## Mixture Problems – 1 Variable or 2?

*September 10, 2010 at 10:30 am* *
1 comment *

In elementary algebra, should you cover mixture problems with linear equations, or should you wait until you cover systems of linear equations? Or maybe in both places? Oddly enough, this is an issue of contention among many instructors. I wait until systems of linear equations, and I’ll explain why.

I used to cover mixture problems in both topics, figuring that students could be introduced to the topic while covering linear equations and they could master mixture problems while working with systems of equations. The problem is that my students rarely understood why they set up their equations as they did, they just mimicked the steps. This reinforced the idea that many beginning developmental students have – “I’ll never understand this stuff!” That’s not what we want our students to be thinking.

Another problem is that I already cover so many types of word problems in linear equations that it really just ends up being overload.

- 7 more than 3 times a number is 43. Find the number.
- One number is 3 less than twice another number. If 4 times the smaller number is added to 3 times the larger number, the sum is 41. Find the numbers.
- The length of a rectangle is 5 inches more than its width. If the perimeter of the rectangle is 46 inches, find the dimensions of the rectangle.
- The sum of 4 consecutive odd integers is 232. Find them.

Do I really need to add mixture problems, and interest problems, to this list? I take a lot of time covering the previous word problems so that students develop an understanding of how to be a problem solver, and I may not be able to be so thorough if I add mixture problems into the mix here.

I also find that students understand mixture problems so much better when we get to 2 variables – “let *x* represent the volume of solution 1 and *y* represent the volume of solution 2″ seems so much clearer. Then students can focus on finding two equations to relate *x* and *y*.

We all know that the two techniques are essentially the same. When we solve mixture problems with a single linear equation we are essentially using the substitution method for solving a system of two linear equations. It’s just that, in my opinion, students truly understand the topic better when it is addressed with systems of equations.

So, where do you like to cover these topics? Please share your experience and thoughts by leaving a comment, or reaching me through the contact page at my web site – georgewoodbury.com.

(* Quick Note:* The school/publishing schedules were a little hectic this week, so I apologize for falling off schedule. I’ll be back on track next week.)

-George

*I am a math instructor at College of the Sequoias in Visalia, CA. Each Wednesday (typically) I post an article related to General Teaching on my blog. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.*

Entry filed under: General Teaching, Math. Tags: algebra, developmental math, education, elementary algebra, george woodbury, interest problems, linear equations, Math, mixture problems, NADE, systems of linear equations, teaching, woodbury, word problems.

1.j edward ladenburger | September 10, 2010 at 11:55 amI agree with your adding mixture problems to general systems of equations problems — as I really believe strongly in showing students how many different sounding problems can be grouped together based on the mathematical form which describes them. As a physics instructor, I find that students are often impressed with the variety of different systems which can be modeled with second order, linear, homogeneous, ordinary differential equations [simple harmonic oscillators] –they begin to appreciate more fully the power of mathematics !