## Exploring Equations Further

• Lesson
9-12
1

Starting from the concrete notion of weights and balance and moving to symbolic expressions and representations of functions, this i-Math investigation has focused on some of the issues that arise along the way. In this part, this connection is extended to functions. More sophisticated tools allow for a greater diversity of investigations.

Students should use technological tools to represent and study the behavior of polynomial, exponential, rational, and periodic functions, among others. They will learn to combine functions, express them in equivalent forms, compose them, and find inverses where possible. As they do so, they will come to understand the concept of a class of functions and learn to recognize the characteristics of various classes.

Principles and Standards for School Mathematics
Algebra Standard 9-12, Chapter 7, p. 297

Starting from the concrete notion of weights and balance and moving to symbolic expressions and representations of functions, this i-Math investigation has focused on some of the issues that arise along the way.

The tools in Parts I-III illustrate one possible transition to more general-purpose tools. In Part III, the connection was made between the question of equality of symbolic expressions and the use of graphical or numerical representations. In this part, this connection is extended to functions. More sophisticated tools allow for a greater diversity of investigations. [See the E-example Exploring Linear relationships for a related investigation].

 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 Y= window, set y1 = ax + b and y2 = bx + a. Make sure to check the plot boxes at the left for both functions.
2. Examine the graph of these functions. Change the values of the parameters a and b using the sliders. 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 ab, and c?

Discussion

This i-Math 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 in 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) or in creating equivalence classes such as 5 = 17 (mod 12).

If you find this investigation interesting, you might also enjoy the E-example Exploring Linear Relationships which in effect explores the equation ax + b = (a + c)x + (b + c).

This lesson was developed by Gary Martin, Auburn University, and Brian Keller, Michigan State University.

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### Learning Objectives

Students will be able to:

• Investigate equivalence and systems of equation

### 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.