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## Exploring Equations

• Lesson
9-12
1

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.

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?

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?

### Discussion

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).

Extension

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.