# Infinite Decimals

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Lesson Plans and Worksheets for Grade 8
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Examples, solutions, and videos to help Grade 8 students know the intuitive meaning of an infinite decimal explain why the infinite decimal $$0.\overline 9$$ is equal to 1.

### New York State Common Core Math Grade 8, Module 7, Lesson 7

Lesson 7 Student Outcomes

• Students know the intuitive meaning of an infinite decimal.
• Students will be able to explain why the infinite decimal $$0.\overline 9$$ is equal to 1.

Lesson 7 Summary

An infinite decimal is a decimal whose expanded form and number line representation are infinite.

Lesson 7 Classwork

Opening Exercises 1–4
1. Write the expanded form of the decimal 0.3765 using powers of 10.
2. Write the expanded form of the decimal 0.333333... using powers of 10.
3. What is an infinite decimal? Give an example.
4. Do you think it is acceptable to write that 1 = 0.9999... ? Why or why not?

Example 1
The number 0.235 on the number line:
Example 2
The number 5/6 = 0.833333... = $$0.8\overline 3$$ on the number line:

Exercises 5–10
5. a. Write the expanded form of the decimal 0.125 using powers of 10.
b. Show on the number line the representation of the decimal 0.125.
c. Is the decimal finite or infinite? How do you know?

6. a. Write the expanded form of the decimal 0.3875 using powers of 10.
b. Show on the number line the representation of the decimal 0.3875.
c. Is the decimal finite or infinite? How do you know?

7. a. Write the expanded form of the decimal 0.777777... using powers of 10.
b. Show on the number line the representation of the decimal 0.777777...
c. Is the decimal finite or infinite? How do you know?

8. a. Write the expanded form of the decimal $$0.\overline {45}$$ using powers of 10.
b. Show on the number line the representation of the decimal $$0.\overline {45}$$.
c. Is the decimal finite or infinite? How do you know?

9. Order the following numbers from least to greatest: 2.121212, 2.1, 2.2, $$2.\overline {12}$$
10. Explain how you knew which order to put the numbers in.

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