1. Activating Prior Knowledge –
Find the decimal expansion of the following:
0.6
.875
0.555…
.6875
Tie to LO

2. Objective:
Today, we will show that every real number with a repeating decimal expansion is a rational number

3. Concept Development CFU
Module 7 LSN 10: Converting Repeating Decimals to Fractions
Concept Development
In lesson 8 we learned that every rational number, i.e. fraction, has a decimal expansion with a repeating pattern.
But is the converse true? That is, can every decimal expansion with an infinite repeating pattern, be written as a fraction?
Let’s start observing the effect of multiplying decimals by powers of 10. Consider for example, the finite decimal If we multiply by we get:
CFU

4. Concept Development: CFU
Module 7 LSN 10: Converting Repeating Decimals to Fractions
Module
Pg. 50
Concept Development:
Example 1: There is a fraction with an infinite decimal expansion of Find the fraction.
Let 𝑥=0. 81 = …
Any ideas where to begin?
Let’s try multiplying 𝑥= by powers of 10.
𝑥= …
10𝑥= …
100𝑥= …
1000𝑥= …
CFU

5. Concept Development: CFU
Module 7 LSN 10: Converting Repeating Decimals to Fractions
Module
Pg. 50
Concept Development:
Example 1: Let’s take a closer look at 100𝑥= …
Do you notice anything?
So the infinite repeating decimal …= 9 11
100𝑥= …
Since 𝑥= …
That means…
100𝑥=81+𝑥
−𝑥 −𝑥
99𝑥=81 99 99
𝑥= 81 99
= 9 11
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6. Concept Development: Do any of these seem to be helpful? CFU
Module 7 LSN 10: Converting Repeating Decimals to Fractions
Module
Pg. 52
Concept Development:
Example 2: Could it be that is also a fraction?
Let’s try the same trick and multiply by powers of 10.
𝑥= …
Do any of these seem to be helpful?
10𝑥= …
100𝑥=
1000𝑥= …
What if I asked a separate question: is … the decimal expansion of a fraction?
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7. Concept Development: CFU
Module 7 LSN 10: Converting Repeating Decimals to Fractions
Module
Pg. 50
Concept Development:
Example 2: Could it be that is also a fraction?
Since we know … is the decimal expansion of some fraction, let’s let 0.888…= 𝑎 𝑏 . Now are any of the previous powers of 10 helpful?
We now know that 100𝑥= …, so we can write it as 100𝑥=213+ 𝑎 𝑏 .
Let’s do some side work: Let’s let 𝑦=0. 8
10𝑦=8. 8
10𝑦=8+0. 8
10𝑦=8+𝑦
Substitution property
9𝑦=8
Subtract y on both sides
𝑦= 8 9
Divide both sides by 9
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8. Concept Development: CFU
Module 7 LSN 10: Converting Repeating Decimals to Fractions
Module
Pg. 50
Concept Development:
Example 2: Could it be that is also a fraction?
Now that we know 0. 8 = 8 9 , we can write the equation:
100𝑥=
100𝑥= 213∙
100𝑥= 213∙9+8 9
100𝑥= 𝑥=
÷25 ÷25
𝑥= 77 36
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9. Independent Practice CFU
Module 7 LSN 10: Converting Repeating Decimals to Fractions
Module
Pg. 50
Independent Practice
Complete Exercises 1 – 2 in your module. (4 minutes)
Exercise 1:
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10. Independent Practice CFU
Module 7 LSN 10: Converting Repeating Decimals to Fractions
Module
Pg. 51
Independent Practice
Exercise 2:
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11. Independent Practice CFU
Module 7 LSN 10: Converting Repeating Decimals to Fractions
Module
Pg. 53
Independent Practice
Complete exercises 3 – 4 in your module. (6 min)
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12. Independent Practice CFU
Module 7 LSN 10: Converting Repeating Decimals to Fractions
Module
Pg. 53
Independent Practice
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13. Module 7 LSN 10: Converting Repeating Decimals to Fractions
Pg. 54
Lesson Summary
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14. Closure – What did you learn? Why is it important?
Module 7 LSN 10: Converting Repeating Decimals to Fractions
Notes
Closure –
What did you learn?
Why is it important?
How does multiplying a decimal by a power of 10 help to determine the fraction that would produce it?
Homework:
Problem Set, page 54 – 55, problems 1 – 8 all.
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