## Monday, December 6, 2010

### My 6th Grade Version of Algebra

This is probably better fitting for a twitter tweet twit, especially considering how delinquent I've recently been but couldn't pass up sharing a few questions/comments from my sixth graders today.

Ok, so maybe a little context to defend my choice of not being on twitter:

EDIT: Ok, maybe more than just a little context.  If this is too much to read, at least skip down to the end and look at the questions/comments.

My sixth graders just started talking about algebra.  I love this because, at least with the way I approach it, it's the first time that I don't have to do much unteaching (fractions, on the other hand...).

I began by asking them to come up with mathematical questions motivated by a 1x1, 1x2, and 1x3 strip of squares made from Construx.

We shared out.  The good: Lots of kids asked questions about counting the pieces and looking for patterns.  The bad: I was going in this direction regardless of the questions they asked.

Then in groups they played around with this Edges, Faces, and Vertices activity for a while, building models and looking for patterns so that they wouldn't have to build huge models.
The next day I asked them what they thought their new unit was on.  Most guessed geometry.  Others guessed graph theory and combinatorics.  No one used those words.  A few said algebra.  I told them all of these are reasonable choices, but we're going to make the choice to explore algebra.  I then talked ranted for a few minutes about some misconceptions of algebra:
1. It's something you do in 8th grade.
3. It's something you check off the list and don't do again after 8th grade.
4. It's math with letters.
5. It's something my smart friend did in 4th grade.
*6. It's something people use all the time in their jobs.

*There may be some disagreement with this misconception.  It'll have to wait for another post.

I then gave my algebra schpeel.    Talked a little history.  Talked about how the word comes from the Arabic al-jabr roughly meaning restoration written by a guy named al-Kwarismi (where the word algorithm came from) and how these ideas managed to avoid Europe for close to 1500 years.  Spent the few minutes to share the aside that while the mathematics of algebra didn't make it to Europe, the word did as barbers in the medieval times were called algebristas because in addition to cutting hair they were also bone setters. As an aside to my aside, they were also bloodletters which is where the red in the barber pole comes from.  Well, at least that's what the internet tells me.

Then we talked about three aspects of algebra:
1. Finding the secret number or numbers, ie solving algebraic equations.  Told them that the problem
3+?=7 was really not so different from x^2-5x-6=0 except that the latter was harder to find the secret number(s).
2. Describing and using patterns, ie writing algebraic expressions and using formulas.  For these types of problems, it's not about finding secret number(s). Instead, it's about describing relationships and making generalizations.  Hey, we've already done this when we described the number of edges, faces, and vertices in a 1xN strip.
3. Proving equality or inequality, ie manipulating expressions.  Hey, we've already done this when we showed that one students pattern for vertices of 2*(N+1) was the same as another student's pattern of 2*N+2.

I'm sure there are other great ways to think about the subject of algebra.  Feel free to share your own in the comments.  After making a few examples, I left them with 2 questions:

1. So we've already done some of #2 and #3.  What about #1?  Can you make up a problem in the context of edges, faces, and vertices that would use the first aspect of algebra?
2. Someone mentioned that 3+?=7 has a solution of 4.  Does it have any other solutions?  Can you create an algebraic equation that has more than one solution?

Mixed results with #1.  Some kids were right on it and came up with the expected "What size strip has 68 vertices?" along with some other good questions.  Others were pretty confused by the task.
Great results with #2:
0*N=0
(11-11)*N=0
1*N=N
N/N=1
7*N=E where E is even

There were more, and this second question prompted some fabulous questions/comments, the point of this post.  While I've tweaked some of the language to be more concise, these are all from the 11 year olds.

Can I make it so N has to be a whole number?
Can I make it so N has to be an even number?
Is 2+3=x an algebraic equation? (anyone have thoughts on this?)
In the equation 0*N=X, N can be anything but X has to be 0.
Can I make the operation a variable? (I had a hard time containing my excitement)
If variables are used to talk about a bunch of numbers at the same time, can we use variables to talk about a bunch of algebraic expressions at the same time? (more failed attempts to contain my excitement)

1. Avery, This is fantastic! I wonder if I could get my college students asking questions like these. (I start another semester in late January, and I'll try this activity in my beginning algebra course. I think it's great.)

2. History note: There are three main stages of development. Rhetorical, Syncopated and symbolic. I show my 7th graders examples of all three, then I translate a simple rhetorical sentence through the other two.

The rest of the year, we use these at terms to describe what stage we are in with some work and we talk all year about translating from one form or another.

"3+?=7" I frequently use shapes like hearts or clovers or whatever instead of an alpha variable. To use a more syncopated form of algebra I will often but the word "what" or "something" or "the unknown" where you have put a question mark. I want my students to have a better feel for the rich texture of the algebraic variable.

3. While you were failing to contain your excitement (I, btw, reading this, am failing to contain mine) - what did you say / what did others say?

4. If x-3 = 2 is an algebraic equations then why not 2+3=x? At least I would think so according to the definition you are using.

Thank you for taking the mystification out of Algebra. Students don't need any more reasons to think they suck at math.

5. Had your students done any algebra prior to doing this activity?

6. Why you should use Twitter:

7. It would be great if your students got to algebra with the idea that the equals sign describes a relationship :-)
http://www.ncsall.net/fileadmin/resources/fob/2008/fob_9a.pdf has some neat ideas for talking about it in the "part whole" relationship article at the beginning.

8. 2 + 3 = x is definitely, for sure, an algebra equation. The x stands for some number, and depending on what number you use, it may be true or false. If x is 5, the equation is true, otherwise it is false.

I think the only fallacy here is trying to make a distinction between "an equation" and "an algebra equation". They have the same characteristics, except that once variables are included there is a new category of sentence: true, false, and open. An open sentence is true or false depending on the values of its variables. And, of course, equations with variables can be universally true or false:

x + y = y + x
2x + 3 = 2x + 1

Cool stuff for 6th graders to see as they move forward. They may not be doing "algebra" but they are certainly doing the types of thinking associated with algebra, even reasoning about systems and extensions.

9. I like that. I'm stealing "true, false, and open". I've kind of done that a little with proofs, though not terribly explicitly, and without really utilizing the power of "false" as well as I think I could have.

Now, enough with the math blogging and back to the Christmas cookies!