Chapter 6: Rational Number Operations and Properties
6.1 Rational Number Ideas and Symbols
Rational Number Uses and Models
Models for rational numbers
Used to describe a quantity between 0 and 1
identify the whole representing the numeral 1
separate the whole into equal parts
use an ordered pair of integers to describe the portion of the whole under consideration
Identifying the whole and separating it into equal parts
Egg carton fractions
Integer Rods
W
R
L
P
Y
G
K
N
B
E
W
R
L
W
W W
R
W W W
R W
W
R
L
P
http://nlvm.usu.edu/en/nav/topic_t_1.html
Make your own fraction kit
http://education.ucf.edu/mathed/fk.cfm
Using two integers to describe part of a whole
Need more language to describe part-whole relationship
number of pieces of interest vs. number of pieces found in the original whole
Defining Rational Numbers
Definition of a rational number: A number is a rational number if and only if it can be
a
a
represented by a pair of integers, , where b  0 and
represents the quotient a  b
b
b
Using Fractions to Represent Rational Numbers
Fractions and Equivalent Fractions
a
Definition of a fraction: A fraction is a symbol, , where a and b are numbers and b  0.
b
Here, a is the numerator of the fraction and b is the denominator of the fraction
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
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)
Definition of equivalent fractions : Two fractions,
a
c
and , are equivalent fractions if and
b
d
only if ad = bc
Paper folding model
Using fractions to represent rational numbers
every rational number can be represented by an integer in the numerator and the denominator
0.25 1
 in a ratio
sometimes rational numbers are represented by non-integers
0.50 2
a
a ac
The Fundamental Law of Fractions: Given a fraction
and a number c  0, 
b
b bc
Fractions in simplest form
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.
Finding equivalent fractions on the number line
Folding paper
Using a calculator
Using Integer rods
Using Decimals to Represent Rational Numbers
Decimals
Description of a decimal: A decimal is a symbol that uses a base-ten place-value system
with tenths and multiples of tenths to represent rational numbers
decimal point divides the decimal portion of the number from the whole number portion of the
number
Using base ten blocks to explore decimals – see p. 307, 309
Expanded notation
 1  1 
23.85  210   31  8   5

 10   100 
Writing a decimal for a fraction
3
 0.75 - divide 3 by 4 to get the decimal equivalent
4
3 3 x25  75


 0.75 - use the Fundamental Law of Fractions
4 4 x25  100
terminating decimals – rational numbers that have a finite number of decimal places when
4
written as decimals:  0.8
5
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
3
5
 0.454545   0.45
11
Generalization about decimals for rational numbers: Every rational number can be
expressed as a terminating or a repeating decimal
Problems and Exercises p. 311
Home work: 1-3, 7, 8ac, 9abc, 10c, 11, 12, 14, 19, 22