Transcript Comparing and Ordering Rational Numbers

PRE-ALGEBRA
Lesson 5-1 Warm-Up
PRE-ALGEBRA
“Comparing and Ordering Rational
Numbers” (5-1)
What is the a
“multiple”?
Multiple: The multiple of a number is the product of that number and any
nonzero number (when you count by a number, you are finding its multiples)
Example: Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36,….
Example: Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54,….
What is the the
“least common
multiple” or
LCM?
Least Common Multiple (LCM): the smallest multiple shared by all of the
numbers
Example: Common Multiples of 4 are 6 are 12, 24, and 36. The smallest
multiple of both numbers , or Least Common Multiple (LCM) is 12.
How do you find
the LCM?
To find the LCM: 1. list the multiples of both numbers until you find the first one
that they share in common, or 2. multiply the greatest power of the factors the
numbers.
Example: Find the LCM of 18, 27, and 36.
Method 1: List the multiples of each number until you find a common one.
Multiples of 18 are 18, 36, 54, 72, 90, 108,…. Find the multiple of each number.
Stop when you find a multiple the
Multiples of 27 are 27, 54, 81, 108,…
numbers share in common.
Multiples of 36 are 36, 72, 108,….
The LCM of 18, 27, and 36 is 108.
PRE-ALGEBRA
“Comparing and Ordering Rational
Numbers” (5-1)
Method 2: Multiply the greatest power of all factors together.
Example: Find the LCM of 18, 27, and 36.
Create a factor
tree number to
find the prime
factors of each
number.
18 = 3 • 3 • 2 = 32 • 2
27 = 3 • 3 • 3 = 33
36 = 3 • 3 • 2 • 2 = 32 • 22
33 • 22 = 27 • 4 = 108
Write each number in
prime factorization. form
Multiply the
greatest powers
of all factors
together.
Example: Find the LCM of 6a2 and 18a3.
LCM of 6a2 and 18a3 is 18a3.
PRE-ALGEBRA
Comparing and Ordering Rational Numbers
LESSON 5-1
Additional Examples
Today, the school’s baseball and soccer teams had
games. The baseball team plays every 7 days. The soccer
team plays every 3 days. When will the teams have games
on the same day again?
7, 14, 21, 28, 35, 42, . . .
List the multiples of 7.
3, 6, 9, 12, 15, 18, 21, . . .
List the multiples of 3.
The LCM is 21. In 21 days both teams will have games again.
PRE-ALGEBRA
Comparing and Ordering Rational Numbers
LESSON 5-1
Additional Examples
Find the LCM of 16 and 36.
16 = 24
36 = 22 • 32
LCM = 24 • 32
= 144
Write the prime factorizations.
Use the greatest power of each factor.
Multiply.
The LCM of 16 and 36 is 144.
PRE-ALGEBRA
Comparing and Ordering Rational Numbers
LESSON 5-1
Additional Examples
Find the LCM of 5a4 and 15a.
5a4 = 5 • a4
15a = 3 • 5 • a
LCM = 3 • 5 • a4
= 15a4
Write the prime factorizations.
Use the greatest power of each factor.
Multiply.
The LCM of 5a4 and 15a is 15a4.
PRE-ALGEBRA
“Comparing and Ordering Rational
Numbers” (5-1)
How do you
compare
fractions?
To compare fractions, you can: 1. use a number line (numbers to the right are
greater than numbers to the left), or 2. compare the numerators (number of
parts) if the denominator (size of the parts) are equal. So, if the denominators
aren’t the same, you need to change one or more of the fractions into equivalent
fractions with a common denominator.
Method 1: Use a number line.
-1
-1
Example: Compare
and
.
2
10
-1
-1
-1
-1
is
on
the
left
of
,
so

.
2
10
10
2
Method 1: Compare the numerators.
Example: Compare 2 and 3 .
3
4
3 • 4 = 12
Multiply the denominators together to find a
common denominator
2 • 4 = 8_
Write equivalent fractions with a denominator of
3 • 4 = 12
12 and compare the numerators (Hint: Notice
3 • 3 = 9_
that you multiply the each fraction by the
4 • 3 = 12
other fractions denominator)
Since
8
9

, then 3  2 .
12
12
4
3
PRE-ALGEBRA
Comparing and Ordering Rational Numbers
LESSON 5-1
Additional Examples
Graph and compare the fractions in each pair.
a. 7 , 3
8
8
3
8
7
8
3
3
7
is on the left, so < .
8
8
8
1
1
b. – , –
3
6
–1 –1
3
6
1
1
– 1 is on the right, so – 6 > – 3 .
6
PRE-ALGEBRA
“Comparing and Ordering Rational
Numbers” (5-1)
What is the “least Least Common Denominator (LCD): the LCM of two or more denominators (in
other words, the smallest common denominator)
common
denominator”
(LCD)?
Example:
List the multiples of each denominator
until you find a multiple that is shared
by both numbers (LCM). LCM = 36
Rewrite the fractions into equivalent
fractions with a denominator of 36
(The LCD is 36). Then, compare the
numerators.
Since
16
15

, then 4  5 .
36
36
9
12
PRE-ALGEBRA
Comparing and Ordering Rational Numbers
LESSON 5-1
Additional Examples
6
The softball team won
of its games and the
7
7
hockey team won 9 of its games. Which team won the
greater fraction of its games?
Step 1
Step 2
Find the LCM of 7 and 9.
7 = 7 and 9 = 32
LCM = 7 • 32 = 63
Write equivalent fractions with a denominator of 63.
6 • 9 54
=
7 • 9 63
7 • 7 49
=
9 • 7 63
Step 3
Compare the fractions.
54
49
6
7
>
, so
>
63
63
7
9
The softball team won the greater fraction of its games.
PRE-ALGEBRA
Comparing and Ordering Rational Numbers
LESSON 5-1
Additional Examples
Order 3 , 1 , and 2 from least to greatest.
7
4
3
3
3 • 12
36
=
=
7
7 • 12
84
1
1 • 21
21
=
=
4
4 • 21
84
The LCM of 7, 4, and 3 is 84.
Use 84 as the common denominator.
2
2 • 28
56
=
=
3
3 • 28
84
21
< 36 < 56 , so 1 < 3 < 2 .
84
84
84
4
7
3
PRE-ALGEBRA
Comparing and Ordering Rational Numbers
LESSON 5-1
Lesson Quiz
Find the LCM of each pair of numbers.
1. 8, 6
2. 12, 16
24
48
3. Compare and order 3 , – 8 , and – 3 from least to greatest.
16
10
16
– 8 < – 3 < 3
10
16
16
PRE-ALGEBRA