1150 day 4

Whole-number operations

Why do we use a base-ten number system?

1

2

3

4

5

6

7

8

9

10

Grouping in base ten

14

Base-ten number system

Base-ten place value

______ ______ ______ ______ ______

110102

100103

1000104

10000

2 4 1 2 3

24123

24123 = 2·10000 + 4·1000 + 1·100 + 2·10 + 3·1expanded form

Allowable digits:0, 1, 2, 3, 4, 5, 6, 7, 8, 9

How would we count if we only had one hand?

1

2

3

4

10

Base-five number system

Grouping in base five

24five

Base-five number system

Base-five place value

______ ______ ______ ______ ______

1552

2553

12554

625

2 4 1 2 3

24123five

24123five = 2·625 + 4·125 + 1·25 + 2·5 + 3·1= 1250 + 500 + 25 + 10 + 3= 1788ten

Allowable digits:0, 1, 2, 3, 4


______ ______ ______

1552

25

2 0 3

203five = 2·25 + 0·5 + 3·1= 50 + 0 + 3= 53ten

203five = _______ ten

Base-four number system

Base-four place value

______ ______ ______ ______ ______

1442

1643

6444

256

2 0 2 1 3

20213four

20213four = 2·256 + 0·64 + 2·16 + 1·4 + 3·1= 512 + 0 + 32 + 4 + 3= 551ten

Allowable digits:0, 1, 2, 3

Base-seven number system

Base-seven place value

______ ______ ______ ______ ______

1772

4973

343

74

2401

2 6 1

261seven

261seven = 2·49 + 6·7 + 1·1= 98 + 42 + 1= 141ten

Allowable digits:0, 1, 2, 3, 4, 5, 6

Base-twelve number system

Base-twelve place value

______ ______ ______ ______ ______

112122

144123

1728124

20736

2 T E

2TEtwelve

2TEtwelve = 2·144 + 10·12 + 11·1= 288 + 120 + 11= 419ten

Allowable digits:0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E

281ten = ________five


______ ______ ______ ______

1552

2553

12554

625

______ 2

31

250

281

1

6

25

31

1

1

5

6

1

2111

97ten = ________three

Base-three place value

______ ______ ______ ______

1332

933

2734

81

______1

16

81

97

1

7

9

16

2

1

6

7

1

10121

0

281ten = ________twelve

Base-twelve place value

______ ______ ______

112122

144123

1728

______

137

144

281

1

5

132

137

E 5

1E5

Addition Algorithms

15 + 16

Using Base-ten Blocks

31

Expanded Algorithm

15+ 16

11 (add ones)20 (add tens)31

Traditional Algorithm1

15+ 16

13

Scratch Addition2

1 32 66 4

+ 3 9241

3

2

1

4 7 85 9 79 8 3

+ 6 7 97

5

7

2

8

6

3

3

3

2

1

72

Base 4 Addition

+ 0 1 2 3

0

1

2

3

Allowable digits: 0, 1, 2, 3

0 1 2 3

1 2 3

4ten = __ __ four

4 1

1 0

10

2 3 10 115ten = __ __ four

4 1

1 1

3 10 11

6ten = __ __ four

4 1

1 212

+ 0 1 2 3

0

1

2

3

0 1 2 3

1 2 3 10

2 3 10 11

3 10 11 12

23four

+ 32four

1four

1

12

203four

+ 133four

2four

1

100

1

23five

+ 14five

2five

1

4

7ten = __ __ five

5 1

1 2

275eight

+ 327eight

4eight

1

62

1 12ten = __ __ eight

8 1

1 4

10ten = __ __ eight

8 1

1 2

T9twelve

+ 75twelve

2twelve

1

16

14ten = __ __ twelve

12 1

1 2


12 1

1 6

Subtraction Algorithms “Take away”

32 – 15

Using Base-ten BlocksStart with 32

Take away 15

17 left

Traditional Algorithm

32- 15

7

2 1

1

Subtraction

32four

- 13four

3four

2 6

1

526seven

- 461seven

5seven

4 9

3

T53twelve

- 528twelve

7twelve

154

25

Whole Number multiplication

3 x 5 = 15

factors product

Models for whole-number multiplication

3 x 5 = 15

Repeated addition model

“three fives”

5 + 5 + 5 = 15

DESE MAP 4th grade Released item, 2004


3 x 5 = 15

Array model (grid model)

3

5

Count intersections


3 x 5 = 15

Area model

3

5

Count rectangles

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, …}

b) {0, 1}

c) {0, 1, 2}

Closed

Closed

Not closed (2 x 2 = 4)

Properties of whole-number multiplication

Commutative property of multiplication

a x b = b x a3 x 5 = 5 x 3

Associative Property of multiplicationa x (b x c) = (a x b) x c5 x (2 x 7) = (5 x 2) x 7

Identity Property of multiplication

a x 1 = a7 x 1 = 7

1 is the multiplicative Identity

Zero multiplication property

a x 0 = 07 x 0 = 0

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

3

4 + 1

= 3 + 3

4 1

Multiplication Algorithms

36 x 52

Partial Sums 36x 52

1260

30015001872

Connection to Algebra:(36)(52)

= (30 + 6)(50 + 2)= 1500 + 60 + 300 + 12= 1872

2 x 6

2 x 30

50 x 6

50 x 30


36x 52

20

1

70

3

181872

Lattice Multiplication

36 x 52

3 6

5

2

15

30

06

12

27

8

1

1872


93 x 83

9 3

8

3

72

24

27

09

91

7

7

7719

1

23four

x 32four

202four

6ten = __ __ four

4 1

1 2

1

5ten = __ __ four

4 1

1 111

9ten = __ __ four

4 1

2 1

1

2

8ten = __ __ four

4 1

2 0

20212


23four x 32four

2 3

3

2

12

21

10

12

22

1

2

2122four

1

5ten = __ __ four

4 1

1 1

6ten = __ __ four

4 1

1 2

9ten = __ __ four

4 1

2 1

4ten = __ __ four

4 1

1 0

4Ttwelve

x E2twelve

808twelve


12 1

1 8

1


12 1

9 22

9


12 1

4 5

45E54

Whole Number Division

6 3 = 2

dividend quotientdivisor

Division is related to multiplication:

6 3 = 2 if 6 = 3 · 2

Models for whole-number division

Set (partition) model

6 3 = 2

6 objects

Divide into 3 equal sets

2 objects in each set

Set (partition) model

6 2 = 3

6 objects




Missing Factor Model

6 3 =

6 = 3 ·

= 2


Repeated Subtraction Model

6 3 = ?

6 3 3 = 0

2 subtractions

Remainders (using a set model)

7 3

7 3 = 2 R 1

7 objects



1 object left over

Division Algorithms

Repeated Subtraction

6 678- 600 100 sixes

78- 60 10 sixes

18- 18 3 sixes

0

678 6 = 113

113 sixes

32four 2four

Set model

= 13four


2four 32four

- 2 2

- 12 0

13

1

2four x 2four = 10four

2four x 3four = 12four

101five 12322five

- 101 22

- 202 20

-2020

12

2

101five

x 2202five

2

2

Order of OperationsParenthesesExponentsMultiplication / Division (Left to Right)Addition / Subtraction (Left to Right)

Simplify 15 + 6 · 4 – 10 = 15 + 24 – 10 = 39 – 10 = 29

Date post:	18-Jul-2015
Category:	Education
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1150 day 4

Education