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Exploring Equations

  • Lesson
Gary Martin
Auburn University

 In this lesson, students use their knowledge of weights and balance, symbolic expressions, and representations of functions to link all three concepts.

Note: This activity can be completed by students in pairs or individually in front of computers.

Try this Task! 

appicon Activity: Balance Scale with Expressions

When is ax + b = bx + a? (Use + for addition, - for subtraction, * for multiplication, / for division, and ^ to raise an expression to a power.)

  1. In the red (y1) window, set y1 = ax + b, and in the blue (y2) window, set y2 = bx + a. Choose constants for a and b.
  2. Examine the graph of these functions. Change the values of the parameters a and b multiple times. What do you notice?
  3. What conclusions can you make about the statement ax + b = bx + a?

Think about this Situation 

Use the tool above to investigate this equation graphically and numerically for different values of a and b.

  1. What do you notice about the intersection of y1 = ax + b and y2 = bx + a?
  2. What does this tell you about the equation ax + b = bx + a?
  3. What can you say about the equation ax + b = bx - a?
  4. What other similar questions could be explored using three parameters a, b, and c?


This investigation has focused on equations as statements of numeric equality or inequality between two objects. The progression of tasks moves from a concrete notion to thinking of an equation as stating a relationship between two symbolic expressions and how this relationship can be investigated using graphical or numerical representations. Of course, this investigation illuminates only a portion of the role of equivalence in mathematics.

The equals sign has many other uses and interpretations. Each use provides an alternative viewpoint on the concept of equality, a different way in which mathematical objects can be equivalent. For example, the equals sign is used:

  • defining functions, such as f(x) = x + 1.
  • in assignments such as I = I + 1 in computer programming.
  • in creating structures such as x2 + y2 = r2 (circle).
  • in creating equivalence classes such as 5 = 17 (mod 12).


If you find this investigation interesting, you might also enjoy other explorations which examine the equation ax + b = (a + c)x + (b + c).

Questions for Students

Refer to the Instructional Plan.

Learning Objectives

Students will:
  • Investigate equivalence and systems of equation
  • Identify and use functions

NCTM Standards and Expectations

  • Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations.
  • Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP7
    Look for and make use of structure.