**Lesson 5.3** Exploring Repeating Decimals

**ALCOS Standards** • PRE-ALGEBRA • DAILY PACING GUIDE • 2ND QUARTER

**Objectives** 1. Students will be able to distinguish between terminating and repeating decimals. 2. Students will be able to identify rational and irrational numbers. • Students will be able to convert decimals to fractions. • Students will be able to identify patterns in repeating decimals.

**Problem of the Day** • What are the decimal representations of 1/33, 2/33, 3/33, and 4/33? • Find other fractions with decimal representations the same as those of 1/33, 2/33, 3/33, and 4/33.

**Answers to POD** • 1/33=.030303… 2/33=.060606… 3/33=.090909… 4/33=.121212… B) 2/66=.030303… 4/66=.060606… 6/66=.090909… 8/66=.121212…

**Review-Problem 5.2** • Write each fraction as a decimal. Tell whether the decimal is terminating or repeating. If the decimal is repeating, tell which digits repeat. A) 2/3 B) 3/8 C) 5/6 D) 35/10 E) 8/99 *Counting decimal places (power of 10) Place Value: Hands-on role play activity-Counting the Rice http://arcytech.org/java/b10blocks/counting.html

**Problem 5.2** Answers: • 2/3 =0.666… R • 3/8 =0.375 T • 5/6 =0.8333… R • 35/10 =3.5 T • 8/99 =0.080808… R

**Repeating Decimals** What is a fraction representation of the decimal 0.333…? A) 3/10 B) 1/3 C) 33/100 *Journal: Explain how you got your answer and why the other two are incorrect.

**Repeating Decimals** Correct answer: B)0.333… equals 1/3 Discuss journal entry.

**Repeating Decimals** How could you write the decimal 0.454545… as a fraction?

**Repeating Decimals** In today’s lesson, we will look for patterns that can help us write repeating decimals as fractions. Let’s begin working on the problem (5.3) and follow-up in pairs (pgs. 57-58). Students will break off into pairs.

**Explore** Problem 5.3 Page 57 A, B, C, D *Transparency

**Explore** *A fraction with a denominator of 9 has a decimal representation with a one-digit repeating pattern, or repetend, and the repetend is the number in the numerator. *A fraction with a denominator of 99 has a decimal representation with a two-digit repetend: a 0 followed by the number in the numerator if that number is less than 10; the number in the numerator if that number is equal to or greater than 10.

**Explore** *A fraction with a denominator of 999 has a decimal representation with a three-digit repetend: two 0’s followed by the number in the numerator if that number is less than 10; one 0 followed by the number in the numerator if that number is equal to or greater than 10 but less than 1000; the number in the numerator if that number is equal to or greater than 100.

**Explore** Problem 5.3 Follow Up Page 58 1, 2, 3

**Follow Up 5.3** Answers: • 1/99=0.0101… 2/99=0.0202… 3/99=0.0303… A fraction with a denominator of 99 is equal to a repeating decimal. The repeating pattern has two digits.

**Follow Up 5.3** • 1/999=0.001001… 2/999=0.002002… 3/999=0.003003… 10/999=0.010010… A fraction with a denominator of 999 is equal to a repeating decimal. The repeating pattern has 3 digits.

**Follow Up 5.3** • a) 0.333…=1/3 b) 0.050505…=5/99 c) 0.454545…=45/99 d) 0.045045…=45/999 e) 10.1212…=10 12/99 f) 3.999…=3 9/9, or 4

**Repeating Decimals** • Discuss the patterns in the decimals. *Question: What is the decimal representation for 19/9? For 20/9? *Write 19/9 as a mixed number. *Write 20/9 as a mixed number.

**Repeating Decimals** Question: What fraction is equivalent to 4.111111…? Answer: (write as a fraction and mixed number)

**Repeating Decimals** Answers: 4.111…=37/9 mixed number: 4 1/9

**Rational / Irrational Numbers** Did you know? Refer to text page 58. *http://www.eduplace.com/math/mathsteps/7/a/index.html

**Rational/Irrational Numbers** Math Video Clip: http://www.youtube.com/watch?v=OjjcF0t6CZ0

**ACE Problems** Page 59 # 5 - 8 Pages 61-62 # 26 - 30

**Homework** ACE Problem #25

**Assessment** Teacher Observation Class Discussion / ACE Problems Mathematical Reflections Page 63

**Additional Resources/Activities** *Interactive Fractions to Decimals http://www.learner.org/channel/courses/learningmath/number/session7/part_a/rings.html *A Well Planned Diet http://www.glencoe.com/sec/math/msmath/mac04/course2/webquest/unit3.php/ *Interactive activities: www.aplusmath.com/ *Interactive activities: www.aaamath.com/ *Interactive Baseball Game: http://www.funbrain.com/math/