1. Integers – Learning Outcomes
Identify and list integers.
Use the number line to order integers.
Add, subtract, multiply, and divide integers.
Use the number line to illustrate integer arithmetic.
Solve problems about integers.

2. Identify and List Integers
Integers are the set of all whole numbers.
ℤ={…, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …}
Recall the natural numbers:
ℕ= 1, 2, 3, 4, 5, 6, …
The natural numbers are all integers.
We can say that the natural numbers are a subset of the integers:
ℕ∈ℤ

3. Identify and List Integers
Outcome check: can you identify the integers? -4
3.14 9
1 2 42
Outcome check: list any ten integers and check with the person beside you.

4. Use the Number Line to Order Integers
Extend the natural number line to the left.
This is where the rest of the integers are found.
The number line gets more negative as you continue left.
Which is greater: 7 or -2?
Which is greater: -2 or 0?
Which is greater: 7 or -7?

5. Use the Number Line to Order Integers
The symbols “>” and “<“ are used to compare numbers.
e.g. 5<9 means that 5 is less than 9.
e.g. 9>5 means that 9 is greater than 5.
The arrow points to the lesser number.
Outcome check: Use the “<“ or “>” symbols to compare the following pairs of numbers:
2, 3
6, 10
12, 0
2, -1
-15, 15
2, -3
-1000, 5
12, 6

6. Use the Number Line to Illustrate Arithmetic
Calculate each of the following and express it on a big enough number line:
3 + 5
3 + -5
-3 + 5
-3 – 5

7. Use the Number Line to Illustrate Arithmetic
Outcome check: Calculate each of the following:
-4 + 3
2 + 4
5 – 6
-2 – 3
-4 + 5
-3 – 2
4 – 6
-5 + 5
1 – (-4)

8. Use the Number Line to Illustrate Arithmetic
Outcome check: The table below shows the average temperature in Finland last year:
Month
Jan
Feb
Mar
Apr
May
Jun
Temp -3 -4 -1 5 9 14
Jul
Aug
Sept
Oct
Nov
Dec 17 16 11 6 1 -2
By how much did the temperature change:
from May to June?
from November to December?
from March to May?
from January to March?

9. Multiply Integers Copy this table into your copybook and complete it.
4×3 12
3×3
2×3
1×3
0×3
−1×3
−2×3
−3×3
−4×3
What is the pattern as you go down the table?

10. Multiply Integers Copy this table into your copybook and complete it.
4×5 20
3×5
2×5
1×5
0×5
−1×5
−2×5
−3×5
−4×5
What is the pattern as you go down the table?
From these two tables, come up with a rule about multiplying a negative number and a positive number.

11. Multiply Integers Copy this table into your copybook and complete it.
4×−3
-12
3×−3
2×−3
1×−3
0×−3
−1×−3
−2×−3
−3×−3
−4×−3
What is the pattern as you go down the table?

12. Multiply Integers Copy this table into your copybook and complete it.
4×−5
-20
3×−5
2×−5
1×−5
0×−5
−1×−5
−2×−5
−3×−5
−4×−5
What is the pattern as you go down the table?
From these two tables, come up with a rule about multiplying two negative numbers together.

13. Multiply and Divide Integers
positive x positive = positive
positive x negative = negative
negative x positive = negative
negative x negative = positive
The rules for signs are the same for dividing two integers.

14. Multiply and Divide Integers
Outcome check: Calculate:
5 x 6
-5 x 6
5 x -6
-5 x -6
(10)(4)
(-2)(16)
7(-8)
(-15)(-6)
(-5)(6)(-3)
(-16)(-2)(-5)
15 ÷ 3
-24 ÷ 6
-39 ÷ -13
21 ÷ -7
16 ÷ 4
-99 ÷ -33
-66 ÷ 11
144 ÷ -12
-16 ÷ -8
25 ÷ -5

15. Raise Integers to Powers
Recall that multiplication is repeated addition:
e.g = 3 x 5
There is a shorthand for repeated multiplication, called powers.
e.g. 3 x 3 x 3 x 3 x 3 = 35
e.g. 2 x 2 x 2 x 2 = 24
Calculate: 23 34
(-5)2
(-3)3
(-4)4
(-2)5

16. Use the Order of Operations
GEMDAS is used as normal for indices.
G – groups (operations inside brackets, roots, modulus, numerators, denominators, …) are always strongest.
E – exponents (powers, roots) are next.
MD – multiplication, division are the same strength.
AS – addition, subtraction are the same strength.
If a tie would cause different answers, work from left to right.

17. Use the Order of Operations
Outcome check: Calculate:
2×3+3× −4
4×5+2× −3
3× −2 +5×4
3× −2 +5× −3
2× × 3−2
2× 5−8 +3× 7−11
5× 2−3 +6× 4−1
5× 2−8 +3× 1−6
5 3− −2 +5
2 8− −4 +4 −
−2 3 − 5 2
1 2 × −3 2
−5 2 × 2 2
2 −7−2
3 − −5 5−2 3
49 −3−4 2
6 −1002÷ ×3