1. Notes
7th Grade Math
McDowell
Chapter 3

2. Order of Operations 9/11 arenthesis xponents ultiplication ivision
PEMDASLR
Please Excuse My Dear Aunt Sally’s Last Request
arenthesis
xponents
ultiplication
ivision
ddition
ubtraction
eft
ight

3. Example 2+4 2 Simplify the top of the fraction 1st 6 2 Then divide 3
Parenthesis
Not just parenthesis
Any grouping symbol
Brackets
Fraction bars
Absolute Values
2+4 2
Example
Simplify the top of the fraction 1st 6 2
Then divide 3

4. Simplify all possible exponents

5. Do multiplication and division in order from left to right
Don’t do all multiplication and then all division
Remember division is not commutative

6. Do addition and subtraction in order from left to right
Don’t do all addition and then all subtraction
Remember subtraction is not commutative

7. You Try
– 5  2
48  8 – 1
3[ 9 – (6 – 3)] – 10

8. Base Exponent Exponents 9/11 Show repeated multiplication baseexponent
The number being multiplied
The number of times to multiply the base
Exponent

9. Example 2³
2 x 2 x 2
4 x 2 8

10. Exponential Notation Expanded
When a repeated multiplication problem is written out long
3 x 3 x 3 x 3
Exponential
Notation
When a repeated multiplication problem is written out using powers 34

11. Example
(-2)²
-2 x –2 4
-2²
-1 x 2²
-1 x 2 x 2
-1 x 4 -4

12. Examples
(12 – 3)²  (2² - 1²)
(-a)³ for a = -3
5(2(3)² – 4)³

13. Scientific Notation 9/14 Powers Of Ten Factors 10 10x10 10x10x10
Product
100
1,000
10,000
Power
101
102
103
104
# of 0s 1 2 3 4

14. Fill in the chart You Try Factors 10x10x10x10x10x10x10 Product
10,000,000
100,000,000,000,000
Power
107
1010
# of 0s 10 14

15. A short way to write really big or really small numbers using factors
Scientific
Notation
A short way to write really big or really small numbers using factors
Looks like:
2.4 x 104

16. 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

17. 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

18. Move the decimal 6 hops to the right 4.6 x 106
Convert between scientific notation and expanded notation
Example
Move the decimal 6 hops to the right
4.6 x 106
Rewrite

19. You Try
Write in expanded notation
2.3 x 103
5.76 x 107
Answers
2,300
57,600,000

20. Convert between expanded notation and scientific notation
Example
13,700,000
Figure out how many hops left it takes to get a factor between 1 and 10
1.3,700,000
Rewrite: the number of hops is your exponent
1.3 x 107

21. You Try
Write in scientific notation
340,000,000
98,200
Answers
3.4 x 108
9.82 x 104

22. Prime Numbers Factor Trees and GCF 9/15
Integers greater than one with two positive factors
1 and the original number
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . .

23. 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,

24. Steps Factor Trees A way to factor a number into its prime factors
Is the number prime or composite?
Steps
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?

25. Example prime or composite even or odd divide by 3 3 33
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

26. Example prime or composite even or odd divide by 2 2 6
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

27. You Try
Find the prime factors of 8
2. 15
3. 82
4. 124
5. 26

28. GCF 9/15 GCF Greatest Common Factor
the largest factor two or more numbers have in common.

29. 3. Pick out the prime factors that match
1. Find the prime factors of each number or expression
Steps to
Finding
GCF
2. Compare the factors
3. Pick out the prime factors that match
4. Multiply them together

30. 126 130 2 63 75 2 15 21 5 3 5 3 3 7 The common factors are 2, 3 2 x 3
Find the GCF of 126 and 130
Example
126
130 2 63 75 2 15 21 5 3 5 3 3 7
The common factors are 2, 3
2 x 3
The GCF of 126 and 130 is 6

31. You Try
Work Book
p 47 # 1-9
p 48 # rd

32. LCM 9/16 LCM Least common multiple
The smallest number that is a multiple of both numbers

33. 1. Make a multiplication table for each number
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

34. The LCM of 8 and 3 is 24 Example Find the LCM of 8 and 31 2 3 4 5 6 7 8 16 24 32 40 48 56 64 9 12 15 18 21
1. Make a mult table
2. Compare
3. Find the smallest match
The LCM of 8 and 3 is 24

35. You Try
Find the LCM between
2 and 5
9 and 7

36. Simplifying Fractions 9/16
Simplest form
When the numerator and denominator have no common factors

37. Simplifying fractions
1. Find the GCF between the numerator and denominator
2. Divide both the numerator and denominator of the fraction by that GCF

38. 28s Prime factors: 2, 2, 7 52s Prime factors: 2, 2, 13 28 52  4 = 7
Example
Simplify 28 52
Use a factor tree to find the prime factors of both numbers and then the GCF
28s Prime factors: 2, 2, 7
52s Prime factors: 2, 2, 13
GCF: 2 x 2 4 28 52
 4
= 7 13

39. Write each fraction in simplest form 27/30 12/16
You Try
Write each fraction in simplest form
27/30
12/16

40. Fractions that represent the same amount
Equivalent fractions
Fractions that represent the same amount
½ and 2/4 are equivalent fractions

41. Making
Equivalent
Fractions
1. Pick a number
2. Multiply the numerator and denominator by that same number 5 8
x 3
= 15 24

42. Find 3 equivalent fractions to 6 11
You Try
Find 3 equivalent fractions to 6 11

43. 1. Simplify each fraction
Are the
Fractions
equivalent?
1. Simplify each fraction
2. Compare the simplified fraction
3. If they are the same then they are equivalent

44. You try
Work Book
p 49 #1-17 odd

45. Least common Denominator 9/17
When fractions have the same denominator

46. 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!

48. Make each fraction have a common denominator 5/6, 4/9
Example
Make each fraction have a common denominator
5/6, 4/9
Find the LCM of 6 and 9
Multiply to change each denominator to 18
5 x 3
6 x 3
= 15 18
4 x 2
9 x 2
= 8 18

49. What are the least common denominators? ¼ and 1/3 5/7 and 13/12
You try
What are the least common denominators?
¼ and 1/3
5/7 and 13/12

50. 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)

51. Project Group work! Get into groups of 3 or 4
Pick 3 or 4 different fractions
Each person make a picture of their fraction
Get together as a group and put the fractions/pictures in order from least to greatest
Project

52. You try
Workbook
p 51 #1-17 odd
p 52 #3-36 3rd

53. Mixed Numbers and Improper Fractions 9/18
When the numerator is bigger than the denominator 7 4
Represents more than 1

54. You try

55. 1 + ¾ 1¾ 7 1¾ = 4 Mixed The sum of a whole number and a fraction
Numbers
The sum of a whole number and a fraction
1 + ¾ 1¾ 7 4
1¾ =

56. Multiply, Add, keep the Denominator
MAD face
Converting
A Mixed
Number
To an
Improper
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

57. 6 x 5 + 3 6
30 + 3 6 33 6

58. Convert to an improper fraction
You try
Convert to an improper fraction 1. 2.

59. 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

60. Convert 26 to a mixed number 3
Example
Convert 26 to a mixed number 3 8
R 2
3 26
-24 2

61. Convert each improper fraction to a mixed number
You try
Convert each improper fraction to a mixed number
1. 14 3
2. 25 5

62. Fractions and Decimals 9/21
Terminating
Decimal
a decimal that ends
1.25
Repeating
Decimal
When the same group of numbers continues to repeat forever
4.3

63. Divide the numerator by the denominator
Converting
Fractions To
decimals
Divide the numerator by the denominator 5 16
Insert the decimal and some place holders to divide

64. You try
Convert each fraction to a decimal. Determine if the decimal is a terminating decimal or a repeating decimal 3 5 6

65. Converting
decimals to
fractions
Remember place values

66. Find the place value of the terminating decimal
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

67. The 5 is in the thousandths place so 1000 is the denominator
Example
0.925
925
1000
Simplify
925  25
1000  25 37 40

68. Convert each decimal to a fraction
You try
Convert each decimal to a fraction
0.05
4.7
0.84

69. Also known as the counting numbers
Number Sets 9/22
Whole
Numbers
0, 1, 2, 3,
Natural
Numbers
for short
Also known as the counting numbers
1, 2, 3, 4, . . .

70. Numbers that can be written as fractions
Integers
Positive and negative whole numbers
for short
. . . –2, -1, 0, 1, 2, . . .
Rational
Numbers
Numbers that can be written as fractions
for short
½, ¾, -¼, 1.6, 8, -5.92

71. You Try
Copy and fill in the Venn Diagram that compares Whole Numbers, Natural Numbers, Integers, and Rational Numbers
Whole #s

72. 1. Change each number to a decimal and compare
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

73. Order from least to greatest
You Try
Order from least to greatest
2.7, -0.3, -4/11
-5/6, 2.2, -0.5
2.56, -2.5, 24/10