Beautiful Math: Mixture Problems in Algebra One

Wednesday, February 3, 2016

Mixture Problems in Algebra One

This intiative by #MTBoS has been great - I've been better about blogging and visually documenting what I do in class. And it's been great fun to read other blog posts on the weekly initiatives. This latest directive is to blog about a lesson. This is mostly what my blog is about - blogging on lessons I do and the interactive student notebook materials I use/create.

Teaching mixture problems using systems of equations can be challenging because it can be difficulty to visualize what they are about. I'm all about visualizing and understanding concepts (for myself and for my students). This lesson sets the stage for building understanding about the concepts underlying mixture problems.

The class is Algebra One, honors level, ninth grade. We've been working with systems of equations for about a week and a half now. Students learned about systems last year in 8th grade, mostly focusing on solving systems using graphs. I started this unit by focusing on creating models for situations with two conditions and two unknowns (see this blog post). We then learned how to solve the algebraic models without graphing and I incorporated applications problems as I went. Mixture problems are their own little category of applications though, so I introduce those separately.

This year I decided to introduce mixtures visually with something all students love - cereal! Last year my student teacher and I talked about this idea and she implemented it in her classes (thank you Tess!) so I took her good work and tweaked it a bit for my class this year.

I set up a little "Research & Development Lab" in the back of my classroom:

Those ziploc bags each hold an ounce of the labeled cereal (about a heaping cup of each).

And provided some facts that our research team is using:

The question was how many ounces of each type (Kix & Cocoa Puffs) would have to be in this blend to achieve the size and cost we are aiming for? Students saw that this was a situation with two unknowns and two conditions. So it was time to organize this information into a system of equations (all this comes from student responses):

The tricky bit of course is in setting up the equation that represents the cost. Students have to remember that the sum represents the cost of each individual type of cereal (cost per ounce times number of ounces) so that must be equal to the cost of the new cocoa kix cereal box (cost per ounce times the total number of ounces in our prototype box).

Once we got this all set up students solve the system. Now the fun part - we actually measured out a "box" of this special blend together. And then students got to sample the cereal (yum). I gave everyone a small cup full of the cereal to munch on.

So how did this work? Well it was certainly fun and I have a bright bunch of kids this year so they were able to create the system pretty easily. I handed out the cereal samples for students to munch on while we did some problems together but there were only 10 minutes left to class and I think everyone was too excited snacking to really think about any more problems. I started to work on an ISN foldable looking at different types of mixture problems but gave that up pretty quickly. Instead I gave them two problems to try for homework and figured the next day we would pull things together.

DAY #2 on Mixtures

Before I went over homework and got any feedback from students I had them work in small groups at the board on mixture problems similar in nature to what we did in class with the cereal. After the group solved theirs they had to switch and analyze the solution and work of the problem next to theirs.

This worked really well and students said they were understanding mixture problems better.

Then we focused on mixture problems that contain percents (including interest problems).

After that we used our ISN to summarize our learning with the left page containing a foldable showing 4 examples that the students set up systems for...