Intent

Students compute the total length of shoelaces their families will need on their travels, which creates a context for bringing out methods of symbolic algebra—in particular, the use of an algebraic expression and the process of substitution.

Mathematics

Students extend their work with variables and algebraic expressions in Family Constraints to examine how to use variables to represent complex arithmetic algorithms. Students are introduced to the formal process of substitution into and evaluation of algebraic expressions, emphasizing the two steps in the process.

Progression

Students work in small groups on their own Overland Trail families, while listening to classmate’s methods for figuring the total length of shoelaces needed. Students share verbal and written statements about procedures for calculating shoelace length. The teacher leads a discussion to convert the generalization to an algebraic expression and test it by substitution.

Approximate Time

25 minutes

Classroom Organization

Individuals, in collaboration with group members, followed by whole-class discussion

Materials

Information about each group’s four families

Doing the Activity

Transition students from their work on Planning for the Long Journey by asking whether anyone thought about some of the more mundane supplies the travelers would need. In this activity, students figure the total length of shoelaces their families may need along the trail.

This is a good opportunity to assess how group interactions are progressing. Each group member should arrive at the same solution to Questions 1–3, but a different solution to Question 4. (FYI, a woman needs 224 inches of shoelace, a man 256 inches, and a child 144 inches.)

You might interrupt groups to encourage students to check in with each other and to listen to the reasoning behind each student’s solution to Question 4. Ask, What similarities do you see among your methods for Question 4? Conversation about their methods will likely help students answer Question 5 prior to the class discussion.

Encourage groups that finish early to figure the amount of shoelace needed for another person’s family, or the whole group’s families, or the entire class’s families. They might also write a general rule—in written form, as a set of steps, or algebraically—to calculate the amount of shoelace needed for a family of any size.

Discussing and Debriefing the Activity

Have two or three students present their work on Question 4. Since each student is discussing a different Overland Trail family, their numeric answers will differ.

To transition to Question 5, ask a presenter, Describe in words how you found your answer to Question 4.

Question 5 is really the heart of the activity, as it leads to the generalization into variables. You might have a couple of additional volunteers show their descriptions. It’s useful for students to see more than one description, even if the descriptions reflect the same arithmetic process. You will probably get descriptions like the following.

Multiply the number of women by 224, the number of men by 256, and the number of children by 144, and then add these products.

Ask students whether they see an opportunity to abbreviate these verbal descriptions by using variables. Does anyone have a suggestion for abbreviating these verbal descriptions by using variables? If necessary, remind students of how they used variables in Family Constraints. Insist that students give clear definitions for the letters they introduce, such as, “W represents the number of women in the family” instead of “W equals women.”

The class should be able to come up with an expression like 224W + 256M + 144C for the amount of shoelace needed for a family with W women, M men, and C children.

Post this expression and the equivalent verbal description, which will be a useful reminder for students, particularly during the activity Ox Expressions.

Remind students that such a combination of numbers and letters is called an algebraic expression. Review the term coefficient and remind students of the convention to omit the multiplication sign in expressions like 224W. Take a moment to ensure that students understand that these abbreviations are simply notational conventions; that is, agreements among mathematicians to write things a certain way. There is nothing inherently wrong about placing a multiplication sign between a number and a variable or about placing a variable in front of the coefficient—it is simply not the common way of communicating rules in mathematics. You might mention order of operations as another example of a mathematical convention.

Provide some examples to illustrate how a general algebraic expression is used to evaluate a particular case. For example, ask students to use the expression 224W + 256M + 144C to find the amount of shoelace needed for a family with four women, six men, and five children.

It is often helpful to identify two stages in the process.

The two stages are sometimes called substitution and evaluation, although either term can refer to the overall process.

Key Questions

What similarities do you see between each of the methods for answering Question 4?

How much shoelace did your Overland Trail family need?

Describe in words how you found your answer to Question 4.

Does anyone have a suggestion for abbreviating these verbal descriptions by using variables?

How can you use the expression 224W + 256M + 144C to find the amount of shoelace needed for a family with four women, six men, and five children?

Supplemental Activities

Pick Any Answer (reinforcement) provides a simple context for which students can create an algebraic expression. It might be an appropriate activity to use in connection with students’ work here with variables.

Substitute, Substitute (reinforcement) provides examples through which students can strengthen their understanding of the basic ideas of substitution. The activity uses a variety of phrases for referring to the process. One question uses substitution to reinforce the idea of combining terms.