0 Introduction

There are many ways of learning multiplication, such as "long multiplication" or the "box method" or the "lattice method". In these notes, I will present another way of learning multiplication, which is closer to how I understand it. Learning it in this way will lead in to multiplication of polynomials.

1 Multiplication as repeated addition

Multiplication is commonly introduced as “ repeated addition.'' For example, if you wanted to calculate the product \(3 \times 4\), you could calculate the sum \(3+3+3+3\) or \(4+4+4\). Either way, the answer is 12. This is an example of the commutative property, which says that the order does not matter. A (somewhat silly) way of remembering this property is that in a commute, someone goes back and forth between two places, usually work and home. So they are essentially switching between two places. To say that \(3\times 4 = 4 \times 3\) means that the numbers can switch between the two places.

It is easier to understand the commutative property symbolically. If we want to add \(4+4+4\), we could take a row of four boxes and stack those three rows into a rectangle, like this:

If we rotate the rectangle, then the number of boxes is the same. But now it is four rows of three boxes, instead of three rows of four boxes.

So this means that \(3+3+3+3=4+4+4\). As a side note, if the two numbers are the same, like in \(3 \times 3\), then we can describe this as &ldquo 3 squared". This is because the rectangle then becomes a square.

2 Two-digit multiplication

When multiplying two-digit numbers, it becomes too tedious to count the boxes. For example, if we want to calculate \(13\times 11\) then the rectangle looks like this:

There is a way to make this simpler! First, write 13 as 10+3 and 11 as 10+1. In other words, we are splitting up by digits. One advantage is that multiplying numbers ending in 0 is easy: you just append the total number of zeroes at the end. For example, since \(2\times 3=6\) then to find \(20\times 300\), you just append three zeroes to 6, to get 6000. For the problem \(11\times 13\) the rectangle looks like this.

We do not actually need to draw all the boxes once we know how many there are, so the grid can look like this:

Next, we need to add the numbers together. It is easiest to group them by how many zeroes are at the end. Visually, this means going across diagonally:

So the answer is 143.

Next, try it out yourself! The webpage will load random numbers every time, so you can reload to try it again. In this case, the random number generator is asking you to multiply and . In the box below, do not use commas in your answers. I will update the code later to remove the commas.

Next, add the numbers together. Enter the numbers one digit at a time. The grey buttons at the top row will let you write a 1 if you need to carry the 1. Click on it again to remove the 1.

3 Arbitrary-digit multiplication

The method generalizes to any number of digits. Try it out! In this case, the random number generator wants you to multiply and . Again, do not use commas in your answers. I will update the code later to remove the commas.

After you click the button, correct boxes will be highlighted in green and incorrect boxes will be highlighted in red.

Next, add the numbers together. Again, the grey buttons at the top will toggle a 1 if you need to carry the 1.

Extra credit: what is Taylor Swift’s favorite number? \[ \answer {13} \]