Lesson Plan

The Squeeze Theorem!

How do we define f(x) = b^x for all real numbers x?
Dena L.
Classroom teacher
Diamond Bar High School
Diamond Bar, United States
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My Grades 9, 10, 11, 12
My Subjects Math

Students will be able to use rational values of x to place f(x)=b^x for positive real b and an irrational value of x within a known open interval. They will also be able to decrease the size of the interval, keeping the function value inside of it, essentially making the interval as small as they wish to make it.

The students will explain how this method works as a way to define the exponential function for all real numbers.

Grades 9 – 10
All Notes
Teacher Notes
Student Notes

1 What we do not know.

Activity: Conversing

Students should be arranged in small groups.

Use this discussion to increase the awareness of the importance of asking WHY in mathematics.

Use a method of randomly choosing students to decide who will share their answers with the class.

Student Instructions

In your small group discuss these questions.  I will call on one of you to share your group's answers to the class.

  1. What is the domain of f(x)=3^x?
  2. What does 3^5.2 mean?
  3. What does 3^(root 2) mean?
  4. What is the difference between 5.2 and root 2?

What is a question we should ask about f(x)?


2 Exploration: using what we do know to move forward

Activity: Exploring

Have your students access the document linked in their instructions for explore the relationship between the function defined for all rational numbers to the function defined for all real numbers.  It is important to ask them why it is essential that we know that the graph of the exponential function is either always increasing or always decreasing.  Why is it important that we know it is continuous?

Student Instructions

Use the following document to explore how we may use what we know about exponential expressions with rational exponents to find, or at least approximate, the value of these expressions with irrational exponents.  CLICK HERE


3 Guided practice

Here you may have your students use my Desmos graph that illustrates the squerze theorem or you may have them design their own.

Student Instructions

Use the following link to see an interactive, graphical representation of the squeeze theorem.  CLICK HERE

4 Independent Practice and Closure

Activity: Other — Writing

This activity should begin with a short small-group discussion.  The students should compose their responses independently.

Student Instructions

Answer question 1 and then choose one of the other questions/statements to address.  Each of your responses should be one paragraph of 4 to 6 sentences.

  1. What is another mathematics concept that we have studied for which you are unsure of why the concept works and why it makes sense?
  2. Explain how the squeeze theorem helps us find the value of 3^(root 2).
  3. Why is it important to know that the exponential function is always increasing or always decreasing?
  4. We have talked about the idea of a limit, which you will learn more about in calculus.  How do you think the idea of a limit applies to the squeeze theorem?
  5. Describe a real-life application of the squeeze theorem.