1. Whole-number
operations

2. Why do we use a base-ten number system?

3. 1

4. 2

5. 3

6. 4

7. 5

8. 6

9. 7

10. 8

11. 9

12. 10

13. Grouping in base ten
14

14. Base-ten number system
24123 Allowable digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Base-ten place value
2 4 1 2 3
______ ______ ______ ______ ______
104 103 102 10 1
10000 1000 100
24123 = 2·10000 + 4·1000 + 1·100 + 2·10 + 3·1
expanded form

15. How would we count if we only had one hand?

16. 1

17. 2

18. 3

19. 4

20. 10
Base-five number system

21. Grouping in base five
24five

22. Base-five number system
Allowable digits:
24123five 0, 1, 2, 3, 4
Base-five place value
2 4 1 2 3
______ ______ ______ ______ ______
54 53 52 5 1
625 125 25
24123five = 2·625 + 4·125 + 1·25 + 2·5 + 3·1
= 1250 + 500 + 25 + 10 + 3
= 1788ten

23. 203five = _______ ten
Base-five place value
2 0 3
______ ______ ______
52 5 1
25
203five = 2·25 + 0·5 + 3·1
= 50 + 0 + 3
= 53ten

24. Base-four number system
Allowable digits:
20213four 0, 1, 2, 3
Base-four place value
2 0 2 1 3
______ ______ ______ ______ ______
44 43 42 4 1
256 64 16
20213four = 2·256 + 0·64 + 2·16 + 1·4 + 3·1
= 512 + 0 + 32 + 4 + 3
= 551ten

25. Base-seven number system
Allowable digits:
261seven 0, 1, 2, 3, 4, 5, 6
Base-seven place value
2 6 1
______ ______ ______ ______ ______
74 73 72 7 1
2401 343 49
261seven = 2·49 + 6·7 + 1·1
= 98 + 42 + 1
= 141ten

26. Base-twelve number system
Allowable digits:
2TEtwelve 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E
Base-twelve place value
2 T E
______ ______ ______ ______ ______
124 123 122 12 1
20736 1728 144
2TEtwelve = 2·144 + 10·12 + 11·1
= 288 + 120 + 11
= 419ten

27. 2111
281ten = ________five
Base-five place value
2 1 1 1
______ ______ ______ ______ ______
54 53 52 5 1
625 125 25
281 31 6
250 25 5
31 6 1

28. 10121
97ten = ________three
Base-three place value
1 0 1 2 1
______ ______ ______ ______ ______
34 33 32 3 1
81 27 9
97 16 7
81 9 6
16 7 1

29. 1E5
281ten = ________twelve
Base-twelve place value
1 E 5
______ ______ ______ ______
123 122 12 1
1728 144
281 137
144 132
137 5

30. Addition Algorithms
15 + 16
Using Base-ten Blocks
31

31. Expanded Algorithm Traditional Algorithm
1
15 15
+ 16 + 16
11 (add ones) 31
20 (add tens)
31

32. Scratch Addition
3 2
2
1 3 4 7 8
2 6 52 98 75
61 4 3 91 86 3
+ 3 92 + 6 73 9 7
14 2 27 3 7

33. Base 4 Addition Allowable digits: 0, 1, 2, 3
+ 0 1 2 3 1 0
4ten = __ __ four
0 0 1 2 3 4 1
1 1 2 3 10
1 1
5ten = __ __ four
2 2 3 10 11 4 1
3 3 10 11 12
1 2
6ten = __ __ four
4 1

34. 1 + 0 1 2 3
23four 0 0 1 2 3
+ 32four 1 1 2 3 10
12 1four 2 2 3 10 11
3 3 10 11 12
11
203four
+ 133four
10 0 2four

35. 1 1 2
7ten = __ __ five
23five 5 1
+ 14five
4 2five
11 1 4
12ten = __ __ eight
275eight 8 1
+ 327eight 1 2
10ten = __ __ eight
62 4eight 8 1

36. 1 1 2
14ten = __ __ twelve
T9twelve 12 1
+ 75twelve
16 2twelve
1 6
18ten = __ __ twelve
12 1

37. Subtraction Algorithms “Take away”
32 – 15
Using Base-ten Blocks
Start with 32
Take away 15
17 left

38. Traditional Algorithm
2 1
32
- 15
17

39. Subtraction
2 6 4 9 4 15
32four 526seven T53twelve
- 13four - 461seven - 528twelve
1 3four 3 5seven 5 2 7twelve

41. Whole Number multiplication
3 x 5 = 15
factors product

42. Models for whole-number multiplication
3 x 5 = 15
Repeated addition model
“three fives”
5 + 5 + 5 = 15

43. DESE MAP 4th grade Released item, 2004

44. Models for whole-number multiplication
3 x 5 = 15
Array model (grid model)
Count
intersections
3
5

45. Models for whole-number multiplication
3 x 5 = 15
Area model
Count
rectangles
3
5

46. A set is closed under multiplication if the
product of any two numbers in the set is still in
the set.
Which sets are closed under multiplication:
a) {1, 2, 3, 4, …} Closed
b) {0, 1} Closed
c) {0, 1, 2} Not closed (2 x 2 = 4)

47. Properties of whole-number multiplication
Commutative property Identity Property of
of multiplication multiplication
axb=bxa ax1=a
3x5=5x3 7x1=7
1 is the multiplicative
Associative Property Identity
of multiplication
a x (b x c) = (a x b) x c Zero multiplication
5 x (2 x 7) = (5 x 2) x 7 property
ax0=0
7x0=0

48. Distributive Property of Multiplication over Addition
a(b + c) = ab + ac
Rectangle (area) model for the Distributive Property
3(4 + 1) = 3 · 4 + 3 · 1
4 + 1 4 1
3 = 3 + 3

49. Multiplication Algorithms Connection to Algebra:
(36)(52)
36 x 52 = (30 + 6)(50 + 2)
= 1500 + 60 + 300 + 12
= 1872
Partial Sums 36
x 52
12 2x6
60 2 x 30
300 50 x 6
1500 50 x 30
1872

50. Traditional Algorithm
1
3
36
x 52
72
18 0 0
1872

51. Lattice Multiplication
36 x 52
3 6
1 3
1 5 0 5
0 1
8 6 2
2
7 2
1872

52. Lattice Multiplication
93 x 83
9 31
7 2
7 2 4 8
2 0
7 7 9
3
1 9
7719

53. 1
2
23four 1 2
6ten = __ __ four
x 32four 4 1
11 2 1 1
5ten = __ __ four
2010 4 1
212 2four 2 1
9ten = __ __ four
4 1
2 0
8ten = __ __ four
4 1

54. Lattice Multiplication 1 2
6ten = __ __ four
23four x 32four 4 1
2 1
9ten = __ __ four
2 3 4 1
1
1 2
2 2 1 3 1 0
4ten = __ __ four
1 1 4 1
1 0 2 2 1 1
5ten = __ __ four
4 1
2 2
2122four

55. 1
9
4Ttwelve 1 8
20ten = __ __ twelve
x E2twelve 12 1
98
9 2
110ten = __ __ twelve
4520
12 1
45E 8twelve
4 5
53ten = __ __ twelve
12 1

56. Whole Number Division
6 3=2
dividend divisor quotient
Division is related to multiplication:
6 3 = 2 if 6 = 3 · 2

57. Models for whole-number division
Set (partition) model
6 3=2
6 objects 2 objects
in each
set
Divide into 3 equal sets

58. Set (partition) model
6 2=3
6 objects 3 objects
in each
set
Divide into 2 equal sets

59. Models for whole-number division
Missing Factor Model
6 3=
6=3·
=2

60. Models for whole-number division
Repeated Subtraction Model
6 3=?
6 3 3 =0
2 subtractions

61. Remainders (using a set model)
7 3
7 objects 2 objects
in each
set
Divide into 3 equal sets 1 object
left over
7 3=2R1

62. Division Algorithms
Repeated Subtraction 678 6 = 113
6 678
- 600 100 sixes
78
- 60 10 sixes
18
- 18 3 sixes
0 113 sixes

63. 32four 2four = 13four
Set model

64. Traditional Algorithm
13
2four 32four
2four x 2four = 10four
- 2
12 2four x 3four = 12four
- 12
0

65. 12 2
101five 12322five 101five
x2
- 101
202five
222
- 202
20 2
-202
0

66. Order of Operations
Parentheses
Exponents
Multiplication / Division (Left to Right)
Addition / Subtraction (Left to Right)
Simplify 15 + 6 · 4 – 10
= 15 + 24 – 10
= 39 – 10
= 29