Interpreting Exponential Functions

5 min

Narrative

This Warm-up addresses a common confusion among students mistaking the expression 2x2^x for the expression x2x^2. One good way to address this confusion is by evaluating the expressions for different values of xx, which is the path students are likely to take here. Another way to illustrate the distinction is by graphing, though students are less likely to take this approach.

Launch

Display the task for all to see, and ask students to read it quietly to themselves. Ask what they think Lin means by equivalent. They may remember from earlier grades, or they may guess based on contrasting her statement with Diego’s statement. Before they start working, be sure that students understand equivalent to mean equal no matter what value is used in place of xx.

Student Task

Lin and Diego are discussing two expressions: x2x^2 and 2x2^x.

  • Lin says, “I think the two expressions are equivalent.”
  • Diego says, “I think the two expressions are equal only for some values of xx.”

Do you agree with either of them? Explain or show your reasoning.

Sample Response

Agree with Diego. Sample explanation: If we find the value of x2x^2 and 2x2^x for different values of xx, say 0, 1, 2, 3, and 4, we can see that they are not always equal. As the table shows, they are equal only sometimes, for example, when xx is 2 and when xx is 44.

xx x2x^2 2x2^x
1 1 2
2 4 4
3 9 8
4 16 16
5 25 32
10 100 1,024
Activity Synthesis (Teacher Notes)

Consider using a blank table or a completed one, as shown here, to facilitate discussion.

xx x2x^2 2x2^x
1
2
3
4
5
10
xx x2x^2 2x2^x
1 1 2
2 4 4
3 9 8
4 16 16
5 25 32
10 100 1024

Possible questions for discussion:

  • “For what values of xx are the two expressions equal?” (when xx is 2 and when xx is 4)
  • “Besides substituting different values of xx, are there other ways to tell if the two expressions are equal for all values of xx or for only some values of xx?” (One way might be to think about possible values the expressions could take when xx is an odd number. 2x2^x means 2 multiplied by itself xx times. When xx is an odd number, the value of 2x2^x will be even because of the multiplication by 2. But multiplying an odd-number xx by itself as in x2x^2will result in an odd number, so we know that in those instances the expressions are not equal. Another way might be to graph the equations y=x2y=x^2 and y=2xy=2^x on the same coordinate plane and examine the graphs.)
  • “Which expression grows more quickly?” (For positive values of xx, 2x2^x grows much more quickly than x2x^2 does.)
Standards
Addressing
  • A-SSE.A·Interpret the structure of expressions
  • HSA-SSE.A·Interpret the structure of expressions.

15 min

15 min