Year 2 · Unit 1 · Lesson 1 of 5

Calculating With Negative Numbers: Order of Operations With a Twist

About 15 minutes · adding and subtracting negatives · the sign rules for × and ÷ · order of operations with negatives · powers of negative numbers · no calculator · exit quiz

Hook

A freezer is at −18°C. The power goes off, and the temperature rises by 3°C every hour.

What is the temperature after 4 hours?

The rise is 4 × 3 = 12°C, so the temperature is −18 + 12 = −6°C. Notice the order: work out 4 × 3 first, then add it to −18. Order of operations works exactly the same with negative numbers. You just have to watch the signs.

Four jumps of +3 from −18°C
Learn

Master the essentials

SignsAnswerExamples
Samepositive−4 × −3 = 12
−12 ÷ −3 = 4
Differentnegative4 × −3 = −12
−12 ÷ 3 = −4

Same signs → positive. Different signs → negative.

Adding and subtracting

Use a number line. Adding a negative moves left. Subtracting a negative moves right: − (−) is the same as +.

−4 − (−9) = −4 + 9 = 5

Multiplying and dividing

Same signs give a positive answer. Different signs give a negative answer.

−6 × −7 = 42 · −36 ÷ 4 = −9

Order of operations

Brackets, then powers, then × and ÷, then + and −. Negatives follow the same order.

5 + 2 × (−4) = 5 + (−8) = −3

Powers of negatives

An even number of negatives multiplied gives a positive; an odd number gives a negative.

(−3)³ = −3 × −3 × −3 = −27

Brackets matter

(−3)² means −3 × −3 = 9. But −3² means −(3²) = −9: only the 3 is squared.

(−3)² = 9 · −3² = −9

Check the size

Ignore the signs and estimate the size first, then decide the sign.

−42 ÷ 6: size 7, signs different → −7

Worked examples

See it done

1. Work out −3 + 4 × (−2).

  1. Multiply first: 4 × (−2) = −8 (different signs).
  2. Then add: −3 + (−8) = −11.

−11

2. Work out (−5)² − 3 × (−2).

  1. Power first: (−5)² = −5 × −5 = 25.
  2. Then multiply: 3 × (−2) = −6.
  3. Then subtract: 25 − (−6) = 25 + 6 = 31.

31

3. Work out (−18 + 6) ÷ (−3).

  1. Brackets first: −18 + 6 = −12.
  2. Then divide: −12 ÷ (−3) = 4 (same signs).

4

Common mistake: Mixing up (−3)² and −3². The brackets say "square the whole of −3": (−3)² = −3 × −3 = 9. Without brackets, only the 3 is squared and the minus stays: −3² = −9.

PractiseNo calculator

10 questions with instant feedback

Choose the correct answer. Decide the order first, then the signs. Questions 1–3 are warm-ups, 4–7 are at exam standard and 8–10 are challenge questions. After you answer, the feedback shows the full solution.

Score: 0 / 10Answered: 0 / 10

Warm-up (The basic skill: one or two steps.)

1.−7 + 12 =

Not quite. Adding 12 moves right, towards zero and past it.

Solution

  1. Start at −7 on a number line.
  2. Adding 12 moves 12 to the right: 7 steps reach 0, then 5 more.
  3. So −7 + 12 = 5.

Answer: B. 5.

Correct.

Solution

  1. Start at −7 on a number line.
  2. Adding 12 moves 12 to the right: 7 steps reach 0, then 5 more.
  3. So −7 + 12 = 5.

Not quite. Start at −7, not 7.

Solution

  1. Start at −7 on a number line.
  2. Adding 12 moves 12 to the right: 7 steps reach 0, then 5 more.
  3. So −7 + 12 = 5.

Answer: B. 5.

Not quite. Start at −7 and move 12 to the right: you pass zero.

Solution

  1. Start at −7 on a number line.
  2. Adding 12 moves 12 to the right: 7 steps reach 0, then 5 more.
  3. So −7 + 12 = 5.

Answer: B. 5.

2.−6 − (−10) =

Not quite. − (−10) means + 10.

Solution

  1. Subtracting a negative is the same as adding: − (−10) = + 10.
  2. −6 + 10 = 4.

Answer: D. 4.

Not quite. Start at −6, not 6.

Solution

  1. Subtracting a negative is the same as adding: − (−10) = + 10.
  2. −6 + 10 = 4.

Answer: D. 4.

Not quite. −6 + 10 is positive.

Solution

  1. Subtracting a negative is the same as adding: − (−10) = + 10.
  2. −6 + 10 = 4.

Answer: D. 4.

Correct.

Solution

  1. Subtracting a negative is the same as adding: − (−10) = + 10.
  2. −6 + 10 = 4.

3.−6 × (−7) =

Not quite. Same signs give a positive answer.

Solution

  1. Ignore the signs first: 6 × 7 = 42.
  2. Both numbers are negative (same signs), so the answer is positive: 42.

Answer: D. 42.

Not quite. Multiply, don't add.

Solution

  1. Ignore the signs first: 6 × 7 = 42.
  2. Both numbers are negative (same signs), so the answer is positive: 42.

Answer: D. 42.

Not quite. Multiply, don't add.

Solution

  1. Ignore the signs first: 6 × 7 = 42.
  2. Both numbers are negative (same signs), so the answer is positive: 42.

Answer: D. 42.

Correct.

Solution

  1. Ignore the signs first: 6 × 7 = 42.
  2. Both numbers are negative (same signs), so the answer is positive: 42.

Exam standard (Several steps, like real exam questions.)

4.5 + 3 × (−4) =

Not quite. Multiply first: (5 + 3) × (−4) is the wrong order.

Solution

  1. Multiply first: 3 × (−4) = −12 (different signs, negative).
  2. Then add: 5 + (−12) = 5 − 12 = −7.

Answer: C. −7.

Not quite. 3 × (−4) is −12, not +12.

Solution

  1. Multiply first: 3 × (−4) = −12 (different signs, negative).
  2. Then add: 5 + (−12) = 5 − 12 = −7.

Answer: C. −7.

Correct.

Solution

  1. Multiply first: 3 × (−4) = −12 (different signs, negative).
  2. Then add: 5 + (−12) = 5 − 12 = −7.

Not quite. 5 − 12 is −7, not 7.

Solution

  1. Multiply first: 3 × (−4) = −12 (different signs, negative).
  2. Then add: 5 + (−12) = 5 − 12 = −7.

Answer: C. −7.

5.(−2)³ =

Correct.

Solution

  1. (−2)³ = −2 × −2 × −2.
  2. −2 × −2 = 4 (same signs, positive).
  3. 4 × −2 = −8 (different signs, negative).

Not quite. Three negatives multiplied give a negative.

Solution

  1. (−2)³ = −2 × −2 × −2.
  2. −2 × −2 = 4 (same signs, positive).
  3. 4 × −2 = −8 (different signs, negative).

Answer: A. −8.

Not quite. Cubing means × itself three times, not × 3.

Solution

  1. (−2)³ = −2 × −2 × −2.
  2. −2 × −2 = 4 (same signs, positive).
  3. 4 × −2 = −8 (different signs, negative).

Answer: A. −8.

Not quite. Cubing means −2 × −2 × −2.

Solution

  1. (−2)³ = −2 × −2 × −2.
  2. −2 × −2 = 4 (same signs, positive).
  3. 4 × −2 = −8 (different signs, negative).

Answer: A. −8.

6.20 − (−4) × 3 =

Correct.

Solution

  1. Multiply first: (−4) × 3 = −12.
  2. Then subtract: 20 − (−12) = 20 + 12 = 32.

Not quite. Multiply first: (20 + 4) × 3 is the wrong order.

Solution

  1. Multiply first: (−4) × 3 = −12.
  2. Then subtract: 20 − (−12) = 20 + 12 = 32.

Answer: A. 32.

Not quite. − (−4) × 3 = − (−12) = + 12.

Solution

  1. Multiply first: (−4) × 3 = −12.
  2. Then subtract: 20 − (−12) = 20 + 12 = 32.

Answer: A. 32.

Not quite. Multiply before subtracting, and − (−) is +.

Solution

  1. Multiply first: (−4) × 3 = −12.
  2. Then subtract: 20 − (−12) = 20 + 12 = 32.

Answer: A. 32.

7.The temperature is −5°C at 6 a.m. and 9°C at noon. How many degrees has it risen?

Correct.

Solution

  1. The rise is the difference: 9 − (−5).
  2. 9 − (−5) = 9 + 5 = 14.
  3. Check on a number line: 5 up to 0, then 9 more makes 14.

Not quite. Find 9 − (−5), not 9 − 5.

Solution

  1. The rise is the difference: 9 − (−5).
  2. 9 − (−5) = 9 + 5 = 14.
  3. Check on a number line: 5 up to 0, then 9 more makes 14.

Answer: A. 14°C.

Not quite. A rise is positive.

Solution

  1. The rise is the difference: 9 − (−5).
  2. 9 − (−5) = 9 + 5 = 14.
  3. Check on a number line: 5 up to 0, then 9 more makes 14.

Answer: A. 14°C.

Not quite. Find the distance from −5 up to 9.

Solution

  1. The rise is the difference: 9 − (−5).
  2. 9 − (−5) = 9 + 5 = 14.
  3. Check on a number line: 5 up to 0, then 9 more makes 14.

Answer: A. 14°C.

Challenge (Harder problems that mix ideas or need reasoning.)

8.Find the mean of −8, 5, −3, −6 and 2.

Not quite. The total is negative, so the mean is negative.

Solution

  1. Add, keeping the signs: −8 + 5 − 3 − 6 + 2 = −10.
  2. There are 5 numbers.
  3. Mean = −10 ÷ 5 = −2.

Answer: B. −2.

Correct.

Solution

  1. Add, keeping the signs: −8 + 5 − 3 − 6 + 2 = −10.
  2. There are 5 numbers.
  3. Mean = −10 ÷ 5 = −2.

Not quite. That is the total. Divide by 5.

Solution

  1. Add, keeping the signs: −8 + 5 − 3 − 6 + 2 = −10.
  2. There are 5 numbers.
  3. Mean = −10 ÷ 5 = −2.

Answer: B. −2.

Not quite. Keep the signs: the total is −10, not 24.

Solution

  1. Add, keeping the signs: −8 + 5 − 3 − 6 + 2 = −10.
  2. There are 5 numbers.
  3. Mean = −10 ÷ 5 = −2.

Answer: B. −2.

9.−3² + (−3)² =

Not quite. −3² means −(3²) = −9: only the 3 is squared.

Solution

  1. −3² means −(3 × 3) = −9: only the 3 is squared.
  2. (−3)² means −3 × −3 = 9.
  3. −9 + 9 = 0.

Answer: C. 0.

Not quite. (−3)² = −3 × −3 = +9.

Solution

  1. −3² means −(3 × 3) = −9: only the 3 is squared.
  2. (−3)² means −3 × −3 = 9.
  3. −9 + 9 = 0.

Answer: C. 0.

Correct.

Solution

  1. −3² means −(3 × 3) = −9: only the 3 is squared.
  2. (−3)² means −3 × −3 = 9.
  3. −9 + 9 = 0.

Not quite. Squaring means × itself, not × 2.

Solution

  1. −3² means −(3 × 3) = −9: only the 3 is squared.
  2. (−3)² means −3 × −3 = 9.
  3. −9 + 9 = 0.

Answer: C. 0.

10.[(−3) × (−4) − 20] ÷ (−2) =

Not quite. A negative divided by a negative is positive.

Solution

  1. Brackets first. Multiply: (−3) × (−4) = 12 (same signs).
  2. Then 12 − 20 = −8.
  3. Divide: −8 ÷ (−2) = 4 (same signs, positive).

Answer: D. 4.

Not quite. (−3) × (−4) = +12, not −12.

Solution

  1. Brackets first. Multiply: (−3) × (−4) = 12 (same signs).
  2. Then 12 − 20 = −8.
  3. Divide: −8 ÷ (−2) = 4 (same signs, positive).

Answer: D. 4.

Not quite. (−3) × (−4) = +12, and −8 ÷ (−2) is positive.

Solution

  1. Brackets first. Multiply: (−3) × (−4) = 12 (same signs).
  2. Then 12 − 20 = −8.
  3. Divide: −8 ÷ (−2) = 4 (same signs, positive).

Answer: D. 4.

Correct.

Solution

  1. Brackets first. Multiply: (−3) × (−4) = 12 (same signs).
  2. Then 12 − 20 = −8.
  3. Divide: −8 ÷ (−2) = 4 (same signs, positive).

Apply

The golf card

In golf, a score below par is negative and above par is positive. Zara's scores for four rounds were −3, +2, −5 and −1. The lowest score is best.

1. What is her total score for the four rounds?

2. What is her mean score per round?

3. Her friend Tom's total is −4. Who did better, and by how much?

  1. −3 + 2 − 5 − 1 = −7.
  2. −7 ÷ 4 = −1.75 per round.
  3. Zara, because −7 is lower than −4. She is better by −4 − (−7) = 3 shots.

Next: the Lesson 1 exit quiz. Five questions, recorded on your dashboard. Aim for 4 out of 5.

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