## Reflection: Connection to Prior Knowledge Mixture Problems - Section 3: Work Time: Let's See How Far We Get!

The first two problems on Mixture Problems #1 are like the two examples we've already seen, because the unknown for which we're solving is the unit cost of some mixture.  I circulate to make sure that kids can apply what they know so far.  I encourage conversations and for students to help each other out, and I reference the notes.  The second problem is measured in percentages of a ton rather than numbers of pounds, but the structure is the same, and if any students get stuck, I encourage them to lay it out just like I did on the baking flour example, which is still up on the side board.

Starting with problem #3, the unknown changes.  This problem gives the cost of the mix and each of its parts, and asks students to determine how much of one part will have to be added to the other.  It happens to be about a mix of walnuts and peanuts, but the context is less important than the structure.  To illustrate what I mean, I return to the table we've used to think about different movie ticket scenarios, and I edit it for everyone to see, to fit this "Mixed Nuts" problem.  I use what we know to fill it in: that there are three pounds of walnuts, that walnuts cost \$12/lb. and peanuts cost \$3/lb., and then with each row of the table, we see what happens for each (less expensive) pound of peanuts that are added to the mix.  It's obvious immediately that the cost of the mix will decrease as more peanuts are added - it's just like how selling more teen-priced movie tickets decreases the average revenue per ticket.  From here, students can use guess-and-check to keep trying different amounts of peanuts until they get the mix to cost \$6 per pound.

What's important for students to notice is that the constant increase in the number of pounds of peanuts does not result in a linear decrease in the cost the mix.  Kids will want to see a linear pattern and use it, but this doesn't happen.  Here's an opportunity to contrast this problem with previous situations.  To find the average revenue for a movie ticket, we were able to divide by a total of 300 tickets each time.  Now, the total amount of mixed nuts increases for each pound of peanuts that are added, which means that we're dividing by successively greater numbers.  When the divisor is a variable, we don't have a linear function.

All of that provides an opportunity to look at this problem in a different way.  I tell everyone that I'd like to try to solve this problem like we thought about the baking flour problem, and I think-aloud about how to fill in the template when there's an unknown value.  I show students how to fill in what we know and to use a variable for what we don't.  We see that the variable - number of pounds of peanuts, in this case - shows up all over the place.  Noting that if we divide the total cost of the mix, 3x+36, by the total weight of the mix, 3+x, we want to get \$6 per pound, students come to understand that we can solve the equation

(3x+36)/(3+x) = 6

to find a solution.

Students will still use guess and check to solve this problem and others like it, but I'm providing a framework for them to use algebraic strategies on their own terms.

This is about as far as I hope to get with each class today.  If students leave having tried a few problems on their own and having seen the algebraic set up for problem #3, then they can try solve more of these problems tonight.  As our student-directed work sessions and year-end review continue, these mixture problems offer great opportunities for both.

Seeing Structure, and Using it to Solve Similar Problems
Connection to Prior Knowledge: Seeing Structure, and Using it to Solve Similar Problems

# Mixture Problems

Unit 11: Functions and Modeling
Lesson 8 of 13

## Big Idea: Mixture problems are hard, but there are a lot of places they can fit in an Algebra 1 curriculum, so they're the perfect lens for review as the year comes to a close.

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Subject(s):
Math, Algebra, modeling, problem solving, seeing structure, mixture problems, review, Algebra 1, Growth Mindset
43 minutes

### James Dunseith

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