## Exploring Equations

- Lesson

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!

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.)

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

- What do you notice about the intersection of
*y*1 =*ax*+*b*and*y*2 =*bx*+*a*? - What does this tell you about the equation
*ax*+*b*=*bx*+*a*? - What can you say about the equation
*ax*+*b*=*bx*-*a*? - 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
*x*^{2}+*y*^{2}=*r*^{2}(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

- 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.