Adding & subtracting,
one hop at a time.

First, the board itself.

An integer is just a whole number with no fractions or decimals — and crucially, it can be positive, negative, or zero. So the integers are …, −3, −2, −1, 0, 1, 2, 3, … marching off forever in both directions. Lay them out evenly and you get the number line: your hopscotch board.

Zero sits in the middle, like home. Walk right and the numbers get bigger. Walk left past zero and you slip into the negatives, where numbers get smaller the further you go. Here's the part people forget: every square is the same size. The hop from 0 to 1 is the exact same distance as the hop from −2 to −1, or from 6 to 7. One hop is always one square — that even spacing is what makes hopping a reliable way to do arithmetic.

A handy way to feel negatives before we do any maths: think of a thermometer turned sideways. To the right it's a warm +8°C; slide left through 0 and it gets colder — −5°C is below freezing, and −12°C is colder still. The further left, the smaller. Hold that picture; we'll hop along it in a moment.

Right is bigger and warmer. Left is smaller and colder. Zero is home. Every square is the same width.

Add → hop right. Subtract → hop left.

Start with the friendly cases, where every number is positive. Say you're standing on 3 and you add 4. Adding means face right and hop 4 squares: 3 → 4 → 5 → 6 → 7. You land on 7, which is exactly what 3 + 4 = 7 has always told you. Adding has just become walking forward.

Now subtract. You're on 3 and you subtract 4. Subtracting means face left and hop 4 squares: 3 → 2 → 1 → 0 → −1. You sail right past zero and land on −1. So 3 − 4 = −1. This is the moment a lot of people first meet a negative answer and panic — but on the board it's no drama at all. You just had more steps to take than there was room before zero, so you kept walking into the cold side.

Notice the pattern already: the sign of the operation (+ or −) chooses your direction, and the size of the number chooses how many squares. Hold onto those two jobs — direction and distance — because they're about to do all the work.

A negative flips your direction.

Here's where it gets interesting. What if the number you're adding is itself negative? Try 3 + (−5). You're adding, so you'd normally face right — but the number is negative, and a negative says "turn around." So you face left instead and hop 5 squares: 3 → 2 → 1 → 0 → −1 → −2. You land on −2. So 3 + (−5) = −2. Adding a negative quietly behaves exactly like subtracting.

Now the headline trick. What if you subtract a negative? Try 2 − (−4). Subtracting normally faces you left — but the number is negative, so it flips you back to the right. You face right and hop 4 squares: 2 → 3 → 4 → 5 → 6. You land on 6. So 2 − (−4) = 6 — the very same place you'd reach with 2 + 4.

That's the famous result, and now it isn't a magic spell to memorise — it's just two flips cancelling out. Subtracting points you left; the negative flips you back to the right. Two turn-arounds bring you facing forward again, which is the same direction as plain adding.

Subtracting −5 is the same as adding 5, because "subtract" turns you left and the "negative" turns you right back — so you're facing forward, just like adding.

Don't just take my word for it — go drive it yourself. Below is a hopper you control: pick where to stand, pick add or subtract, pick the number (it can be negative), and press Go. Watch which way it turns and count the squares.

Build a calculation. Make it hop.

Choose a starting square, choose + or −, and slide the number (drag it below zero to make it negative). The hopper faces the right way, hops one square at a time, and lands on the answer — which is worked out live, so it's always correct. Try 3 + (−5), then flip it to 2 − (−4) and watch it leap the other way.

You already hop this line every day.

Negative numbers aren't a maths-class invention — they're how we measure anything that can go below a natural zero. And whenever a quantity rises or falls, you're hopping along a number line without thinking about it. Three everyday boards:

And here's where subtracting a negative shows up for real. Suppose the forecast says tonight will be −6°C, but it ends up 4 degrees warmer than forecast. "Warmer" means hopping right, so the actual temperature is −6 + 4 = −2°C. Now flip the question: how much warmer was it than forecast? That's "actual minus forecast," or −2 − (−6). Subtracting that negative forecast flips you right and you hop from −2 up to 4: it was 4 degrees warmer. Whenever you measure "the gap between two things, one of which is negative," you'll find a subtract-a-negative hiding inside — and the hopper handles it the same way every time.

Four moves, two directions.

Every adding-and-subtracting question is secretly one of just four moves — and they collapse into only two directions. Tap a preset calculation below; the matching rule lights up, and the answer is worked out in full so you can see the flip happen. Notice how the two right-hand rules are really the same hop, and so are the two left-hand ones.

The trap: "two minuses make a plus."

You've probably heard the chant "two minuses make a plus." It's catchy, it's sometimes true — and it gets people into real trouble because they fire it off in the wrong place. Let's untangle exactly when it works and when it absolutely doesn't.

Five hops to prove it.

No memorising required — for each one, picture the hopper. Where are you standing, which way does the sign make you face, and how far do you hop?