Download Chapter 1

Chapter 6: Rational Number Operations and Properties
6.1 Rational Number Ideas and Symbols
6.1.1. Rational Number Uses and Models
6.1.1.1. Models for rational numbers
6.1.1.1.1. Used to describe a quantity between 0 and 1
6.1.1.1.2. identify the whole representing the numeral 1
6.1.1.1.3. separate the whole into equal parts
6.1.1.1.4. use an ordered pair of integers to describe the portion of the whole under
consideration
6.1.1.2. Identifying the whole and separating it into equal parts
6.1.1.2.1. Egg carton fractions
6.1.1.2.1.1.
http://pegasus.cc.ucf.edu/~mathed/egg.html
6.1.1.2.1.2.
6.1.1.2.2. Integer Rods
6.1.1.2.2.1.
http://pegasus.cc.ucf.edu/~mathed/crods.html
6.1.1.2.2.2.
http://arcytech.org/java/
6.1.1.2.3. Make your own fraction kit
6.1.1.2.3.1.
http://pegasus.cc.ucf.edu/~mathed/fk.html
6.1.1.2.4.
6.1.1.3. Using two integers to describe part of a whole
6.1.1.3.1. Need more language to describe part-whole relationship
6.1.1.3.2. number of pieces of interest vs. number of pieces found in the original whole
6.1.2. Defining Rational Numbers
6.1.2.1. Definition of a rational number: A number is a rational number if and only if it
a
a
can be represented by a pair of integers, , where b  0 and
represents the
b
b
quotient a  b
6.1.3. Using Fractions to Represent Rational Numbers
6.1.3.1. Fractions and Equivalent Fractions
a
6.1.3.1.1. Definition of a fraction: A fraction is a symbol, , where a and b are
b
numbers and b  0. Here, a is the numerator of the fraction and b is the
denominator of the fraction
6.1.3.1.2. Proper fraction: when the numerator of the fraction is less than the
denominator of the fraction and both the numerator and the denominator are
integers
6.1.3.1.3. Improper fraction: when the numerator of the fraction is greater than the
denominator of the fraction (fractions with non-integers in the numerator or
denominators are also improper)
a
c
6.1.3.1.4. Definition of equivalent fractions : Two fractions,
and , are equivalent
b
d
fractions if and only if ad = bc
6.1.3.1.5. Paper folding model
6.1.3.2. Using fractions to represent rational numbers
6.1.3.2.1. every rational number can be represented by an integer in the numerator and
the denominator
0.25 1
 in a
6.1.3.2.2. sometimes rational numbers are represented by non-integers
0.50 2
ratio
6.1.3.3.
The Fundamental Law of Fractions: Given a fraction
a
and a number c  0,
b
a ac

b bc
6.1.3.4. Fractions in simplest form
6.1.3.4.1. Description of the simplest form of a fraction: a fraction representing a
rational number is in simplest form when the numerator and the denominator are
both integers that are relatively prime and the denominator is greater than zero.
6.1.3.4.1.1.
Finding equivalent fractions on the number line
6.1.3.4.1.2.
Folding paper
6.1.3.4.1.3.
Using a calculator
6.1.3.4.1.4.
Using Integer rods
6.1.4. Using Decimals to Represent Rational Numbers
6.1.4.1. Decimals
6.1.4.1.1. Description of a decimal: A decimal is a symbol that uses a base-ten placevalue system with tenths and multiples of tenths to represent rational numbers
6.1.4.1.2. decimal point divides the decimal portion of the number from the whole
number portion of the number
6.1.4.1.3. Using base ten blocks to explore decimals – see p. 287-8
6.1.4.2. Expanded notation
 1  1 
6.1.4.2.1. 23.85  210   31  8   5

 10   100 
6.1.4.3. Writing a decimal for a fraction
3
6.1.4.3.1.  0.75 - divide 3 by 4 to get the decimal equivalent
4
3 3 x25 75
6.1.4.3.2. 

 0.75 - use the Fundamental Law of Fractions
4 4x25 100
6.1.4.3.3. terminating decimals – rational numbers that have a finite number of
4
decimal places when written as decimals:  0.8
5
6.1.4.3.4. repeating decimals – rational numbers that have an infinite number of
decimal places filled by the same number or a fixed number of digits repeated
over an infinite number of decimal places:
1
 0.33333   0. 3
6.1.4.3.4.1.
3
5
 0.454545  0.45
6.1.4.3.4.2.
11
6.1.4.3.4.3.
Generalization about decimals for rational numbers: Every
rational number can be expressed as a terminating or a repeating decimal
6.1.5. Problems and Exercises p. 291
6.1.5.1. Home work: 1-3, 7, 8ac, 9abc, 10c, 11, 12, 14, 19, 22