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Extending to Symbols

  • Lesson
6-8
1
Algebra
Gary Martin
Auburn University

In this investigation, students learn about the notion of equivalence in concrete and numerical settings. As students begin to use symbolic representations, they use variables as place holders or unknowns. This investigation illustrates the continued transition from the concrete balance view of equivalence to a more abstract view.  

In this investigation, students learn about the notion of equivalence in concrete and numerical settings. As students begin to use symbolic representations they use variables as place holders or unknowns. As their use of symbolic representation grows, they need to engage in reasoning about the symbolic representation. For example, in addition to experience with equivalent representations that arise from a contextual situation, students should explore questions which arise within the symbolic representation such as:

  • Under what conditions are two symbolic expressions equal?
  • When is an equation an identity?
  • What operations within the algebraic system preserve solutions or equality?

One method of investigating these questions is through the connection of symbolic representations to other representations such as tables or graphs. This investigation illustrates the continued transition from the concrete balance view of equivalence to a more abstract view.

Try This Task! 

appicon Pan Balance - Expressions 

Use the interactive tool above to explore the following questions. Use the following symbols:

+ for addition
- for subtraction
* for multiplication
/ for division
^ to raise an expression to a power

Students should perform the following steps and answer the questions which follow.

  1. Enter the expressions x + x and 2*x into the boxes on the pans.

    Change the value of x using the slider or by clicking and holding the mouse button down while dragging the cursor in the graph window.

    What do you notice as you change the value of x?

  2. What do you notice about the behavior of the balance when you use the two expressions 7 – x and x – 7? What does this behavior correspond to in the graph?
  3. What is an expression which is never equal to 7 – x? How do you know from the graph that these expressions are never equal?

Think About These Situations 

For each of the following sequences, find two expressions such that as x increases, the position of the balance moves according to the sequence.

For example, in (i) / - \ , find two expressions so that when your cursor is at the left edge of the graph the left side is down (/), then ,as you move your cursor to the right, there is a place where the sides are balanced (-), and then, as you continue to move your cursor to the right, the right side goes down (\).

(i). / – \     (ii). \ – / – \     (iii). \ / (Never balanced)

Use the pan balance to test expressions that you think might equal (balance) the following expressions. Use ^ to raise a number, variable, or expression to a power.

  • x * (x + 1)
  • (x + 1) / x 
  • x / (x + 1)
  • (x + 1) / (x + 1)

Discussion 

The three questions in the Try This Task! illustrate how the tasks investigated using numbers generalize to broader expressions within algebra. The first question raises issues of equivalent expressions. When written as an equation x + x = 2*x, the statement of this relationship is an identity. The second question illustrates that an equation might be valid for some but not all x values, and the third question illustrates that an equation might not be valid for any x values at all.

Typically, students initially encounter symbolic equations in the context where a variable is a place holder and the variable is 'solved for'. The expression balance highlights that an equation can be thought of as a relationship between two symbolic expressions. This method of thinking about equations helps students to make the transition to exploring the solutions of equations either graphically or numerically.

The tools provide connections from the concrete experience of balance to abstract investigations using symbols and multiple representations. These tools are only meant to aid in the transition to more sophisticated tools.

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Learning Objectives

Students will:

  • Investigate the equivalence of two symbolic expressions
  • Transition from numeric to algebraic notation

Common Core State Standards – Mathematics

Grade 6, Expression/Equation

  • CCSS.Math.Content.6.EE.A.3
    Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Grade 6, Expression/Equation

  • CCSS.Math.Content.6.EE.A.4
    Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.

Grade 6, Expression/Equation

  • CCSS.Math.Content.6.EE.B.6
    Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

Grade 6, Expression/Equation

  • CCSS.Math.Content.6.EE.B.7
    Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

Grade 7, Expression/Equation

  • CCSS.Math.Content.7.EE.A.2
    Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that ''increase by 5%'' is the same as ''multiply by 1.05.''

Grade 8, Expression/Equation

  • CCSS.Math.Content.8.EE.B.5
    Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP1
    Make sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP4
    Model with mathematics.
  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP7
    Look for and make use of structure.