Intent

In this activity, similar to their work in Shoelaces, students put arithmetic processes into words.

Mathematics

Students work through and describe the arithmetic used to solve several problems, setting the stage for writing algebraic expressions to describe situations.

Progression

After working on their own, students focus on the verbal descriptions of their computations. Class discussion is oriented once again, as in Shoelaces, to translating verbal descriptions into algebraic expressions. This activity leads to more practice with substitution in To Kearny by Equation and sets the stage for students to focus on the meaning of algebraic expressions in Ox Expressions.

Approximate Time

10 minutes for introduction

20 minutes for activity (at home or in class)

15 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Doing the Activity

Introduce the activity, reminding students to use the new data that are given, rather than the information in Shoelaces.

Discussing and Debriefing the Activity

You might begin by having students share the questions they made up for Question 2 and try to answer each other’s questions. Meanwhile, ask two or three students to put their verbal descriptions from Question 1 on transparencies.

The main focus of the discussion should be on turning students’ verbal descriptions into algebraic expressions. For example, suppose one student’s description for Question 1 begins, “You take the 150 wagon trains times the 25 families per train times the length of shoelace per family. The amount per family is twice the amount per man plus the amount per woman plus three times the amount per child.” Ask students to rewrite this sentence without reference to specific numbers. For example, the first sentence would come out something like, “Multiply the number of wagon trains by the number of families per train by the length of shoelace per family.”

To write this as an algebraic expression, students need to choose a letter to represent each quantity, such as T for the number of wagon trains, F for the number of families per wagon train, and S for the total amount of shoelace needed per family. The total amount of shoelace needed for an entire year would then be represented by an expression like TFS.

Emphasize the difference between an object and the number of such objects and the importance of being precise about what a letter represents. For instance, in the example above, F does not represent a family, or the number of people in a family, but the number of families in a wagon train.

As time allows, build on the discussion by having students share some of the questions they wrote for Question 2 and their verbal descriptions from Question 3, and having the class develop algebraic expressions to go with these questions.

Key Questions

Describe in words how you found the answer.

How can you rewrite this sentence without reference to specific numbers?

How can you use variables to rewrite this sentence as an algebraic expression?

Supplemental Activity

From Numbers to Symbols and Back Again (extension) uses formulas from two settings from earlier units (The Game of Pig and Patterns) as the context for work with substitution. Students will have to guess and test solutions to Questions 1c and 2b, as the equations are quadratic. Furthermore, the solution to Question 1c is not integral (or even rational), so students will have to determine a reasonable approximation.