 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum  MENU        A B C D E  Notes for Session 2, Part C Note 8

We've been using variables to describe patterns concisely, and some would claim that this is a move toward algebra. But what is algebra?

Groups: Discuss this question in small groups, then share answers with the whole group.

Reflect on this statement from mathematician Zal Usiskin that describes what algebra is: "Algebra is not easily defined." In the first session, we clearly used algebraic thinking to describe the pattern in the Eric the Sheep problem, but we didn't use an equation with variables. When we focus on the concept of variable, we can see that variables have many facets, as illustrated in the following examples:

 1 A = LW 2 40 = 5x 3 sin x = cos x * tan x 4 1 = n * (1/n) 5 y = kx

Each of these has a different feel. Can you explain the differences?

If you get stuck, here's a way to sort through these examples:

 1 Example 1 is a formula. 2 Example 2 is an equation or open sentence to solve. 3 Example 3 is an identity that is true for any value of x, other than when cos x = 0. 4 Example 4 is a property that is true for all n not equal to 0. 5 Example 5 is an equation representing a direct variation function, where it is implied that k is a constant and y and x are variables.

Each of these has a purpose in the study of algebra, and to quote again from Usiskin:

"Purposes for algebra are determined by, or are related to, different conceptions of algebra, which correlate with the different relative importance given to various uses of variables."

Also, consider this quote by Bob Davis: "Algebra is the way we talk about what numbers do when we don't know what the numbers are. "

Now go on to Usiskin's four conceptions of algebra. Read through the conceptions as described in the course materials.

Groups: Discuss the first three conceptions (saving the fourth for the last session) and the examples contained in the descriptions. Group members can help each other understand the different representations in the examples -- for instance, the representation of "even numbers" as "2a" and "2b." Note 9 Groups: Work in small groups on Problems C1-C3. Spend some time sharing responses. Some people may need help in using symbols to describe their work to explain what the symbols are showing.   Session 2: Index | Notes | Solutions | Video

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