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Transcript
Chapter 5: Rational Numbers as Fractions 5.1 The set of rational numbers 5.1.1. Vocabulary 5.1.1.1. rational numbers – elements of the set of numbers of the form a , where b ≠ 0 b a and a and b are integers: Q = { | a ∈ I and b ∈ I and b ≠ 0} b a 5.1.1.2. numerator – for , the integer in the top: a b a 5.1.1.3. denominator – for , the integer in the bottom: b b a 5.1.1.4. proper fraction – a fraction , where 0 ≤ | a | ≤ | b | b a 5.1.1.5. improper fraction – in general a fraction , where | a | ≥ | b | > 0 b 5.1.1.6. equivalent fractions – when one fraction is a multiple of another fraction 5.1.1.7. equal fractions – same as equivalent fractions 5.1.1.8. simplifying fractions – eliminating common factors from the numerator and the denominator 5.1.1.9. simplest form – when all common factors from the numerator and denominator have been removed; we say that the numerator and the denominator are relatively prime when a fraction is in simplest form 5.1.1.10. lowest terms – same as simplest form 5.1.1.11. denseness – there are no holes in the number line, the number line is said to be dense because in between any two numbers is another number such that for any numbers a and b, there is a number c such that a < c < b. 5.1.1.12. Trichotomy principle – for any numbers a and b, one and only one of the following must be true: a > b, a < b, or a = b. 5.1.1.13. Now try this 5-1 p. 301: Discuss in your groups 5.1.1.14. Now try this 5-2 p. 301: Discuss in your groups 5.1.2. Equivalent or Equal Fractions a an 5.1.2.1. Fundamental Law of Fractions: = , a, b, and n can be any numbers but b ≠ b bn 0 and n ≠ 0 5.1.2.2. Now try this 5-3 p. 302: Discuss in your groups 5.1.2.3. With pattern blocks 5.1.2.3.1. http://arcytech.org/java/patterns/patterns_j.shtml 5.1.2.3.2. Good for increasing spatial sense and making connections 5.1.2.3.3. 5.1.2.4. Base ten blocks 5.1.2.4.1. http://arcytech.org/java/b10blocks/b10blocks.html 5.1.2.4.2. Good for exploring decimal equivalents 5.1.2.4.3. 5.1.2.5. Cuisenaire rods 5.1.2.5.1. http://arcytech.org/java/integers/integers.html 5.1.2.5.2. Good for exploring fractional equivalents 5.1.2.5.3. 5.1.3. Simplifying factions a is in simplest form if a and b b have no common factor greater than one, that is if a and b are relatively prime 5.1.3.2. If and only if the fraction is in simplest form, then the GCF = 1 5.1.3.3. We do NOT reduce fractions, they do NOT get any smaller after removing common factors 5.1.4. Equality of fractions a an is any fraction, then there exists fractions in the form that are equivalent 5.1.4.1. If b bn to the original fraction a an 5.1.4.2. is a multiple of bn b a an 5.1.4.3. = b bn a c and are equal if, and only if, ad = bc 5.1.4.4. Property: two fractions b d 5.1.5. Ordering rational numbers a c 5.1.5.1. Theorem 5-1: If a, b, and c are integers and b > 0, then > if, and only if, a > b b c 5.1.5.2. Now try this 5-5 p. 307: Discuss in your groups a c 5.1.5.3. Theorem 5-2: If a, b, and c are integers and b > 0, d > 0, > if, and only if, ad b d > bc 5.1.5.4. Now try this 5-6 p. 308: Discuss in your groups 5.1.6. Denseness of Rational numbers a c 5.1.6.1. Property: given rational numbers and , there is another rational number b d between these two numbers 5.1.6.2. Now try this 5-7 p. 308: Discuss in your groups 5.1.3.1. Definition of simplest form: a rational number a c and be any rational numbers with positive denominators b d a c a a+c c a c where < , then < < . (Hint: If < , then ad < bc. Now add ab to b d b b+d d b d a a+c both sides and use this to show that < . Then finish the proof in a similar b b+d way.) 5.1.6.4. Now try this 5-8 p. 309: Discuss in your groups 5.1.7. Ongoing Assessment p. 310 5.1.7.1. Home work: 1, 2ace, 3ac, 5ac, 6ac, 10ac, 11ac, 13aceg, 19, 21, 24ac, 25a 5.1.6.3. Theorem 5-3: Let