Invent Arithmetic

You may already add 478 and 356 on paper: start at the right, write a digit, carry a 1. But is that the only way? Is it even the best way? Some people add from the left. Some add in their head faster than you can write. Some use tricks that look like cheating.

How many ways to add are there? Why does every one of them give the same answer? And when you multiply 23 by 64, what is the shifted row in the middle really doing?

In this module, you build your own ways to add, take away and multiply, test them, and find out what they have in common. Some of what you find has a name, and a proof you can follow.

By the end, you will have a handful of ways to calculate, and you will be able to say why each one works.

What you will do

• Break it up: add by breaking numbers into parts, and find out what to do when a part will not fit
• The short way: find shorter ways to add, on paper and in your head
• Any order you like: you jumped numbers around to add them faster; was that allowed?
• One for ten: take away more than you have on one line
• Rows of pebbles: count a tray of eggs without counting them one by one
• Cut the rectangle: multiply numbers too big to lay out
• Crosswise: find a faster way to multiply, hidden in the zeros
• Double it: multiply the Egyptian way, when all you can do is double and add
• The fair: check a supplier’s bill, set a price for your stall, change it when the day goes badly, and find out whether you made a profit

Each problem builds on the last. The pages are interactive: you tap, move and write, then answer a question or two about what you noticed. This module follows on from Invent Place Value.