Different Variable Usages

14 Jan

I remember there being a time when I had identified at least 3 different ways to use a variable, but for now, I can only think of two.

  1. Using variables to represent an unknown.
  2. Using variables as a pattern summary.

Well, w/ Common Core right around the corner, here are a couple of activities I loved, but rarely found the time to do well in the past. However, with my double Algebra this year, I have twice as much time w/ them to go over the same material so I got to work on these much more thoroughly… and I LOVE ’em even more!! I truly believe the lack of understanding variables in the different ways is what makes Algebra hard and not something that should make solving problems easier.

1. Variables to represent an unknown.

Usually used in solving simple equations. x + 5 = 9. “Some number plus 5 will give me 9”. I did a LOT more mental math to teach this part and I really do believe it went significantly better than ever before. Even up to equations like -2x + 6 = 0.

Another big way of using variables as an unknown that we neglect often, is in word problems. So this year, I have used a couple of worksheets that my former professor created which contains just word problems. After going over them pretty thoroughly for the first time this year, I fell in love with it even more because my students came up w/ different ways of writing the equations! Here we come, Common Core!

One example:
Lunch Money. Daisy’s mom gives her lunch money every weekday. On Monday and Tuesday she gave her an extra dollar. The rest of the week, Daisy received the regular amount. For one week, she received a total of $42 for lunch. How much money does Daisy get for lunch each day normally?
Here are the 3 different answers my students came up with:
Once I got them to start drawing pictures to represent the problems prior to this, they did this one on their own. =)
Execution Tips: Only pass out the first two pgs of Amazing Stories first (no answers). We did the problems, one by one, up to “Pocket Change.” Starting “Lunch Money” (the posted example), they were supposed to do it by themselves.
Once they started working by themselves, after a few minutes, I projected all the possible answers so that they had to find their equation (or the equivalent) up on the board. They got a copy of the answers in order to finish the worksheet at home for homework.
The next day, I gave them the follow-up worksheet of “Life of a High School Student” for them to get a few more practice problems on their own, without having a bank of answers.
I saved a few problems from “Life of a HS Student” to use as an exit ticket and on future quizzes.
                 2. Variables as a pattern summary
This is the type of variable usage where all those darn rules come from. For example, if 2^3 * 2^5 = 2^8
4^9 * 4^7 = 4^16
what does x^m * x^n = ?
Anyways, here is the activity on practicing this type. Again, I love it all the more due to the multiple solutions that are possible. I copied down my entry below.
Materials: Cubes. Preferably not too small. And the worksheets.
Students are then asked to build certain objects that build on top of each other and asked how many cubes it would take to build a structure of 10 layers… then 100 layers.
For example:
First question after this would include: “How many cubes will it take for 10 cows? Explain.
Next they would be asked, “How many cubes will it take for 100 cows? Explain.
You can then summarize the patterns by writing an equation, with the input (x) representing how many cows, in this case, and the output (y), representing the number of cubes.
TimingWe started Pattern 1 and 2 in the last half hour of one class, continued the next day for about an hour, and then they were doing Pattern 5 by themselves by the next day’s 50 minute class. Amazing how many students who do not understand Algebra actually are really good at figuring these puzzles out.
Anyways, to get a better feel, you have to take a look at the multiple solutions that these problems have to offer. I have written out what my students had come up with this year. Yep, these were my intensive students coming up with these different solutions. We also took time to go over how to write a good explanation also. I went around giving them feedback and having them re-write a lot of what they had written…
Again, here are the links to the two types of variable practice.
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