# Systems of Equations and Context

Disclaimer: not a coding lesson! Just a reflection on teaching a notoriously tricky Algebra I concept.

In my Algebra I class, we’re learning about systems of equations. I had a career as a software engineer before teaching, and as I tell the students often, systems are actually a concept I used every so often as an engineer. Sometimes you have multiple variables or constraints that you have to meet at the same time, and modeling them as an algebraic system is helpful. I found, however, that knowing systems are useful doesn’t translate to easy teaching or learning.

I spent 5 or so lessons going through the usual order of solving-by-graphing, solving-by-substitution, solving-by-elimination. I used the Illustrative Mathematics lessons available online. They are decent lessons, but I could tell the pace was leaving some students behind. By the time I assigned practice problems, maybe a third of the class had a decent grasp of an algebraic way to solve a linear system and the other 2/3 were struggling. And, as you can imagine, the kids who didn’t understand showed me by misbehaving – fun.

What do you normally do in this situation? Over time I have learned the best approach is to back up without making it seem like you’re backing up. Increase the problem-solving load while you decrease the procedural load. Lesson planning is creative problem solving.

Sometimes my friends post math memes on Facebook – picture puzzles that are actually systems of equations. They’re fun. I did some searching for “algebra picture puzzles math” and followed the rabbit hole to pinterest boards that hosted lots of them. One site you can mine for picture puzzles is brainfans.com which is where I captured the ones I used in my lesson.

I gave the students a couple of picture puzzles to work through in small groups, along with a couple of word problems I made up about purchasing food. Here are the ones I used. I purposefully chose puzzles that always had 2 or more variables in each equation, so you needed to use the concept of elimination or substitution, you couldn’t just solve for one variable in a single equation.

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Dawn went to a burger stand on Saturday and bought 5 cheeseburgers and 2 fries. She spent $21.24. On Sunday, she was still hungry and she went back to the same burger stand. This time she bought 2 cheeseburgers and 2 orders of fries for $16.14. How much are the cheeseburgers and fries?

CCCCC + FF = $21.24

CC + FF = $16.14

Dawn went to a smoothie shop on Monday and bought 6 large smoothies and 2 small smoothies for $61.64. They were so delicious, on Tuesday she went back and bought 7 large smoothies and 4 small smoothies for $82.33. How much are the large and small smoothies?

LLLLLL + SS = $61.64

LLLLLLL + SSSS = $82.33

These were VERY accessible to the kids. The students that already had a good concept of solving systems modeled the picture puzzles as equations and solved them formally. The students that were having a tough time with it used less-formal approaches that still used the idea of substitution or elimination.

For example, from the first two equations using the cars, they could tell the yellow race car was worth 2 more than the blue race car, using the concept of elimination. Then, in the third equation using the cars, they could reason out that “x + 2 + x” was equal to 32 and decide the final value of the blue race car – basically using the concept of substitution.

I could tell students got the “cheeseburger” problem correct when they shouted across the room “Why are your cheeseburgers so cheap and why do the fries cost so much?” Ha! I love gourmet fries!

In the “smoothie” word problem, most groups struggled with it at first – even those that understood symbolic equation solving so far. So I gave them a tiny hint – I asked them what would happen if they doubled the first order. How many large smoothies would that be? How many small smoothies? And the price now? And how is this new order different from the 2nd order? And every single group of students said “oh” and finished independently. Context matters!

To finish the day, we did the Noah’s Ark problem which I found a long time ago on Julie Reulbach’s blog. It uses the same concepts, substitution and elimination, and it’s just as much fun with 9th graders as it is with young kids!

The students’ assessment is to write their own picture puzzle and word problem, complete with solution, for others to solve. We’ll swap them next week!

I enjoyed this SO much more than teaching systems the old fashioned way, and the students had fun problem-solving instead of continuing to learn procedures. Math class was fun, and avoidance / bad behavior was almost completely absent today.

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