## Friday, September 26, 2014

### Discovering a Variable

I wanted to see if I could get students to "create" or "discover" the idea of a variable.  To try this, I completed the following in my ESL (English as a Second Language) math class.

We first started with a discussion around language, and how math is the "Universal Language".  Next, we talked about "What is the best way to learn a language?"  The students agreed that we should learn how to translate from our language into math would be a great start.  I then told them how I once ordered 2 pepperoni pizzas and 3 Hawaiian pizzas and it cost \$70.00, and I asked the class if there is a way we could translate this into math?  One student came up and wrote,
2 Pepperoni + 3 Hawaiian = \$70.00
We then had a discussion how, currently, we would not be able to deduce how much each pizza cost, however this would count as a translation.  I then asked how would you translate "4 Pineapple Pizzas, 3 bottles of Coca-Cola, and 1 Meat lovers, costing \$92.00"? Another student came to the board and wrote
4 Pineapple + 3 Coca Cola + 1 Meat Lovers = 92
The class again agreed this was sufficient.  At this time, a student in the back was getting irritated at how easy and time consuming this one.  I asked him to go to the front and in front of everyone translate "3 super size fries, 2 Extra large Coca-cola, and 1 double, extra bacon, cheeseburger costs \$21".  He let out a big "UGH!", and asked me to repeat.  As I repeated he wrote...

3 F + 2 C + 1 CB = 21

He looked at me with a smile, and some of his classmates started to laugh.  I then told him "I said supersize fries, not Fs", which he responded with "Yeah this F is supersize fries".  We then had a dialogue around what CB could mean.  After some time, a student asked "Could that be Cheese times burgers?",  and almost immediately a student yelled "but C is coca-cola, so coca-cola times burger?". The student, at the board then changed his answer to

3 F + 2 C + 1 B = 21

I then wrote on the board

3D +2C = 13

and asked "What does that mean?".  The answers ranged from "3 Dogs and 2 cats cost 13" to "3 bags of dill pickles and 2 bags of cheetos is \$13".  We then decided, as a class, that it is important to create a legend at the top.  Therefore we went back and wrote legends such as "F is Super size fries, C is extra large Coca-Cola..."

This was my attempt at students creating their own knowledge of variables.

1. Thanks for sharing this interesting conversation! Can I ask you a follow up question? How would your class "encode" the sentence: A bacon-cheeseburger costs three times more than a small fries?

I'm asking because sometimes when students talk a lot about sentences like

2 Pepperoni + 3 Hawaiian = \$70.00

and encode that as

2P + 3H = 70

they don't think about P and H as variable quantities standing for the price of the types of pizza, they think of them as unit labels for the 2 and 3. Like they might think of 3m + 5m = 8m, or 3ft + 5in = Caesar's height.

It can come back to bite them when translating sentence like A bacon-cheeseburger costs three times more than a small fries or dealing with problems like: John mows lawns for \$11 per hour and trims hedges for \$15 per hour. If he mows lawns for 5 hours and trims hedges for 3 hours, how much money does he make? Now write a rule that expresses John's total income as a function of the number of hours he mows lawns and trims hedges. Students are just as likely to write 5L and 3H because he mows 3 lawns and trims 5 hedges. They know that the 5 and 3 vary, but they think the letters are just labels, not placeholders for varying quantities.

Curious to hear your thoughts on this!

2. Do you love Coca-Cola or Pepsi?