standard_description: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
problem_activity: Activity
Students extend their understanding of exponents to include negative exponents and explain patterns in the placements of the decimal point when a decimal is multiplied by 10 or
\(\frac{1}{10}\)
. Students use repeated reasoning to recognize that negative powers of 10 represent repeated multiplication by
\(\frac{1}{10}\)
and generalize to the rule
\(10^{\text- n} = \frac{1}{10^n}\)
(MP8).
A table is used to show different representations of decimals, fractions, and exponents. The table is horizontal to mimic the structure of decimals. As students work, notice the strategies they use to go between the different representations of the given powers of 10. Ask students who use contrasting strategies to share later.
Launch
Tell students to complete the table one row at a time to see the patterns most clearly. Ask a student to read the first question aloud. Select a student to explain the idea of a “multiplier” in this context. Give students 5–7 minutes to work. Follow with a whole-class discussion.
Student Facing
Complete the table to explore what negative exponents mean.
[IMG]
As you move toward the left, each number is being multiplied by 10. What is the multiplier as you move right?
How does a multiplier of 10 affect the placement of the decimal in the product?  How does the other multiplier affect the placement of the decimal in the product?
Use the patterns you found in the table to write
\(10^{\text -7}\)
as a fraction.
Use the patterns you found in the table to write
\(10^{\text -5}\)
as a decimal.
Write
\(\frac{1}{100,000,000}\)
using a single exponent.
Use the patterns in the table to write
\(10^{\text -n}\)
as a fraction.
Student Response
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Anticipated Misconceptions
Some students may think that, for example,
\(\frac{1}{1,000,000} = 10^{\text-7}\)
because the number 1,000,000 has 7 digits. Ask these students if it is true that
\(\frac{1}{10} = 10^{\text-2}\)
.
Activity Synthesis
One important idea is that multiplying by 10 increases the exponent, thus multiplying by
\(\frac{1}{10}\)
decreases the exponent. So negative exponents can be thought of as repeated multiplication by
\(\frac{1}{10}\)
, whereas positive exponents can be thought of as repeated multiplication by
\(10\)
. Another key point is the effect that multiplying by 10 or
\(\frac{1}{10}\)
has on the placement of the decimal.
Ask students to share how they converted between fractions, decimals, and exponents. Record their reasoning for all to see. Here are some possible questions to consider for whole-class discussion:
“Do you agree or disagree? Why?”
“Did anyone think of this a different way?”
“In your own words, what does
\(10^{\text- 7}\)
mean? How is it different from
\(10^7\)
?”
Introduce the visual display for
\(10^{\text- n} = \frac{1}{10^n}\)
and display it for all to see throughout the unit. For an example that illustrates the rule, consider displaying
\(10^{\text- 3} = \frac{1}{10} \boldcdot \frac{1}{10} \boldcdot \frac{1}{10} = \frac{1}{10^3}\)
.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections. For example, annotate the patterns students notice to help synthesize how negative exponents can be thought of as repeated multiplication by
\(\frac{1}{10}\)
, whereas positive exponents can be thought of as repeated multiplication by
\(10\)
.
Supports accessibility for: Visual-spatial processing
Conversing, Representing, Writing: MLR2 Collect and Display.
During the whole-class discussion, capture the vocabulary and phrases students use to describe their strategies for converting between fractions, decimals, and exponents. Listen for students who make connections between repeated multiplication and the placement of the decimal point. Record this language onto a visual display that can be referenced in future discussions. This routine provides feedback to students that supports sense-making and increases meta-awareness of mathematical language.
Design Principle(s): Support sense-making; Maximize meta-awareness


aligns: no