1. 6 Chapter Rational Numbers and Proportional Reasoning
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2. NCTM Standard: Rational Numbers as Fractions
K–2: Children should understand and represent commonly used fractions.
3–5: Children should develop an understanding of fractions to include them as parts of unit wholes, parts of a collection, as locations on number lines, and as divisions of whole numbers.
6–7: Children work flexibly with fractions to solve problems, compare and order fractions, and find their locations on a number line. They understand and use ratios and proportions to represent quantitative relationships. (p.33)
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3. 6-1 The Set of Rational Numbers
Equivalent or Equal Fractions
Simplifying Fractions
Equality of Fractions
Ordering Rational Numbers
Denseness of Rational Numbers
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4. Numerator, Denominator
Definition
Rational numbers
The set of numbers Q such that
Numerator, Denominator
numerator
denominator
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5. Uses of Rational Numbers
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6. Rational Number Models
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Meaning of a Fraction
To understand the meaning of any fraction, using
the parts-to-whole model, we must consider each of the following:
The whole being considered.
The number b of equal-size parts into which the whole is divided.
The number a of parts of the whole that we are selecting.
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Definition
Proper fraction
A fraction where
Examples are
Improper fraction
A fraction such that
Examples are
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Number Line Model
What numbers are represented on the number line?
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10. Equivalent or Equal Fractions
Equivalent fractions are numbers that represent the same point on a number line.
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Fraction Strips
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12. Fundamental Law of Fractions
Let be any fraction and n a nonzero whole
number, then
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Example 6-1
Find a value for x such that
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14. Simplifying Fractions
A rational number is in simplest form if b > 0 and GCD(a,b) = 1; that is, if a and b have no common factor greater than 1.
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Example 6-2
Write each of the following fractions in simplest form if they are not already so: a. b. c. d.
cannot be simplified
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Example (continued) e. f. g.
cannot be simplified
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Equality of Fractions
Show that
Method 1: Simplify both fractions to the same simplest form.
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Equality of Fractions
Method 2: Rewrite both fractions with the same least common denominator.
LCM(42, 35) = 210, so
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Equality of Fractions
Method 3: Rewrite both fractions with a common denominator.
A common multiple of 42 and 35 is 42 · 35 = 1470.
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Equality of Fractions
Two fractions and , d  0 are equal if and only if ad = bc.
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21. Ordering Rational Numbers
If a, b, and c are integers and b > 0, then if and only if a > c.
If a, b, c, and d are integers
and b > 0, d > 0, then
if and only if ad > bc.
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22. Denseness of Rational Numbers
Given rational numbers there is another rational number between these two numbers.
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Example 6-3a
Find two fractions between
Because is between
Now find two fractions equal to respectively, but with greater denominators.
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24. Example 6-3a (continued)
are all between so they are between
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Example 6-3b
Show that the sequence is increasing.
Because the nth term of the sequence is the next term is
We need to show that for all positive integers n,
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26. Example 6-3b (continued)
The inequality is true, if, and only if,
which is true for all n.
Thus, the sequence is increasing.
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27. Denseness of Rational Numbers
Let be any rational numbers with positive denominators, where Then
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