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1.1: Variable Expressions

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Learning Objectives

At the end of this lesson, students will be able to:

  • Evaluate algebraic expressions.
  • Evaluate algebraic expressions with exponents.

Vocabulary

Terms introduced in this lesson:

algebra
generalize
variables
equations
substitution (evaluation, “plugging-in”)
values
solution

Teaching Strategies and Tips

Although the properties of real numbers are not covered until the chapter Real Numbers, some basic working assumptions must be made in this chapter. Addition and multiplication involving negative numbers, for example, will probably not be new to the majority of your students, but it is necessary nonetheless for the completion of the Review problems. A comprehensive assessment prior to the end of the first week of classes is suggested so that you can gauge how much time you will need to cover Equations and Functions and Real Numbers.

Unknowns are used as placeholders in equations like 4x + 3 = x - 2 and formulas like A = 2\pi r. Using unknowns offers great advantages. It prevents having to solve problems “from scratch.” They allow the general formulation of arithmetical laws such as a+b = b+a and a \times b = b \times a for all real numbers a and b. Variables are used as shorthand in functional relationships such as, “Let x represent the number of cricket chirps heard in a 15 \;\mathrm{minute} interval. Then the temperature outside is given by the function T(x) = mx + b, where T represents the outside temperature.”

The difference between equations and expressions is important. Basically, one has an equals sign and the other one does not; equations can be solved and expressions are evaluated. It will help the students to see several examples side-by-side:

Equations: -x + 2 = 7, \ x + 2y = x - 1, \ x = 7

Expressions: -x+2, \ 2x + 2y, \ \pi r^2

Coefficients represent repeated addition and exponents represent repeated multiplication.

The process of evaluation is also called substitution. It may be explained to the students that since the variable is standing in place of all values (or some unknown value), substitution is an occasion to replace the letter with a value. Concerning notation: It is important for students to understand that parentheses are at their disposal at all times because they make calculations more explicit.

Several examples can be presented, each involving a different aspect of the evaluation process. For example:

  • One instance of the variable: Evaluate -3x + 2 when x = -2
  • Two or more instances of the variable: Evaluate -3x + 7-x + 1 when x = -2
  • Two or more variables: Evaluate -3x + 7y + 1 when x = -2 and y = 4
  • Evaluation involving negative numbers and fractions: Evaluate \frac{-3}{4}x + 7 when x =-4

Examples from geometry and physics allow the students to see the use and application of substitution and even provide motivation in other directions. See lesson Examples 5 and 7.

It is helpful to include words and phrases associated with functions in your lesson. Establishing familiarity with words and phrases like, “input and output,” “…in for x,” “out of the expression comes... ,” “value,” “…the expression evaluated at x,” etc., will help provide students with a foundation for the upcoming lesson on functions and function notation.

Word or story problems are important to establish right from the beginning.

Error Troubleshooting

Teachers are advised not to allow notation to become an obstacle for their students from the beginning. Writing should be in-line and organized and maintain a logical flow. Students may misread their work if their writing is illegible. Z’\mathrm{s} can look like 2’\mathrm{s}, +’\mathrm{s} like lower-case t’\mathrm{s}, tiny negatives disappear into the paper, other stray marks become minus signs, etc.

Students often compress their answers on paper, especially when they are used to writing across instead of in a downward fashion. 3x-1=8=3x=9=x=3. The equals sign is being used in at least two ways: equality between quantities, and equivalence (or implication) between equations.

When evaluating algebraic expressions, students can easily lose track of negatives.

Evaluate 1-x, where x = -1.

Evaluate 1-x^2, where  x = -1.

Evaluate 1-\frac{x-3}{-3}, where x = -1.

The first example involves a variable with a negative “out in front” and the value being substituted is negative. The second involves the vulnerable combination of negatives and exponents. The third involves negatives and fractions. As a check, ask your students if they can tell the difference between

-1^2 \ \ \ \text{and} \ \ \ (-1)^2

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