*Expressions* There are four types of

expressions Addition: x + 5, or 5 + x Subtraction: x – 3, or 3 – x

6th Grade Math Notes

*One Step Equations* The goal of an equation is to isolate (get it

by itself) the variable (the unknown value..usually is represented by an “x”).

In order to do that you must do the inverse (opposite operation) which means instead of adding you subtract and instead of multiplying you divide.

*Example #1*

*Check*Instead of x put your answer (4). 4 + 3 does equal 7 so your answer is correct!

*Example #2*

*Check*Instead of x put your answer (13). 13 - 8 does equal 5 so your answer is correct!

*Example #3*

*Check*Instead of x put your answer (7). -7 x -8 does equal 56 so your answer is correct!

*Example #4*

*Check*Instead of x put your answer (24). 24 ÷ 8 does equal 3 so your answer is correct!

*Two Step Equations* The goal of an equation is to isolate (get it

by itself) the variable (the unknown value is always represented by a letter)

In order to do that you must do the inverse (opposite operation) which means instead of adding you subtract and instead of multiplying you divide.

*Example #1*

*Check*Instead of x put your answer (9). (3 x 9) + 7 does equal 34 so your answer is correct.

*Rules*1) Perform the inverse operation2) Check your answer (in order to check you

must put your answer in place of the variable to see if the equation holds true)

One Step Equations

Two Step Equations

*Rules*1) Perform the inverse operation (undo

addition or subtraction first)2) Then undo multiplication or division3) Check your answer (in order to check you

must put your answer in place of the variable to see if the equation holds true)

Rule 1 (undo +/-)

Rule 2 (undo x/÷)

*Example #2*

*Check*Instead of x put your answer (3). (7 x 3) + 1 does equal 22 so your answer is correct.Expressio

ns*Rules*

1) Instead of the variable put the number that the problem tells you it equals.

2) Solve the expression

*Expressions* There are four types of

expressions Addition: x + 5, or 5 + x Subtraction: x – 3, or 3 – x

*Rules*1) Instead of the variable put the number

that the problem tells you it equals.2) Solve the expression

*Example #2*

X - 2

Instead of x put 8

8 - 2 = 6

Answer = 6

*Example #1*

X + 5

Instead of x put 8

8 + 5 = 13

Answer = 13

*Example #3*

5x + 2

Instead of x put 8

(5 x 8) + 2 = 42

Answer = 42

*Example #4*

x/2 - 1

Instead of x put 8

(8/2) - 1 = 3

Answer = 3

*Rules*1) Underline the place value2) Circle the # on the right3) Round up the circled # is 5 or greater.

When you round up the underlined # goes up one # higher

4) Round down if the circled # is 4 or less. When you round down the underlined # stays the same

5) Everything after the underlined # then becomes zeros.

Rounding

*Example #1* (round to nearest thousand)

5,678

5, 6 78 (round up since 6 is 5 or higher)

6,000 (5 becomes a 6 and everything else zeros)

*Example #2* (round to nearest hundredth)

.562

.5 6 2 (round down since 2 is 4or lower)

.560 (the 6 stays the same and everything else zeros)Place

Value

Multiplication

*2 digit multiplication (34 x 23 or 345 x 56)* With 2 digit multiplication use Only Pizza

Tastes Astonishing or OPTA. O = (Ones place) First multiply the # in

the ones place P = (Place Holder) Put one place holder T = (Tens Place) Next multiply the # in the

tens place A = (Add) Then add all of the numbers up

*Multiplying with Decimals Do not line up decimals!!! Multiply Add all of the numbers Count the decimals

*Example #1*

Directions: Solve the expression when x = 8

*example* what is the unit rate if 8 apples cost $16

To find your answer divide (16/8). The unit rate ends up being $2/apple.

*2 digit multiplication (34 x 23 or 345 x 56)* With 2 digit multiplication use Only Pizza

Tastes Astonishing or OPTA. O = (Ones place) First multiply the # in

the ones place P = (Place Holder) Put one place holder T = (Tens Place) Next multiply the # in the

tens place A = (Add) Then add all of the numbers up

*Multiplying with Decimals Do not line up decimals!!! Multiply Add all of the numbers Count the decimals

*Example #1*

*Dividing w/ a decimal in the dividend Bring the decimal up in the dividend Divide You cannot have a remainder..if there is a

remainder you must add a zero, drop it, and continue dividing

*Dividing w/ a decimal in the dividend and divisor

Move the decimals over first Bring up the decimal in the dividend Divide You cannot have a remainder..if there is a

remainder you must add a zero, drop it, and continue dividing

*Example #1**Example #1*

*Rules*1) Line up the decimals!!!2) Add or Subtract

If you have a whole # you must turn it into a decimal first (5 = 5.00, 7 = 7.00)

Adding & Subtraction Decimals

*Ratios* Ratios explain how for every x units of one

item there are x units of another. For example a ratio of 3 to 5 would mean

for every 3 apples there are 5 oranges. A ratio of 2:6 would mean for every 2 girls

there are 6 boys. Rations can be expressed three different

ways: 1 to 3, 1:3, and 1/3.

*Unit Rates* Unit rates compare unlike units to 1 unit. For example how much one item costs,

how many miles a car travels in 1 hour, or how many laps a person runs in 1 day.

Unit Rates

Division

Ratios

Multiplying Fractions

*Rules* Change all mixed #’s and whole #’s into

improper fractions The denominators do not have to be the

same Simply multiply across (numerator x

numerator…denominator x denominator) Make sure your answer is a proper

fraction and in simplest form..

*Rules* If there are mixed #’s or whole #’s you

must first change them into improper fractions

Change the division sign into a multiplication sign

Use the reciprocal or flip the second fraction upside down

Multiply Make sure your answer is a proper

fraction and in simplest form

*Rules* If the denominators are not the same then

you must first make them the same Add or subtract (borrow if you have do) Make sure your answer is a proper

fraction and in simplest form.

Subtracting Fractions w/ borrowing

There is not fraction on top to subtract 1/3

You must borrow from the 6 which makes the whole # a 5. When you borrow you are borrowing a whole #. Since the denominator in 1/3 is a 3 you are borrowing 3/3.

Now you can subtract & make sure you answer is in simplest form

In this case you can’t subtract 3/12 from 5/12 so you have to borrow which makes the whole # 10 a 9. Again look at the denominator which is a 12 so you are borrowing 12/12. Add 12/12 to 3/12 and you get 15/12. Now you can subtract and make sure your answer is in simplest form

Dividing Fractions

Adding & Subtracting Fractions

Prime Factorization

1) List all of the factors for each #2) Find the greatest factor that the #’s both have

Example: 12 = 12, 1, 2, 6, 3, 4 16 = 1, 16, 2, 8, 4

GCF = 4

Greatest Common Factor (GCF)

Factors, Multiples, Prime #’s, & Composite #’s

*Prime Numbers* Prime #’s are those that have only 2

factors like 5, 7, 3, 11.*Composite Numbers*

Composite #’s are those that have more than 2 factors like 4, 6, 8, 10.

Prime and Composite Numbers

1) List all of the multiples for each #2) Find the smallest multiple that the #’s both have

Example: 3 = 3, 6, 9, 12, 15, 18, 21, 24 5 = 5, 10, 15, 20, 25, 30, 35

LCM = 15

Factors: the 2 #’s you multiply to get the product. In 2 x 3 = 6, 2 & 3 are the factors.Factors of 8 = 8, 1, 4, 2

Multiples: Multiples of 8 = 8, 16, 24, 32, 40, 48 etc..Multiples of 3 = 3, 6, 9, 12, 15, 18, 21 etc..

Prime #’s: Have only 2 factorsComposite #’s: Have more than 2 factors

1) In the box out of the box (numerator in, denominator out)2) Whole #: Quotient Numerator: Remainder Denominator: Divisor

3) Make sure your answer is in simplest form

1) Multiply the denominator and the whole # together2) Add the answer from step one to the numerator3) The denominator from the mixed # will be the denominator in your improper fraction

Least Common Multiple (LCM)

Improper fractions into mixed #’s

1) A proper fraction is one where the numerator is smaller than the denominator: 1/3, 3/5, or 6/82) An improper fraction is one where the numerator is larger than the denominator: 5/3, 8/2, or 13/43) A mixed # will have a whole # and a fraction: 3 2/4, 5 1/6, or 7 4/9.

Ordered Pairs (x,y)

1) Probability is always a fraction2) Numerator: what you want to get3) Denominator: Total possible outcomes

Example: Probability of landing on a 3Answer: 1/8

Example: Probability of landing on an even #Answer: 4/8 = 1/2

Probability

Proper Fractions, Improper Fractions, Mixed #’s

Mixed #’s into Improper Fractions

1) To find the probability of 2 or more events u must add the probabilities of each event together & then simplify

Example: Probability of landing on a 3 then a 4. P(3,4)Answer: 1/8 + 1/8 = 2/8 = 1/4

Example: Probability of landing on an even # then 3. P(even#,3)Answer: 4/8 + 1/8 = 5/8

Adding Probabilities

Mode: The # that occurs the most often (there can be no mode or several modes)Example: 5,6,2,6,3,4,9 mode = 6 Example: 4,5,2,1,4,6,5 mode = 4, 5

Mean: 1) add all of the 3’s2) divide by how many #’s there are3) no remainders you must add a decimal and bring it up4) round your answer to the nearest tenth

Median:1) Put the #’s in order from least to greatest2) Cross the #’s out two at a time from the outside3) If there r 2 #’s left, add them together and divide by 2 (no remainders and round to nearest tenth)

Range: Subtract the largest # from the tiniest #

Outlier: The # in a set of #’s that does not belong, it either way larger or smaller than the rest of the #’s. ex: 2,4,5,6,22…outlier would be 22.

1) If the integers have the same sign (either both positive or negative) then you just addExample: (+5) + (+4) = +9 Example: (-4) + (-3) = -7

2) If the signs of the integers are different (one is positive and one negative) then you must subtract. The larger # goes on top and use the sign of the larger #.Example: (-10) + (+3) = -7Example: (+15) + (-6) = +9

1) Change the subtraction sign into a plus sign2) Change the sign of the 2nd integer3) use addition rules to addExample: (-10) - (+3) = (-10) + (-3) = -13

Example: (+7) – (+2) = (+7) + (-2) = +5

1) If the signs of the integers are the same then your answer is always positive (+)Example: (+5) x (+4) = +20 Example: (-12) ÷ (-3) = +4

2) If the signs of the integers are different then your answer is always negative (-)Example: (-10) x (+3) = -30Example: (+54) ÷ (-6) = -9

Median, Range, OutlierMode & Mean

Area = Length x Width (A=lw)(area is always squared)Example:

Area = Length x WidthArea = 14 x 7Area = 98cm²

Area = 1/2 x base x height (A= 1/2bh)(area is always squared)

Example:

Adding Integers Subtracting Integers

Area of a rectangle or squareMultiply & Dividing Integers

Area = base x height (A= bh)(area is always squared)

Example:

Area of a triangle Area of a parallelogram

Area = 1/2 x base x height (A= 1/2bh)(area is always squared)

Example:

Area = base x height (A= bh)(area is always squared)

Example:

Area = TTr² (TT = 3.14)(area is always squared)

Example:

Area = TTr²Area = 3.14 x 6²(6 x 6)Area = 3.14 x 36Area = 113.04m²

D=12mm

Area = TTr² (TT = 3.14)(area is always squared)

Example:

Area = TTr²Area = 3.14 x 4²(4 x 4)Area = 3.14 x 16Area = 50.24m²

C = TT x Diameter (TTd)(TT = 3.14)(area is always squared)

Example:

C = TTdC = 3.14 x 12C = 37.68m

D=12mm

C = TT x Diameter (TTd)(TT = 3.14)(area is always squared)

Example:

C = TTdC = 3.14 x 8C = 25.12m

Circles Area of a Circle Area of a Circle

Volume = (area of the base) x height (volume is always cubed)

Example:

Volume = (area of the base) x heightVolume = (length x width) x heightVolume = (8 x 3) x 12Volume = 288in³

Volume = (area of the base) x height (connects the bases) (volume is always cubed)

Example: Volume = (area of the base) x heightVolume = (1/2 x base x height) x heightVolume = (1/2 x 6 x 3) x 12Volume = 9 x 12Volume = 108cm³

Volume of a rectangular prismCircumference of a Circle

Circumference of a Circle

Volume of a triangular prism

Volume = (area of the base) x height (connects the bases) (volume is always cubed)

Example:

Volume of a cylinder

Volume = (area of the base) x height (volume is always cubed)

Example:

Volume = (area of the base) x heightVolume = (TTr²) x heightVolume = (3.14 x 4²) x 12Volume = 50.24 x 12Volume = 602.88in³

Volume of a cylinder

Volume = (area of the base) x height (volume is always cubed)

Example:

Volume = (area of the base) x heightVolume = (TTr²) x heightVolume = (3.14 x 4²) x 12Volume = 50.24 x 12Volume = 602.88in³

1)Brackets + Parenthesis first2) exponents3) multiply or divide from left to right4) add or subtract from left to right

Example:(48 ÷ 8) x 5 + 3²6 x 5 + 3²6 x 5 + 930 + 9Answer: 39

Order of operations

1) Change the percent into a decimal (add a decimal to the end of the number and then move the decimal to the left twice)2) Larger # goes on top, multiply, count decimals

Example: 25% of 35 = .25 x 35 = 8.75

Example: 34% of 125 = 125 x .34 = 42.5

Percent (%) of a # (of = multiply)

1) Change the tax from a percent into a decimal (add a decimal to the end of the # and then move the decimal to the left twice) 7% = .072) then multiply the tax by the subtotal (amount you are spending). This gives you the tax amount.25 x .07 = $1.75

3) To find the total cost you must then add the tax amount to your subtotal25.00 + 1.75 = $26.75

Tax & Total Cost

Example:Subtotal: $25

Tax : 7%

Tax Amount: $1.75

Total Cost: $26.75

1) Change the discount from a percent into a decimal (add a decimal to the end of the # and then move the decimal to the left twice) 25% = .252) then multiply the discount by the subtotal

Discount & Sale Price

Example:Subtotal: $52

Discount : 25%

4m x

8m 6m

8x = 24

X = 24/8

X = 3 meters

1) Change the discount from a percent into a decimal (add a decimal to the end of the # and then move the decimal to the left twice) 25% = .252) then multiply the discount by the subtotal

Example:Subtotal: $52

Discount : 25%

1) Change the tip from a percent into a decimal (add a decimal to the end of the # and then move the decimal to the left twice) 15% = .152) then multiply the tip by the subtotal (amount you are spending). This gives you the tip amount.85 x .15 = $12.75

3) To find the total cost you must then add the tip amount to your subtotal85.00 + 12.75 = $97.75

Tip & total Cost

Example:Subtotal: $85

Tip : 15%

Tip Amount: $12.75

Total Cost: $97.75

If Alex ran 12 miles in 3 hours then how many did he run in 1 hour?

*in order to solve this problem you must use a proportion.*When using a proportion the labels in the numerators must be the same as well as the label in the numerators.*Once your proportion is set up you must then cross multiply (always start with the variable) and solve accordingly.

With Shapes*When doing proportions with shapes…each side of the proportion corresponds with the exact same side on each shape.*Same shape has to be on top and bottom of the proportion then cross multiply and then solve

Proportions

X miles 12 miles

1 hour 3 hours

3x = 12

X = 12/3

X = 4 miles

=

The 3 angles of a triangle add up to 180°̊

Example:

1) add the 2 angles together 36 + 57 = 932) Then subtract your answer from 180 180 – 93 = 87

Missing Angles: Triangles

The 3 angles of a triangle add up to 180°̊

Example:

1) Subtract 180 - 40

Missing Angles: Triangles

40°

x

x

=

x

The 3 angles of a triangle add up to 180°̊

Example:

1) add the 2 angles together 36 + 57 = 932) Then subtract your answer from 180 180 – 93 = 87

The 3 angles of a triangle add up to 180°̊

Example:

1) Subtract 180 - 40

The 4 angles of a quadrilateral add up to 360°

Example:

1) add the 3 angles together 50 + 50 + 130 = 2302) Then subtract your answer from 360 360 – 230 = 1303) missing angle = 1304) If your answer is correct then all 4 angles should add up to 360

Missing Angles: Quadrilaterals

The 4 angles of a quadrilateral add up to 360°

Example:

1) add the 2 angles together 48 + 64 = 1122) Then subtract your answer from 360 360 – 112 = 2483) 248 is the total of both missing angles together4) To find each angle you must divide by 2248 ÷ 2 = 1245) See each missing angle is 124°6) If your answer is correct then all 4 angles should add up to 360

Missing Angles: Quadrilaterals

48° 64°

xx

1) add all of the sides2) line up the decimals if there are any

Example:

Perimeter = add all sidesPerimeter = 1 + 5 + 4 + 2 + 7Perimeter = 19

Perimeter

1) In the box out of the box (numerator in, denominator out)2) Add a decimal to the end of the # (dividend) and bring it up3) There are NO remainders, if you have a remainder then you must add a zero, drop it, and continue dividing*To change a fraction into a % you must change it to a decimal then to a %

Fractions into Decimals

Example: 1.251) the # before the decimal is always the whole #2) look at the last #..the 5 is in the hundredths place so the fraction would be:1 25/100 = 1 1/4

Example: .81) the 8 is in the tenths place so the fraction would be:8/10 = 4/5

Example: .051) the 5 is in the hundredths place so the fraction is: 5/100 = 1/20

Decimals into fractions (always simplify)

1) If there is no decimal in the percent then you must add a decimal at the end of the percent 45% =

Percents into Decimals

1) Move the decimal twice to the right .38 = 38%

Decimals into Percents

1) If there is no decimal in the percent then you must add a decimal at the end of the percent 45% =

1) Move the decimal twice to the right .38 = 38%

1) Two angles that add up to 180°

Supplementary Angles

1) Two angles that add up to 90°

Complementary Angles

1) Angles that are opposite to each other (are always congruent)

Vertical Angles

Types of Triangles

Quadrilaterals Polygons

1) List all of the factors of the numerator2) Start w/ the largest factor and see which one you can divide the denominator by3) Once u find the factor you must divide both the numerator and denominator by it 4) Repeat until u can no longer simplify anymore

Simplifying Fractions

Comparing Fractions

1) U can only compare fractions when the denominators are the same2) If the denominators are different then make them the same 3) Compare the fractions

Example 2: how many 4 letter codes can u make from the letters a,b,c,d,e (repeat letters can be used)?

Since there r repeat letters…once a letter is chosen it can be used again (so a code of aaaa would be allowed)

For the 1st letter in the code u can choose from 5 letters

For the 2nd u can choose from 5 letters The 3rd 5 letters and the 4th 5 letters So your permutation would look like this:

5 x 5 x 5 x 5 = 625 possible codes

Example 2: how many ways can u give 3 cans to 8 kids?

Number Lines

Least Greatest

1) U can only compare fractions when the denominators are the same2) If the denominators are different then make them the same 3) Compare the fractions

1) Plug in known values into the formula then solve the equation.

Ex: Erik ran 24 miles at a rate of 4 miles per hour. How long did it take him to run 24 miles?

D = R x T

24 = 4T

24/4 = T

6 hours = T

Distance = Rate x Time

Least to Greatest

1) If there r fractions u must change them all to decimals first.2) Line up your decimals and compare your #’s (be very careful if there are negative integers)3) Remember that the # furthest left on the # line is always the smaller #.4) Write down the original #’s for your answer.

Example (least – Greatest) 1/2, .67, 2, 3/4 1/2, .67, 3/4 = .50, .67, 2.00, .75 Put in order= .50, .67, .75, 2.00 Answer = 1/2, .67, 3/4, 2

Permutations

If a problem is a permutation then the order of the items matters.

For example if you were to make a 4 letter code from the letters a, b, c, d, e……then the code abcde is totally different than the code abced.

Therefore in this case order matters and the problem would be a permutation.

Example 1: how many 4 letter codes can u make from the letters a,b,c,d,e (no repeat letters can be used)?

Since there r no repeat letters…once a letter is chosen it cannot be used again (so a code of aaaa would not be allowed)

For the 1st letter in the code u can choose from 5 letters

For the 2nd u can choose from 4 letters The 3rd 3 letters and the 4th 2 letters So your permutation would look like this:

5 x 4 x 3 x 2 = 80 possible codes

Combinations

If a problem is a combintation then the order of the items does not matter.

Probability w/replacement

Probability with replacement means that once an item is picked it is then put back into the sample space so that it can be picked again

Example 1: What the probability of picking a black marble and then a stripped marble (p) black,strpiped w/ replacement

The probability of getting a black marble = 1/6

The probability of getting a stripped marble = 3/6

Add the probabilities for each event 1/6 + 3/6 = 4/6 = 2/3

Probability w/o replacement

Probability without replacement means that once an item is picked it is not put back into the sample space so that it cannot be picked again

Example 1: What the probability of picking a

Example 2: how many ways can u give 3 cans to 8 kids?

Shirts Sizes

Red Small

Blue Medium

White Large

X-Large

If a problem is a combintation then the order of the items does not matter.

Probability without replacement means that once an item is picked it is not put back into the sample space so that it cannot be picked again

Example 1: What the probability of picking a

Figuring out percents (out of)

1) Change to a fraction2) Change the fraction into a decimal3) Change the decimal into a percent

Example 1: Find the % for 1 out of 41) change to a fraction = 1/42) change to a decimal = .253) change to a percent = 25%

Probability (Finding all Possible Combinations): List, Tree Diagram, Counting Principal

Counting Principal1) Multiply the total # of choices from each category2) 3(shirts) x 4 (sizes) = 12 possible combinations

List1) In a list you do just that make a list of all of the possible combinations..this is also usually done at the end of a tree diagram.

Ex.Red, small blue, small white, smallRed,medium blue,medium white,mediumRed,large blue, large white,largeRed,xlarge blue,xlarge white,xlarge

Tree Diagram

Graphing Equations

Ex: Graph the equation y = 2x

1) Make a table…in place of x pick 3 #’s usually pick 3’s that r easy to calculate…in this case we will pick 1, 2, -2

X Y

1

2

-2

2) Fill out the table by solving the equation with each value u have for x (y = 2 x 1…so y = 2)

X Y

1 2

2 2

-2 -4

3) Plot each set of points on a graph…connect your dots and then label your equation on the line.

y = 2x

Words and their operation

Addition Subtraction Multiplication DivisionSum Difference Of Each

Deposit WithdrawAbove Below

Forward BackwardIncrease Decrease

Gain LossAltogether

Hidden Integers

You can divide any number by…1) 2 if the number is even2) 3 if the individual numbers in the number add up to a multiple of 33) 5 if the number ends in a 5 or 04) 10 if the number ends in a 05) if you can divide by 2 then always try a 4 and an 86) if you can divide by 3 three then always try a 6 and a 9

1) Change the rate from a percent into a decimal (add a decimal to the end of the # and then move the decimal to the left twice) 4% = .042) then multiply the principal by the rate. 52 x .05 = 2.08

3) Now multiply that answer by the time.2.08 x 3 = $6.24

Calculating Interest…..Interest = Principal x rate x time

Example:Principal (amount u have in bank): $52

Rate (interest rate): 4%

Time (amount of time interest is accruing): 3 years

Interest (amount gained over x amount of time): ?

1) Change the rate from a percent into a decimal (add a decimal to the end of the # and then move the decimal to the left twice) 4% = .042) then multiply the principal by the rate. 52 x .05 = 2.08

3) Now multiply that answer by the time.2.08 x 3 = $6.24

Example:Principal (amount u have in bank): $52

Rate (interest rate): 4%

Time (amount of time interest is accruing): 3 years

Interest (amount gained over x amount of time): ?