Without losing track of the signThe sign in front of a number: + for positive, − for negative..
This booklet helps you learn about negative numbersA number less than zero, written with a minus sign in front: −5, −10, −0.3., step by step.
What’s a negative number?
A negative numberA number less than zero, written with a minus sign in front: −5, −10, −0.3. is a number LESS than zero. We show negatives by putting a minus signThe sign in front of a number: + for positive, − for negative. in front.
Example: −5 is 5 less than zero. −3 is 3 less than zero.
Negative numbersA number less than zero, written with a minus sign in front: −5, −10, −0.3. go to the LEFT of zero on a number lineA horizontal line where numbers are marked in order, negatives on the left, positives on the right.:
… −4 −3 −2 −1 0 1 2 3 4 …
You see negative numbersA number less than zero, written with a minus sign in front: −5, −10, −0.3. in real life on cold-weather forecasts (−3°C), bank accounts that are overdrawn (−£50), submarine depth readings (−200 m), and football goal differences.
What you’ll learn
We start with putting negatives on a number lineA horizontal line where numbers are marked in order, negatives on the left, positives on the right. and saying which is bigger. Then we addPutting things together: 3 + 4 = 7. and subtractTaking away: 7 − 4 = 3. with negatives, and finally multiplyCombining groups, written as × or *. 4 × 3 = 12. and divideSplitting into equal groups, written as ÷. 12 ÷ 3 = 4. with them.
How to use the booklet
Work through the questions in order. If you get stuck, that’s OK — stop there. Where you stop tells your tutor exactly where you need help.
Lesson 1
The number line
Locate negative numbersA number less than zero, written with a minus sign in front: −5, −10, −0.3. and read off which is bigger or smaller at a glance.
A number lineA horizontal line where numbers are marked in order, negatives on the left, positives on the right. is the only picture you need for negative numbersA number less than zero, written with a minus sign in front: −5, −10, −0.3.. Zero sits in the middle. Positive numbersA number greater than zero. 5, 10, 0.3 are positive numbers. stretch out to the right. Negative numbersA number less than zero, written with a minus sign in front: −5, −10, −0.3. stretch out to the left.
−5
−4
−3
−2
−1
0
1
2
3
4
← smaller
larger →
→Numbers further right are bigger.
←Numbers further left are smaller.
That’s it. Every comparison and ordering question in this paper falls out of those two facts.
✔Sense-check. Picture the two numbers on the line. The one further right is bigger. If your answer disagrees with that picture, it’s wrong.
Try one
Which number is 3 places to the left of 1?
Show answer
Start at 1 and step left three times.
1 → 0 → −1 → −2.
Answer: −2. ✓
Lesson 2
Comparing and ordering
Decide which of two negatives is bigger, and put a list of negatives in order.
This is where pupils typically slip. With positive numbersA number greater than zero. 5, 10, 0.3 are positive numbers., “bigger digit means bigger number”. With negatives, this flips: −8 has a bigger digit than −5, but −8 is smaller than −5.
✔Closer to zero is greater. Among negatives, the one nearer to zero on the number lineA horizontal line where numbers are marked in order, negatives on the left, positives on the right. is the larger value.
Watch out — the bigger-digit trap
−8 is smaller than −5, even though 8 is bigger than 5. The minus signThe sign in front of a number: + for positive, − for negative. flips the intuition. Always picture the number lineA horizontal line where numbers are marked in order, negatives on the left, positives on the right..
Worked example
Order from smallest to largest: −3, 2, −8, 0, −1
Place each number on the number lineA horizontal line where numbers are marked in order, negatives on the left, positives on the right. in your head.
−8 is furthest left, so it’s the smallest.
2 is furthest right, so it’s the largest.
Fill in the middle by walking left-to-right on the line.
Answer−8, −3, −1, 0, 2
Try one
Which is greater: −3 or −7?
Show answer
On the number line, −3 is to the right of −7.
Further right means greater.
So −3 is greater. ✓
Lesson 3
Adding and subtracting
Handle the four cases — including when two signs sit next to each other.
Adding moves right on the number lineA horizontal line where numbers are marked in order, negatives on the left, positives on the right.. Subtracting moves left. That covers the easy cases.
Worked example A — simple cases
−8 + 12
Start at −8 on the number lineA horizontal line where numbers are marked in order, negatives on the left, positives on the right..
Adding 12 means moving right 12 places.
Count: −8 → −7 → ... → 4.
Answer4
The hard cases are when two signs sit next to each other, like −3 + −4 or 12 − −6. To handle these confidently, here’s a second mental picture alongside the number lineA horizontal line where numbers are marked in order, negatives on the left, positives on the right..
The counter picture
Picture negative numbersA number less than zero, written with a minus sign in front: −5, −10, −0.3. as red counters and positives as blue counters. One red and one blue cancel to zero (they pair off and disappear). What’s left over is your answer.
−3 + 5:−−−++++++
pair off:−−−++++++3 pairs cancel; 2 blue left over → +2
−3 + −4:−−−+−−−−
total:−−−−−−−No blues to pair with — 7 red counters → −7
⚠Adding red counters never produces blues. Two negatives added together stay negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3..
Watch out — the most-mis-remembered Year 7 rule
−3 + −4 is NOT +7. It is −7.
Some pupils memorise the phrase “two negatives make a positiveA number greater than zero. 5, 10, 0.3 are positive numbers.” and apply it everywhere. That phrase is unsafe. It works for some situations (multiplying, and subtracting a negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3.) but it fails for additionPutting things together: 3 + 4 = 7.. Picture the counters instead — you can’t make blues out of reds.
Subtracting a negative: the two signs cancel
→→Wherever you see two minus signs side by side, they cancel and become a single plus.
Worked example B — subtracting a negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3.
12 − −6
Spot the two minus signs sitting side by side.
Replace them with a single plus: 12 + 6.
AddPutting things together: 3 + 4 = 7. as normal.
Answer18
The counter picture works here too: subtracting a negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3. means removing a red counter from the pile. Taking away a red is the same as adding a blue — the pile gets more positiveA number greater than zero. 5, 10, 0.3 are positive numbers. either way.
The two cases side by side
Adding a negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3.
−3 + −4
Two minus signs separated by a plus. Just keep going left on the number lineA horizontal line where numbers are marked in order, negatives on the left, positives on the right..
= −7
Subtracting a negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3.
12 − −6
Two minus signs side by side. They cancel to a plus.
= 18
Try one
Work out 14 − −7.
Show answer
Two minus signs side by side cancel to a plus.
So 14 − −7 becomes 14 + 7.
= 21. ✓
Lesson 4
Multiplying and dividing
Apply the signThe sign in front of a number: + for positive, − for negative. rules — and understand why they work, so you don’t forget them under pressure.
✔Same signs → positiveA number greater than zero. 5, 10, 0.3 are positive numbers.. Different signs → negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3.. (Same rule for ÷.)
If you only remember the rule, you’ll get the questions right. But you’ll forget it under exam pressure unless you understand whynegativeA number less than zero, written with a minus sign in front: −5, −10, −0.3. × negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3. is positiveA number greater than zero. 5, 10, 0.3 are positive numbers.. The next paragraph shows you that it isn’t a magic trick.
Why negative × negative = positive: spot the pattern
Start with multiplicationCombining groups, written as × or *. 4 × 3 = 12. you already know. Look down this list — each row’s answer is 3 smaller than the row above:
3 × 3 = 9
3 × 2 = 6
3 × 1 = 3
3 × 0 = 0
3 × (−1) = −3
3 × (−2) = −6
3 × (−3) = −9
The pattern continues naturally. Each row subtracts another 3. That gives us positiveA number greater than zero. 5, 10, 0.3 are positive numbers. × negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3. = negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3. — without needing a separate rule, just by extending what we already know.
Now do the same trick from the other direction. Each row’s answer is 3 bigger than the row above:
3 × (−3) = −9
2 × (−3) = −6
1 × (−3) = −3
0 × (−3) = 0
(−1) × (−3) = +3
(−2) × (−3) = +6
(−3) × (−3) = +9
✔The pattern forcesnegativeA number less than zero, written with a minus sign in front: −5, −10, −0.3. × negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3. to be positiveA number greater than zero. 5, 10, 0.3 are positive numbers.. It isn’t a rule someone made up — it’s the only way multiplicationCombining groups, written as × or *. 4 × 3 = 12. can behave consistently.
The sign rules in one place
+ × + = +3 × 4 = 12
+ × − = −3 × (−4) = −12
− × + = −(−3) × 4 = −12
− × − = +(−3) × (−4) = +12
DivisionSplitting into equal groups, written as ÷. 12 ÷ 3 = 4. follows the same rules: same signs give a positiveA number greater than zero. 5, 10, 0.3 are positive numbers. answer, different signs give a negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3. answer.
Worked example
−15 ÷ −3
Check the signs. Both are negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3. — same signs.
Same signs → positiveA number greater than zero. 5, 10, 0.3 are positive numbers. answer.
DivideSplitting into equal groups, written as ÷. 12 ÷ 3 = 4. the numbers: 15 ÷ 3 = 5.
Answer+5
A second picture: each negative is a flip
If the pattern feels abstract, here’s another way to see it. Think of each negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3.signThe sign in front of a number: + for positive, − for negative. as a direction-reverser:
0 flips → positiveA number greater than zero. 5, 10, 0.3 are positive numbers.
1 flip → negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3.
2 flips → back to positiveA number greater than zero. 5, 10, 0.3 are positive numbers.
3 flips → negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3. again
An even number of flips returns you to positiveA number greater than zero. 5, 10, 0.3 are positive numbers.. An odd number leaves you negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3.. This is the picture that explains cubes in the next lesson.
Try one
Work out −8 × −3.
Show answer
Same signs (both negative) give a positive.
Multiply the numbers: 8 × 3 = 24.
= 24. ✓
Lesson 5
Powers of negatives — the bracket trap
Tell the differenceTaking away: 7 − 4 = 3. between (−3)² and −3² — the single most-mis-answered Year 7 negatives question.
This is the single most-asked, most-mis-answered Year 7 topic involving negatives. Look at these two expressions:
With bracket
(−3)² = 9
The bracket binds the minus to the 3 first. So you square −3: (−3) × (−3) = +9.
Without bracket
−3² = −9
BIDMAS does the index first. So 3² = 9, then the minus signThe sign in front of a number: + for positive, − for negative. is applied: −9.
Same digits, oppositeThe number on the other side of zero with the same distance. The opposite of −7 is +7. signs — just because of one pair of brackets. This is the BIDMAS priority rule. The bracket forces the minus to bind first; without the bracket, the index wins.
?When you see a minus signThe sign in front of a number: + for positive, − for negative. next to a power, always ask: are the brackets there or not?
Cubes (and odd powers)
This is where the “each negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3. is a flip” picture from Lesson 4 earns its keep. (−3)³ means three negatives multiplied together: flip, flip, flip. The first two flips cancel out (back to forward), the third flip reverses you again. So (−3)³ ends up negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3..
Worked example
(−3)³
Write it out: (−3) × (−3) × (−3).
First pair: (−3) × (−3) = +9 (same signs → positiveA number greater than zero. 5, 10, 0.3 are positive numbers.).
Now multiplyCombining groups, written as × or *. 4 × 3 = 12. by the third: 9 × (−3) = −27 (different signs → negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3.).
Answer−27
ΣCount the negatives. Even count → positiveA number greater than zero. 5, 10, 0.3 are positive numbers. answer. Odd count → negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3. answer.
Watch out — cubing a negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3.
(−3)³ is not +27. Three flips don’t cancel cleanly; one flip is always left over.
Try one
Work out (−7)².
Show answer
Squaring means −7 × −7.
Same signs give a positive.
= 49. ✓
Words you’ll meet in the questions
fall by / falls
the temperature or value goes down. Subtract.
rise by / rises
the temperature or value goes up. Add.
overdraft
when you owe money to the bank. Your balance is a negative number.
deposit
putting money into your account. Add.
below sea level / depth
a position below the surface of the sea. Use a negative number for depth.
Recap before the practice paper
Number lineA horizontal line where numbers are marked in order, negatives on the left, positives on the right..Negative numbersA number less than zero, written with a minus sign in front: −5, −10, −0.3. sit to the left of zero. Further left = smaller.
Comparing. Closer to zero is greater. −8 < −5 because −8 sits further left.
Adding a negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3.. −3 + −4 = −7. Reds + reds = more reds (never blues). Stays negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3..
Subtracting a negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3.. 12 − −6 = 12 + 6 = 18. Two minus signs side by side cancel to a plus.
Multiplying/dividing. Same signs → +. Different signs → −. The pattern of multiplicationCombining groups, written as × or *. 4 × 3 = 12. forces this — it isn’t a separate rule.
Squaring/cubing. Count the negatives: even count → positiveA number greater than zero. 5, 10, 0.3 are positive numbers. answer; odd count → negativeA number less than zero, written with a minus sign in front: −5, −10, −0.3. answer. (−3)² = 9. −3² = −9. (−3)³ = −27. Always check the brackets.
Part Two
Now try the practice questions
45 questions across three sections, 30 minutes when you’re ready. The step-by-step walkthroughs unlock after you mark.
Instructions
Type each numeric answer in the box. For sign questions, use a minus sign (e.g. -7). For radio questions, tap your choice.
You can start the 30-minute timer or work untimed.
When finished, click Mark My Work. Your answers will then be locked, the correct answers will appear under each wrong question with the working, and you will get 5 similar practice questions inline for each one you missed.
Timed practice · 30 minutes
When you’re ready, start the timer
This practice is designed to take 30 minutes. Your tutor will see how long you took. You can work untimed if you prefer.
Section A — Basic skill
Twenty short questions covering number line, comparison, ordering, four operations, and powers with negatives.
Q1.
Work out 1 − 3.
Your answer:(1 mark)
Start at 1 on the number line. Subtracting 3 moves you 3 places left: 1 → 0 → −1 → −2.
Watch out: the trap answer is 2 (ignoring the sign and just doing 3 − 1). Subtraction can give a negative answer when the second number is bigger.
Q2.
Which is greater: −3 or −7?
Your answer:(1 mark)
On the number line, −3 sits to the right of −7. Numbers further right are greater.
Watch out: the trap is to say −7 because 7 is bigger than 3 — but with negatives, the larger the digit, the smaller the value.
Q3.
Which symbol fills the gap: −5 ___ −2?
Your answer:(1 mark)
−5 is further left than −2 on the number line, so −5 is smaller. The correct symbol is <.
Q4.
From the list −3, 2, −8, 0, −1, what is the smallest number?
Your answer:(2 marks)
The smallest number is furthest left on the number line. That is −8.
Watch out: the trap is −1 (thinking "smallest digit"). Negative numbers flip the intuition.
Q5.
Order largest first: −2, −8, −1.5, −0.5, −3. What is the largest?
Your answer:(2 marks)
Among negatives, closer to zero is greater. −0.5 is closest to zero, so it is the largest.
Watch out: trap is −8 (treating "bigger digit" as bigger value). With negatives, that intuition flips.
Q6.
Work out −8 + 12.
Your answer:(1 mark)
Start at −8 and move right 12 places: −8 → −7 → ... → 4.
Or: 12 + (−8) = 12 − 8 = 4.
Q7.
Work out −5 − 4.
Your answer:(1 mark)
Start at −5 and move left 4 places (subtraction moves left): −5 → −6 → −7 → −8 → −9.
Q8.
Work out −3 + −4.
Your answer:(2 marks)
"Adding a negative" is the same as subtracting: −3 + (−4) = −3 − 4 = −7.
Watch out: the trap is +7 (applying "two negatives make a positive"). That rule applies to multiplying/dividing, NOT to adding two negatives. Two negatives added together stay negative.
Q9.
Work out 12 − −6.
Your answer:(2 marks)
Subtracting a negative is the same as adding: 12 − (−6) = 12 + 6 = 18.
Watch out: the trap is 6 (treating the two minus signs as one). Two minus signs next to each other turn into a plus.
Q10.
Work out −27 − −3.
Your answer:(2 marks)
Two minus signs become a plus: −27 − (−3) = −27 + 3 = −24.
Watch out: the trap is −30 (treating −−3 as −3 and subtracting).
Q11.
Work out 14 − −7.
Your answer:(2 marks)
− and − make +. So 14 − (−7) = 14 + 7 = 21.
Q12.
Work out 5 × −2.
Your answer:(1 mark)
Positive × negative = negative. 5 × 2 = 10, with a minus sign: −10.
Q13.
Work out −8 ÷ 8.
Your answer:(1 mark)
Negative ÷ positive = negative. 8 ÷ 8 = 1, with a minus sign: −1.
Watch out: the trap is −24 (forgetting the sign rule).
Q15.
Work out −15 ÷ −3.
Your answer:(2 marks)
Negative ÷ negative = positive. 15 ÷ 3 = 5.
Q16.
Work out −32 ÷ −8.
Your answer:(2 marks)
− ÷ − = +. So −32 ÷ −8 = 32 ÷ 8 = 4.
Q17.
Work out (−7)².
Your answer:(2 marks)
The bracket binds the minus to the 7 first. (−7)² = (−7) × (−7) = 49.
Watch out: the trap is −49. That answer comes from −7² (no bracket), where BIDMAS squares first then applies the minus.
Q18.
Work out −5² (no bracket around the −5).
Your answer:(2 marks)
No bracket, so BIDMAS does the index first. 5² = 25, then the minus sign is applied: −25.
Watch out: trap +25 if you treat this as (−5)². The bracket changes everything. With no bracket, the index wins (BIDMAS), then the minus sign is applied to the result.
Q19.
Work out (−3)³.
Your answer:(2 marks)
(−3)³ = (−3) × (−3) × (−3). First two give +9, then 9 × (−3) = −27.
Watch out: odd powers of a negative stay negative. Even powers (squared, fourth) become positive. The trap is +27 (over-applying the "two negatives positive" rule).
Watch out: trap 11 (dropping the negative sign during substitution: 2 × 3 + 5 = 11).
Q40.
The formula F = m × a is used in science. Find F when m = 5 and a = −3.
Your answer:(2 marks)
Substitute the values into the formula: F = 5 × (−3) = −15. The negative sign in a carries through to F.
Watch out: trap +15 if you drop the minus sign during substitution.
Section C (ii) — Cross-link word problems stretch
Five real-life problems combining negatives with another topic. The toughest in the paper.
Q41.
Three winter days have temperatures −2°C, −5°C, 7°C. What is the mean temperature, in °C?
Your answer:(3 marks)
Mean = total ÷ count. Total: −2 + (−5) + 7 = −7 + 7 = 0. Then 0 ÷ 3 = 0°C.
Watch out: dropping the negatives gives (2 + 5 + 7) ÷ 3 = 4.67 — ignores the cold days.
Q42.
A submarine starts at −60m. It rises 4m every minute for 5 minutes. Then it descends a further (−3)² m. What is its final depth, in metres? (Use a negative number.)
Your answer:(3 marks)
Rise: 4 × 5 = 20m, so position becomes −60 + 20 = −40m.
Watch out: trap 11 if you ignore the starting overdraft and just do 15 − 4.
Q44.
Yara's score is x. She loses 5 points, then her score is doubled. Given x = −2, what is her final score?
Your answer:(3 marks)
Step 1 (loses 5): x − 5 = −2 − 5 = −7.
Step 2 (doubled): −7 × 2 = −14.
Watch out: trap −9 if you double first and then subtract: (−2 × 2) − 5 = −4 − 5 = −9. Order of operations matters: do the "loses 5" first because the question says so.
Q45.
A weather model predicts: tomorrow's temperature = today's − 8°C. If today's temperature is 4°C and the same model runs again the next day, what is the temperature two days from now, in °C?
Your answer:(3 marks)
Day 1: today = 4°C. Tomorrow: 4 − 8 = −4°C.
Day 2: −4 − 8 = −12°C.
Watch out: trap −4 (only doing one step).
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