I. INTRODUCTION
Students life experiences include fractions,1/5
or a pie; three kids share two oranges, etc.
The set of rational numbers, denoted Q, is the set of all numbers of the
form a/b where a and b
are integers and b ≠ 0. The number a is called the ____________________
and the number b is called the _____________________.
We can write this using set builder notation as:
A fraction is any number of the form
a/b where a and b are any numbers (not
necessarily integers),
with
b ≠ 0. (Note that all rational numbers are fractions, but not all fractions are
rational numbers.)
Examples: Classify as fraction, rational number or both.
VENN DIAGRAM:
OF THE REAL
NUMBERS
N: Natural Numbers
W: Whole Numbers
I: Integers
Q: Rational Numbers
R: Real Numbers
TRUE OR FALSE:
II. MODELS FOR FRACTIONS
A. COLORED REGION ( or AREA) MODEL
What fraction is shaded? __________
What fraction is shaded? __________
Shade ¾.
B. SET MODEL
What fraction of the balls are shaded?
Shade ⅓ of the balls.
C. NUMBER LINE MODEL
Shade 4/5 of the unit segment
Place the fol lowing rational numbers on a number line at
the proper intervals:
D. FRACTION STRIP MODEL (See Appendix A-29 in the
Activities Book)
Find strips that can be folded into parts so that the resulting strip is equal
in length to the
given fraction. Folds may be made ONLY on the lines of the strips.
Proper Fractions Improper Fractions
A fraction a/b where
Simply put, a proper fraction re presents a number
greater than _______ and less than _________
Examples:
A fraction a/b where
Simply put, an improper fraction represents a number
greater than or equal to __________
and less than or equal to _________
Examples:
Fraction Activity with Fraction Circles (T101 –
Fractions Activity)
III. EQUIVALENT OR EQUAL FRACTIONS
Notice that the value of the fraction does not change if
its numerator and denominator are
multiplied ( or divided ) by the same non zero whole number.
FUNDAMENTAL LAW OF FRACTIONS
Let a/b be any fraction and n a nonzero
whole number, then  |
We use this law to “build up” fractions (ie. common
denominators) and also to simplify (ie. reduce )
them.
Find the value for x such that :
IV. SIMPLIFYING FRACTIONS
DEFINTION OF SIMPLEST FORM
A rational number a/b is in simplest
form if a and b have no common factor greater than 1, that
is, a and b are relatively prime (that is, the GCD (a, b) = 1). |
Express in Simplest Form
V. EQUALITY OF FRACTIONS
We can use three methods to de termine if two fractions are
equal.
A. Simplify (reduce) both fractions
B. Rewrite both fractions using the Least Common
Denominator (the LCM of the
denominators )
C. Rewrite both fractions with any common denominator
This last method leads us to the following property:
| PROPERTY Two fractions
a/b and
c/d are equal if, and only if, ad = bc |
Use the above property to determine if the following
fractions are equal:
VI. ORDERING FRACTIONS
For some students comparisons are instinctive , based upon life experiences.
A. Like Denominators : It is simple to compare
fractions that have like denominators; just
compare the numerators. The one with the greater numerator is the greater
fraction.
Insert <, >, or =.
B. Unlike Denominators
It is a bit more tricky when the denominators are not the same.
1. AREA MODEL 
2. MAKE SAME DENOMINATORS AND COMPARE
3. USING CROSS- PRODUCTS
| THEOREM
If a, b, c, and d, are integers and b > 0, d > 0,
then
 if, and only if, ad > bc |
EXAMPLE 1:
Order the following fractions from least to greatest :
EXAMPLE 2
VII. DENSENESS OF RATIONAL NUMBERS
DENSENESS PROPERTY FOR RATIONAL NUMBERS
Given rational numbers a/b and
c/d, there is another rational number
between
these two numbers. |
There is no “next” rational number after a given rational
number. Rational numbers are not
discrete
Find one rational number between
2/3 and 1/2.
Find two rational numbers between
7/18 and 1/2.
QUESTION? How many rational numbers lie between ¼ and ¾
(or between any two rational
numbers)?
