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To Kearny by Equation

Module by: Interactive Mathematics Program

Intent

This activity creates an opportunity for further development of symbolic algebra.

Mathematics

Students begin with a formula for profit and are asked to explain why it makes sense. They then adjust the formula to represent half the total profit, a process that can lead to two equivalent expressions and create a context for bringing forth the idea of equivalence and the distributive property. Students conclude the activity with substitution and evaluation practice.

Progression

Students work individually and in their groups on the task, coming together as a class to review the ideas of equivalence and the distributive property.

Approximate Time

25 minutes

Classroom Organization

Individuals, then small groups, followed by a brief whole-class discussion

Materials

Information about each group’s four families

Doing the Activity

Locate Kearny, Nebraska, on the map so students can connect this activity to their journey westward from Westport. Also locate the Kansas River, in Topeka, Kansas, along the route to Kearny. If a student (or you) has ever visited Kansas or Nebraska, talk about their geography. Remind students that 150 years ago, there were no cars, highways, or bridges. Students might speculate about how supplies, wagons, and animals were transported across rivers.

Introduce the activity, letting students know that Joseph and Louis Papan were real individuals.

As groups work on Questions 3 and 4, watch for common sources of confusion that may warrant a class discussion. Disagreements about the numeric answers can arise from at least two sources:

  • Incorrect values substituted for the individual variables—for example, students may have forgotten to account for some families having more than one wagon
  • Arithmetic mistakes, possibly due to misuse of order-of-operations rules

Discussing and Debriefing the Activity

In the discussion, take advantage of the opportunity to review, once again, the distributive property.

For Question 2, students may have identified two ways to split the profit.

  • Split the $2 per wagon to get $1 for each brother, and, separately, split the 30¢ per hour pay so they each pay 15¢ per hour. This gives the expression W – 0.15H for each brother’s profit.
  • Split the overall profit between the brothers so each gets 2W−0.3H22W−0.3H2 size 12{ { {2W-0 "." 3H} over {2} } } {}.

If both methods arise (you may want to suggest the one that does not), use the opportunity to speak about the distributive property. The equivalent results illustrate that W – 0.15H and 2W−0.3H22W−0.3H2 size 12{ { {2W-0 "." 3H} over {2} } } {} must represent the same amount and so must be equivalent expressions.

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