Categories

## How does 5i^2 become 4?

This come via Lois Burke on twitter, and immediately Max shows up with a possible explanation.

Dave has a different idea. Maybe the student was thinking in words — “5 and minus 1” — and this turns into its homonym “5-1.”

Personally, what I have the easiest time imagining is that the student just had “combine 5 and -1” on their mental ledger. When it came time to address that ledger, there was so much other stuff they were paying attention to that they slipped into the most natural sort of way to combine numbers they had, which is adding. (I like the metaphor of slipping. You’d very rarely see a kid slip in the other direction — from 5 + (-1) to 5 x (-1) — I think. There is a direction to this error.)

Here are the activities we came up with to help develop this sort of thinking in class. Ideas for improvement? More ideas? Other explanations of the student’s thinking?

UPDATE:

Pam Harris has an idea:

Love it. Here’s a digital version.

John Golden point out that there might be issues with the Which One Doesn’t Belong puzzle, so I offer this as an alternative.

John also offers a different problem string: “I’d be curious to see 5+i, 5+i^2, 5+i^3, 5+i^4, 5i, 5i^2, 5i^3, 5i^4.”

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## Complex Number Mistakes Are Often Algebra Mistakes

One of my little obsessions is teaching complex numbers, but it’s really hard to find genuine instances of complex number mistakes. You start looking around at the most pernicious complex number mistakes, and they’re a lot like this one here: essentially algebraic. These mistakes, to my mind, are indistinguishable from the sorts of mistakes you’d expect from $(4-6x)^2.$

That observation is probably helpful in itself, though. The mistakes that we see from kids working with complex numbers are essentially algebraic mistakes. That means that kids aren’t really seeing much of a difference between the algebra that they’re usually asked to do and their work with complex numbers. Complex number arithmetic is just algebra with a twist.

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## Negative Square Roots

Thanks to Tina, we’ve got this great example of a tiny little error that crops up during complex numbers. Here’s my take on it: there’s no way that this kid would make this mistake if their problem was just “Simplify the square root of negative 4.”  When the skill is laid forth is such a direct way, it’s very clear what the student is supposed to do. But when the skill is embedded in a much more complex problem, the student “handled” the negative root by realizing that this was a context that deserved a complex number. Happy and satisfied that they noticed and “handled” every aspect of the problem, the student moved on.

I like calling these sorts of mistakes “local maxima” mistakes, and I think they’re fairly common. To me, the importance of these sorts of mistakes is that they reveal the problem with testing any skill in isolation of others. I’m <i>absolutely sure</i> that this student could simplify the square root of negative four if plainly asked to. But  that didn’t mean that this student was able to use that skill in this context, when there are many more things to juggle.

To me, this means that you can’t really assess any individual skill in that sort of isolation. Instead, I’d prefer an assessment system that gives students a bunch of chances to use a skill — unprimed — in the context of a fairly difficult problem. If the student can simplify negative radicals in 3-4 more involved problems, then I’m pretty confident that this kid has that skill down.

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## Adding and Subtracting Complex Numbers

What make you of this?

Thanks again, Tina!

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## Imaginary Roots

This is tough stuff.  How would you help this student?

(Thanks Tina!)

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## Complex Division

Thanks to John Burk for the submission. He writes, “I know this student wouldn’t simplify (6+2x)/(3-x) this way.”

So: if the student wouldn’t simplify (6+2x)/(3-x) this way, then how did the student end up messing it up for (6+2i)/(3-i)?

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## Complex Roots and Quadratics

Here’s the work:

Can you make sense of it? What’s the cause of this mistake? How could you help?

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## From complex roots to a quadratic equation

How did this student end up with the answer he did?  What does it seem that he knows about complex numbers?

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## Adding Complex Numbers

What is it about complex numbers that this student finds confusing? Do you think he would make an identical mistake on $(12 + 3x) + (2 - 3i)$? Why or why not?