Year 7 Mathematics · Number and Algebra · 8m read

Integers and the Number Line

Ever wondered why your bank account can go negative, or how weather apps show temperatures below zero? Integers are the complete set of whole numbers that includes negatives, zero, and positives — and they work together on a number line in surprising ways. Getting comfortable adding, subtracting, multiplying, and dividing them is essential for everything from algebra to real-world situations like calculating profit and loss or understanding elevation changes.

Learning objective: Add, subtract, multiply and divide integers including negative numbers.

By the end of this lesson you can:

• I locate positive and negative integers on a number line.
• I add and subtract integers using sign rules.
• I multiply and divide integers knowing the sign result.

Integers and sign rules

INTEGERS = whole numbers (positive, negative and zero): ..., −3, −2, −1, 0, 1, 2, 3, ...

NUMBER LINE:
← −5 −4 −3 −2 −1 0 1 2 3 4 5 →
Negative ←────────────────────────── Positive

ADDITION and SUBTRACTION:
• Adding a negative = subtracting: 5 + (−3) = 5 − 3 = 2
• Subtracting a negative = adding: 5 − (−3) = 5 + 3 = 8
TRICK: Two signs next to each other:
Same signs → POSITIVE: 5 − (−3) → +3 → add
Different signs → NEGATIVE: 5 + (−3) → −3 → subtract

MULTIPLICATION and DIVISION sign rules:
(+) × (+) = + (−) × (−) = + (SAME signs → positive result)
(+) × (−) = − (−) × (+) = − (DIFFERENT signs → negative result)
Same rule applies for division.

−4 × −3 = +12
−4 × 3 = −12
−12 ÷ −4 = +3
12 ÷ (−4) = −3

Temperature analogy. It's −5°C. The temperature rises 8 degrees. −5 + 8 = 3°C. It's now 3°C. Negative numbers ARE real — they're just "below zero".

Integer sign rules cheatsheet

• Double negative: 5 − (−2) = 5 + 2 = 7. Remove double negative → becomes addition.
• Same × Same = +: Negative × negative is ALWAYS positive. (−3)(−4) = 12.
• Mixed = −: One positive, one negative → answer is NEGATIVE. (−3)(4) = −12.

Integer sign rules

• Same signs: (+)(+) = +, \;(-)(-) = +
• Different signs: (+)(-) = -
• Subtracting a negative: a - (-b) = a + b

Quick reference

| Operation | Rule | Example |
|---|---|---|
| Add a negative | Same as subtract | 5 + (−3) = 2 |
| Subtract a negative | Same as add | 5 − (−3) = 8 |
| Multiply/divide same signs | Positive | (−4) × (−2) = 8 |
| Multiply/divide different signs | Negative | (−4) × 2 = −8 |

Worked example (Easy)

Calculate: −8 + 5 − (−3)

1. Rewrite double negative. −8 + 5 − (−3) = −8 + 5 + 3
2. Work left to right. −8 + 5 = −3. Then −3 + 3 = 0.

Answer: 0

Worked example (Hard)

Evaluate: $(-3) \times 4 + (-2)^2 \times (-5) - (-6)$

1. Indices first. $(-2)^2 = (-2) \times (-2) = 4$. Same signs → positive.
2. Multiplication (left to right). $(-3) \times 4 = -12$. Then $4 \times (-5) = -20$.
3. Rewrite expression. $-12 + (-20) - (-6) = -12 - 20 + 6$
4. Add/subtract left to right. $-12 - 20 = -32$. Then $-32 + 6 = -26$.

Answer: $-26$

Common mistakes

• Thinking $(-2)^2$ and $-2^2$ are the same $(-2)^2 = 4$ (the negative is inside the brackets and gets squared). $-2^2 = -4$ (only 2 is squared, then the negative is applied). Fix: Brackets matter. $(-2)^2 = +4$ but $-2^2 = -(2^2) = -4$. Always check if the negative sign is inside or outside the brackets.
• Subtracting in the wrong direction on the number line Students think $-3 - 5 = -2$ instead of $-8$, confusing subtraction direction. Fix: Subtracting a positive moves LEFT on the number line. $-3 - 5$: start at $-3$, go left 5 → land on $-8$.
• Forgetting that negative × negative = positive The double-negative rule feels counterintuitive, so students default to a negative answer. Fix: Same signs → positive. Different signs → negative. Chant it: "same-same-positive, different-negative."

Why does negative times negative equal positive?

This is one of the most frequently asked questions in Year 7 maths, and the answer reveals something beautiful about mathematical consistency.

Think of multiplication as repeated addition. $3 \times 4 = 4 + 4 + 4 = 12$. Now consider $3 \times (-4) = (-4) + (-4) + (-4) = -12$. Multiplying by a positive number and a negative number gives a negative result — that makes intuitive sense.

But what about $(-3) \times (-4)$? Consider the pattern:
$3 \times (-4) = -12$
$2 \times (-4) = -8$
$1 \times (-4) = -4$
$0 \times (-4) = 0$
$(-1) \times (-4) = ?$

Each step, the answer increases by 4. Following the pattern: $(-1) \times (-4) = +4$. And $(-2) \times (-4) = +8$, $(-3) \times (-4) = +12$.

The rule exists because it's the only result that keeps the number system consistent. If negative × negative gave a negative, basic algebra would break. Mathematics requires this rule to remain logical — it's not arbitrary, it's necessary.

Practice questions

1. Calculate: (−4) × (−7)

   1. −28
   2. +28
   3. −11
   4. +11

   (1 marks)

2. Calculate: −3 − (−9) + (−4)

   (2 marks)

Summary

• Locate positive and negative integers on a number line.
• Add and subtract integers using sign rules.
• Multiply and divide integers knowing the sign result.