While I’m not teaching algebra this year, David Cox’s post on mixture problems caught my eye. I, like many, had a whole lot of trepidation around teaching mixture problems.
My trajectory went something like this:
Year 1 of teaching 8th grade algebra:
Me, “I have absolutely no idea how to solve these problems so I’ll never get my kids to understand them. I’m way behind in the pacing guide, so I’m just going to skip them”
(Mind you, my year 1 of teaching algebra was my year 8 as a teacher, so the fact that I flat-out skipped teaching something because of my own fears and lack of content knowledge is actually quite bothersome)
Year 2: Me, “I know how to DO these problems, but I don’t really understand their ins and outs. I’ll give the class 2 options: Work in a small group of students where I’ll show you something new and challenging, or stay in your regular groups and you can work on catching up on your missing work.
Year 3: I fess up to my colleague Marlo that I am terrified to teach mixture problems to an entire class of 8th graders, as I don’t understand them well myself and can’t figure out how to make them accessible to all students. Marlo has recently been to a summer PD where she played around with using a see saw method. It’s documented in this great video by the Teaching Channel. I realized there was hope for me! I still didn’t teach mixture problems to the whole class, but I had a lot more confidence in teaching them to small groups of students who wanted an extra challenge.
Year 4: I’m ready to help ALL students learn mixture problems. After some initial work with kool-ade and water, all students understand the idea that you’ll need MORE of the solution which is closer in percent to the desired mixture percentage. As a class, we have the routine of putting a large colored star next to the solution which we’ll need more of. I find that this makes a huge difference in students not mixing up the quantities since they are inversely proportional. We talk about two ways of modeling these types of problems:
I never taught systems of equations as a solution strategy to these problems and last year I’d say that at least 75% of my students could solve mixture problems correctly. For a very heterogeneous group of 8th graders, 75% success is pretty good.
One of my 7th grade algebra students read about systems of equations in the textbook and came in to ask me about it. He thought it was way easier, so I decided, weeks after the fact, to teach this solution strategy only to my 7th graders. A few loved it, but the vast majority stuck to the proportions model even though they were quite proficient at solving systems of equations.
Is the way I model it just another trick to memorize? Maybe. It’s not as visual as David’s GeoGebra applet, but using colors (which my students always do), it felt like a decent model. I’m curious to hear reactions.