Equivalent Equations

5 min

Narrative

The purpose of this Warm-up is to help students recall what it means for two expressions to be equivalent. The given expressions are in forms that are unfamiliar to students but are not difficult to evaluate for integer values of the variable. This is by design—to pique students' curiosity while keeping the mathematics accessible.

Launch

Arrange students in groups of 2. Assign one partner the first expression and the other partner the second expression.

Student Task

Your teacher will assign you one of these expressions:

 n2−92(4−3)or(n+3)⋅n−38−3⋅2\dfrac {n^2 - 9}{2(4-3)}\qquad \text{or} \qquad (n+3)\boldcdot \dfrac {n-3}{8-3\boldcdot2}

Evaluate your expression when nn is:

  1. 5
  2. -1
  3. With your partner, select 2 other values to try. Consider different kinds of numbers like fractions, decimals, large numbers, small numbers, negative numbers, and so on.

Sample Response

For both expressions, the values are:

  1. 8
  2. -4
  3. Sample response: For n=0.5n = 0.5, the expression is -4.375. For n=100n = 100, the expression is 4,995.5.
Activity Synthesis (Teacher Notes)

Ask a few students from each group for their results. Then, ask students what they wonder about the results. Students are likely curious if the values of the two expressions will be the same for other values of nn. If they noticed that all the given values of nn are odd numbers, they might wonder if even values of nn would give the same result. If time permits, consider allowing students to try evaluating the expressions using a value of their choice.

Discuss questions such as:

  • "Were you surprised that these expressions have the same result for different values of nn?" (No, because using distribution in the denominator results in the same values, so the other parts with the nn may do the same thing.)
  • "If (or when) you tried using other values of nn, what did you find?" (They were the same for all the different kinds of numbers we tried.)
  • "Do you think that the two expressions will have the same value no matter what value of nn is used? How do you know?" (Yes, because the distributive property shows that the numerators are equivalent and so are the denominators.)

Tell students that it would be impossible to check every value of nn to see if the expressions would give the same value. There are, however, ways to show that these expressions must have the same value for any value of nn. We call expressions that are equal no matter what value we use for the variable equivalent expressions.

Remind students that in middle school they had seen simpler equivalent expressions. For example, they know that 3(x+5)3(x + 5) is equivalent to 3x+153x + 15 by the distributive property (without trying different values of xx).

Explain to students that they'll learn more about how to identify or write equivalent expressions—and about equivalent equations—in this unit.

Anticipated Misconceptions

When evaluating their expression, some students may perform the operations in an incorrect order. For example, when finding the value of 8−3⋅28-3\boldcdot 2 in the second expression, they may find 8−38-3 and then multiply by 2. Ask them whether the subtraction or multiplication should be performed first. Remind them about the order of operations as needed.

Standards
Building On
  • 6.EE.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). <em>For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.</em>
  • 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). <span>For example, the expressions <span class="math">\(y + y + y\)</span> and <span class="math">\(3y\)</span> are equivalent because they name the same number regardless of which number <span class="math">\(y\)</span> stands for.</span>
Building Toward
  • A-REI.A·Understand solving equations as a process of reasoning and explain the reasoning
  • HSA-REI.A·Understand solving equations as a process of reasoning and explain the reasoning.

15 min

15 min