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Bell Work:
Write the first 10 prime numbers.
Answer:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
LESSON 10:
RATIONAL NUMBERS
EQUIVALENT FRACTIONS
In the loop is a set of whole
numbers.
Whole Numbers
0, 1 , 2, 3, ….
Is the sum of any two whole numbers
also a whole number?
Is the product of any two whole
numbers also a whole number?
We say that the set of whole
numbers is closed under addition
and multiplication because every
sum or product is a whole number.
Referring to our illustration we
might say that we can find any
sum or product of whole numbers
within the whole numbers loop.
Let us consider subtraction.
If we subtract any two whole
numbers, is the result a whole
number?
3 – 1 is a whole number, but 1 – 3
= -2 which is not a whole number
and therefore outside the loop.
Whole Numbers
0, 1, 2, 3…….
Integers
Integers includes the whole numbers
as well as the negatives of whole
numbers.
The set of integers is closed under
addition, subtraction and
multiplication. Every sum, difference,
and product can be found inside the
integers loop.
Now we will consider division.
If we divide any two integers, is
the result an integer?
The divisions 1÷2, 3÷2, 3÷4 and
many others all have quotients
that are fractions and not
integers. These quotients are
examples of rational numbers.
Whole Numbers
0, 1, 2, 3……
Integers -2
Rational Numbers ½
Rational Numbers*: All numbers
that can be written as a ratio of
two integers.
⅗ and 37 are rational numbers.
√2 and π are not rational numbers.
On the number line, rational
numbers include the integers as
well as many points between the
integers.
Every integer can be expressed as a
ratio of two integers. Therefore,
every integer is a rational number.
Only some rational numbers can be
expressed as a whole number or a
negative of a whole number.
Therefore, only some rational
numbers are integers.
Example:
Describe each number as a whole
number, an integer, or a rational
number. Write every term that
applies. Then graph on a number
line.
-3
2
¾
Answer:
-3: integer and rational number
2: whole number, integer, and
rational number
¾: rational number
One way to express a rational
number is as a fraction. Many
different fractions can name the
same number. These are
examples of equivalent fractions.
4/8
3/6
2/4
1/2
Equivalent Fractions*: Different
fractions that name the same
amount.
2/4 = ½ = 4/8 = 3/6 = 5/10
We can reduce fractions by
removing pairs of factors that the
numerator and denominator have
in common. By doing this we get
the fraction in simplest form.
12/18 = 4/6 = 2/3
We divide 12/18 by 6 to get 2/3 or
the simplest form. This means that 6
is the greatest common factor.
Greatest Common Factor*: The
largest whole number that is a factor
of every number in the set.
In this lesson we will practice
reducing fractions by first writing
the prime factorization of the
terms of the fraction. Then we
remove pairs of like terms from
the numerator and denominator
and simplify.
Example:
Using prime factorization, reduce
72
108
Answer:
72 = 2 x 2 x 2 x 3 x 3
108 2 x 2 x 3 x 3 x 3
= 2/3
Each identical pair of factors from
the numerator and denominator
reduces to 1 over 1.
Notice that a fraction equals 1 if
the numerator and denominator
are equal. Multiplying by a fraction
equal to 1 does not change the
size of the fraction, but it changes
the name of the fraction.
What property of multiplication
are we using when we multiply by
a fraction equal to 1?
Answer:
Identity Property of Multiplication
Example:
Write a fraction equivalent to ½
that has a denominator of 100.
Answer:
50/100
Improper Fraction*: A fraction
equal to or greater than 1.
5/2, 4/4, 10/3, 12/6
Mixed Number*: A whole number
plus a fraction.
3 ½, 6 ¾,
2 2/3
Example:
Express each improper fraction as
a whole or mixed number.
10/3
12/6
Answer:
10/3 = 3/3 + 3/3 + 3/3 + 1/3
= 3 1/3
12/6 = 6/6 + 6/6
=2
HW: Lesson 10 #1-30
Due Next Time