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Mathematics

Negative Numbers Practice Questions, From Ordering to Two Minus Signs

Twelve questions on ordering, temperature change and all four operations, with the two-minus-signs rule shown on a number line before it turns into a shortcut.

The short answer

Negative numbers get smaller as their digits get larger, so −11 is colder than −7. Adding moves you right along the number line and subtracting moves you left, which is why −5 − (−8) means starting at −5 and moving 8 to the right, giving 3.

The questions

  1. Question 1Foundation2 marks

    Put these temperatures in order, coldest first: 3, −7, 0, −2, 5, −11.

    Show the working

    −11, −7, −2, 0, 3, 5

    1. On a number line the negatives sit to the left of zero.
    2. Among the negatives, the larger the digit the further left it sits, so −11 is coldest and −2 is warmest.
    3. Then 0, then the positives 3 and 5.
    4. Order: −11, −7, −2, 0, 3, 5.
  2. Question 2Foundation2 marks

    The temperature in Murree is −4 °C. It rises by 9 degrees. What is the new temperature?

    Show the working

    5 °C

    1. Start at −4 and move 9 to the right.
    2. 4 of those steps reach 0.
    3. The remaining 5 steps carry on to 5.
    4. New temperature: 5 °C.
  3. Question 3Foundation1 marks

    Work out −6 + 10.

    Show the working

    4

    1. Start at −6 and move 10 to the right.
    2. 6 steps reach 0.
    3. 4 steps remain, landing on 4.
  4. Question 4Foundation1 marks

    Work out 3 − 8.

    Show the working

    −5

    1. Start at 3 and move 8 to the left.
    2. 3 steps reach 0.
    3. 5 steps remain, carrying past zero to −5.
  5. Question 5Core2 marks

    The temperature falls from 6 °C to −9 °C. By how many degrees has it fallen?

    Show the working

    15 degrees

    1. Count in two stages through zero.
    2. 6 °C down to 0 °C is 6 degrees.
    3. 0 °C down to −9 °C is 9 degrees.
    4. 6 + 9 = 15 degrees. The answer is a distance, so it is not written as −15.
  6. Question 6Core2 marks

    Work out −5 − (−8).

    Show the working

    3

    1. Subtracting means moving left, so subtracting a negative reverses it and moves right.
    2. Start at −5 and move 8 to the right.
    3. 5 steps reach 0; 3 steps remain.
    4. −5 − (−8) = 3.
  7. Question 7Core1 marks

    Work out −7 × 4.

    Show the working

    −28

    1. Multiply the digits: 7 × 4 = 28.
    2. There is one negative factor, which is an odd number of them.
    3. So the answer is negative: −28.
    4. It can also be read as four lots of −7, which is −28.
  8. Question 8Core1 marks

    Work out −36 ÷ (−9).

    Show the working

    4

    1. Divide the digits: 36 ÷ 9 = 4.
    2. There are two negatives, an even number.
    3. So the answer is positive: 4.
    4. Check by multiplying back: −9 × 4 = −36.
  9. Question 9Stretch3 marks

    Work out −5² and (−5)², and explain why they differ.

    Show the working

    −5² = −25 and (−5)² = 25

    1. In −5² the index applies only to the 5, so it means −(5 × 5) = −25.
    2. In (−5)² the brackets make −5 the thing being squared: (−5) × (−5) = 25.
    3. Two negative factors give a positive product.
    4. The brackets are the entire difference, which is why they should always be written.
  10. Question 10Stretch3 marks

    At 06:00 the temperature was −3 °C. By 14:00 it had risen 11 degrees, and by 23:00 it had fallen 16 degrees from its 14:00 value. What was the temperature at 23:00?

    Show the working

    −8 °C

    1. −3 + 11 = 8, so at 14:00 it was 8 °C.
    2. 8 − 16: move 16 to the left from 8.
    3. 8 steps reach 0, and 8 more carry on to −8.
    4. Temperature at 23:00: −8 °C.
  11. Question 11Stretch2 marks

    Work out (−4) × (−3) × (−2).

    Show the working

    −24

    1. (−4) × (−3) = 12, since two negatives give a positive.
    2. 12 × (−2) = −24.
    3. Count the negative factors instead: there are three, an odd number.
    4. So 4 × 3 × 2 = 24 becomes −24.
  12. Question 12Stretch3 marks

    a = −3 and b = 5. Work out a² − 4ab.

    Show the working

    69

    1. a² = (−3)² = 9. The value of a includes its sign, so it is squared with it.
    2. 4ab = 4 × (−3) × 5 = −60.
    3. a² − 4ab = 9 − (−60).
    4. Subtracting −60 adds 60: 9 + 60 = 69.

Where these go wrong

  • Ordering −11 above −7 because 11 is bigger than 7.
  • Applying "two minuses make a plus" to −5 + (−8) and answering 3 instead of −13, because there are two minus signs visible.
  • Turning −5 − (−8) into −13 by treating the double sign as a single subtraction.
  • Finding the fall from 6 °C to −9 °C by subtracting digits and answering 3 degrees instead of counting through zero to 15.
  • Writing −5² = 25 by squaring a minus sign that is not inside brackets.

The number line before the rule

"Two minuses make a plus" is true in the two places it applies and false almost everywhere a struggling pupil applies it. Before it is allowed to become a shortcut, it should be seen happening: subtracting means moving left, so subtracting a negative reverses that and moves right. Standing at −5 and moving 8 to the right lands on 3, and that is the whole of −5 − (−8) = 3.

The reason to insist on the number line first is the error it prevents. A pupil holding only the slogan will apply it to −5 + (−8) as well, because there are two minus signs on the page, and answer 3 instead of −13. On a number line there is nothing to confuse: adding a negative is a move to the left, so you arrive at −13.

Difference is a distance; the sign is a direction

Temperature questions ask two different things and pupils often answer the wrong one. "How much colder is Murree than Karachi" wants a distance, which is never negative. "What is the new temperature" wants a position, which can be.

The safe method for a difference across zero is to count in two steps through zero rather than to subtract the digits. From 6 °C down to 0 °C is 6 degrees; from 0 °C down to −9 °C is another 9; total 15. Subtracting 9 from 6 to get 3 is the standard wrong answer, and counting through zero makes it impossible.

Counting the minus signs in a product

For multiplication and division the rule is genuinely mechanical: work with the digits, then count how many negative factors there were. An even number of them gives a positive answer, an odd number gives a negative one. (−4) × (−3) × (−2) has three, so the answer 24 becomes −24.

The one place this is worth slowing down is the difference between −5² and (−5)². In −5² only the 5 is squared and the minus stays outside, so the answer is −25. In (−5)² the brackets make −5 the thing being squared, so the answer is 25. Calculators disagree with each other on how they read the first of these, which is a reason to write the brackets rather than to trust the machine.

Common questions

Is it true that two minuses always make a plus?

Only in two situations: subtracting a negative, and multiplying or dividing an even number of negatives. It says nothing about −5 + (−8), which stays firmly negative at −13. Taught as a slogan without those limits it produces more wrong answers than right ones, which is why the number line comes first.

How do I explain −7 × 4 to a child who accepts −7 + −7 but not the rule?

Keep it as repeated addition for as long as it works. −7 × 4 is four lots of −7, which is −28, and a child can walk that out on a number line. The rule about counting negative factors is worth introducing once (−7) × (−4) appears, where repeated addition no longer has anything to model.

Why does my calculator give 25 for −5² when the book says −25?

Because calculators differ in how they read an unbracketed minus in front of a power — some treat it as part of the number, some as a subtraction applied afterwards. Convention in written maths is that the index binds to the 5 only, giving −25. Write the brackets you mean and the disagreement disappears.

Which year do children start work with negative numbers?

In England, pupils meet negative numbers in context in Year 4 and 5 and are expected to calculate intervals across zero by Year 6, which is why temperature questions dominate at primary level. The four operations with negatives are then developed through Years 7 and 8 in secondary school.

Sources

  1. National curriculum in England: mathematics programmes of study — Department for Education

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