Download Chapter 5: Rational Numbers as Fractions

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