# To Accumulate a Rate — Integrate!

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# Basic Algebra in Calculus

I had a student in Calculus today ask why $\left(3x\right)^2=9x^2$

I said, “that’s a fair question, let’s check it out.”
As I wrote the following stuff on the board I was explaining how squared means to multiply by itself. I also talked about 3x being 3 times x. She had a quizzical look on her face so I continued with another example. Finally she said, “I don’t want to hold everyone up so don’t worry about it.” I said, “this is kind of important and they don’t have anything better to do.” So, I did one more example and she said, “I still don’t get how the 3 becomes a 9.” I finally showed $\left(3x\right)^2\neq 3x^2$. I’m still not sure she gets the idea, and I’m really not sure what else I can do to help her. She has been doing ok with the Calculus concepts. A student teacher in the building suggested either $\left(ab\right)^2$ or $\left(2\cdot 3\right)^2$ to use as examples. These seem like a good next attempt, but what in the world is going on that a Calculus student doesn’t understand this basic algebra concept even after a straightforward explanation?

Here are my two main questions:

How in the world did she make it to this point?

What does she fundamentally not understand?

### 3 Comments on “Basic Algebra in Calculus”

1. Paul Gitchos
November 2, 2012

Thanks for posting. I, too, find my calculus class to be filled with constant, ongoing basic algebra review. I have now accepted this as part of the deal, and next year I intend to formalize it, so that the basics of algebra are listed, reviewed, and assessed throughout the course. Sam Shah’s posts on Algebra bootcamps in Calculus have been inspiring.
As to your two main questions, 1) I wonder that sometimes too, and 2) I wonder how she perceives the x vs the 3 – if she is aware that adding the exponent of 2 represents a multiplication and not just a notational move – and how clear she is on what 3^2 by itself (with no variables around) means. These kind of wonderings can keep me up at night. It can be unsettling to work with a student who can differentiate a polynomial, or even use the chain rule, without attaching meaning to these more basic algebraic – and even arithmetic – objects.
I’m really looking forward to seeing what other responses you receive to your post.

• kaleb40
November 2, 2012

I have done some Algebra Bootcamp stuff, but this seemed elementary and I didn’t include it. I guess I’ve been doing more of a Math Analysis and Trigonometry Bootcamp. It would be nice to have a system of Algebra review tied into entry tasks, or something. I’m now thinking that she either just has no concept of multiplication beyond memorization, or she was just having a brain fart moment. I think that I need to be more allowing of my students having a forgetful moment here and there. Thanks for the response.

2. Fawn Nguyen (@fawnpnguyen)
December 26, 2012

Hi Kaleb. Back in September, I wrote this post http://fawnnguyen.com/2012/09/22/20120922.aspx. At the bottom, I mentioned the (3x)^2 and drew out the square to show it. Would this have helped the student?

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This entry was posted on November 1, 2012 by in Algebra, Calculus and tagged , , , , .