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# 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.”

## 4 replies on “How does 5i^2 become 4?”

Lois Burkesays:

Michael – you are amazing! This is absolutely what I LOVE about Twitter. I throw it out there and all these ideas come rushing back! Thank you! You definitely have my brain going.

In the number 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

Could the second one just be: some people mistakenly say 5i^2=4. What are they thinking and what is the mistake? (Basically give them the teacher problem of this post, I guess.)

I think the WODB is a nice way to maybe get students looking at each other’s reasoning. You could always go to an unclaimed corner. What reasons do you imagine for -8 (TR)?

Michael Pershansays:

Hmm, I definitely thought I had some clever reason for picking -8 last night, but now I can’t remember it.

Would a different set of numbers be better for the “Which One Doesn’t Belong” activity?

Lois Burkesays: