Lesson 18

Bases and Exponents

These materials, when encountered before Algebra 1, Unit 5, Lesson 18 support success in that lesson.

18.1: Math Talk: Different Bases (5 minutes)

Warm-up

This warm-up is a slightly simpler version of the warm-up in the associated Algebra 1 lesson. It prompts students to use properties of exponents to identify equal expressions. This skill will be critical as students investigate different ways to write expressions for compounded interest. When students notice that they can rewrite an expression by substituting an equal expression or using a property, they are noticing and making use of structure (MP7).

Launch

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

Student Facing

Decide if each expression is equal to \(9^{16}\).

\((9^{8})^{8}\)

\((9^4)^4\)

\((3^2)^{16}\)

\(3^{32}\)

Student Response

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

Activity Synthesis

Pay particular attention to the last expression, \(3^{32}\). In order to identify this as equal to \(9^{16}\) students need to work backward and write this as \((3^2)^8\), or work forward and rewrite \(3^{32}\) as \(3^{2 \boldcdot 8}\). The third expression is intended to facilitate this thinking. All of these problems rely on an important property of exponents, \((x^a)^b=x^{ab}\).

18.2: What’s the Factor? (20 minutes)

Activity

The purpose of this activity is to highlight how the factor changes when you consider different-sized intervals. The different-sized intervals are related to the compounding intervals that students see in the associated Algebra 1 lesson. By using an additional row of the given tables, students are encouraged to reason about the factors and anticipate additional entries by thinking in terms of exponential expressions and the properties of exponents.

Launch

Provide access to calculators. Invite students to work through the first problem and then pause, asking at least one student to share their response.

Ask students to think about how they know the factor from step 7 to 10 is 27. If no students mention it, point out that each number is 10 times a power of 3. \(10 \boldcdot 3^0, 10 \boldcdot 3^1, 10 \boldcdot 3^2, 10 \boldcdot 3^3\), and so on. To find the factor from one number to another, they likely divided. For example, for the factor from step 1 to 4, they might have divided 810 by 30. In other words, \(\frac{10 \boldcdot 3^4}{10 \boldcdot 3^1}\), which equals \(3^3\).

Arrange students in groups of 2. Ask students to work through the remaining questions, pausing to check in with their partner as they go.

Student Facing

  1. Refer to the first table.
    step 0 1 2 3 4 5 6
    value 10 30 90 270                       
    expression \(10\boldcdot3^0\) \(10\boldcdot3^1\) \(10\boldcdot3^2\)                                                
    1. Predict the value in steps 4, 5, and 6.
    2. By what factor does the value change . . .
      1. from step 1 to step 4?
      2. from step 3 to step 6?
      3. Conjecture about the factor from step 7 to step 10.
    3. By what factor does the value change . . .
      1. from step 0 to step 5?
      2. from step 1 to step 6?
      3. Conjecture about the factor from step 10 to step 15.
  2. Refer to the second table.
    step 0 1 2 3 4 5 6
    value 3 6 12 24      
    expression \(3\boldcdot2^0\)                                                                                
    1. Predict the value in steps 4, 5, and 6.
    2. By what factor does the value change . . .
      1. from step 1 to step 3?
      2. from step 3 to step 5?
      3. Conjecture about the factor from step 10 to step 12.
    3. By what factor does the value change . . .
      1. from step 0 to step 3?
      2. from step 2 to step 5?
      3. Conjecture about the factor from step 10 to step 13.
  3. Refer to the third table.
    step 0 1 2 3 4 5 6
    value 2,048 1,024 512        
    expression                                                                                               
    1. Predict the value in steps 4, 5, and 6.
    2. By what factor does the value change . . .
      1. from step 1 to step 3?
      2. from step 3 to step 5?
      3. Conjecture about the factor from step 10 to step 12.
    3. By what factor does the value change . . .
      1. from step 0 to step 3?
      2. from step 2 to step 5?
      3. Conjecture about the factor from step 10 to step 13.

Student Response

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

Activity Synthesis

Ask selected students to share their responses. If it doesn’t come up, demonstrate that each entry can be rewritten using an exponential expression that fits a certain pattern, and this insight can help us understand and describe the pattern. For example, each entry in the last table can be written \(2048 \boldcdot \left(\frac12 \right)^x\), where \(x\) is the step number.

Here are some possible questions for discussion:

  • “How do you calculate the factor from one entry to the next?” (Think about what I would multiply the first entry by to get the second entry. Or, divide the second entry by the first.)
  • “How does this factor change when looking at two steps? Three steps?” (The factor ends up being two or three powers of the same base.)
  • “Some of the questions in this activity ended up having the same answer. Why?” (When you divide powers of the same base, you end up subtracting the exponents. Since the exponents were the same distance apart, we ended up with the same answer.)

18.3: Rewriting Expressions (20 minutes)

Activity

The purpose of this activity is to practice using the property \((x^a)^b=x^{ab}\) to rewrite expressions.

Launch

Arrange students in groups of 2, or allow them to work individually. Provide access to calculators so that students can evaluate things like \(1.2^3\). Encourage them to use the calculator to check their reasoning, rather than just trying a bunch of numbers.

Student Facing

  1. For each given expression, decide what to write in the box to create equal expressions.
    given expression equal expression 1 equal expression 2
    \(5\boldcdot10^8\) \(5\boldcdot100^\boxed{\phantom{3}}\) \(5\boldcdot\boxed{\phantom{3}}^2\)
    \(7\boldcdot16^9\) \(7\boldcdot\boxed{\phantom{3}}^{4\boldcdot9}\) \(7\boldcdot4^\boxed{\phantom{3}}\)
    \((0.25)^3\) \((0.5)^\boxed{\phantom{3}}\) \(\boxed{\phantom{3}}^1\)
    \(3\boldcdot(1.2)^6\) \(3\boldcdot1.44^\boxed{\phantom{3}}\) \(3\boldcdot1.728^\boxed{\phantom{3}}\)
    \(6\boldcdot0.09^{10}\) \(6\boldcdot\boxed{\phantom{3}}^5\) \(6\boldcdot0.3^\boxed{\phantom{3}}\)

     

  2. Write at least 3 new expressions that are equal to \(4 \boldcdot 27^6\).

Student Response

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

Activity Synthesis

Display the correct answers and invite students to check their work. Encourage them to discuss any incorrect answers with a partner. Select one or more examples and ask a student to share how they reasoned their way toward a correct answer, providing a justification for each step. For example, these steps might be listed:

\(7 \boldcdot 16^9\)

\(7 \boldcdot (2^4)^9\) by replacing the 16 with an equal expression, \(2^4\) , and then

\(7 \boldcdot 2^{4 \boldcdot 9}\)  by using a property of exponents to rewrite \((2^4)^9\) as \(2^{4 \boldcdot 9}\).