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.”

Categories

## (16x^2)^3/4 = 1024x^11/4

OK OK OK I think I’ve got where 1024 comes from but what is going on with that 11?

Update: I think banderson2 nails it in the comments. “It comes from the power of 2. 2 = 8/4 so 8/4 + 3/4 is 11/4.”

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## Arithmetic Series and Subtracting Signed Numbers

What feedback would you write on this kid’s paper? Why?

(Thanks, KN!)

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## Sum of an Infinite Series

Here are the results from the 59 students who answered this question on an exam:

How would you give feedback to the students who wrote “infinity”?

Imagine that you were to give feedback to the students who wrote “-3/7.” What feedback would you give?

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## Exponentials and Not-Quite Exponentials

The submitter directs us to 2a:

This student has gotten something very right, no? What does she know, and how would you build on it to help her with this sort of problem?

(Thanks Zach P!)

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## The Derivatives of e^x and ln(x)

What you need to know is that the student work is the stuff typed in red, and that this came on a take-home quiz:

Why does this student think that ln(e) = x^1. Or did I get that wrong? Are there ideas that “feel similar” that he’s confusing, or is it something else?

(Thanks Taylor!)

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## Seeing Exponentials Where They Aint

Did this kid just get excited by a coincidence? Or is there something deeper going on here?

(Thanks Tina!)

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## “y=x/(x+1) has got to be a line.”

We’ve been studying graphs of rational functions in Precalculus.

Me: “Take 1 minute with your group: what will the graph of y = x/(x+1) look like?”

One group, during discussion, asserted that it had to be a line, using a sort of process of elimination: it’s not a parabola, it’s not cubic, it’s not a hyperbola.

Interesting, right? Why does this seem like a linear equation? I guess that it sort of looks like one…

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## Finding Inverse Functions

What can we say that this student does or does not understand about inverse functions?

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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.