# When algebra is easier than subtraction

(Maybe because I cut school a lot in second grade, but) I’ve always found subtracting to be disproportionately difficult compared to other arithmetic stuff (maybe logarithms are up there, too). To the point that even algebra is simpler. Which, about half a year ago led me to realizing the following hack…

Since I can see the answer to something like “3345 + x = 86744 solve for x” right away (it’s 83399)… that means that whenever I see 86744–3345, I should think of it as “what, plus 3345, becomes 86744?”

WTF! I’m 41! I couldn’t’ve, you know, come up with this sooner? Instead of, when confronted with 86744–3345 or any other subtraction, being all “Uh…carry the… one?”

It’s not just about the carry flag, this goes for easy stuff too like 17–3. Three plus blank is seventeen? Obviously fourteen! Seventeen minus three… it’s fourteen. But it takes me longer.

Now, most peeps have probably memorized
[any single digit] [any operand] [any single digit] and so have I. So I’m not actually struggling with seven minus four but I’ll use it as an example to model what’s going on in my proverbial pea brain.

Let’s say I don’t know jack and I’m back to “Alice has seven apples” levels of thinking. She loses four? I can take away apples gradually until I’ve taken away four and keep track of how many apples I’ve taken away. That is the subtraction way. Which, if I’ve really lost it, is four subtractions. counting backwards 6, 5, 4, 3 (which is the answer) while simultaneously keeping track of an incrementing mental counter 1, 2, 3, 4 (so I know when to stop).

Instead, with the solve for blank way… it’s “Alice has seven apples. This morning she had four. How many did she gain during the day?” both the addend and the incrementing counter are going forward. 5, 6, 7 (which is when I need to stop) and 1, 2, 3 (which is the answer).

In this case it happens to be fewer steps, three vs four, but that’s just happenstance, it’s just as often the other way around.

The real reasons it’s easier are:

• Going forward is slightly easier than going backwards (not by a lot).
• More isomorphic with addition (four plus three? Exact same operation, 5, 6, 7 and 1, 2, 3—I just return the tally instead of the counter).
• Both counters going in the same direction is way easier. That’s the biggest of these three factors.

We don’t have to go one at a time, either. Let’s say it’s 41 + blank is 174. I can start by adding a hundred to both the tally and the counter. Then, I can see that adding fifty more would overshoot… how about twenty? Not enough. So, make that thirty. Wow, the tally is at 171, almost there… and three more makes the tally 174 and the counter 133, which is our answer. My brain can even be chaotic and jumbled, adding things out of order. The tally might jump from 41, 44, 144, 174 and also doing the same operations to the counter: 0, 3, 103, 133.

Now, I’m not suggesting we start by teaching kids to literally rewrite
7 – 4 to 4 + _ = 7, or, worse, 4 + x = 7. Instead, just changing how we state subtraction. Instead of saying

Alice had seven apples but she lost four, how many does she have?

maybe it’s gonna be easier saying

Alice had seven apples but now she has four, how many did she lose?

Still the same order of operands, still the same 7 – 4, but just with new semantics.

The first phrasing leads me to 6, 5, 4, 3 and 1, 2, 3, 4.
The second phrasing leads me to 5, 6, 7 and 1, 2, 3.

I don’t know, maybe everyone else already does it this way and it’s just me that’s been doing subtraction backwards all these years. That’d explain a lot.