and denominators of two fractions. For example, you could use

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1 Learning Goal denominators. Open Question and denominators of two fractions. For example, you could use and 0. Make sure that some of your fractions are improper and some are proper. Make sure that some of your fractions use the same numerator and some do not. Tell which of your fractions is greater and how you know. them. Tell how you know which fraction is greater each time. September 011 Marian Small, 011 Fractions (IS)

2 Think Sheet Pairs of fractions are either equal or one fraction is greater than the other fraction. We can decide which statement is true by using a model, by renaming one or both fractions, or by using benchmarks. Using a Model and, we can use a picture that shows the two fractions lined up, so we can see which extends farther. _ < _ 1 of counters is counters, so of counters is 1 counters. 1 of counters is counters, so of counters is counters. is more than 1, so >. September 011 Marian Small, 011 Fractions (IS)

3 Renaming Fractions To rename a fraction, we can think about how to express the fraction as an equivalent fraction or equivalent decimal. Equivalent Fractions Two fractions are equivalent, or equal, if they take up the same part of a whole or wholes. For example, =. Each section of the model is split into two sections in the model. So, there are twice as many sections in the second model, and twice as many are shaded. The numerator and denominator have both doubled. We can multiply the numerator and denominator by any amount (except 0) and the same thing happens as in the example above. There are more sections shaded and more sections unshaded, but the amount shaded does not change. For example, if we multiply by : = = 9 1. or If we multiply by : = = 0 0. Common Denominators and Common Numerators Renaming fractions to get common denominators or common numerators is helpful when comparing fractions. If we compare fractions with the same denominator, the one with the greater numerator is greater. If we compare fractions with the same numerator, the one with the lesser denominator is greater. To determine if or is more, we might use common denominators: = 1 = 1 = = 9 = 1 = 1 = 1 1 = 1 1 = 1 Since = 1 and = 1, is a common denominator. Since 1 < 1, then <. = 1 = = 9 = 1 = 1 Since = 1 and =, is a common numerator. 1 1 > 1, and >. September 011 Marian Small, 011 Fractions (IS)

4 Decimals Another way to rename a fraction is to represent it as a decimal. For example, =. Using a calculator, we see that = 0. The reason a = a b is because we can imagine a objects, each being shared b by b people. Each person gets one section of each object, so altogether each gets a sections of size 1 b ; that is a b. For example, if four people share three bars each one gets one fourth from each of the whole bars and that is. We can use decimal renaming to compare fractions, such as to. Since = 0. and = 0., we see that <. Using Benchmarks We can use benchmarks to compare fractions. them to 1. For example, 1 1 is more than 1, and 1 is less than 1, so 1 >. greater than one half and one is not. We can mentally rename one half to compare the other fractions to it. For the fraction, since one half is, is more than one half. For the fraction, since one half is, is less than one half. So, >. them to whole numbers other than 1. For the fraction 11, since is, 11 is more than. For the fraction, since is, is less than. So, 11 is more than. 9 September 011 Marian Small, 011 Fractions (IS)

5 1. What fraction comparison is being shown? a) c) d). Draw a picture to show why this statement is true. a) = 1 =. Tell or show why is not equivalent to 9.. a) 9 or 9 9 or 11 September 011 Marian Small, 011 Fractions (IS)

6 . How might you rename one or both fractions as other fractions to make it easier to compare them? Tell how it helps. a) and and 1 c) 9 and 1 1 d) and 1. a) Why does it make sense that 9 = 9? What are the decimal equivalents of and 9? c) How could you use decimal equivalents to compare and 9? 11 September 011 Marian Small, 011 Fractions (IS)

7 . How can each of these fraction pairs be compared without renaming them as other fractions or as decimals? a) and and 1 c) 1 and 1 d) 1 and 9 e) and 9 f) 9 and 1. Describe a real-life situation when you might make each comparison: a) to 1 to 1 9. A fraction with a denominator of is between one fraction with a denominator of and one fraction with a denominator of. Fill in all 9 blanks to show three ways this could be true. < < < < < <. a) List all the fractions with a denominator of between and. List all the fractions with a denominator of between and. c) Is it possible to list all the fractions between and? Explain. 1 September 011 Marian Small, 011 Fractions (IS)

Section 3.2 Comparing and Ordering Fractions and Decimals. 1. Model fractions and/or decimals using blocks, fraction pieces, pattern blocks, etc.

$Section 3.2 Comparing and Ordering Fractions and Decimals. 1. Model fractions and/or decimals using blocks, fraction pieces, pattern blocks, etc.$ Section 3.2 Comparing and Ordering Fractions and Decimals We will use several methods to compare and order fractions: 1. Model fractions and/or decimals using blocks, fraction pieces, pattern blocks, etc.