Activating Prior Knowledge – Find the decimal expansion of the following: 1. 9 15 2. 7 8 0.6 .875 3. 5 9 4. 11 16 0.555… .6875 Tie to LO
Objective: Today, we will show that every real number with a repeating decimal expansion is a rational number
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 1.2345678. If we multiply by 10 5 we get: CFU
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 0. 81 . Find the fraction. Let 𝑥=0. 81 =0.818181818… Any ideas where to begin? Let’s try multiplying 𝑥=0. 81 by powers of 10. 𝑥=0.818181818… 10𝑥=8.18181818… 100𝑥=81.8181818… 1000𝑥=818.181818… CFU
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=81.8181818… Do you notice anything? So the infinite repeating decimal 0.818181818…= 9 11 100𝑥=81+0.8181818… Since 𝑥=0.818181818… That means… 100𝑥=81+𝑥 −𝑥 −𝑥 99𝑥=81 99 99 𝑥= 81 99 = 9 11 CFU
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 2.13 8 is also a fraction? Let’s try the same trick and multiply by powers of 10. 𝑥=2.138888… Do any of these seem to be helpful? 10𝑥=21.38888… 100𝑥=213.8888 1000𝑥=2138.8888… What if I asked a separate question: is 0.8888… the decimal expansion of a fraction? CFU
Concept Development: CFU Module 7 LSN 10: Converting Repeating Decimals to Fractions Module Pg. 50 Concept Development: Example 2: Could it be that 2.13 8 is also a fraction? Since we know 0.8888… 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=213+.8888…, 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 CFU
Concept Development: CFU Module 7 LSN 10: Converting Repeating Decimals to Fractions Module Pg. 50 Concept Development: Example 2: Could it be that 2.13 8 is also a fraction? Now that we know 0. 8 = 8 9 , we can write the equation: 100𝑥=213+ 8 9 100𝑥= 213∙9 9 + 8 9 100𝑥= 213∙9+8 9 100𝑥= 1925 9 𝑥= 1925 900 ÷25 ÷25 𝑥= 77 36 CFU
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: CFU
Independent Practice CFU Module 7 LSN 10: Converting Repeating Decimals to Fractions Module Pg. 51 Independent Practice Exercise 2: CFU
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) CFU
Independent Practice CFU Module 7 LSN 10: Converting Repeating Decimals to Fractions Module Pg. 53 Independent Practice CFU
Module 7 LSN 10: Converting Repeating Decimals to Fractions Pg. 54 Lesson Summary CFU
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. CFU