Year 7 Negative Numbers: How to Make Sense of the Minus Signs

Why minus signs can be confusing

A minus symbol can indicate a negative number, subtraction or the opposite of a quantity. In −4 − (−7), these meanings appear together. Memorising rules without understanding them makes it easy to get lost.

England's national curriculum introduces negative numbers in primary school and develops arithmetic with positive and negative integers during Key Stage 3 (Years 7–9). Schools may teach the content in different orders. The explanations below suit students beginning secondary mathematics.

1. Begin with a number line

Values increase to the right and decrease to the left. Although 6 is larger than 2, −6 < −2 because −6 lies farther left. A temperature of −6°C is colder than −2°C.

Ask: “Which is greater, −3 or −8?” A student who points to the number line and explains the position understands more than someone who guesses from the digits.

2. Add positive and negative numbers

For −4 + 7, start at −4 and move seven places right. You land at 3.

For −4 + (−7), adding negative seven moves seven places left. The result is −11.

The number being added tells you the direction of movement. This is clearer than applying a vague rule about signs.

3. Subtracting a negative

Consider −4 − (−7). Subtracting a negative means adding its opposite:

−4 − (−7) = −4 + 7 = 3.

Similarly, 5 − (−2) = 5 + 2 = 7. This is not a trick in which minus signs disappear at random. Subtraction is rewritten as addition of the additive inverse.

4. Why a negative times a negative is positive

Watch the pattern as the second factor decreases by one:

• (−3) × 3 = −9

• (−3) × 2 = −6

• (−3) × 1 = −3

• (−3) × 0 = 0

• (−3) × (−1) = 3

• (−3) × (−2) = 6

The product increases by three each step. Continuing the consistent pattern gives (−3) × (−4) = 12. This also agrees with the distributive property of multiplication.

Division uses corresponding sign rules: (−24) ÷ (−6) = 4, because 4 × (−6) = −24.

5. Apply it to everyday life

The morning temperature is −5°C and the afternoon temperature is 3°C. The change is final minus initial:

3 − (−5) = 8°C.

The temperature rose by eight degrees: five degrees to reach zero, then three more.

A bank account balance is −£12. If £20 is deposited, the new balance is −12 + 20 = £8. The deposit clears the debt and leaves eight pounds.

6. Keep the order of operations

For 6 − 2 × (−4), multiply first: 2 × (−4) = −8. Then 6 − (−8) = 14.

Brackets also change powers: (−3)² = 9, but under the usual convention −3² = −9 because the exponent is evaluated before the leading minus. Clear brackets make the intended meaning unambiguous.

Five original practice questions

1. Calculate −7 + 12. Answer: 5. Move twelve steps right from −7.

2. Calculate −2 − (−9). Answer: 7. Rewrite as −2 + 9.

3. Calculate (−5) × (−6). Answer: 30. Two negative factors have a positive product.

4. Calculate 4 − 3 × (−2). Answer: 10. Multiply first to get −6, then 4 − (−6) = 10.

5. A temperature rises from −8°C to +6°C. How much does it rise? Answer: 14°C, because 6 − (−8) = 14.

When a child makes an error, identify whether the problem involves ordering numbers, interpreting subtraction, applying multiplication rules or following the order of operations. Each needs a different explanation.

A ten-minute home practice routine

Spend two minutes comparing negative numbers, three minutes drawing number-line jumps, three minutes exploring multiplication patterns and two minutes on a temperature or money problem. Accuracy and explanation matter more than speed.

Ask your child: “Why does −4 − (−7) equal 3?” A child who can explain the opposite of a negative has a stronger foundation for later algebra than one who only repeats “two minuses make a plus”.

Official references:

• Department for Education: Mathematics national curriculum, Key Stage 3.

• Department for Education / NCETM: Key Stage 3 mathematics guidance.

• NCETM: Negative numbers, counting, comparing and calculating, for prerequisite number-line ideas.

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