Intent

In this activity, students reexamine the questions raised in Following Families on the Trail and apply their new understanding of equations and equation-solving techniques to those questions. This activity concludes the part of the unit devoted to solving equations algebraically and emphasizing the connection to other types of solution methods. It will help you evaluate students’ ability to represent situations using linear equations as well as their facility with solving them.

Mathematics

Students create equations that were originally derived in Following Families on the Trail and solve them using the algebraic methods they have been developing. Each situation had previously been modeled by two linear functions and solved by determining the coordinates of the point of intersection of their graphs.

Progression

Students work on the activity in small groups. Subsequent class discussion reemphasizes important ideas from the past several activities and helps students to confirm their understanding of methods for solving simple linear equations.

Approximate Time

25 minutes

Classroom Organization

Groups, followed by whole-class discussion

Materials

Students’ work from Following Families on the Trail

Doing the Activity

Tell students that in this activity they will return once more to a previous activity and solve a question already asked, this time using the algebraic techniques they have recently studied.

Ask students to work in their groups to complete this activity. Give some groups transparencies to prepare to present either Question 1 or 2.

If groups have difficulty setting up an equation, help them focus on two expressions that must be set equal. What two things are you trying to set equal? They could write this in words; for example,

Buck’s distance from Green River = Wood’s Distance from Green River

If necessary, you might ask a follow-up question like, What arithmetic do you have to do to figure the Buck family’s distance from Green River after 5 days? Then explain that if students generalize that procedure for x days, they will have an expression for the left side of their equation.

Discussing and Debriefing the Activity

Although Question 1 may seem straightforward, a presentation will provide a good opportunity to summarize ideas about both creating and solving equations. After the presentation, ask, In what ways does your solution to this algebraic equation relate to the graphs you built in Following Families on the Trail?

Encourage students to turn back in their notes to these graphs, either individually or in their groups, and look for connections. After a couple of minutes, ask for a volunteer to address your question.

On the surface, Question 2 appears quite similar, but the fact that the two families consume coffee at the same rate may create confusion. Students are likely to arrive at the equation 100 – 5x = 70 – 5x, but become perplexed when they try to solve it.

This is a good time to look back at the graphical representation of the situation. Students likely saw that the graphs for the two families were parallel lines, meaning the families would never have the same amount of coffee. Similarly, there is no value for x that gives 100 – 5x and 70 – 5x the same value.

Help students see that using the usual algebraic methods yields 100 = 70, and explain that this essentially means there is no solution to the original equation.

Key Questions

What two things are you trying to set equal?

What arithmetic do you have to do to figure the Buck family’s distance from Green River after 5 days?

In what ways does your solution to this algebraic equation relate to the graphs you built in Following Families on the Trail ?