Why Mixture Problems, Why Now?
The purpose of this lesson is to introduce mixture problems as a modeling application and a review task. We're about halfway through this four-week review unit. After a few days of open-ended work time in the classroom and in the computer lab, plus some group problem solving, I take the beginning of today's class to focus everyone on one task.
There are two reasons that mixture problems are happening right now, near the end of this Algebra 1 course. First of all, there's the practical reason: it didn't happen earlier in the year. When we created equations at the start of the year, or during the lines unit, or during systems of equations, I skipped these sorts of problems. I really try to meet my students where they are, and not to force a topic if it doesn't seem to fit. Of course, this can change: if I see that next year these lessons make sense during the systems unit, for example, I might teach them then.
With all of that in mind, the other reason to teach mixture problems right now is that it's a terrific review topic. Defining variables and writing equations, using guess and check, seeing structure, and solving systems of equations can all come into play here. Students will also see how problems that seem different can be very similar; the ability to see the connections between different but related problems is an important takeaway in any math course.
Today's Opener Sets the Stage
Before we jump full-on into mixture problems, which are notorious among students for being annoying and difficult to grasp, I want to give kids a chance to think about what happens when we mix two items. I also take the opportunity to raise engagement by referencing a movie that a lot of kids are talking about. Today's opener is on the first slide of the lesson notes, and it's more of a mixture situation than a problem. I expect it to be simple, so I give students a minute or two to come up with their answers.
Students may need a little clarification on the second question, "What is the average revenue per ticket?" This is what I really want to explore as today's class begins. It's easy enough to figure out that the movie theater will bring in
$8(100) + $12(200), or $3200
if they sell 100 teen tickets and 200 adult tickets. (Here I also like to propose that it's good business for a theater to charge less for teens, who will often spend more on concessions than adults.) Then, to find the average revenue per ticket, we can divide:
$3200/300 to get $10.67 per ticket.
When everyone sees that, I flip to the second slide and say, "What will happen to the average revenue per ticket if more teen tickets are sold? What about less?" I facilitate a full-class discussion about these questions as we fill in the chart on the second slide. Counting by 50's in # of teen tickets column and keeping the total number of tickets at 300, we fill in the rest accordingly. Everyone is quickly able to generalize that if you sell more of a less-costly item, then both total revenue and average revenue will decrease. The opposite to that scenario - more of the more expensive item raises the average revenue - is also true.
With this background knowledge in mind, we'll have a reference point as we dig into some mixture problems.
Mixture Problem Example: Baking Flour
We move straight into the "Mixture Problem Example" on the third slide of the lesson notes. Note how his problem is nearly identical to the movie tickets opener. The difference is that in this situation, the mix is something that can actually be bought and sold for the resulting price. No one actually pays the average cost of a movie ticket (unless only one sort of ticket is sold), but baking flour can be made and sold for any price, depending on how much of each ingredient go into it. Just like we did on the opener, we work together to fill out the table on the fourth slide. By now, I hope that kids have a foundation on which to build as they try these problems on their own.
I tell students that I have some quick notes for them, and I create these notes on the side board. As problems get more difficult, this structure will become more important, as you'll see in the next section of this lesson. I pause for a moment to take clarifying questions, and then I move on.
Tangible Demonstration: Mixing the Kool-Aid
At this point, kids are feeling good and making connections for themselves. I run a brief demonstration to help the ideas stick. I got this idea from a video by Marlo Warburton over on Teaching Channel, and I substituted two colors of Kool-Aid because I thought it was fun. I take out three jars: one that is full of water and smaller empty containers. I prepare one batch of Kool-Aid in one jar, and a different color in another. Then I ask students what will happen if I add more water to one jar. Or a little bit of red to the water that's left over. Then, what will happen if I mix the two colors. I just pour from one jar to another at the suggestions of my students, and we all watch what happens.
It's most important to try not to editorialize here, but to give kids space to notice on their own that this is like the Baking Flour problem. And they will! Kids shout out, "Hey, this is just like what we were just doing with the flour!" I tell them they're right. There's no need to attach numbers to any of this - my only purpose in mixing the Kool-Aid is to provide a visual example for what we've just seen, and that it's a real idea. As students practice problems like these on their own, I'll want them to notice the relationships between similar problems with different context. That's what kids are learning to do as they watch the colors mix and think about the two problems they've seen.
New Assignments: Mixture Problems
I tell students that their year-ending work time will continue now, and that a great option is to try solving some mixture problems like those they've just seen. I've prepared two new handouts Mixture Problems #1 and Mixture Problems #2 that form a sequence of problems with a sequence of different unknown values. I encourage everyone to try the first set. The second is totally optional, just like every other mastery assignment that's currently available to kids.
Like I noted at the end of the previous section, I'll be satisfied if today's class ends with a conversation about different ways to solve problem #3 on the first mixture problems assignment. Moving forward, and back into several consecutive days of work time, these Mixture Problems assignments will be added to the array of work options available to students. I encourage everyone to try the first one, but if they're committed to working on something else, then I let them run with that. Other students might finish an earlier assignment before picking these up later, and that's great too.
I'll continue to provide notes like these (Mixture Problems #2, problem 1) that incorporate the structures I share with students today. Students will continue to use what they've seen and to create their own strategies. Take a look at the work I've included here to get an idea of the various approaches students take. If you haven't already, I encourage you to read through the two mixture assignments to see how the problems change over the course of the work.