Systems of Equations Update

A few days ago I posted about the puzzle-based lesson we did to introduce solving systems of equations. I was filled with optimism and convinced that this was going to be the best thing I’ve done all year.

That optimism was rather premature.

Honestly, I should have done the puzzles myself before giving them to my students to run wild. I should have looked for patterns, tried to find more than one way to solve them, and tried to anticipate the methods my students would use. I should have focused on what hints I would give my kids in class to get them moving on the right track without walking them through the process myself, or giving them the “answer”. I say all this because I was very disappointed to find that as soon as students got through the first two puzzles (which were the easiest), many of them started guessing-and-checking, a process that is just not going to work when we get to actual systems that could have negative or fractional solutions.

Also, remember the two girls that I was so proud of, because they didn’t ask me for help right away, even though they had absolutely no idea what they were doing? They spent the next two days asking for “hints” constantly, but then being completely unable to do anything with my hint after I gave it. I basically had to walk them through the process to find the first shape in all of the harder puzzles. And this is after I split them up, so they were working with people who should have been more helpful to each of them.

I still think the lesson is sound. I just need to approach it differently next year. I started to emphasize that students needed to be thinking about GROUPS of shapes instead of individual shapes; for example, if the circle and triangle TOGETHER make 30, and the circle+triangle+square make 45, we can figure out the square, NOT the circle and the triangle. They really got stuck on this. They could see that the circle+triangle made 30, but they really wanted to use this information to figure out the circle and/or the triangle. I don’t think any of my students were able to see that they could use what they knew to figure out a different shape. (This is what I mean by guess-and-check – they would assume the circle and triangle were each 15, and then change the numbers gradually to a point where it worked. This will eventually get the job done, but it’s extremely slow and inefficient. I didn’t realize until Thursday that this is why Puzzle #3 had taken so long for them to do on Wednesday.)

They’re not seeing patterns and relationships between the shapes or groups of shapes. They were still seeing them as individual shapes.

Some thoughts for next year:

  • Go through an example before they begin to demonstrate the kind of thinking I want to see from them. (Either make up my own example(s) or use some or all of the odd-numbered puzzles and have them complete the even-numbered puzzles on their own. Since each odd/even pair is very similar in structure, this might be more effective.)
  • Tell them before they begin that they are not allowed to guess-and-check. It’s a viable problem-solving strategy for many situations, but this is not one of those situations.
  • Emphasize the idea of GROUPS of shapes, and have them use colored pencils or draw circles around those groups to they can visualize them. This way I can better see the strategies that they are using as well.
  • Prepare my hints better before I begin the lesson. Decide what to recommend students look at as the very first strategy. (The reflection questions on the bottom of each page actually help with this, so a valid hint on Puzzle #3 in particular was to read the question. The hypothetical kid is wrong, but the question points out a relationship that can be used to figure out a shape.)
  • Also decide what to do next if they don’t see the relationship they are suppose to see, or understand what it means. (For example, if a row containing 2 circles and a triangle makes 27, and a column containing 2 circles, a triangle, and a square makes 41, a few students COULD NOT make the jump to see that the difference is the square, which means the difference in the numbers is the value of the square.) I struggle to know exactly how much help to give, so when they are totally clueless, I often hand-hold way too much.

My roommate also teaches Algebra at the same school I do, and she is planning to introduce this lesson next week to her students. (She had a schedule conflict this week, so she is doing next week what I did this week, and vice versa.) She and I already discussed this lesson in some detail last night as we drove home, and I’ll probably give her this list of things I would do differently. Maybe if she has a more successful experience, we can both fine-tune the lesson more for next year.

This post, and that last sentence in particular, sound more downcast than I actually feel right now. (In all fairness, that may be because this is the first day I’ve been able to just relax in 3 or 4 weeks, and I’m so excited that nothing can get to me right now.) I’m glad we spent our time on these puzzles, and I plan to keep using the idea of the shapes as we delve into algebraically solving a system in the next few weeks.


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