Equivalent Exponential Expressions

10 min

Teacher Prep
Setup
2 minutes of quiet work time, followed by a whole-class discussion.

Narrative

In this Warm-up, students take two numbers to different powers and look for patterns. The first number is a whole number, 3, and the second is its reciprocal, 13\frac{1}{3}. The goal is for students to notice that when a fraction is raised to a positive exponent, its value decreases as the exponent increases. Aside from the presence of exponents, these observations are largely a review of work from grade 5.

As students complete the table, monitor for those who can describe some of the following patterns:

  • The values in the 3x3^{x} column increase as the exponent increases.
  • The values in the (13)x(\frac13)^x column decrease as the exponent increases.
  • The values in the (13)x(\frac13)^x column are reciprocals of the values in the corresponding row of the 3x3^x column.

Launch

Give students 2 minutes of quiet work time, followed by a whole-class discussion.

Student Task

Find the values of 3x3^x and (13)x\left(\frac13\right)^x for each value of xx. What patterns do you notice?

xx 3x3^x (13)x\left(\frac13\right)^x
1
2
3
4

Sample Response

xx 3x3^x (13)x\left(\frac13\right)^x
1 3 13\frac13
2 9 19\frac19
3 27 127\frac{1}{27}
4 81 181\frac{1}{81}

Sample responses:

  • The values in the 3x3^x column increase as the exponent increases.
  • The values in the 3x3^x column are multiplied by 3 each time you go down a row.
  • The values in the (13)x\left(\frac13\right)^x column decrease as the exponent increases.
  • The values in the (13)x\left(\frac13\right)^x column are multiplied by 13\frac13 each time you go down a row.
  • The values in the (13)x\left(\frac13\right)^x column are reciprocals of the values in the corresponding row of the 3 column.
Activity Synthesis (Teacher Notes)
Display the table for all to see. Ask students to share their responses and record them in the table. Ask selected students to share the patterns they noticed in the table and ask others to explain why they think these patterns happen. If the ideas described in the Student Response do not arise from students during this discussion, bring those ideas to students’ attention.
Standards
Addressing
  • 6.EE.1·Write and evaluate numerical expressions involving whole-number exponents.
  • 6.EE.2.c·Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). <em>For example, use the formulas V = s³ and A = 6 s² to find the volume and surface area of a cube with sides of length s = 1/2.</em>
  • 6.EE.A.1·Write and evaluate numerical expressions involving whole-number exponents.
  • 6.EE.A.2.c·Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). <span>For example, use the formulas <span class="math">\(V = s^3\)</span> and <span class="math">\(A = 6 s^2\)</span> to find the volume and surface area of a cube with sides of length <span class="math">\(s = 1/2\)</span>.</span>

10 min

15 min