# Lesson 1

Properties of Exponents

## 1.1: Which One Doesn’t Belong: Exponents and Equations (5 minutes)

### Warm-up

This warm-up prompts students to compare four equations related to exponents. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

### Launch

Arrange students in groups of 2–4. Display the equations for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning as to why a particular item does not belong, and together, find at least one reason each item doesn’t belong.

### Student Facing

A. \(2^3 = 9\)

B. \(9 = 3^2\)

C. \(2 \boldcdot 2 \boldcdot 2 \boldcdot 2 = 16\)

D. \(a \boldcdot 2^0 = a\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as base, repeated factor, and exponent. Also, press students on unsubstantiated claims.

## 1.2: Name That Power (10 minutes)

### Optional activity

This activity is optional because it revisits below grade-level content. In this activity, students decide which values of an exponent will make an equation true. The word *solve* is purposely avoided to encourage students to look for and make use of the structure of exponential expressions rather than try to perform algebraic manipulations (MP7).

### Launch

*Representation: Internalize Comprehension.*Activate or supply background knowledge. Provide students with access to a three-column chart. Remind students that exponents provide a special way of writing repeated multiplication. Invite students to write the expression on the left of the equation in the first column, the expanded form of the expression in the second column, and the expression on the right of the equation in the third column to support students in finding patterns.

*Supports accessibility for: Visual-spatial processing; Organization*

### Student Facing

Find the value of each variable that makes the equation true. Be prepared to explain your reasoning.

- \(2^3 \boldcdot 2^5 = 2^a\)
- \(3^b \boldcdot 3^7 = 3^{11}\)
- \(\frac{4^3}{4^2} = 4^c\)
- \(\frac{5^8}{5^d} = 5^2\)
- \(6^m \boldcdot 6^m \boldcdot 6^m = 6^{21}\)
- \((7^n)^4 = 7^{20}\)
- \(2^4 \boldcdot 3^4 = 6^s\)
- \(5^3 \boldcdot t^3 = 50^3\)

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

If students have trouble getting started, encourage them to write out the repeated factors—for example, writing \(2^3\) as \(2 \boldcdot 2 \boldcdot 2\). They may need to experiment with different values of variables in order to see patterns.

### Activity Synthesis

The purpose of discussion is to review these exponent rules:

\(\begin{align} b^m \boldcdot b^n &= b^{m+n} \\ \left(b^m\right)^n &= b^{m \boldcdot n} \\ \frac{b^m}{b^n} &= b^{m - n} \\ a^n \boldcdot b^n & = (a \boldcdot b)^n \end{align}\)

Each of these exponent rules is an equation in which the expression on the left uses two different exponents and the expression on the right uses only a single exponent. For each exponent rule, display only the left side, and ask students what the exponent rule is for writing an equivalent expression that uses only a single exponent.

## 1.3: The Power of Zero (15 minutes)

### Optional activity

This activity is optional because it revisits below grade-level content. In this activity, students use repeated reasoning to recognize that \(2^0=1\) and that negative integer powers of 2 represent repeated factors that are the reciprocal of the base, \(\frac12\) (MP8). Students then apply the same reasoning to find the value of other expressions involving integer exponents less than 1.

### Student Facing

- Use exponent rules to write each expression as a single power of 2. Find the value of the expression. Record these in the table. The first row is done for you.
expression power of 2 value \(\frac{2^5}{2^1}\) \(2^4\) 16 \(\frac{2^5}{2^2}\) \(\frac{2^5}{2^3}\) \(\frac{2^5}{2^4}\) \(\frac{2^5}{2^5}\) \(\frac{2^5}{2^6}\) \(\frac{2^5}{2^7}\) - What is the value of \(5^0\)?
- What is the value of \(3^{\text{-}1}\)?
- What is the value of \(7^{\text{-}3}\)?

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

Explain why the argument used to assign a value to the expression \(2^0\) does not apply to make sense of the expression \(0^0\).

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

For division to make sense with exponents, negative integer exponents must represent repeated factors that are the *reciprocal* of the base, and an exponent of 0 must mean the expression has a value of 1. The key idea is to frame the discussion of 0 and negative integer exponents in terms of comparing numerator and denominator in a division of powers of the same base. Here are some questions for discussion:

- “How can you tell \(\frac{2^5}{2^3}\) is greater than 1 without doing any computation?” (There are more factors that are 2 in the numerator, so the numerator is greater than the denominator.)
- “How can you tell that \(\frac{2^5}{2^7}\) is less than 1 without doing any computation?” (There are fewer factors that are 2 in the numerator, so the numerator is less than the denominator.)
- “How can you tell that \(\frac{2^5}{2^5}\) is equal to 1 without doing any computation?” (The numerator and denominator are identical, so the expression has a value of 1.)
- “Looking at \(2^{\text-5}\), what does the exponent -5 mean?” (In a fraction with powers of 2 in the numerator and denominator, there are 5 more factors in the denominator than in the numerator.)
- “Looking at \(2^{0}\), what does the exponent 0 mean?” (In a fraction with powers of 2 in the numerator and denominator, the numerator and denominator have the same number of factors, so the value is 1.)
- “What changes if the base is 5 or 3 or \(b\) instead of 2? What stays the same?” (The bases are different numbers, but the structure of the division is exactly the same. Negative exponents still correspond to more factors in the denominator, and an exponent of 0 still corresponds to a value of 1.)

*Speaking: MLR8 Discussion Supports.*As students share their interpretation of the exponent -5 in \(2^{\text- 5}\) and the exponent 0 in \(2^0\), press for details by asking students to provide examples of fractions with powers of 2 in the numerator and denominator that are equivalent to \(2^{\text-5}\) or \(2^0\). Show concepts multi-modally by writing each factor in the fractions to show that there are five more factors in the denominator than in the numerator or the same number of factors in the numerator and denominator. This will help students make sense of 0 and negative integer exponents.

*Design Principle(s): Support sense-making*

*Engagement: Develop Effort and Persistence.*Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . . .”

*Supports accessibility for: Language; Social-emotional skills*

## 1.4: Matching Exponent Expressions (15 minutes)

### Optional activity

This activity is optional because it revisits below grade-level content. In this activity, students build fluency by matching equivalent expressions that involve exponents. This task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

Listen for conversations students have comparing \((\text-4)^2\) and \(\text{-}4^2\).

### Launch

Arrange students in groups of 2. Tell students that once they identify a match, they should check with their partner. If there is disagreement, work to reach agreement before moving on to identify another match.

*Conversing: MLR8 Discussion Supports.*Arrange students in groups of 2. Invite students to take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “ ____ and ____ are equal because…” and “I noticed _____, so I matched….” Encourage students to challenge each other when they disagree. This will help students clarify their reasoning about equivalent expressions that involve exponents.

*Design Principle(s): Support sense-making; Maximize meta-awareness*

*Engagement: Develop Effort and Persistence.*Encourage and support opportunities for peer collaboration. When students share their work with a partner, display sentence frames to support conversation such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . . .”

*Supports accessibility for: Language; Social-emotional skills*

### Student Facing

Sort expressions that are equal into groups. Some expressions may not have a match, and some may have more than one match. Be prepared to explain your reasoning.

- \(2^{\text{-}4}\)
- \(\frac{1}{2^4}\)
- \(\text{-}2^4\)
- \(\text{-}\frac{1}{2^4}\)
- \(4^2\)
- \(4^{\text{-}2}\)
- \(\text{-}4^2\)
- \(\text{-}4^{\text{-}2}\)
- \(2^7 \boldcdot 2^{\text{-}3}\)
- \(\frac{2^7}{2^{\text{-}3}}\)
- \(2^{\text{-}7} \boldcdot 2^{3}\)
- \(\frac{2^{\text{-}7}}{2^{\text{-}3}}\)
- \((\text-4)^2\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Select students to share their reasoning about \(\text{-}4^2\) versus \((\text-4)^2\). Ask groups to share which expressions they thought were equivalent and to explain their reasoning. To involve more students in the conversation, consider asking:

- “Who can restate \(\underline{\hspace{0.3 in}}\)’s reasoning in a different way?”
- “Did anyone have the same strategy but would explain it differently?”
- “Did anyone use a different way to decide that expressions matched?”
- “Does anyone want to add on to \(\underline{\hspace{0.3 in}}\)’s strategy?”
- “Do you agree or disagree? Why?”

## Lesson Synthesis

### Lesson Synthesis

In this lesson, students reviewed exponent rules for the cases in which the exponents are integers:

\(\begin{align}b^m \boldcdot b^n &= b^{m+n} \\ \left(b^m\right)^n &= b^{m \boldcdot n} \\ \frac{b^m}{b^n} &= b^{m-n} \\ b^{\text-n} &= \frac{1}{b^n} \\ b^0 &= 1 \\ a^n \boldcdot b^n &= (a \boldcdot b)^n \end{align}\)

Arrange students in groups of 2 and ask them to come up with at least 3 expressions that are equivalent to \(27x^{12}\) using the exponent rules. Invite groups to share their expressions and explain which rules they used to think of them. Record their thinking for all to see.

## 1.5: Cool-down - The Power of Negative Thinking (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

Exponent rules help us keep track of a base’s repeated factors. Negative exponents help us keep track of repeated factors that are the *reciprocal* of the base. We can define a number to the power of 0 to have a value of 1. These rules can be written symbolically as:

\(\begin{align}b^m \boldcdot b^n &= b^{m+n} \\ \left(b^m\right)^n &= b^{m \boldcdot n} \\ \frac{b^m}{b^n} &= b^{m-n} \\ b^{\text-n} &= \frac{1}{b^n} \\ b^0 &= 1 \\ a^n \boldcdot b^n &= (a \boldcdot b)^n \end{align}\)

Here, the base \(b\) can be any positive number, and the exponents \(n\) and \(m\) can be any integer.