Something goes away

Erklärus had lined up five little meteorites on the table. A tiny, drawn black hole at the edge of the table slowly pulled two of them towards itself and swallowed them.

Erklärus: “Five meteorites. The black hole swallows two of them. How many are left?”
Liri: “One, two, three... three are left!”
Erklärus: “Exactly. That is a subtraction. Something gets less – and you count what is left.”
Liri: “That’s harder than adding on. When you take away, you somehow have to think backwards.”
Erklärus: “Many people feel that way. Counting backwards feels stranger than counting forwards. But with a little practice it gets easy.”

🎈 Taking away with pictures

Six soap bubbles float in front of the window. Two of them pop straight away.

🫧 🫧 🫧 🫧 🫧 🫧

Two are crossed out – they have popped.

6 − 2 =

🌠 Bridging through ten – the biggest hurdle

14 stars are shining in the sky. A cloud drifts past and covers 6 of them. How many can still be seen?

Erklärus: “The trick: first go back to 10. 14 − 4 = 10. But we wanted to subtract 6, not just 4 – so there are 2 still to go. Then: 10 − 2 = 8.”
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🔵 First 4 stars are covered (14 − 4 = 10)  ·  🔴 then 2 more (10 − 2 = 8)  ·  🟡 the remaining 8 stars can still be seen!

14 − 6 =
🌌 Subtraction in the universe

A starry sky shows 17 bright stars – if 9 of them disappear behind clouds, only a few are left.
Without subtraction, astronomers couldn’t work out how much fuel is left after part of a journey.

❌ Common mistake

When bridging through ten, people often forget how much they have already subtracted.

Example: 14 − 6. Anyone who stops at 14 − 4 = 10 and forgets that 2 more are still to come (because 6 should be subtracted, not just 4) ends up with the wrong answer.

Tip: always ask yourself: “How much have I already subtracted – and how much is still missing from the number I was supposed to subtract?”

➖ Exercise 1
A spaceship has taken 9 photos. 3 of them are blurry and are deleted. How many are left?
☄️☄️☄️☄️☄️☄️☄️☄️☄️

The crossed-out meteorite photos are deleted.

🌠 Exercise 2 – Bridging through ten
17 stars are shining in the sky. 9 of them fade in the morning light. How many can you still see? (Hint: go back to 10 first!)
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The crossed-out stars fade in the morning light.

🚀 Exercise 3
A rocket has 12 engines. 5 of them are detached after launch. How many stay on the rocket?
🪨🪨🪨🪨🪨🪨🪨🪨🪨🪨🪨🪨

The crossed-out engines are detached.

What is left

Liri had got used to thinking backwards. It no longer felt strange – just like a different direction.

Liri: “Subtracting is like telling a journey backwards.”
Erklärus: “Nicely put. And when you bridge through ten, you simply make a stopover – just like in addition, only the other way round.”

Liri wrote that on her star map too – you never know which sum will still be needed on the way home.

“Subtraction: minuend − subtrahend = difference. When bridging through ten, first go back to 10, then subtract the rest.”
What you learned today
Minuend − subtrahend = difference
Bridging through ten: first back to 10, then subtract the rest
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Chapter 4 mastered!

From taking away with your hands to mental arithmetic – Erklärus and Liri can now subtract up to 20.

Erklärus’s notebook – page 4:
“Subtraction: minuend − subtrahend = difference. When bridging through ten (e.g. 14−6): first back to 10, then subtract the rest. Counting backwards feels unfamiliar – but with practice it’s easy.”
← Back to overview 4 – Subtraction, Part 1