10. Mixing Solutions

Among the great word problems of elementary algebra are those that involve mixing solutions. For example, “How many ounces of alcohol must be added to 20 ounces of a 30% alcohol solution in order to obtain a solution that is 40% alcohol?” Or, “How many ounces of a 70% alcohol solution should be added to 20 ounces of a 40% alcohol solution in order to obtain a solution that is 50% alcohol?”

Such problems can be modeled well by using colored beads that can be obtained from various craft stores. You want a good supply of at least two colors, such as white and red. Do not use round (spherical) beads, … you’d soon have beads dangerously rolling all over the floor! Instead, there are nice “Y-shaped” beads that will serve quite well. Be sure, however, that both the white and red beads are identical (size, shape, and weight) except for color. In addition, you want a good electronic scale that can measure to a tenth or hundredth of a gram, because it is best and easiest to prepare mixes by weight. You’ll also need a bunch of plastic bags in which to mix the beads.

These are great activities. Pretending that white beads are water and red are alcohol, even such a simple notion (to us!) as comparing two “solutions”—say, one 60 grams and the other 200 grams—for which both are 20% red, will help students understand the very notion of percent. Once the various resources (lots of beads, containers to store the separate colors, plastic bags to mix them, and an electronic scale to weigh them) are made available, there are a myriad of mixing solution word problems that can be modeled.

So, we have:

Q. Various real world questions, such as “How many grams of a 70% red solution should be added to 200 grams of a 40% red solution in order to obtain a solution that is 52% red?”

R. that require the use of mathematics to resolve, and

V. the solution (no pun intended!) is verifiable by actually making the mixture. Once done, the red and white beads can be separated and weighed to see how well the students performed.

Collaborative School Activities for the Teaching and Learning of Mathematics