Modeling Linear Equations

“I have a certain amount of nickels and dimes that add up to $2.10.”

“Sam bought a number of tacos and burritos. The tacos were $2 and the burritos were $3. Sam spent $24.”

“Janice joined a gym with a sign-up fee of $40 and a monthly rate of $25.”

I put these three statements on the board and told my Algebra II students to generate a table of values and graph the values.

Some students were instantly able to use their “real-life” math knowledge to start working. Others needed a push and were then able to start. Some needed calculators to work out the values, but all 90 students were, with minimal prompting, able to use that part of their brain that had nothing to do with school to create a list of possibilities. Only a few questioned the lack of an “answer”, and none needed the explanation more than once.

We did this for three days. Over half my students were able to determine the slope of the line and link it back to the word problem without my expressly teaching it. Some of them improved as time went on; few of them are completely incapable of linking the word model to an equation. With discussion, most students began to realize that models with two changing numbers adding up to a constant (nickels and dimes, tacos and burritos) had a negative slope, because an increase in one led to a decrease in the other.

Then, for two days, I gave them problems in equation form:

“y = 3x+2”

“2x + 5y = 45”

“3x – 2y = 6”

“y = -.5 + 50”

They had to generate a table of values and graph the line. When some of them had difficulties, I pointed out the links to word models (what if you were buying burritos for $2 and 6-packs for $5–how much money did you have to spend?) and they got it right away. The subtraction models were the most difficult (which I expected, and didn’t emphasize).

And then, for a couple days, I mixed it up–gave them word models and equations.

I began writing the answer to the most common question in huge letters on the white board: YES, YOU CAN JUST PICK ANY NUMBER.

We’ve just spent two days doing the same thing with word models that provide two points but no relationship.

“Janice joined a gym that had a monthly rate and a signup fee. After three months, she’d paid $145. After 6 months, she paid $190.”

“Brad buys grain for his livestock on a regular cycle and re-orders it when he runs out. After 3 days, he had 72 pounds left. After 7 days, he had 24 pounds left.”

So they have to create the table, find the slope from the table and graph it. Day 2, they had to do that while also answering questions (“What was the signup fee? What was the monthly rate? How much grain did Brad use daily? When would he have to reorder?)

At no point during these two weeks did I work problems algebraically. Some students did, but in all cases I encouraged them to think about other ways to work the data.

They improved dramatically. I gave them a quiz, and all but a few students were able to do the problems with a minimum of questions, although they needed lots of reassurance. “Okay, I think I know what’s going on, so there must be something wrong.” A few kids gave me the “I have no idea how do to this” and I was pretty brutal in my lack of sympathy because they were the kids who don’t pay attention.

I spent the entire first semester teaching them linear and quadratic equations–graphing, systems, solving, factoring, the works. Algebra II is a course designed for students who don’t want or aren’t ready to move to Algebra II/Trigonometry–or who failed that and need a third year. So the first semester is a rehash of Algebra I. I covered it, they all learned a lot–and yet, the semester final was dismal. Some of it was Christmas crazy, and then I wasn’t happy with the test. But nonetheless, they should have done better.

So I mulled this over Christmas break. All but a few of my students are juniors or seniors. Some will be taking a proficiency exam in a few months. Others will be taking a proxy for the exam in their state tests. I’ve always been more focused on their college tests than their knowledge of second year algebra. I want my students to test out of remedial math, or spend as little time as possible in it.

That’s why I’ve decided to spend a month helping them use their math knowledge–the knowledge they see as entirely separate from algebra and geometry, their “real-life” knowledge–to model data.

If this works properly, the strongest students will have a much deeper understanding of the equations and how they relate to the data. The weaker students will be able to work problems using their inherent math ability, rather than struggling to turn the problem into an abstract representation they don’t recognize.

I’m going to finish up linear equations with maps, to give them a better understanding of midpoint, distance, parallel and perpendicular.

For more samples and boardwork, see Modeling Linear Equations, part 3
Then it’s onto quadratics.

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