Integers include positive and negative whole numbers. Absolute value is the distance from zero. Addition and subtraction on a number line involve moving left or right. Multiplication follows patterns based on sign. Division is the inverse of multiplication regarding sign. A number is divisible by another if it can be written as a product of an integer multiple. Divisibility tests identify patterns in a number's digits.

1. Integers and
Divisibility

2. Counting (natural) numbers 1, 2, 3, …
Whole numbers 0, 1, 2, 3, …
Integers … -3, -2, -1, 0, 1, 2, 3, …

3. Absolute value - the distance between a
number and zero
|4| = 4
| 2| = 2

4. Models for integer addition
Number Line Model
2 + –3 = 1
1+ 2= 3

5. Charged field (chip) model for Addition
2+ 3= 5 3+3= 0
- - - +
- - - +
- - +
1+ 2= 1
Two numbers are
+ - additive inverses if
- their sum is zero.

6. Charged field (chip) model for Subtraction
“Take away”
2 ( 1) = 1
Start with 2 negatives
- Take away 1 negative
1 negative left
-
1 ( 2) = 3
Start with 1 positive
+ Take away 2 negatives
3 positives left

7. 2 1= 3
- Start with 2 negatives
Take away 1 positive
- 3 negatives left

8. Number Line model for subtraction “Walk
direction to face the line”
Negative numbers “face left”
Positive numbers “face right”
how to walk
Subtraction “walk backward”
Addition “walk forward”
2 1= 3
2 + ( 1) = 3
-1 -2
Relating subtraction
to addition:
a – b = a + (– b)

9. Number Line model for subtraction
direction to face how to walk
Negative numbers Subtraction (backward)
Positive numbers Addition (forward)
2 ( 3) = 1
-(-3)
-2
2 + (3) = 1

10. Integer Multiplication
Charged Field (Chip) Model
--
3( 2) = 6
--
Add 3
groups
of two
negatives
--
3(2) = 6 --
-- ++
Take away of two ++ --
3 groups positives ++

11. Number Line model for multiplication
arrows
Negative numbers
Positive numbers
-1 -1 -1
3( 1) = 3
3 groups Of -1 arrows
-1 -(-1)-(-1) (-1)
-
3( 1) = 3
Reverse
Of -1 arrows
3 groups

12. Integer Multiplication – An investigation of patterns
3·3= 9 3
3· 3= 9 +3
3·2= 6 2· 3= 6
3 +3
3·1= 3 1· 3= 3
3 +3
3·0= 0 0· 3= 0
3 +3
3· 1= 3 1· 3= 3
3 +3
3· 2= 6 2· 3= 6
3· 3= 9 3 3· 3= 9 +3
Positive · Positive = Positive Positive · Negative = Negative
Positive · Negative = Negative Negative · Negative = Positive
Same Signs – Positive answer
Different Signs – Negative answer

13. Multiplication and Division
a · b = c means c b=a
Example: 3 · 4 = 12 means 12 4 = 3
Integer Division
pos · pos = pos so pos pos = pos
pos · neg = neg so neg neg = pos
neg · pos = neg so neg pos = neg
neg · neg = pos so pos neg = neg
Sign rules for division are identical to multiplication

14. Using the Difference of Squares formula to multiply
(a + b)(a – b) = a2 – b2
Multiply 42 · 38 Multiply 107 · 93
=(40 + 2)(40 – 2) =(100 + 7)(100 – 7)
= 40 2 – 22 = 1002 – 72
= 1600 – 4 = 10000 – 49
= 1596 = 9951

16. Divisibility
If a and b are integers, then b divides a if
there is an integer c such that a = b · c
Why?
Does 3 | 12 Yes Because 12 = 3 · 4
Does 6 | 12 Yes Because 12 = 6 · 2
Does 24 | 12 No Because 12 = 24 · integer

17. Divisibility tests
2 Even number (ends in 0, 2, 4, 6, 8)
3 Sum of digits is divisible by 3
4 Last two digits divisible by 4
5 Ends in 0 or 5
6 Divisible by both 2 and 3
7 Cross out, double, subtract
8 Last three digits divisible by 8
9 Sum of digits divisible by 9
10 Ends in 0
11 Difference of alternate digits (ocean waves)

18. Test 5182 for divisibility by
Why?
2 Yes 5182 is even
3 No 5 + 1 + 8 + 2 = 16, and 3 | 18
4 No 4 | 82
5 No 5182 does not end in 0 or 5
6 No Not divisible by both 2 and 3

19. Test 5182 for divisibility by 5182
Why? -4
7 No 7 | 43 514
-8
8 No 8 | 182
43
9 No 5 +1 + 8 + 2 = 16 and 9 | 16
10 No 5182 does not end in 0
13
11 No 11 | 10 5182 13 – 3 = 10
3

20. Test 3,885,840 for divisibility by
Why?
2 Yes 3,885,840 is even
3 Yes 3+8+8+5+8+4+0=36, and 3 | 36
4 Yes 4 | 40
5 Yes 3,885,840 ends in 0
6 Yes Divisible by both 2 and 3

21. Test 3,885,840 for divisibility by 3,885,840
Why?
7 Yes 7 | 21 -0
388,584
8 Yes 8 | 840 -8
38,850
9 Yes 9 | 36 -0
3,885
10 Yes Ends in 0 -10
378
11 No 11 | 2 19 -16
21
3885840
17 19 – 17 = 2