Adding and subtracting integers

Plus goes right, minus goes left

It was −3 °C and it warmed by 5 degrees. How much is it now? You have 20 Kč of debt and you earn 50. How much is left? Both are adding integers — and both you can handle with one picture: stepping along the line.

The rules are two and they are fair:

Adding = a step to the right. −3 + 5 means: stand on −3 and take 5 steps to the right. You land on 2.

Subtracting = a step to the left. 1 − 4 means: stand on 1 and go 4 steps to the left. You land on −3. Yes — subtracting can now end below zero, and that is fine! You spend more than you have, and you are in debt.

In time you will start seeing a shortcut through money: pluses are earnings, minuses debts. −20 + 50 is “debt twenty, earnings fifty” → 30 left. And −20 − 30 is “debt and more debt” → −50.

The line is certainty, money is speed. Learn both — the line at the start, money for life.

Adam keeps a pocket-money notebook. State: −20 Kč (“I owe Sofie for a ticket”). Today he got 50 Kč from grandma. “So I have… fifty?” Sofie raises an eyebrow. “And my twenty-koruna coin?” Adam draws a line on the edge of the notebook: he stands on −20, fifty steps to the right… he passes zero after twenty steps and the remaining thirty he walks to +30. “I have thirty. The twenty went to you.” He writes: −20 + 50 = 30. Sofie nods: “Debts do not vanish. But the line is fair — first it wipes the debt, only then savings grow.”

💡

The result −20 + 50 you estimate with a fight of signs: the stronger is 50 (earnings), so the result will be positive. Then just 50 − 20 = 30.

Stepping and the shortcuts that come from it

The base: the line and steps.

Stand on the first number.
Plus → go right, minus → go left.
As many steps as the second number says. Where you land, that is the result.

Examples by stepping: −3 + 5 = 2 (from frost into plus), 1 − 4 = −3 (through zero into debt), −2 − 3 = −5 (deeper into frost).

Shortcut 1: the same signs add. Debt + debt = a bigger debt: −20 − 30 = −50. Earnings + earnings = more money: 20 + 30 = 50. Add the numbers, the sign stays.

Shortcut 2: different signs fight. −20 + 50: against each other stand debt 20 and earnings 50. The bigger wins (50 → the result is positive) and the difference is left: 50 − 20 = 30. And with 20 − 50 debt wins: −30.

Shortcut 3: minus and minus right after each other = plus. 5 − (−3) read as “take a debt of 3 from me” — and taking someone’s debt means adding: 5 + 3 = 8. A double minus flips into plus.

When you are not sure, go back to stepping. The line never lies — shortcuts are just sped-up steps.

Temperature: −6 + 9

Morning −6 °C, in the day it warmed by 9 degrees.

Stepping: I stand on −6, I go 9 steps to the right. I use six steps on the way to zero, 3 are left → I land on +3.

By shortcut: different signs fight: 9 (warm) against 6 (frost). 9 wins, 9 − 6 = 3 is left, result +3 °C.

Both paths, the same result. In the afternoon it is three degrees above zero.

A double debt: −15 − 25

Filip owes Ema 15 Kč and borrows another 25 from Jonáš.

The same signs (both minuses) add: 15 + 25 = 40, the minus sign stays.

−15 − 25 = −40.

By stepping to check: I stand on −15 and go 25 steps to the left — deeper into debt, I land on −40. Filip owes 40 korunas in total. Two debts never cancel — they add.

A tricky double minus: 7 − (−3)

I read: “from seven take away minus three”. Taking away a debt = adding!

7 − (−3) = 7 + 3 = 10.

A story: you have 7 Kč and a friend scratches a debt of 3 Kč that you have with him. Your wealth grew to 10.

And another game: −2 + (−4)? Adding a debt = more debt: −2 − 4 = −6. A bracket with a sign is always “translated” first: − (−) → +, + (−) → −.

⚠️

−20 + 50 is not −70 or −30. When the signs differ, the numbers do NOT add — they fight: the bigger wins and the difference is left (30, positive, because earnings won). You only add with the same signs. In doubt always step along the line.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Start with ten. When it goes well, add more.

Add and subtract integers — step along the line, or count through debts and earnings. Write a negative result with a minus, for example -12.

Practise the questions

On paper

The line from the last lesson on the table — you can step every question.

Work out by stepping: −4 + 7; 3 − 8; −5 − 6; −9 + 9. By each one draw an arrow along the line.
Work out with shortcuts and mark which you used: −12 − 18; −25 + 40; 15 − 40; 6 − (−4).
Adam’s notebook: he started with −20 Kč, got +50, bought a snack for −35, found +10. How much does he have at the end? Write the running state after each change.
A lift stands on floor 3, goes down 5 floors, up 2 and down 4. On which floor does it end? (Underground floors are negative.)

Now you 💪

I can step on the line any adding and subtracting of integers.
I use the shortcuts: same signs add, different ones fight.
I know that subtracting a negative number means adding: 7 − (−3) = 10.

✅

Done when: You add and subtract integers by stepping along the line and with sign shortcuts, and you keep a running state like in a pocket-money notebook.

What to take from this lesson

Plus = steps to the right, minus = steps to the left. The line never lies.
Same signs: add the numbers, the sign stays (−20 − 30 = −50).
Different signs: they fight — the bigger wins, the difference is left (−20 + 50 = 30).
Minus minus = plus: 7 − (−3) = 10. Taking a debt means adding.
Subtracting may end below zero — that is not a mistake, that is debt.