0.3 x 0.4
Show 3 tenths (3
out of 10) of
each of the 4
tenths.

73. Array Model 0.3 x 0.4
Think, “3 tenths
of the 4 tenths is
the same as 12
hundredths of
the whole.”
0.3 x 0.4 = 0.12

74. Sometimes you may feel like this on the inside…

75. But this is what you let them see…

76. Just keep smiling!!!

77. WANT TO KNOW MORE?
Email me – kpkimrimbey@gmail.com
Visit www.kpmathematics.com
Sign up for newsletter

78. THANK YOU!
Kimberly Rimbey, Ph.D.
KP Mathematics
623-297-7373
www.kpmathematics.com
kpkimrimbey@gmail.com