Transcript Document

Notes
7th Grade Math
McDowell
Order of Operations
9/11
PEMDASLR
Please Excuse My Dear Aunt Sally’s Last Request
Parenthesis
– Not just parenthesis
• Any grouping symbol
– Brackets
– Fraction bars
– Absolute Values
Example
2+4
2
6
2
3
Simplify the top of the
fraction 1st
Then divide
Exponents
Simplify all possible exponents
Multiplication
And
Division
Do multiplication and division
in order from left to right
Don’t do all multiplication
and then all division
Remember division is not
commutative
Addition
And
Subtraction
Do addition and subtraction in
order from left to right
Don’t do all addition and then
all subtraction
Remember subtraction is not
commutative
You Try
1. 3 + 15 – 5  2
2. 48  8 – 1
3. 3[ 9 – (6 – 3)] – 10
4. 16 + 24
30 - 22
Exponents
Exponents
9/11
Show repeated multiplication
baseexponent
Base
The number being multiplied
Exponent
The number of times to
multiply the base
Example
2³
2x2x2
4x2
8
Expanded
Notation
When a repeated
multiplication problem is
written out long
3x3x3x3
Exponential
Notation
When a repeated
multiplication problem is
written out using powers
34
Example
(-2)²
-2 x –2
4
-2²
-1 x 2²
-1 x 2 x 2
-1 x 4
-4
Examples
(12 – 3)²  (2² - 1²)
(-a)³ for a = -3
5(2(3)² – 4)³
Scientific Notation
Powers
Of
Ten
9/14
Factors
10
10x10
10x10x10
10x10x10x10
Product
10
100
1,000
10,000
Power
101
102
103
104
# of 0s
1
2
3
4
You Try
Fill in the chart
Factors
10x10x10x10x10x10x10
Product
10,000,000
Power
107
# of 0s
100,000,000,000,000
1010
10
14
Scientific
Notation
A short way to write really big
or really small numbers using
factors
Looks like:
2.4 x 104
One factor will always be a
power of ten: 10n
The other factor will be less than
10 but greater than one
1 < factor < 10
And will usually have a decimal
The first factor tells us what the
number looks like
The exponent on the ten tells us
how many places to move the
decimal point
Example
Convert between scientific notation
and expanded notation
4.6 x 106 Move the decimal 6
hops to the right
4.600000 Rewrite
4600000
You Try
Write in expanded notation
1. 2.3 x 103
2. 5.76 x 107
Answers
1. 2,300
2. 57,600,000
Example
Convert between expanded notation
and scientific notation
13,700,000 Figure out how many hops
left it takes to get a factor
between 1 and 10
1.3,700,000
1.3 x 107
Rewrite: the number
of hops is your
exponent
You Try
Write in scientific notation
1. 340,000,000
2. 98,200
Answers
1. 3.4 x 108
2. 9.82 x 104
Factor Trees and GCF9/15
Prime
Numbers
Integers greater than one with
two positive factors
1 and the original number
2, 3, 5, 7, 11, 13, 17, 19,
23, 29, . . .
Composite
Numbers
Integers greater than one
with more than two
positive factors
4, 6, 8, 9, 10, 12, 14, 15,
18, 20, 21, 22, 24, . . .
Factor
Trees
Steps
A way to factor a number into its
prime factors
Is the number prime or composite?
If prime: you’re done
If Composite:
Is the number even or odd?
If even: divide by 2
If odd: divide by 3, 5, 7, 11,
13 or another prime number
Write down the prime factor and
the new number
Is the new number prime or
composite?
Example
Find the prime factors of
99
prime or composite
even or odd
divide by 3
3
33 prime or composite
even or odd
divide by 3
3
11 prime or
composite
The prime factors of 99: 3, 3, 11
Example
Find the prime factors of
12
prime or composite
even or odd
divide by 2
2
6 prime or composite
even or odd
divide by 2
2
3 prime or
composite
The prime factors of 12: 2, 2, 3
You Try
Find the prime factors of
1. 8
2. 15
3. 82
4. 124
5. 26
GCF
GCF
9/15
Greatest Common Factor
the largest factor two or
more numbers have in
common.
Steps to
Finding
GCF
1. Find the prime factors of
each number or
expression
2. Compare the factors
3. Pick out the prime
factors that match
4. Multiply them together
Example
Find the GCF of 126 and 130
126
2
130
63
3
75
2
21
15
5
5
3
7
The common factors are 2, 3
2x3
The GCF of 126 and 130 is 6
3
You Try
Work Book
p 47 # 1-9
p 48 # 3-33 3rd
LCM
LCM
9/16
Least common multiple
The smallest number that
is a multiple of both
numbers
Steps
To
Find
LCM
1. Make a multiplication table
for each number
2. Compare the multiplication
tables
3. Pick the smallest number
that both (all) tables have
Example
Find the LCM of 8 and 3
1 2
3
4
5
6
7
8
8 16
24
32
40
48
56
64
1. Make a
mult table
2. Compare
3 6
9
12
15
18
21
24
3. Find the
smallest
match
The LCM of 8 and 3 is 24
You Try
Find the LCM between
1. 2 and 5
2. 9 and 7
Simplifying Fractions
Simplest form
9/16
When the numerator and
denominator have no
common factors
Simplifying
fractions
1. Find the GCF between the
numerator and denominator
2. Divide both the numerator
and denominator of the
fraction by that GCF
Example
Simplify
28
52
28s Prime factors: 2, 2, 7
52s Prime factors: 2, 2, 13
28  4 = 7
52  4 13
Use a factor tree
to find the prime
factors of both
numbers and
then the GCF
GCF: 2 x 2
4
You Try
Write each fraction in
simplest form
1. 27/30
2. 12/16
Equivalent
fractions
Fractions that represent the
same amount
½ and 2/4 are equivalent
fractions
Making
Equivalent
Fractions
1. Pick a number
2. Multiply the numerator and
denominator by that same
number
5 x 3 = 15
8 x 3 24
You Try
Find 3 equivalent fractions
to
6
11
Are the
Fractions
equivalent?
1. Simplify each fraction
2. Compare the simplified
fraction
3. If they are the same then they
are equivalent
You try
Work Book
p 49 #1-17 odd
Least common Denominator
9/17
Common
When fractions have the same
Denominator denominator
Steps to
Making
Common
Denominators
1. Find the LCM of all the
denominators
2. Turn the denominator of
each fraction into that LCM
using multiplication
Remember: what ever you
multiply by on the
bottom, you have to
multiply by on the top!
Example
Make each fraction have a
common denominator
5/6, 4/9 Find the LCM of
6 and 9
6 12 18 24 30 36 42 48
9 18 27 36 45 64 73 82
Multiply to change
5 x 3 = 15
each denominator
6 x 3 18
to 18
4x2=8
9 x 2 18
You try
What are the least
common denominators?
1. ¼ and 1/3
2. 5/7 and 13/12
Comparing
And
Ordering
fractions
Manipulate the fractions so
each has the same
denominator
Compare/order the fractions
using the numerators (the
denominators are the same)
Project
Group work!
1. Get into groups of 3
or 4
2. Pick 3 or 4 different
fractions
3. Each person make a
picture of their
fraction
4. Get together as a
group and put the
fractions/pictures in
order from least to
greatest
You try
Workbook
p 51 #1-17 odd
p 52 #3-36 3rd
Mixed Numbers and Improper Fractions 9/18
Improper
fractions
When the numerator is bigger
than the denominator
7
4
Represents more than 1
You try
http://www.youtube.com/user/M
athRaps#play/uploads/3/VZQ
Dvb5Yjvw
Mixed
Numbers
The sum of a whole number
and a fraction
1+¾
1¾
7
1¾ =
4
Converting
A Mixed
Number
To an
Improper
MAD face
Multiply, Add, keep the Denominator
Multiply the denominator by
the whole number then add
the product to the numerator
That is the new numerator—
keep the old denominator
6x5+3
6
30 + 3
6
33
6
You try
Convert to an improper
fraction
1.
2.
Converting
An
Improper
To a
Mixed #
Divide the numerator by the
dominator
The quotient is the whole
number
The remainder is the new
numerator
Keep the same denominator
Example
Convert 26 to a mixed number
3
8 R2
3 26
-24
2
You try
Convert each improper
fraction to a mixed
number
1. 14
3
2. 25
5
Fractions and Decimals
9/21
Terminating
Decimal
a decimal that ends
Repeating
Decimal
When the same group of numbers
continues to repeat forever
1.25
4.33333333333
4.3
Converting
Fractions
To
decimals
Divide the numerator by the
denominator
5
16
Insert the
decimal and
some place
holders to divide
You try
Convert each fraction to a
decimal. Determine if the
decimal is a terminating
decimal or a repeating
decimal
1. 3
5
2.
1
6
Converting
decimals
to
fractions
Chart
Remember place values
Converting
decimals
to
fractions
Find the place value of the
terminating decimal
Place the numbers after the
decimal over the place value
Keep whole numbers as whole
numbers
Simplify the fraction to lowest
terms
Example
0.925
The 5 is in the thousandths
place so 1000 is the
denominator
925
1000
925  25
1000  25
37
40
Simplify
You try
Convert each decimal to a
fraction
1. 0.05
2. 4.7
3. 0.84
Number Sets
Whole
Numbers
Natural
Numbers
9/22
0, 1, 2, 3, . . .
for short
Also known as the counting
numbers
1, 2, 3, 4, . . .
Integers Positive and negative whole numbers
for short
. . . –2, -1, 0, 1, 2, . . .
Rational Numbers that can be written as
Numbers fractions
for short
½, ¾, -¼, 1.6, 8, -5.92
You Try
Copy and fill in the Venn Diagram that
compares Whole Numbers, Natural
Numbers, Integers, and Rational
Numbers
Whole #s
Ordering
Rational
Numbers
Two Options
1. Change each number to a
decimal and compare
2. Write each number as a
fraction with a common
denominator and compare
You Try
Order from least to greatest
1. 2.7, -0.3, -4/11
2. -5/6, 2.2, -0.5
3. 2.56, -2.5, 24/10