Subtraction Calculation Investigation

In the middle of a lesson on subtraction, I stumbled across a nice looking calculation. Later, I investigated variations of it and spotted some nice patterns.

Below, you can the first slide of a PowerPoint I've put together, which can be found here.

There are five solutions, each of which has \(R=3\): \(51-15=36, 62-26=36, 73-37=36, 84-48=36, 95-59=36\)

Let's change it slightly. If the visible digit was 5 instead of 6, how would that change the answer?

There are four solutions, each of which has \(R=4\): \(61-16=45, 72-27=45, 83-38=45, 94-49=45\) 

What about if the digit were 4, 3 and so on? 

The pattern continues with \(R\) and the visible digit summing to 9, and the number of solutions being 1 less than the visible digit.

What about the other direction? What about the digit being 7, 8 and so on?

The pattern continues up to 8, but the visible digit being 9 requires \(R=0\) which isn't allowed. Similarly, the digit being 0 requires \(W=S\) resulting in 00.

I've already explained the pattern, but the PowerPoint has a slide prompting students to try identify the pattern. There's also a nice discussion about extrapolation here (which is why I started at 6 and worked outwards!)

Can you prove that \(R\) and the visible digit always sum to 9?

This is a nice little proof for students with an understanding of how to convert numbers like \(AB\) into \(10A+B\).

If \(W=S\) then the result is 0, so reject that idea. If \(W<S\) then the result is negative, so reject this too. We now know that \(W>S\).

Looking at the ones column, \(W>S\) so one of the tens needs exchanging to ten ones. In old money, we need to borrow from the \(W\).

The calculations become:

Ones column: \(10+S-W=X \implies 10-X=W-S\)
Tens column: \((W-1)-S=R \implies W-S=R+1\)

Equating the \(W-S\) gives \(10-X=R+1\implies R=9-X\) as needed.