Lesson 3

Properties of Exponents

These materials, when encountered before Algebra 1, Unit 5, Lesson 3 support success in that lesson.

3.1: Math Talk: Comparing Expressions (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for evaluating expressions with exponents. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to make sense of expressions with exponents. The expressions were chosen to highlight some nuances such as \(2^3 \ne 2 \boldcdot 3\), and \(5^2 \neq 2^5\). Additionally, students can recall that a number raised the first power equals itself. The last question is meant to highlight that when multiplication and division appear in the same expression, there are different ways to write the operations with the same result.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Student Facing

Compare each pair of expressions. Are they equal? If not, which is greater?

\(2^3\) and \(2 \boldcdot 3\)

\(5^2\) and \(2^5\)

\(100^1\) and \(1^{100}\)

 \(3 \boldcdot \frac{8}{2}\) and \(\frac{8 \boldcdot 3}{2}\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”

3.2: Reviewing the Properties of Exponents (25 minutes)

Launch

Arrange students in groups of 2 or allow them to work individually.

Student Facing

  1. Complete the table to explore patterns in multiplying powers of 3. 

    expression expanded single power of 3
    \(3^2 \boldcdot 3^4\) \((3\boldcdot3)\boldcdot (3\boldcdot3\boldcdot3\boldcdot3)\) \(3^6\)
    \(3^5\boldcdot3^2\)    
    \(3^3\boldcdot3^6\boldcdot3^2\)    
     \(3^{17}\boldcdot3^{41}\) (you can skip this box)  
  2. Use any patterns you found to write an expression equivalent to \(3^a \boldcdot 3^b\)

  3. Use your rule to write \(3^4 \boldcdot 3^0\) with a single exponent. What does this tell you about the value of \(3^0\)?

  4. Complete the table to explore patterns in dividing powers of 3. 

    expression expanded single power of 3
    \(3^6 \div 3^4\) \(\frac{333333}{3333}= \frac{3333}{3333} \boldcdot 3 \boldcdot 3 = 1 \boldcdot 3 \boldcdot 3\) \(3^2\)
    \(3^7 \div 3^2\)    
    \(3^5 \div 3^1\)    
    \(3^{100}\div3^{98}\) (you can skip this box)  
  5. Use any patterns you found to write an expression equivalent to \(3^a \div 3^b\)

  6. Use your rule to write \(3^7 \div 3^0\) with a single exponent. What does this tell you about the value of \(3^0\)?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to summarize what they learned or remembered from completing this activity. If desired, create a semi-permanent display that students can refer to throughout the unit, showing: \(x^a \boldcdot x^b = x^{a+b}\) \(\frac{x^a}{x^b} = x^{a-b}\) \(x^0 = 1\)

3.3: Use Your Powers! (15 minutes)

Activity

The purpose of this activity is for students to apply the exponent rules to evaluate expressions or write them with fewer terms.

Launch

Arrange students in groups of 2. Explain how the row game works, and demonstrate if needed.

Student Facing

Write each expression using a single exponent. One partner works only on Set A, the other partner works only on Set B. In each row, you should get the same answer. Pause after each problem to check if you got the same answer as your partner. If not, work together to check each other’s work and come to agreement. 

  Set A Set B
row 1 \(3^2\boldcdot3^7\) \(3^5\boldcdot3^4\)
row 2 \(3^0 \boldcdot3^{101}\) \(3^{99} \boldcdot 3^2\)
row 3 \(x^{20} \boldcdot x^{17}\) \(x^{37} \boldcdot x^0\)
row 4 \(\frac{3^{10}}{3^4}\) \(\frac{3^8}{3^2} \)
row 5 \(\frac{b^{19}}{b^{10}}\) \(\frac{b^{100}}{b^{91}}\)
row 6 \(3^0 \boldcdot 3^{10}\) \(\frac{3^{12}}{3^2}\)
row 7 \(\frac{a^{20}}{a^3}\) \(a^{17} \boldcdot a^0\)
row 8 \(\frac{7^{15}}{7^0 \boldcdot 7^{10}}\) \(\frac{7^9 \boldcdot 7^2}{7^6}\)
row 9 \(m^4 \boldcdot \frac{m^{12}}{m^9}\) \(\frac{m^{17}}{m^{10}} \boldcdot m^0\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Much discussion takes place between partners. Invite students to share the expressions they wrote.

  • “Describe any difficulties you experienced and how you resolved them.”
  • “Which ones were tricky? Explain why.”
  • “Did you need to make adjustments in your work? What might have caused an error? How did you resolve it?”