Mathematics 8 · Handsworth 2026–27 · Unit 1

Integers

Ms. Ras Mulinta
Handsworth Secondary
Guided notes

Integers are the whole numbers and their opposites: … −3, −2, −1, 0, 1, 2, 3 … They describe anything with a direction as well as a size — temperature above and below zero, depth below sea level, money owed rather than held. You met them in Grade 7; this year you use them constantly, inside equations, graphs and measurement. Each idea has a worked example, then a few problems to try. Answers are at the end of each section so you can check yourself right away.

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The toolkit words we will use

An integer is a whole number, positive, negative or zero. Two integers are opposites if they are the same distance from zero on opposite sides: the opposite of −7 is 7. The size of an integer is how far it sits from zero, ignoring the sign, so −12 has a bigger size than −2 even though it is the smaller number. On a number line, numbers get larger as you move right.
Reading the sign
A minus sign in front of a number tells you which side of zero it is on: −5 is five steps left of zero.
A minus sign between two numbers is an instruction to subtract: 8 − 5.
Brackets keep them apart, which is why we write 8 + (−5) rather than 8 + −5.
Same symbol, two jobs. Read which one you are looking at.

1 · Comparing & ordering further right is larger

Put the numbers on a number line and read left to right. Every negative is less than zero, and every negative is less than every positive. Careful: with negatives, the bigger the size, the smaller the number. −12 is less than −2.
Worked example — ordering
Order −7, 4, 0, −3 from smallest to largest.
Place them: −7 is furthest left, then −3, then 0, then 4.
−7, −3, 0, 4

Check yourself

  1. Put in order, smallest first: 3, −8, 0, −1, 5
  2. Which is larger, −12 or −2?
  3. Write the opposite of −7, and of 11
Answers
1. −8, −1, 0, 3, 5    2. −2 — it sits further right on the number line    3. 7 and −11

2 · Adding integers zero pairs, then two cases

One positive and one negative make a zero pair: together they are worth nothing, so they cancel. That single idea gives the whole rule. Same signs: add the sizes and keep the sign. Different signs: subtract the sizes and keep the sign of whichever is bigger.
Worked example — same signs
−3 + (−8): both negative, so nothing cancels.
Add the sizes: 3 + 8 = 11. Keep the negative.
= −11
Worked example — different signs
−9 + 4: nine negatives and four positives.
Four zero pairs cancel, leaving 9 − 4 = 5 negatives.
= −5
Worked example — they cancel exactly
−15 + 15: every negative finds a partner.
= 0

Check yourself

  1. −6 + (−5)
  2. 8 + (−3)
  3. −14 + 14
Answers
1. −11    2. 5    3. 0

3 · Subtracting integers add the opposite

Rewrite every subtraction as an addition before you solve it. To subtract, add the opposite. Once it is an addition, you are back in section 2 and only ever using the two cases there.
Worked example — subtracting a positive
5 − 9 becomes 5 + (−9).
Different signs: 9 − 5 = 4, and the negative is bigger.
= −4
Worked example — subtracting a negative
3 − (−8) becomes 3 + 8.
Two minus signs side by side turn into a plus.
= 11
Worked example — both negative
−6 − (−10) becomes −6 + 10.
Different signs: 10 − 6 = 4, and the positive is bigger.
= 4

Check yourself

  1. 4 − 9
  2. −2 − 6
  3. 5 − (−7)
Answers
1. −5    2. −8    3. 12

4 · Multiplying & dividing count the signs

Same signs give a positive. Different signs give a negative. The same rule covers multiplying and dividing, so learn it once. With more than two numbers, count how many are negative: an even count gives a positive, an odd count gives a negative.
Why a negative times a negative is positive
−3 × 3 = −9,   −3 × 2 = −6,   −3 × 1 = −3,   −3 × 0 = 0.
Each step up adds 3 to the answer. Carry the pattern one step further.
−3 × (−1) = 3
Worked examples
−9 × (−8): same signs, so positive. 9 × 8 = 72.
12 × (−5): different signs, so negative. 12 × 5 = 60.
−56 ÷ 8: different signs, so negative. 56 ÷ 8 = 7.
72,   −60,   −7
More than two numbers
−2 × 3 × (−5): two negatives, an even count.
(−4) × (−2) × (−6): three negatives, an odd count.
= 30   and   = −48

Check yourself

  1. −6 × 7
  2. −48 ÷ (−6)
  3. −2 × (−3) × (−4)
Answers
1. −42    2. 8    3. −24

5 · Order of operations BEDMAS still rules

Brackets, then exponents, then multiply and divide left to right, then add and subtract left to right. Integers change none of that. Watch the brackets around a negative base: (−5)2 is not the same as −52.
Worked example — multiply before add
−4 + 3 × (−5): do the multiply first.
3 × (−5) = −15, so −4 + (−15).
= −19
Worked example — brackets first
(−6 + 2) × (−7): the bracket gives −4.
(−4) × (−7): same signs, so positive.
= 28
The bracket trap
(−5)2 means (−5) × (−5) = 25.
−52 means −(5 × 5) = −25, because only the 5 is squared.
(−5)2 = 25 but −52 = −25

Check yourself

  1. −3 + 4 × (−2)
  2. (−8 + 3) × 2
  3. (−4)2
Answers
1. −11    2. −10    3. 16

If you want to go further

Every rule here comes back in Unit 3 when you solve two-step equations, and again when you plot points below the axis. If you want a challenge: can two negative integers ever add to a positive? Can two negative integers ever multiply to a negative? Explain each answer in one sentence.

Mathematics 8 · Unit 1 · Integers · Guided notes · Ms. Ras Mulinta · Handsworth Secondary 2026–27 · No calculator.