Don’t Translate English to Math Backwards

This is Day 30 of 31 Days of Math Learning Success. Follow all the days here and check out others that are writing for 31 days here.

Translating english sentences into their math counterparts can be tricky. Especially for the ones that always want to end up backwards!Often, the idea that there can be a wide range of translations of one text doesn’t occur to people – or that a translation could be bad, very bad, and unfaithful to the original.
~Lydia Davis

When we say “five more than eleven,” we often write 5+11.

But really, eleven was there first. And then five more was added to it.

So technically, we should write 11+5.

But who cares, right?

Addition is commutative, so it doesn’t matter. It’s just semantics, as they say.

But what about, “five less than eleven”?

Is that 5-11? Does the order matter this time?

Easy with numbers. Sometimes easy with letters.

This is pretty straight up when we deal with plain old numbers. But what happens when we go over to letters (variables)?

Suppose you have y cookies. And I have x more cookies than you. (Notice I get more cookies – I love doing the writing!)

Then I have y + x cookies.

If you have y zits and I have x less zits than you, I have y - x zits.

The order followed the logic in these. It was all good.

But sometimes not so easy with letters.

Let’s rephrase it.

x more than y” gives us x+y (following the order in the statement) or y+x (following the logic). The commutative property holds. Great.

x less than y” is a bit different. If you follow the order and notice the math keyword “less,” you’ll write x-y.

But that’s not what it is. Look back at 11 and 5, and you’ll see that “x less than y” should be written y-x.

Multiplication and division are squirrelly, too.

How about x is three times bigger than y? This often leads people (me included) to write 3x = y.

But consider it with numbers: 15 is three times bigger than 5. That’s totally true. But is 3 \cdot 15 = 5? Heavens no!

You can still go in order, just make sure the equal sign goes where “is” is.

x is three times bigger than y
x = 3 \cdot y
And look at the fraction one: x is one third the size of y. I often jump in with \frac{1}{3} \cdot x = y. But it’s the other way.

Again if you make sure to put the “=” where “is” is, you’re good.

x is one third the size of y
x=\frac{1}{3}\cdot y

Test with numbers!

Make sure you test with numbers. Swap fun numbers like 5 and 7 with the variables. (Don’t use “nice” numbers like 1 and 0 – they come out right at all the wrong times.)

And test those numbers in the English sentence, too. Your numbers should make sense in both places!


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