Lesson 5: Negative Exponents with Powers of 10

Negative Exponents with Powers of 10

5 min

Teacher Prep

Setup

1 minute of quiet work time per problem , followed by partner discussion and then a whole-class discussion.

Narrative

This Math Talk focuses on strategies and fluency regarding exponents and place value. It encourages students to think about the relative values of powers of 10 and to rely on what they know about exponents to mentally solve problems. The ideas elicited here will be helpful later in the lesson when students investigate negative exponents.

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Task

Find the value of $x$ mentally.

$\dfrac{100}{1} = 10^x$
$\dfrac{100}{x} = 10^1$
$\dfrac{x}{100} = 10^0$
$\dfrac{100}{1,000} = 10^{x}$

Sample Response

$x=2$ . Sample reasoning: Dividing 100 by 1 gives 100, and 100 is equal to $10^2$ .
$x = 10$ . Sample reasoning: $10^1$ is equal to 10, and 100 divided by 10 is also equal to 10.
$x = 100$ . Sample reasoning: $10^0$ is equal to 1, so the left side of the equation must be $\frac{100}{100}$ .
$x = \text- 1$ . Sample reasoning: On the left side of each of the four equations, 100 is being divided by a larger and larger power of 10. On the right side of each equation, the exponent is decreasing by a value of 1 each time. This pattern would suggest that the missing value of $x$ is -1.

Activity Synthesis (Teacher Notes)

To involve more students in the conversation, consider asking:

“Who can restate $\underline{\hspace{.5in}}$ ’s reasoning in a different way?”
“Did anyone use 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?”
“What connections to previous problems do you see?”

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I

\underline{\hspace{.5in}}

because . . . .” or “I noticed

\underline{\hspace{.5in}}

so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing

Standards

Addressing

8.EE.1·Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3-5 = 3-3 = 1/3³ = 1/27.
8.EE.A.1·Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, $3^2\times3^{-5} = 3^{-3} = 1/3^3 = 1/27$.

15 min

Knowledge Components

All skills for this lesson

No KCs tagged for this lesson

Negative Exponents with Powers of 10

5 min

Teacher Prep

Setup

1 minute of quiet work time per problem , followed by partner discussion and then a whole-class discussion.

Narrative

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Task

Find the value of $x$ mentally.

$\dfrac{100}{1} = 10^x$
$\dfrac{100}{x} = 10^1$
$\dfrac{x}{100} = 10^0$
$\dfrac{100}{1,000} = 10^{x}$

Sample Response

$x=2$ . Sample reasoning: Dividing 100 by 1 gives 100, and 100 is equal to $10^2$ .
$x = 10$ . Sample reasoning: $10^1$ is equal to 10, and 100 divided by 10 is also equal to 10.
$x = 100$ . Sample reasoning: $10^0$ is equal to 1, so the left side of the equation must be $\frac{100}{100}$ .
$x = \text- 1$ . Sample reasoning: On the left side of each of the four equations, 100 is being divided by a larger and larger power of 10. On the right side of each equation, the exponent is decreasing by a value of 1 each time. This pattern would suggest that the missing value of $x$ is -1.

Activity Synthesis (Teacher Notes)

To involve more students in the conversation, consider asking:

“Who can restate $\underline{\hspace{.5in}}$ ’s reasoning in a different way?”
“Did anyone use 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?”
“What connections to previous problems do you see?”

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I

\underline{\hspace{.5in}}

because . . . .” or “I noticed

\underline{\hspace{.5in}}

so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing

Standards

Addressing

8.EE.1·Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3-5 = 3-3 = 1/3³ = 1/27.
8.EE.A.1·Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, $3^2\times3^{-5} = 3^{-3} = 1/3^3 = 1/27$.

Negative Exponents with Powers of 10

Warm-upMath Talk: What's That Exponent?5 minKCs

Setup

Narrative

Launch

Student Task

Sample Response

ActivityNegative Exponent Table15 minKCs

ActivityFollow the Exponent Rules15 minKCs

Knowledge Components

Negative Exponents with Powers of 10

Warm-upMath Talk: What's That Exponent?5 minKCs

Setup

Narrative

Launch

Student Task

Sample Response

ActivityNegative Exponent Table15 minKCs

ActivityFollow the Exponent Rules15 minKCs

5 min

15 min

15 min

5 min

15 min

15 min