Start Free
Level 7Grade 7skill: integers_intro· 8 min read

Introduction to Integers (Negative Numbers)

In short: Read every integer as a move on a number line: the sign is a direction, the digits are a distance. Adding a negative walks left (lowers); subtracting a negative reverses to walk right (raises), so −2 − (−11) = 9. For ordering, more negative means further left and smaller, so −20 < −3. Convert units and finish every step before answering.

How to actually handle negative numbers

Stop reading integers as "numbers with a mood." Read every one as a move on a number line: the sign is a direction (right for positive, left for negative) and the digits are a distance. Once you do that, almost every trap on an MMO integers question disappears, because you are no longer memorising sign rules — you are just walking.

Two moves in a row combine. Adding a negative means "then head left," so it lowers your total. Subtracting a negative means "undo a step to the left" — you reverse it and move right, which raises your total. That is the whole reason minus a minus turns into plus, and it is the single fact examiners bait you on most.

For ordering, remember the number line runs coldest-and-lowest on the left. So the integer whose digits look biggest, like −20, is actually the smallest. On a thermometer −20°C is colder than −3°C; on a number line it sits further left. Trust the position, not the size of the digits.

  1. Rewrite the problem as moves: mark the starting point, then each change as left (−) or right (+).
  2. Turn every "− (−n)" into "+ n" before you compute anything — kill the double negative first.
  3. Combine same-direction moves by adding distances; combine opposite moves by subtracting the smaller distance and keeping the bigger one's direction.
  4. For "how much did it change?" always do final − start, then simplify the signs.
  5. When ordering, place every number on the line: more negative = further left = smaller.
  6. Finish the whole journey — check you have applied every step before you write the answer.

The one idea that beats the sign traps Why it works

6 + (−9)
Adding a negative = head left. It lowers the total.

Start at 6, count 9 places left → −3.

−5 − (−8)
Subtracting a negative = undo a left move = move right. It raises the total.

−5 − (−8) = −5 + 8 = 3.

Change = final − start
"How much did it rise/fall?" is always final minus start, signs and all.

Fell from 4 to −6: −6 − 4 = −10 (a fall of 10).

Ordering
More negative = further left = smaller. Big digits after a minus mean colder/lower.

−20 < −15 < −8 < −3.

If you take one thing: treat every integer question as a walk on the number line, not a stack of sign rules to recite. Position tells you which value is smaller (further left) and which way any change carries you. Under time pressure a mental picture survives where half-remembered chants collapse — and a minus in front of a bracketed negative is a reversal, not another subtraction.

Worked examples at exam pace

Five overnight low temperatures were recorded: −12°C, −3°C, −20°C, −8°C and −15°C. List them from coldest to warmest. = −20°C, −15°C, −12°C, −8°C, −3°C (coldest to warmest).
  1. Coldest means furthest left on the number line — the most negative.
  2. Rank by how far left: −20 is furthest, then −15, −12, −8, and −3 is closest to zero.
  3. Do not be fooled by the digits: −20 has the biggest digits but is the smallest value, i.e. the coldest.
At dawn the temperature was −11°C. By noon it had risen to −2°C. By how many degrees did it rise? = It rose by 9°C. (The tempting −2 − 11 = −13 forgets to flip the double negative.)
  1. Rise = final − start = −2 − (−11).
  2. Subtracting a negative reverses it: −2 − (−11) = −2 + 11.
  3. Move 11 right from −2 and you land on 9.
A diver is holding at −16 m. She ascends 75% of the way to the surface, then a current pushes her back down 2.5 m. What is her depth now? = −6.5 m (6.5 m below the surface).
  1. Distance to the surface is 16 m, so 75% of that is 0.75 × 16 = 12 m of ascent.
  2. Ascending moves her right/up: −16 + 12 = −4 m.
  3. The current pushes her down 2.5 m: −4 − 2.5 = −6.5 m.
  4. Do not stop at −4 m — that is only half the journey.
A freezer starts at 5°C and drops 3°C every 30 minutes. What is its temperature after 2 hours? = −7°C.
  1. Convert the time to matching units: 2 hours = 120 minutes = four 30-minute intervals.
  2. Total drop = 4 × 3 = 12°C.
  3. A fall moves left: 5 − 12 = −7°C.
  4. Using '2' (the hours) as the number of drops is the classic unit slip — it gives 5 − 6 = −1°C, which is wrong.
An account is overdrawn at −£80. An interest charge increases the debt by 15%. What is the new balance? = −£92 (the account is £92 overdrawn). Adding a percentage of a negative amount makes it more negative, not less.
  1. The debt is £80, so 15% of it is 0.15 × 80 = £12.
  2. Increasing the debt pushes the balance further left (more negative): −80 − 12.
  3. −80 − 12 = −£92.

The slips that cost the marks

Not flipping a double negative

✗ −2 − (−11) = −13

The student treated − (−11) as "take away 11" and moved left instead of right, so they subtracted when they should have added.

Fix: Rewrite − (−11) as + 11 before touching the numbers: −2 + 11 = 9. Subtracting a negative always raises the total.

sign_error

Stopping halfway through the journey

✗ Diver's depth = −4 m

−4 m is correct after the ascent, but it ignores the 2.5 m the current pushes her back down. It is a right intermediate value, not the final answer.

Fix: Finish every step: after −16 + 12 = −4, apply −4 − 2.5 = −6.5 m. Reread the question and tick off each change.

partial_computation

Adding when the word says "falls"

✗ −4°C falls by 9°C → −4 + 9 = 5°C

"Falls" means the temperature goes down, so it is a leftward move — subtraction — but the student added.

Fix: Match the operation to the direction: a fall subtracts. −4 − 9 = −13°C.

wrong_operation

Treating the percent as the amount

✗ Ascend 75% of 16 m → −16 + 75 = 59 m above the surface

The student used the number 75 directly instead of working out 75% of 16. An answer that flings the diver 59 m into the sky should look impossible.

Fix: Turn the percent into a distance first: 0.75 × 16 = 12 m, so −16 + 12 = −4 m.

percent_as_whole

Taking the percentage of the wrong base

✗ A share falls £50 → £40, a £10 fall → 10 ÷ 40 = 25%

The question asks for the fall as a percentage of the original £50, but the student divided by the new price £40.

Fix: Read which value is the base. "Of the original" means divide by the start: 10 ÷ 50 = 20%.

base_confusion

Misplacing the decimal point

✗ 0.4°C per minute × 15 minutes → 0.6°C

0.4 × 15 = 6.0, but the student slid the decimal point one place and wrote 0.6 (as if 0.4 × 1.5).

Fix: Count places: 0.4 × 15 = 6°C of cooling. Estimate first — 0.4 is nearly a half, and half of 15 is about 7.5, so expect roughly 6, not 0.6.

decimal_point_shift

Mixing minutes and hours

✗ Start 8°C, falls 4°C every 15 min over 1 hour → 8 − 1 × 4 = 4°C

The student multiplied by 1 (the hour) instead of the number of 15-minute intervals, mixing units without converting.

Fix: Convert to one unit: 1 hour = four 15-minute steps, so 8 − 4 × 4 = −8°C.

unit_mismatch

Practise until the sign traps feel obvious

Work through a mixed set of MMO-style integer problems — temperature swings, dives, balances and percentage changes — with the double-negative and ordering traps built in. Each answer explains exactly which slip the tempting wrong option encodes.

Before and after this skill

First, make sure of

Frequently asked questions

Why does subtracting a negative make a number bigger?

Because subtracting means "undo a move." A negative is a step to the left, so undoing it is a step to the right — which raises the total. That is why −2 − (−11) = −2 + 11 = 9. Always rewrite "minus a minus" as a plus before you calculate.

Which is greater, −7 or −3?

−3 is greater. On the number line, −3 sits closer to zero (further right) than −7, and −7 is colder/lower. The bigger the digits after a minus sign, the smaller the number actually is, so −7 < −3.

Is zero positive or negative?

Neither. Zero is the boundary between them — it is the origin on the number line. It is not counted as positive or negative, though it is an integer.

What is the quickest way to add a positive and a negative number?

Ignore the signs, subtract the smaller distance from the larger, then keep the sign of the number that was further from zero. For 6 + (−9): 9 − 6 = 3, and since −9 was further from zero, the answer is −3.

How do I find how much a temperature changed?

Always compute final − start, keeping the signs. From −11°C to −2°C is −2 − (−11) = +9, a rise of 9°C. A positive result is a rise; a negative result is a fall.