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Week 1 BASIC MATH REVIEW
SAT Prep
I BASIC ARITHMETIC
A) Sets means ldquobelongs tordquo or ldquois a member ofrdquo
If C is the set of prime numbers then
= in either one or the other or both
= in both
Given X = 2 3 5 7 Y = 1 3 5 7 9
= 1 2 3 5 7 9
= 3 5 7
17 C
X Y
X Y
B) Signed Numbers and Operations
- + 0 | a | = distance a is from 0 (NEVER NEGATIVE)
| a | = 7 then a =
| a ndash 2 | = 7 then a =
7
2 7 or 2 7 a a
5 or 9
Add + sum totalSub - differenceMult x ( ) productDiv divide quotient
Zero Property if ab = 0 then a = 0 or b = 0
Ex What is the product of the integers -3 -2 -1 hellip3 4 5
0
1( ) 1a
a
ProdQuot of EVEN number of negatives is POSITIVEProdQuot of ODD number of negatives is NEGATIVE
Reciprocals
Sign Changes 2 + (-6) = 2 ndash 6 = -42 ndash (-6) = 2 + 6 = 8
C) INTEGERSNumber ne Integer
Consecutive integers x x+1 x+2 x+3 hellip Consecutive oddeven integers x x+2 x+4 hellip
Ex The sum of three consecutive integers is less than 75 What is the largest possible value of the first of the three consecutive integers 1 2 75 x x x
3 3 75 x3 72x
24x
2 26 x
D) FACTORS
Factors ndash Finite set of values - Know your ldquogoes intosrdquoMultiples ndash Infinite set of values
GCF and LCM ndash use product of primes
GCF = product of common primes(1 time)LCM = product of common primes(1 time) and
uncommon primes
Ex Find the GCF and the LCM of 32 and 44
32
2 16
2 8
2 4
2 2
44
11 4
2 2
32 2 2 2 2 2
44 2 2 11
2 2 4 GCF
2 2 2 2 2 11 352 LCM
E) EXPONENTS and ROOTS
Rules of exponents 3 + 3 + 3 + 3 = 4(3)3(3)(3)(3) = 34
0
1
1
b c b c
cb bc
bb c
c
a
a a
a a a
a a
aa
a
Ex If find the value of x
52 2
5
x
x
2 32x
Ex Which of the following if any are true
10
10
10 10
I -2 0
II - -2 0
III 2 (-2 ) 0
Always Negative ---FALSE
Always Positive ---TRUE
Always Positive ---TRUE
II and III
F) SQUARES and SQUARE ROOTS
Know your squaressquare roots 1-151-225Know exact values always + 2 answers +-
2aa
Ex Find the circumference of a circle whose area is 20π
2A r2
2
20
20
20
2 5
r
r
r
r
2
2 2 5 4 5
C r
C
Rules of radicals
Ex Evaluate the following
A) B)
1
3832
1
1nn
nn
ab a b
a a
b b
aa
a a
3
1 1
2 8 3 8 2
G) PEMDAS
Careful with negatives on calculator
H) INEQUALITIES
or x a - SWITCH SIGN
Ex Solve for a if 5 ndash 3a lt 10
5 3 10a 3 5a
3 5
3 3
a
5
3a
Properties of Inequalities between 0 and 1
1 1 if
if 0 1
1 if 0 1
b c
b ca a
a a a
a aa
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
I BASIC ARITHMETIC
A) Sets means ldquobelongs tordquo or ldquois a member ofrdquo
If C is the set of prime numbers then
= in either one or the other or both
= in both
Given X = 2 3 5 7 Y = 1 3 5 7 9
= 1 2 3 5 7 9
= 3 5 7
17 C
X Y
X Y
B) Signed Numbers and Operations
- + 0 | a | = distance a is from 0 (NEVER NEGATIVE)
| a | = 7 then a =
| a ndash 2 | = 7 then a =
7
2 7 or 2 7 a a
5 or 9
Add + sum totalSub - differenceMult x ( ) productDiv divide quotient
Zero Property if ab = 0 then a = 0 or b = 0
Ex What is the product of the integers -3 -2 -1 hellip3 4 5
0
1( ) 1a
a
ProdQuot of EVEN number of negatives is POSITIVEProdQuot of ODD number of negatives is NEGATIVE
Reciprocals
Sign Changes 2 + (-6) = 2 ndash 6 = -42 ndash (-6) = 2 + 6 = 8
C) INTEGERSNumber ne Integer
Consecutive integers x x+1 x+2 x+3 hellip Consecutive oddeven integers x x+2 x+4 hellip
Ex The sum of three consecutive integers is less than 75 What is the largest possible value of the first of the three consecutive integers 1 2 75 x x x
3 3 75 x3 72x
24x
2 26 x
D) FACTORS
Factors ndash Finite set of values - Know your ldquogoes intosrdquoMultiples ndash Infinite set of values
GCF and LCM ndash use product of primes
GCF = product of common primes(1 time)LCM = product of common primes(1 time) and
uncommon primes
Ex Find the GCF and the LCM of 32 and 44
32
2 16
2 8
2 4
2 2
44
11 4
2 2
32 2 2 2 2 2
44 2 2 11
2 2 4 GCF
2 2 2 2 2 11 352 LCM
E) EXPONENTS and ROOTS
Rules of exponents 3 + 3 + 3 + 3 = 4(3)3(3)(3)(3) = 34
0
1
1
b c b c
cb bc
bb c
c
a
a a
a a a
a a
aa
a
Ex If find the value of x
52 2
5
x
x
2 32x
Ex Which of the following if any are true
10
10
10 10
I -2 0
II - -2 0
III 2 (-2 ) 0
Always Negative ---FALSE
Always Positive ---TRUE
Always Positive ---TRUE
II and III
F) SQUARES and SQUARE ROOTS
Know your squaressquare roots 1-151-225Know exact values always + 2 answers +-
2aa
Ex Find the circumference of a circle whose area is 20π
2A r2
2
20
20
20
2 5
r
r
r
r
2
2 2 5 4 5
C r
C
Rules of radicals
Ex Evaluate the following
A) B)
1
3832
1
1nn
nn
ab a b
a a
b b
aa
a a
3
1 1
2 8 3 8 2
G) PEMDAS
Careful with negatives on calculator
H) INEQUALITIES
or x a - SWITCH SIGN
Ex Solve for a if 5 ndash 3a lt 10
5 3 10a 3 5a
3 5
3 3
a
5
3a
Properties of Inequalities between 0 and 1
1 1 if
if 0 1
1 if 0 1
b c
b ca a
a a a
a aa
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
B) Signed Numbers and Operations
- + 0 | a | = distance a is from 0 (NEVER NEGATIVE)
| a | = 7 then a =
| a ndash 2 | = 7 then a =
7
2 7 or 2 7 a a
5 or 9
Add + sum totalSub - differenceMult x ( ) productDiv divide quotient
Zero Property if ab = 0 then a = 0 or b = 0
Ex What is the product of the integers -3 -2 -1 hellip3 4 5
0
1( ) 1a
a
ProdQuot of EVEN number of negatives is POSITIVEProdQuot of ODD number of negatives is NEGATIVE
Reciprocals
Sign Changes 2 + (-6) = 2 ndash 6 = -42 ndash (-6) = 2 + 6 = 8
C) INTEGERSNumber ne Integer
Consecutive integers x x+1 x+2 x+3 hellip Consecutive oddeven integers x x+2 x+4 hellip
Ex The sum of three consecutive integers is less than 75 What is the largest possible value of the first of the three consecutive integers 1 2 75 x x x
3 3 75 x3 72x
24x
2 26 x
D) FACTORS
Factors ndash Finite set of values - Know your ldquogoes intosrdquoMultiples ndash Infinite set of values
GCF and LCM ndash use product of primes
GCF = product of common primes(1 time)LCM = product of common primes(1 time) and
uncommon primes
Ex Find the GCF and the LCM of 32 and 44
32
2 16
2 8
2 4
2 2
44
11 4
2 2
32 2 2 2 2 2
44 2 2 11
2 2 4 GCF
2 2 2 2 2 11 352 LCM
E) EXPONENTS and ROOTS
Rules of exponents 3 + 3 + 3 + 3 = 4(3)3(3)(3)(3) = 34
0
1
1
b c b c
cb bc
bb c
c
a
a a
a a a
a a
aa
a
Ex If find the value of x
52 2
5
x
x
2 32x
Ex Which of the following if any are true
10
10
10 10
I -2 0
II - -2 0
III 2 (-2 ) 0
Always Negative ---FALSE
Always Positive ---TRUE
Always Positive ---TRUE
II and III
F) SQUARES and SQUARE ROOTS
Know your squaressquare roots 1-151-225Know exact values always + 2 answers +-
2aa
Ex Find the circumference of a circle whose area is 20π
2A r2
2
20
20
20
2 5
r
r
r
r
2
2 2 5 4 5
C r
C
Rules of radicals
Ex Evaluate the following
A) B)
1
3832
1
1nn
nn
ab a b
a a
b b
aa
a a
3
1 1
2 8 3 8 2
G) PEMDAS
Careful with negatives on calculator
H) INEQUALITIES
or x a - SWITCH SIGN
Ex Solve for a if 5 ndash 3a lt 10
5 3 10a 3 5a
3 5
3 3
a
5
3a
Properties of Inequalities between 0 and 1
1 1 if
if 0 1
1 if 0 1
b c
b ca a
a a a
a aa
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
Add + sum totalSub - differenceMult x ( ) productDiv divide quotient
Zero Property if ab = 0 then a = 0 or b = 0
Ex What is the product of the integers -3 -2 -1 hellip3 4 5
0
1( ) 1a
a
ProdQuot of EVEN number of negatives is POSITIVEProdQuot of ODD number of negatives is NEGATIVE
Reciprocals
Sign Changes 2 + (-6) = 2 ndash 6 = -42 ndash (-6) = 2 + 6 = 8
C) INTEGERSNumber ne Integer
Consecutive integers x x+1 x+2 x+3 hellip Consecutive oddeven integers x x+2 x+4 hellip
Ex The sum of three consecutive integers is less than 75 What is the largest possible value of the first of the three consecutive integers 1 2 75 x x x
3 3 75 x3 72x
24x
2 26 x
D) FACTORS
Factors ndash Finite set of values - Know your ldquogoes intosrdquoMultiples ndash Infinite set of values
GCF and LCM ndash use product of primes
GCF = product of common primes(1 time)LCM = product of common primes(1 time) and
uncommon primes
Ex Find the GCF and the LCM of 32 and 44
32
2 16
2 8
2 4
2 2
44
11 4
2 2
32 2 2 2 2 2
44 2 2 11
2 2 4 GCF
2 2 2 2 2 11 352 LCM
E) EXPONENTS and ROOTS
Rules of exponents 3 + 3 + 3 + 3 = 4(3)3(3)(3)(3) = 34
0
1
1
b c b c
cb bc
bb c
c
a
a a
a a a
a a
aa
a
Ex If find the value of x
52 2
5
x
x
2 32x
Ex Which of the following if any are true
10
10
10 10
I -2 0
II - -2 0
III 2 (-2 ) 0
Always Negative ---FALSE
Always Positive ---TRUE
Always Positive ---TRUE
II and III
F) SQUARES and SQUARE ROOTS
Know your squaressquare roots 1-151-225Know exact values always + 2 answers +-
2aa
Ex Find the circumference of a circle whose area is 20π
2A r2
2
20
20
20
2 5
r
r
r
r
2
2 2 5 4 5
C r
C
Rules of radicals
Ex Evaluate the following
A) B)
1
3832
1
1nn
nn
ab a b
a a
b b
aa
a a
3
1 1
2 8 3 8 2
G) PEMDAS
Careful with negatives on calculator
H) INEQUALITIES
or x a - SWITCH SIGN
Ex Solve for a if 5 ndash 3a lt 10
5 3 10a 3 5a
3 5
3 3
a
5
3a
Properties of Inequalities between 0 and 1
1 1 if
if 0 1
1 if 0 1
b c
b ca a
a a a
a aa
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
1( ) 1a
a
ProdQuot of EVEN number of negatives is POSITIVEProdQuot of ODD number of negatives is NEGATIVE
Reciprocals
Sign Changes 2 + (-6) = 2 ndash 6 = -42 ndash (-6) = 2 + 6 = 8
C) INTEGERSNumber ne Integer
Consecutive integers x x+1 x+2 x+3 hellip Consecutive oddeven integers x x+2 x+4 hellip
Ex The sum of three consecutive integers is less than 75 What is the largest possible value of the first of the three consecutive integers 1 2 75 x x x
3 3 75 x3 72x
24x
2 26 x
D) FACTORS
Factors ndash Finite set of values - Know your ldquogoes intosrdquoMultiples ndash Infinite set of values
GCF and LCM ndash use product of primes
GCF = product of common primes(1 time)LCM = product of common primes(1 time) and
uncommon primes
Ex Find the GCF and the LCM of 32 and 44
32
2 16
2 8
2 4
2 2
44
11 4
2 2
32 2 2 2 2 2
44 2 2 11
2 2 4 GCF
2 2 2 2 2 11 352 LCM
E) EXPONENTS and ROOTS
Rules of exponents 3 + 3 + 3 + 3 = 4(3)3(3)(3)(3) = 34
0
1
1
b c b c
cb bc
bb c
c
a
a a
a a a
a a
aa
a
Ex If find the value of x
52 2
5
x
x
2 32x
Ex Which of the following if any are true
10
10
10 10
I -2 0
II - -2 0
III 2 (-2 ) 0
Always Negative ---FALSE
Always Positive ---TRUE
Always Positive ---TRUE
II and III
F) SQUARES and SQUARE ROOTS
Know your squaressquare roots 1-151-225Know exact values always + 2 answers +-
2aa
Ex Find the circumference of a circle whose area is 20π
2A r2
2
20
20
20
2 5
r
r
r
r
2
2 2 5 4 5
C r
C
Rules of radicals
Ex Evaluate the following
A) B)
1
3832
1
1nn
nn
ab a b
a a
b b
aa
a a
3
1 1
2 8 3 8 2
G) PEMDAS
Careful with negatives on calculator
H) INEQUALITIES
or x a - SWITCH SIGN
Ex Solve for a if 5 ndash 3a lt 10
5 3 10a 3 5a
3 5
3 3
a
5
3a
Properties of Inequalities between 0 and 1
1 1 if
if 0 1
1 if 0 1
b c
b ca a
a a a
a aa
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
C) INTEGERSNumber ne Integer
Consecutive integers x x+1 x+2 x+3 hellip Consecutive oddeven integers x x+2 x+4 hellip
Ex The sum of three consecutive integers is less than 75 What is the largest possible value of the first of the three consecutive integers 1 2 75 x x x
3 3 75 x3 72x
24x
2 26 x
D) FACTORS
Factors ndash Finite set of values - Know your ldquogoes intosrdquoMultiples ndash Infinite set of values
GCF and LCM ndash use product of primes
GCF = product of common primes(1 time)LCM = product of common primes(1 time) and
uncommon primes
Ex Find the GCF and the LCM of 32 and 44
32
2 16
2 8
2 4
2 2
44
11 4
2 2
32 2 2 2 2 2
44 2 2 11
2 2 4 GCF
2 2 2 2 2 11 352 LCM
E) EXPONENTS and ROOTS
Rules of exponents 3 + 3 + 3 + 3 = 4(3)3(3)(3)(3) = 34
0
1
1
b c b c
cb bc
bb c
c
a
a a
a a a
a a
aa
a
Ex If find the value of x
52 2
5
x
x
2 32x
Ex Which of the following if any are true
10
10
10 10
I -2 0
II - -2 0
III 2 (-2 ) 0
Always Negative ---FALSE
Always Positive ---TRUE
Always Positive ---TRUE
II and III
F) SQUARES and SQUARE ROOTS
Know your squaressquare roots 1-151-225Know exact values always + 2 answers +-
2aa
Ex Find the circumference of a circle whose area is 20π
2A r2
2
20
20
20
2 5
r
r
r
r
2
2 2 5 4 5
C r
C
Rules of radicals
Ex Evaluate the following
A) B)
1
3832
1
1nn
nn
ab a b
a a
b b
aa
a a
3
1 1
2 8 3 8 2
G) PEMDAS
Careful with negatives on calculator
H) INEQUALITIES
or x a - SWITCH SIGN
Ex Solve for a if 5 ndash 3a lt 10
5 3 10a 3 5a
3 5
3 3
a
5
3a
Properties of Inequalities between 0 and 1
1 1 if
if 0 1
1 if 0 1
b c
b ca a
a a a
a aa
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
D) FACTORS
Factors ndash Finite set of values - Know your ldquogoes intosrdquoMultiples ndash Infinite set of values
GCF and LCM ndash use product of primes
GCF = product of common primes(1 time)LCM = product of common primes(1 time) and
uncommon primes
Ex Find the GCF and the LCM of 32 and 44
32
2 16
2 8
2 4
2 2
44
11 4
2 2
32 2 2 2 2 2
44 2 2 11
2 2 4 GCF
2 2 2 2 2 11 352 LCM
E) EXPONENTS and ROOTS
Rules of exponents 3 + 3 + 3 + 3 = 4(3)3(3)(3)(3) = 34
0
1
1
b c b c
cb bc
bb c
c
a
a a
a a a
a a
aa
a
Ex If find the value of x
52 2
5
x
x
2 32x
Ex Which of the following if any are true
10
10
10 10
I -2 0
II - -2 0
III 2 (-2 ) 0
Always Negative ---FALSE
Always Positive ---TRUE
Always Positive ---TRUE
II and III
F) SQUARES and SQUARE ROOTS
Know your squaressquare roots 1-151-225Know exact values always + 2 answers +-
2aa
Ex Find the circumference of a circle whose area is 20π
2A r2
2
20
20
20
2 5
r
r
r
r
2
2 2 5 4 5
C r
C
Rules of radicals
Ex Evaluate the following
A) B)
1
3832
1
1nn
nn
ab a b
a a
b b
aa
a a
3
1 1
2 8 3 8 2
G) PEMDAS
Careful with negatives on calculator
H) INEQUALITIES
or x a - SWITCH SIGN
Ex Solve for a if 5 ndash 3a lt 10
5 3 10a 3 5a
3 5
3 3
a
5
3a
Properties of Inequalities between 0 and 1
1 1 if
if 0 1
1 if 0 1
b c
b ca a
a a a
a aa
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
Ex Find the GCF and the LCM of 32 and 44
32
2 16
2 8
2 4
2 2
44
11 4
2 2
32 2 2 2 2 2
44 2 2 11
2 2 4 GCF
2 2 2 2 2 11 352 LCM
E) EXPONENTS and ROOTS
Rules of exponents 3 + 3 + 3 + 3 = 4(3)3(3)(3)(3) = 34
0
1
1
b c b c
cb bc
bb c
c
a
a a
a a a
a a
aa
a
Ex If find the value of x
52 2
5
x
x
2 32x
Ex Which of the following if any are true
10
10
10 10
I -2 0
II - -2 0
III 2 (-2 ) 0
Always Negative ---FALSE
Always Positive ---TRUE
Always Positive ---TRUE
II and III
F) SQUARES and SQUARE ROOTS
Know your squaressquare roots 1-151-225Know exact values always + 2 answers +-
2aa
Ex Find the circumference of a circle whose area is 20π
2A r2
2
20
20
20
2 5
r
r
r
r
2
2 2 5 4 5
C r
C
Rules of radicals
Ex Evaluate the following
A) B)
1
3832
1
1nn
nn
ab a b
a a
b b
aa
a a
3
1 1
2 8 3 8 2
G) PEMDAS
Careful with negatives on calculator
H) INEQUALITIES
or x a - SWITCH SIGN
Ex Solve for a if 5 ndash 3a lt 10
5 3 10a 3 5a
3 5
3 3
a
5
3a
Properties of Inequalities between 0 and 1
1 1 if
if 0 1
1 if 0 1
b c
b ca a
a a a
a aa
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
E) EXPONENTS and ROOTS
Rules of exponents 3 + 3 + 3 + 3 = 4(3)3(3)(3)(3) = 34
0
1
1
b c b c
cb bc
bb c
c
a
a a
a a a
a a
aa
a
Ex If find the value of x
52 2
5
x
x
2 32x
Ex Which of the following if any are true
10
10
10 10
I -2 0
II - -2 0
III 2 (-2 ) 0
Always Negative ---FALSE
Always Positive ---TRUE
Always Positive ---TRUE
II and III
F) SQUARES and SQUARE ROOTS
Know your squaressquare roots 1-151-225Know exact values always + 2 answers +-
2aa
Ex Find the circumference of a circle whose area is 20π
2A r2
2
20
20
20
2 5
r
r
r
r
2
2 2 5 4 5
C r
C
Rules of radicals
Ex Evaluate the following
A) B)
1
3832
1
1nn
nn
ab a b
a a
b b
aa
a a
3
1 1
2 8 3 8 2
G) PEMDAS
Careful with negatives on calculator
H) INEQUALITIES
or x a - SWITCH SIGN
Ex Solve for a if 5 ndash 3a lt 10
5 3 10a 3 5a
3 5
3 3
a
5
3a
Properties of Inequalities between 0 and 1
1 1 if
if 0 1
1 if 0 1
b c
b ca a
a a a
a aa
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
Ex If find the value of x
52 2
5
x
x
2 32x
Ex Which of the following if any are true
10
10
10 10
I -2 0
II - -2 0
III 2 (-2 ) 0
Always Negative ---FALSE
Always Positive ---TRUE
Always Positive ---TRUE
II and III
F) SQUARES and SQUARE ROOTS
Know your squaressquare roots 1-151-225Know exact values always + 2 answers +-
2aa
Ex Find the circumference of a circle whose area is 20π
2A r2
2
20
20
20
2 5
r
r
r
r
2
2 2 5 4 5
C r
C
Rules of radicals
Ex Evaluate the following
A) B)
1
3832
1
1nn
nn
ab a b
a a
b b
aa
a a
3
1 1
2 8 3 8 2
G) PEMDAS
Careful with negatives on calculator
H) INEQUALITIES
or x a - SWITCH SIGN
Ex Solve for a if 5 ndash 3a lt 10
5 3 10a 3 5a
3 5
3 3
a
5
3a
Properties of Inequalities between 0 and 1
1 1 if
if 0 1
1 if 0 1
b c
b ca a
a a a
a aa
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
F) SQUARES and SQUARE ROOTS
Know your squaressquare roots 1-151-225Know exact values always + 2 answers +-
2aa
Ex Find the circumference of a circle whose area is 20π
2A r2
2
20
20
20
2 5
r
r
r
r
2
2 2 5 4 5
C r
C
Rules of radicals
Ex Evaluate the following
A) B)
1
3832
1
1nn
nn
ab a b
a a
b b
aa
a a
3
1 1
2 8 3 8 2
G) PEMDAS
Careful with negatives on calculator
H) INEQUALITIES
or x a - SWITCH SIGN
Ex Solve for a if 5 ndash 3a lt 10
5 3 10a 3 5a
3 5
3 3
a
5
3a
Properties of Inequalities between 0 and 1
1 1 if
if 0 1
1 if 0 1
b c
b ca a
a a a
a aa
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
Rules of radicals
Ex Evaluate the following
A) B)
1
3832
1
1nn
nn
ab a b
a a
b b
aa
a a
3
1 1
2 8 3 8 2
G) PEMDAS
Careful with negatives on calculator
H) INEQUALITIES
or x a - SWITCH SIGN
Ex Solve for a if 5 ndash 3a lt 10
5 3 10a 3 5a
3 5
3 3
a
5
3a
Properties of Inequalities between 0 and 1
1 1 if
if 0 1
1 if 0 1
b c
b ca a
a a a
a aa
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
G) PEMDAS
Careful with negatives on calculator
H) INEQUALITIES
or x a - SWITCH SIGN
Ex Solve for a if 5 ndash 3a lt 10
5 3 10a 3 5a
3 5
3 3
a
5
3a
Properties of Inequalities between 0 and 1
1 1 if
if 0 1
1 if 0 1
b c
b ca a
a a a
a aa
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
Properties of Inequalities between 0 and 1
1 1 if
if 0 1
1 if 0 1
b c
b ca a
a a a
a aa
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
A) Fractions and Decimals
ComparingDecimals ndash same of places ndash larger is
larger number
Fractions ndash same denom ndash bigger numer is larger fraction
same numer ndash smaller denom is smaller fraction
II RATIONAL NUMBERS
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
Powers of 10 - move decimal place ndash + exponent ndash move to right
ndash exponent ndash move to the left
Rounding - know rules- do not need to round grid-ins
Operations with fractionsAddsub like denominatorsMultiply numerators and denominatorsDivide multiply by reciprocals
ldquoofrdquo means multiply
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
B) Percents
Ex Find the following
1) 45 of 200
is
of 100
45
200 100
x
100 45(200)x
90x
90
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
2) 90 is 45 of what number
3) 90 is what percent of 200
90
200 100
x
90 45
100
x 45 90 100x
200x
200
200 90 100x
45x
45
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
Percent increasedecrease IncreaseDecrease = ldquoisrdquoOriginal = ldquoofrdquo
C) Ratio and Proportion
Part Part compared to Part Whole
73 710 or 310
Two numbers in a ratio of a to b ndash Apply an x to each
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
Ex Two acute angles of a right triangle are in a ratio of 513 What is the measure of the larger acute angle
Proportions Cross multiply Ex Tommy types 35 words per minute How long would it take Tommy to type 987 words
5 13 90 x x
18 90x
5x 13 13 5 65 x
35 987
1
x
35 987x
282x28 min 12 secx
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
Direct Variation vs Inverse Variation orDirectly Proportional vs Inversely Proportional
x uarr y uarr x uarr y darrx darr y darr x darr y uarr
y = kx y = kx or k = xy
Ex If y is directly proportional to x and y = 7 when x = 2 find the value of y when x is 5
35 k
y kx
7 2 k175y
35y x
35 5y
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
Ex If it takes 4 workers 2 hours to complete a certain job how many minutes would it take 6 workers to complete the same job
480k
k
yx
120(4)k
80y1204
k
480y
x
480
6y
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
III AVERAGES
A) Average = Mean = Sumcount
Know the average then sum = average x count Ex After 5 tests Tony has an 85 average If Tony had an 83
average after his first three tests what was his average on the last two tests
5 5 85 425T 425 24987
2A
3 3 83 249T
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
B) Median = Middle
ARRANGE THEM IN ORDER FIRST
Odd of scores - Find middle Even of scores - Find average of middle 2 rsquos
C) Mode = Number that occurs the MOST
The End
The End