To begin this lesson, display the
Working With Mixtures overhead on the projector. Get students thinking about mixtures and percent concentrations by answering the questions on their own and then discussing with a partner. Review the questions as a class, and cover the following key points:
- The percent concentration of a final mixture is somewhere between the percent concentrations of the two original mixtures. For instance, if some amount of a 40% mixture is combined with some amount of a 60% mixture, the result will have a percent concentration greater than 40% and less than 60%.
- The percent concentration of the result will be closer to the mixture from which a greater quantity was taken. For instance, if a small amount of a 40% mixture is combined with a large amount of a 60% mixture, the result will have a percent concentration closer to 60%, and vice versa.
Distribute the Working With Mixtures activity sheet. In Questions 1‑7, students will predict and test the results of combining two mixtures. Each group will need three empty containers, a measuring cup, a container of red beads, and a container of white beads.
Now, students will work in groups to create two source mixtures from red and white beads, and in turn, put together a final mixture of beads from the source mixtures (Questions 1‑3). To facilitate the process:
- Demonstrate a simple mixture with 1 scoop of each color. Point out that students should assume there are an even number of beads in each scoop.
- Explain that the percent concentration of red beads is the percent of red beads in the mixture. Have students determine the percent concentration of the mixture you just created (50%).
- Explain that after they work with the mixtures, they will be using their new understanding to develop a formula for finding percent concentrations from two source mixtures (you may want to have students pause after Question 7 and discuss their findings).
Students are now ready to create their own mixtures by following the directions on the activity sheet. Observe students as they work. Are their calculations for final mixtures similar to the sampling results? If not, are the beads well-mixed in each container? Are they correctly calculating percents? Are they consistently measuring scoops of beads? Is their math strategy for calculating the final percent concentration correct?
Finally, have students complete and discuss the summary questions at the end of the activity. (These questions appear on the Working With Mixtures overhead.) They should recognize that the final mix will always be weaker than the more‑concentrated mix and stronger than the less‑concentrated mix. If students still struggle with the concept, the diagrams on the last page of the overhead sheet might help to illustrate the concept.
The image above shows that if a "bigger slice" is taken from the mixture with a higher percent concentration, the result will be "pulled" toward the greater percent. Similar images are used on the last page of the overhead.
Mixture problems can be difficult for students, and it helps for them to see a variety of types. Working with mixtures without attempting to solve them algebraically can be valuable in developing conceptual understanding. The Mixtures Applet allows students to informally solve three different types of mixture problems:
- Unknown Percent: A known percent of Pile A is colored. Students must determine how many circles in B to color so the combined mixture has a specific percent concentration.
- Unknown Total: A portion of A is colored. Students must determine how many circles to add to B as well as how many to color so that a) Pile B has a specific percent concentration and b) the combined mixture has a specific percent concentration.
- Unknown Pile: Similar to Unknown Total, students must add and color circles in Pile B, but it is not required that B has a specific percent concentration.
You may wish to allow students some time to explore this applet after the lesson has been completed. In addition to giving them practice with the type of problem exhibited on the activity sheet, it will also give them exposure to the other types of mixture problems.
