Systems of Equations Introduction

Last year, I was really struggling to teach my Algebra 1 students how to solve a system of equations using substitution. This struggle completely flabbergasted me, because substitution always made so much sense to ME and I found elimination to be harder, while my students LOVED elimination and couldn’t comprehend substitution at all.

“What do you mean, ‘substitute’?” “(Seriously? We’ve been substituting things into equations all year! I can’t even say the word properly anymore because I’ve said it so much!)

“So, I put the other equation here?” (Then they replace the y with the whole equation y=3x so now they have two equal signs.) (Their other favorite thing was to leave the original variable in the equation, so now their new equation says y3x.)

They adored elimination. It was their favorite thing ever. They started adding equations together on problems that didn’t even have a system. Of course, they really struggled to remember to multiply the equation before adding when they didn’t have equal coefficients…but their equation-solving skills were so atrocious that they could somehow still manage to lose the variable they had not actually eliminated…and now I’m just getting frustrated all over again so I’m going to go back to my original point.

Ahem. So I was looking around online last year for a different way to approach teaching substitution as a method for solving systems, and I ran across two ideas, both of which I’m using this year. I’m only going to discuss one of them in this post, because I need to go to bed. I’ll write up my thoughts on the other one later, after I actually implement it next week or the week after.

The first idea, which I started in class today, was originally posted on I Hope This Old Train Breaks Down. The idea is that the kids start by solving puzzles involving shapes that represent numbers, gradually work their way up in puzzle difficulty, and eventually transition to using the shapes to represent variables. I loved this idea, because we’re demonstrating the idea of a variable as a placeholder for something we don’t know yet. (I’ve never really understood why they hate using letters for that purpose so much. If I taught pre-algebra in middle school, I would totally use pictures/shapes/random stuff instead of letters to introduce variables, because the letters just really bother them.) The kids have to apply a variety of different types of logic to figure out each shape. In most situations, they don’t actually NEED the whole puzzle to figure out the value for the shape – in most cases, they just needed one or two rows/columns. This teacher, whose information I cannot seem to find on his/her website, also wrote some reflection/processing-type questions into the worksheets, which I kept when I duplicated the idea for my own use.

I’ve been talking about how excited I am to start the unit on solving systems with my kids for weeks. This is probably because it was the only unit I felt was a success last year. (I’ll leave you to draw your own conclusions about how the REST of the year went, if the experience I described above was “successful”.) The kids have been less excited. In fact, they groaned when I mentioned it and said they learned it last year and didn’t like it. “Yeah, yeah,” I said, “that’s ok. Because I’m going to start over from the very beginning, so you can forget everything you learned about systems last year.” (Let’s be honest, they probably weren’t remembering it correctly anyway.)

Another tangent: this is one of the conversations I have with them on a regular basis:

Student: “That’s not what [teacher] said last year!”

Me: “Yes, she did.”

Student: “No, she told us to do [insert totally crazy, invalid thing here]!”

Me: “You may have HEARD that; that does not mean your teacher SAID that.”

They give me weird looks at this, which I ignore.


Back to the puzzle thing. I mentioned it to my students a little over the last couple of days, including showing the worksheet to a few students yesterday. They were skeptical. Actually, I’m being nice. They hated it. They rolled their eyes and insisted that it looked really hard. I rolled my eyes back and largely ignored them. We have a great relationship.

Then I gave them the assignment today. First I put them into groups of two or three – partly random and partly by my assignment – and passed out the first two pages (front and back of one sheet of paper) to each group. Then I stood up in front of the class, briefly explained that each row adds up to the number on the end, each column adds up to the number on the bottom, and your job is to figure out the value of each shape, and said, “Go”.

They dove in. They stared at the page. They discussed. They guessed-and-checked. Each group eventually realized that if 3 triangles make 27 (second row), then each triangle must be 9. (One pair needed a significant amount of assistance to make this connection, but didn’t ask me for another hint the rest of the class period.) They used that knowledge to figure out the other two shapes. The back side of the worksheet was similar in structure, so they whipped through it. They wrote explanations (some of which were terrible so I made them give me more information).

Then I gave them the second sheet, and told them these ones were harder. Some students were glad, because the earlier ones were too easy. Some students gave me that deer-in-the-headlights look because the word “hard” scares the crap out of them, which I ignored. (The biggest difference between first-year-teacher-me and second-year-teacher-me might be my ability to ignore things this year.)

And guess what? They got it. Yes, it took longer. The relationships are harder to see. Only one group finished both pages – most finished the first 3 puzzles. Tomorrow, they’re going to finish the last 3 puzzles and do the last 2 pages where variables get introduced. Those last 2 pages are going to be a quiz grade (because I need more quizzes in their grading system and I feel like it’s nice and authentic).

I consider this activity to be an unqualified success in my class today. Granted, I have an amazing class. They know, I tell them all the time. I believe we have been able to develop a strong mutual respect. They work hard for me because they know that their hard work is important to me, and they want me to be proud of them. I’m trying to focus on that this year, and not their right answers. I think this shift has been helpful in fostering that respect and work ethic, although I know I could be doing better.

My favorite moment: Student A (one of my top performers who lacks some self-confidence in his work) called me over and asked me for a hint. His partner, Student B (the one who rolled his eyes yesterday and complained and told me it was way too hard and he wouldn’t do it) immediately yelled “NO HINTS! I’VE GOT THIS!” and refused to let me say a word. Do you know what that’s called? It’s called PERSEVERANCE, and it’s something I’m not sure I’ve ever seen from this particular student.

My other favorite moment: The two students who couldn’t figure out how to get started were actually trying to figure out a starting point for a really long time before I stepped in to offer assistance. Again, PERSEVERANCE. They didn’t immediately commit my all-time biggest pet peeve, the famous “Miss, I don’t get it”.

I had a great time in Algebra today. I’m pretty sure my students did too. I’m also pretty sure they learned something (I’ll find out tomorrow when they finish). It was a great day.


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