Lesson 1

Reviewing Exponents

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

1.1: Reviewing Exponents (5 minutes)

Warm-up

The purpose of this activity is for students to recall how to interpret expressions that use exponents. This will be useful when students have the opportunity to use exponents to represent a situation in the associated Algebra 1 lesson.

Launch

Before beginning the activity, pose the following question.

“Kiran thinks that \(3^2\) is 9. Han thinks that \(3^2\) is 6. Who do you agree with and why?”

Before students do calculations or any other work, ask the class whether they agree with Kiran or Han. Ask selected students to explain why they chose one and did not choose the other. Ask others if they agree or have something to add.

Display the table for all to see. Explain that the “expanded” column shows the factors being multiplied, the “exponent” column shows how to write the repeated multiplication more succinctly with exponents. Arrange students in groups of 2. After a few minutes of quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they look different. Follow with a whole-class discussion.

Student Facing

Complete the table.

expanded form exponential form
\(2\boldcdot2\boldcdot2\) \(2^3\)
\(3\boldcdot3\boldcdot3\boldcdot3\)  
  \(5^2\)
\(x\boldcdot x\boldcdot x\boldcdot x\boldcdot x\boldcdot x\boldcdot x\)  
  \(y^3\)
  \(\left(x\boldcdot y\right)^2\)

Student Response

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

Activity Synthesis

The goal of this activity is for students to evaluate how well they understand exponential notation. Select students to share their responses. Ask if other groups have a different solution.

1.2: Saving Money (20 minutes)

Activity

The purpose of this activity is for students to recall how to write expressions using exponents. Students apply what they know about exponents to represent money doubling each week. They use multiplication to express a different quantity growing linearly each week. This will be useful when students represent two situations and compare them in the associated Algebra 1 lesson. When students write an expression in terms of \(x\), they express regularity in repeated reasoning (MP8).

Launch

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

Student Facing

Clare has a summer job. She wants to save money to spend on the family vacation at the end of summer. She is going to save \$5 per week for each of the 10 weeks she is working.

Tyler also has a summer job and he, too, would like to save money to spend on the family vacation. He is going to start by saving \$2 the first and second weeks and double the amount he saves each of the other weeks he is working (\$4 the third week, \$8 the fourth week, and so on).

Complete the table showing how much money each of them will have at the end of each week for the 10 weeks.

week 1 2 3 4    5    6    7    8    9    10   \(x \)
Clare 5 10 15                
Tyler 2 4 8                

Student Response

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

Activity Synthesis

Focus the discussion on how students created their expressions for finding the amount of money that Clare and Tyler had at the end of each week. After students share their responses, ask if Clare and Tyler had a reasonable plan for saving money. Ask students to explain their reasoning. Look for things like how it is reasonable for Clare to save \$5 a week from a part time or full time summer job. Assuming a wage of \$10 per hour, Clare is saving money from less than one hour per week. Tyler’s plan to double the amount each week is reasonable for the first five or six weeks. After that, each week gets more unlikely. For week 9, he would have to save approximately 25 hours of wages at \$10 per hour, and for week 10 he would have to save approximately 50 hours of wages at \$10 per hour, which is probably unrealistic for a part-time summer job.

1.3: Identifying Equivalent Expressions (15 minutes)

Activity

In this partner activity, students take turns using exponent rules to analyze expressions and identify equivalent ones. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). The practice will prepare students to compare growth patterns and interpret graphs to answer questions from a context in the associated Algebra 1 lesson. A matching task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

Launch

If desired, demonstrate that expressions can be rewritten by substituting expressions of equal value. For example, if you were looking for an expression equivalent to \(16 \boldcdot 16\), you might try replacing each 16 with the product of two 4’s, like \(4 \boldcdot 4 \boldcdot 4 \boldcdot 4\). This might help you recognize a matching expression like \(4 \boldcdot 4^3\).

Arrange students in groups of 2. Tell students that for each expression in List A, one partner chooses an expression from List A and finds an equivalent expression in List B and List C and explains why they think it is equivalent. The partner’s job is to listen and make sure they agree. If they don’t agree, the partners discuss until they come to an agreement. For the next expression in List A, the students swap roles. If necessary, demonstrate this protocol before students start working.

To encourage reasoning about equivalent expressions, it would be best if students tackled this activity without access to a calculator.

Student Facing

Choose an expression from List A and match it with an equivalent expression from List B and from List C.

  • For each match that you find, explain to your partner how you know it’s a match.
  • For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
  • Switch roles so that your partner chooses a different expression from List A and matches it with an equivalent expression from List B and from List C.

List A

\(8\boldcdot8\boldcdot8\)

\(9\boldcdot27\boldcdot3\)

\(10\boldcdot100\)

\(\frac{1}{4}\boldcdot\frac{1}{4}\boldcdot\frac{1}{2}\)

\(3+3+3+3+3+3\)

List B

\(10^3\)

\(6 \boldcdot 3\)

\(\left(\frac{1}{2}\right)^5\)

\(3^2\boldcdot3^4\)

\(2^3\boldcdot4^3\)

List C

18

\(\frac{1}{32}\)

512

729

1,000

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 how they determined which expressions were matches. Select groups to share their matches and how they sorted their expressions. Ask if other groups have different matches of expressions. Choose as many different groups as time allows. Attend to the language that students use to describe their pairing, giving them opportunities to describe their relationships more precisely. Highlight the use of terms like expanded notation, exponents, factors, multiplication, and addition. The purpose of this discussion is to highlight the meaning of an exponent. Here are sample questions for discussion:

  • “What’s a simpler way to write \(5 \boldcdot 5 \boldcdot 5 \boldcdot 5 \boldcdot 5 \boldcdot 5 \boldcdot 5\) ? What’s a simpler way to write \(5+5+5+5+5+5+5\)?”
  • “Explain how \(3^2 \boldcdot 3^4\) is equivalent to \(3^6\).” (If you expand each exponent, it creates the equivalent expression \(3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3\). The expanded expressions shows that there are actually 6 factors being multiplied, so we can rewrite this as \(3^6\).)
  • For what value of \(x\) will \(x^3\) equal 125?
  • For what value of \(y\) will \(y^3\) equal 64?
  • “What were some ways you handled . . . ?”
  • “Describe any difficulties you experienced and how you resolved them.”