1. Fractions and Rational Numbers
6.1 The Basic Concepts of Fractions and Rational Numbers
6.2 Addition and Subtraction of Fractions
6.3 Multiplication and Division of Fractions
6.4 The Rational Number System
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2. The Basic Concepts of Fractions and Rational Numbers
6.1
The Basic Concepts of Fractions and Rational Numbers
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3. THE MEANING OF A FRACTION
To interpret the meaning of any fraction
we must:
• agree on the unit;
understand that the unit is subdivided
into b parts of equal size;
understand that we are considering a of
the parts of the unit.
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4. • The integer a is called the numerator of the fraction.
DEFINITION: FRACTION
A fraction is an ordered pair of integers a and b, b ≠ 0, written or a/b.
• The integer a is called the numerator of
the fraction.
The integer b is called the denominator
of the fraction.
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5. • What is the unit? (the whole) Into how many equal parts has the unit
MODELS FOR FRACTIONS
A physical or pictorial representation of a fraction must clearly answer the following questions:
• What is the unit? (the whole)
Into how many equal parts has the unit
been subdivided? (the denominator)
How many of these parts are under
consideration? (the numerator)
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6. MODELS FOR FRACTIONS: COLORED REGIONS
A shape is chosen to represent the unit and is then subdivided into subregions of equal size.
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7. MODELS FOR FRACTIONS: THE SET MODEL
Each subset A of U corresponds to the fraction
of the apples have worms.
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8. MODELS FOR FRACTIONS: FRACTION STRIPS
The unit is defined by a rectangular strip of cardstock. A set of fraction strips typically contains strips for the denominators 1, 2, 3, 4, 6, 8, and 12.
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9. MODELS FOR FRACTIONS: THE NUMBER-LINE
Fractions can be modeled by subdividing the unit interval into equal parts determined by the denominator and then counting off the number of those parts determined by the numerator.
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10. THE FUNDAMENTAL LAW OF FRACTIONS
Let be a fraction. Then
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11. THE CROSS-PRODUCT PROPERTY OF EQUIVALENT FRACTIONS
The fractions are equivalent if,
and only if, ad = bc. That is,
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12. FRACTIONS IN SIMPLIEST FORM
A fraction is in simplest form if a and b have no common divisor larger than 1 and b is positive.
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13. Example 6.3 Finding Common Denominators
Find equivalent fractions to
with a common denominator of 12.
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14. DEFINITION: RATIONAL NUMBERS
A rational number is a number that can be represented by a fraction , where a and b are integers, b ≠ 0.
Two rational numbers are equal if, and only if, they can be represented by equivalent fractions.
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15. Example 6.4 Representing Rational Numbers
How many different rational numbers are given in this list of five fractions?
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16. Addition and Subtraction of Fractions
6.2
Addition and Subtraction of Fractions
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17. DEFINITION: ADDITION OF FRACTIONS
Let two fractions have a
common denominator. Then their sum is the fraction given by
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18. MODELING ADDITION OF FRACTIONS WITH COLORED REGIONS
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19. MODELING ADDITION OF FRACTIONS WITH THE NUMBER-LINE
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20. MODELING ADDITION OF FRACTIONS WITH UNLIKE DENOMINATORS
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21. A mixed number can always be rewritten in the standard form
MIXED NUMBERS
A mixed number can always be rewritten in the standard form
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22. MIXED NUMBERS
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23. Example 6.7 Working with Mixed Numbers
a. Give an improper fraction for
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24. Example 6.7 Working with Mixed Numbers
b. Give a mixed number for
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25. DEFINITION: SUBTRACTION OF FRACTIONS
Let be fractions.
Then
if, and only if,
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26. MODELING SUBTRACTION OF FRACTIONS WITH FRACTION STRIPS
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27. Multiplication and Division of Fractions
6.3
Multiplication and Division of Fractions
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28. DEFINITION: MULTIPLICATION OF FRACTIONS
Let be fractions.
Then their product is given by
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29. Example 6.10 Calculating Products of Fractions
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30. Example 6.12 Multiplying Fractions on the Number Line
Illustrate why with a number-line diagram.
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31. THE INVERT-AND-MULTIPLY ALGORITHM FOR DIVISION OF FRACTIONS
where
Note that this is a process for dividing fractions, not a definition of division.
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32. Example 6.15 Dividing Fractions
Compute. a. b.
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33. The Rational Number System
6.4
The Rational Number System
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34. DEFINITION: NEGATIVE OR ADDITIVE INVERSE
Let be a rational number.
Its negative, or additive inverse,
written is the rational number
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35. Example 6.18 Subtracting Rational Nmbers
Compute. a. b.
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