In this activity, students write equations, express them in equivalent form, and graph them. Two important concepts are reinforced in the process: abstracting a problem to an algebraic equation and reading numeric information from a graph. Students convert a linear equation into Y= form, conducive to use on the graphing calculator. The activity sets the stage for students to make meaning of graphical solutions to systems of equations.

Students examine possible solutions to a linear equation that involves two variables and explore how to express that equation by giving one variable in terms of the other. They also solve equations for one variable in terms of the other, based on reasoning in the problem context. Finally, they graph linear conditions on the calculator and use this tool to find solutions to an equation.

Students will work on the activity in groups and then share their results with the class. This activity sets the stage for lending graphical meaning to solving systems of equations. Students will revisit this situation in More Fair Share on Chores.

30 minutes

Groups, followed by whole-class discussion

Read the introduction as a class. If you have a map, you might point to the split between the California and Oregon Trails.

You may want to have a volunteer offer a sample solution for Question 1 as a way to get the class started. You may need to prompt students to consider values other than whole numbers.

Circulate as groups work. In Question 2, students are asked to express a problem condition by using an equation. Although this may seem straightforward, translating words into symbols can be a complex process. If an entire group is stumped, remind them of the strategy of identifying and writing in words the arithmetic used. In this case, it is likely that students are performing an arithmetic operation like “multiply the girls’ hours by 2 and the boys’ hours by 3, add the results, and see if the total equals 10.”

If groups are stuck on Question 3 or 4, focus them on the arithmetic. If each girl’s shift was four hours, how would you find the length of each boy’s shift? Help them to translate their verbal reply into symbolic form—an equation like B=10−2G3B=10−2G3 size 12{B= { {"10"-2G} over {3} } } {}.

Begin the discussion by asking students to compile their possible answers to Question 1 into an In-Out table. Because students will be expressing the boys’ shift length, B, in terms of the girls’ shift length, G, the girls’ shift length should go in the In column. Have students plot these points.

If students had trouble with Question 2, ask them to use a number pair from the table and express in words how it gives the desired total of ten hours. They will probably reply by saying something to this effect: “Multiply the length of a girl’s shift (such as 1212 size 12{ { {1} over {2} } } {}) by 2 and the length of a boy’s shift (such as 3) by 3 and add.” Essentially, they have given you the equation; they just need to restate it symbolically.

Help students to see that the purpose of Questions 3 and 4 is to express the equation in the form B= so they can graph it on the calculator. Ask, Why might you want to express one variable in terms of the other? If you know the length of a girl’s shift, how would you use it to find the length of a boy’s shift? Otherwise, they could stick with the equation 2G + 3B = 10, which is a more natural way to think about the problem.

Here are some observations that might arise in the discussion.

Post the equation from Question 4 so students can refer to it in the next activity.

If each girl’s shift was four hours, how would you find the length of each boy’s shift?

How can you check whether a number pair fits the condition?

Why might you want to express one variable in terms of the other?