Intent

Students work with a given set of variables to generate meaningful algebraic expressions. The approach emphasizes the context in which the expressions emerge, giving students a meaningful connection to the symbols, equivalence, and symbol manipulation.

Mathematics

Students have created algebraic expressions to describe calculation procedures. Now they will move in the other direction, interpreting algebraic expressions in terms of the concepts that the variables represent. Students explain and evaluate algebraic expressions, generate meaningful algebraic expressions, and interpret algebraic expressions using summary phrases. The equivalent meaning of the expressions developed with a given summary phrase will help students to recognize that the expressions are equivalent, offering another opportunity to identify the distributive property at work.

Progression

Students work in groups to create meaningful algebraic expressions and describe them with summary phrases. The activity concludes with sharing findings and discussing the experience of writing algebraic expressions and summary phrases.

Approximate Time

30 minutes

Classroom Organization

Groups, followed by whole-class discussion

Doing the Activity

This activity asks students to write as many different meaningful algebraic expressions as possible for a given set of variables. The key idea is to give contextual meaning to algebraic expressions, rather than seeing them simply as descriptions of a set of arithmetic operations.

Have students read the activity. Discuss the idea of a summary phrase for describing an algebraic expression, emphasizing that summary phrases should be as concise as possible. For instance, using the example given in the activity, the summary phrase for the expression FC is “the number of children traveling in a train” rather than “the number of families traveling in a train times the number of children in a family.”

Also emphasize, as mentioned in the activity, that not every algebraic expression has meaning in the given context. You might illustrate with some additional examples or ask students to create one or two of their own.

Discussing and Debriefing the Activity

If the students in a group seem overwhelmed after working a bit, ask them to begin by considering just three symbols: M (the number of men in a family), W (the number of women in a family), and C (the number of children in a family). Then ask what the expression M + W + C represents in terms of the problem (the total number of people in a family). Similarly, ask about the expression M + W (the number of adults).

You can then introduce other variables, such as F, B, and D. Ask, What questions can you express using these variables?The group can brainstorm to come up with questions that use those variables, such as “How much water do the women in the wagon train drink altogether on the trip?”

It is unlikely that the whole class will need such support to move forward. If several groups are stuck, though, another strategy is to call up a representative from each group to create an algebraic expression for the summary phrase, “The amount of water the women in the wagon train will drink altogether on the trip.”

You might notice students using the numeric values in their work. Some students are quite productive this way, using the numbers to confirm that their expressions make sense. Others might be tricked into thinking that simply because the numbers “work” the expressions make sense.

Bring students together to create a class list of meaningful “ox expressions.” You might ask each group to give one of its expressions together with a summary phrase. Other groups can then challenge the meaningfulness of the phrase or propose an improvement.

One error students often make is to neglect to say “number of” in defining variables. For example, they might say “F is the families in a wagon train” rather than “F is the number offamilies in a wagon train.” You may want to pay attention to this.

After generating a long list of expressions, you may want to discuss these questions.

Which letters appear most frequently in the expressions? Why is that?

What determines whether an expression has meaning or not?

Do you think you found all the meaningful expressions? Why or why not?

Look for opportunities to point out the use of the distributive property, possibly posting any examples.

Key Questions

What is a summary phrase for the expression FC ?

What does M + W + C represent in terms of the problem?

What questions can you express using these variables?

What is one of your expressions, and what is the summary phrase that goes with it?

Which letters appear most frequently in the expressions?

What makes an expression meaningful?

Do you think you found all the meaningful expressions?