Teaching CCSS.Math.Content.HSA-CED.A.3

Instructions for You

Objective: This activity's all about taking one big chunk of information and running through different ways of representing it. A situation that starts out in plain-old English is shifted into ye-old algebra, and then shifted again, into a brand-new graph.

What's more, students get to see how these representations are helpful in learning more about the situation. When trying to find solutions to a problem, finding points on a shaded region or running values through an inequality is a heck of a lot easier (and more effective) than just guessing.

Activity Length: This activity will take one class period
Activity Type: Individual
Materials Needed: Graph paper, rulers, and colored pencils for each student

Step 1: Your students will be working with the following scenario:

The science club has decided to sell Bobblehead Einstein figures and Minecraft posters, to raise money for their next round of after-school experiments. They can't spend more than $500 on figures and posters. The Einstein figures cost $3.50 each and the posters cost $5 each. Write an inequality that explains this situation, and define the variables.

The first step for the students is to pick out an inequality from this mass of words. It should look a little something like this:

3.50e + 5p ≤ 500

e stands for the number of Einstein figures they can buy and p stands for the number of Minecraft posters they can buy. Seems logical enough to us, but it's fine if they choose some different letters.

Step 2: Students will then rearrange this inequality into slope-intercept form. It doesn't really matter which one they choose as their independent variable, just as long as they rearrange things properly. Kinda like one of these:

e ≤ 142.9 − 1.4p
p ≤ 100 – 0.7e
Step 3: Right around now seems like a good time to have students graph their inequality, so that's just what we'll have them do. Regardless of which variable is independent, the graph will have a negative slope, a solid line, and be shaded everywhere less than the line. The scale on their grid should be appropriate for the numbers involved, and the axes should be labeled. Also, the graph should have a title, too; "Lord Defender of the Realm" is already taken (and not that applicable here, to boot).
Step 4: Ask the students to come up with ten different combinations of Bobblehead Einsteins and Minecraft posters that would satisfy the inequality. This shouldn't be too hard; they should all fall in the shaded region of the graph, while also not being something silly, like a negative number of Bobbleheaded Einsteins.

Step 5: Once everyone finishes, discuss the following points as a class:

1. What's a practical scale to use for the axes of our graph?
2. Why can't any of the points lie within the unshaded region of the graph?
3. What does the shaded region mean?
4. When putting the inequality into slope-intercept form, how does our choice of independent variable affect the graph and its interpretation?

Here are a few thoughts on how these questions might be answered:

1. For the scale on the graph, you should probably be thinking in terms of some multiple of ten (e.g. 30s, 50s). After all, we've got a lot of bobbleheads and posters to sell.
2. The unshaded region represents all the points where the inequality doesn't hold. For our inequality, that's all of the impossible-to-hit sales figures; the science club won't pay less than $500 for 5 million Einsteins to sell.
3. The shaded region is all the points that do satisfy the inequality. Simple as that. For our inequality, it's all of the combinations of bobbleheads and posters that cost less than 500 smackeroos.
4. The graphs will certainly look different; take a look at their respective y-intercepts if you want proof. The x- and y-axes will mean different things depending on whether we isolated e or p. What doesn't change, though, are our results. If 5 Einsteins and a poster is a solution to one graph, it will definitely be a solution for the other one, too.

Instructions for Your Students

This activity is all about taking some information about a situation in plain-old English, then translating it into mathier language, then translating it again, into a purely visual graph. Hopefully it'll work out better than when a good book is adapted into a not-so-great movie which is adapted into a bewildering piece of performance art. (Call us purists, but we don't think The Hobbit works as modern dance.)

Step 1: All right, here's the situation:

The science club has decided to sell Bobblehead Einstein figures and Minecraft posters, to raise money for their next round of after-school experiments. They can't spend more than $500 on figures and posters. The Einstein figures cost $3.50 each and the posters cost $5 each. Write an inequality that explains this situation, and define the variables.

Step 2: Rearrange this inequality in slope-intercept form. Of course, this isn't the usual slope-intercept form we normally use, since we've got an inequality this time around.

Step 3: Graph this inequality on your graph paper. Give it proper shading and a dotted or solid line—the works—using your colored pencils. The scale on your grid paper must be appropriate for the occassion, and the axes should be labeled. Don't forget a title, too; "Lord Defender of the Realm" is already taken (and not that applicable here, to boot).
Step 4: Come up with ten different combinations of Einstein Bobbleheads and Minecraft posters that the science club could buy that would satisfy the inequality. Then make like a pirate and mark these points on the graph. Yarr har harr!

Step 5: As a class, discuss the following points:

1. What is a practical domain and range for this inequality?
2. Why can't any of the points lie within the unshaded region of the graph?
3. What does the shaded region mean?
4. When putting the inequality into slope-intercept form, how does our choice of independent variable affect the graph and its interpretation?