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Number Systems

Expressions and Equations

Ratios and Proportions

Geometry

Statistics and Probability

Fractions

Long Division

Decimals

Integers

Rational Numbers

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Reciprocal = when you “flip” the fraction

The numerator goes to the bottom

The denominator goes to the top

To divide fractions, multiply by the reciprocal of the divisor.

Must change mixed numbers to improper fractions before do

anything.

Top number = numerator

Bottom number = denominator

𝐷 ÷ 𝑆 = 𝑄

D = Dividend (first number)

S = Divisor (second number)

Q = Quotient (answer)

𝑎

𝑏÷

𝑐

𝑑=

𝑎

𝑏×

𝑑

𝑐=

𝑎𝑑

𝑏𝑐

𝐷 ÷ 𝑆 = 𝑄

D = Dividend (first number)

S = Divisor (second number)

Q = Quotient (answer)

𝐷 ÷ 𝑆 𝑆 𝐷

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𝐷 ÷ 𝑆 = 𝑄

D = Dividend (first number)

S = Divisor (second number)

Q = Quotient (answer)

𝐷 ÷ 𝑆 𝑆 𝐷

To add and subtract:

Line up the decimal points.

Add 0s if needed.

Add and subtract as normal.

Just bring the decimal down.

To multiply:

Line numbers up along the right.

Move the decimal(s) all the way to the right in all

numbers.

Multiply as normal.

Move the decimal back to the left as many times as you

moved them in the original numbers.

Dividing Decimals

𝐷 ÷ 𝑆 = 𝑄

D = Dividend (first number)

S = Divisor (second number)

Q = Quotient (answer)

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To divide a decimal by a whole number:

Bring the decimal up.

Divide as normal.

To divide a whole number by a decimal:

Move the decimal in the divisor all the way to the right.

Add the same number of 0s as you moved in the divisor, and put the decimal at the end.

Divide as normal.

Bring the decimal up – that’s where it goes in your answer.

To divide a decimal by a decimal:

Move the decimal in the divisor all the way to the right.

Move the decimal in the dividend the SAME number of times.

Divide as normal.

Bring the decimal up – that’s where it goes in your answer.

Dividing Decimals

𝐷 ÷ 𝑆 = 𝑄

D = Dividend (first number)

S = Divisor (second number)

Q = Quotient (answer)

Dividing Decimals

𝐷 ÷ 𝑆 = 𝑄

D = Dividend (first number)

S = Divisor (second number)

Q = Quotient (answer)

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Dividing Decimals

𝐷 ÷ 𝑆 = 𝑄

D = Dividend (first number)

S = Divisor (second number)

Q = Quotient (answer)

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Number lines have negatives on the left and positives

on the right, with 0 in the middle.

A positive is always greater than a negative.

Two negatives make a positive because the first

negative means “opposite of” and the second negative

is a negative sign.

Ex: − −3 = the opposite of −3=3

Whole numbers

Zero

Opposites of whole

numbers

Terminating fractions/decimals become whole numbers

when divided out

Repeating fractions/decimals repeat the same set of digits

with no end

Non-repeating fractions/decimals do not end but do not

repeat

**These are NOT rational numbers. Whole numbers

Zero

Opposite of whole numbers

Fractions and decimals that terminate or repeat

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Elevation/Altitude

Above sea level=positive

At sea level=zero

Below sea level=negative

Money

Deposit=positive

Withdrawal=negative

Temperature

Degrees above zero=positive

Degrees below zero=negative

Real-World Situations

Numerical Expressions

Algebraic Expressions & Properties

Equations

Inequalities

Relationships

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Order of Operations

P.E.M.D.A.S.

Parentheses (or any grouping symbol, such as brackets and

absolute value bars)

Exponents (the little numbers in the top right corner)

Multiply and Divide from LEFT TO RIGHT

Add and Subtract from LEFT TO RIGHT

You can find equivalent expressions (expressions with the

same value) using properties

Distributive

8 3 + 4 = 8 3 + 8 4 = 24 + 32

Commutative

9 + 7 = 7 + 9

Expressions with only

numbers and math

symbols

No equal sign

Can use letters to stand for unknown numbers

Those letters can have numerical values

You can substitute those values to evaluate (solve) an

expression

You can find equivalent expressions (expressions with the

same value) using variables and properties

Distributive

8 𝑥 + 4 = 8 𝑥 + 8 4 = 8𝑥 + 32

Commutative

y + 7 = 7 + 𝑦

Numbers, math symbols, variables

Variables are letters the represent unknown numbers

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Use the opposite sign but the same number to solve

the equation

Solve an equation by doing the same thing to both

sides of the equal sign

Go in the opposite order of the Order of Operations

Add and subtract first

Then multiply and divide

Then eliminate grouping symbols Two expressions set equal

to each other

Equal sign in the middle

Add: 9 + 𝑥 = 14

Subtract 9 from both sides

𝑥 = 5

Subtract: 𝑚− 24 = 62

Add 24 to both sides

𝑚 = 86

Multiply: 7𝑏 = 35

Divide both sides by 7

𝑏 = 5

Divide: 𝑘

3= 12

Multiply both sides by 3

𝑘 = 36

Divide with fraction: 3

4𝑤 = 15

Multiply by sides by the reciprocal: 4

3

𝑤 = 15 ×4

3=

15×4

1×3=

60

3= 20

How to solve one-step

equations

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When writing inequality, two expressions are not equal

If the first is greater than the second, use > symbol.

If the first is less than the second, use the < symbol.

If you solve the inequality and they are not equal, use the

≠ symbol.

When graphing, use an open circle and an arrow:

Arrow to the left means less than

Arrow to the right means greater than Two unequal expression being compared with a comparison symbol

Greater than (>)

Less than (

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Rates/Rates

Unit Rate/Price

Speed

Percents

Measurement Conversions

– Ratios can be written in 3 ways:

– 𝑥 ∶ 𝑦

– 𝑥 𝑡𝑜 𝑦

–𝑥

𝑦

– There are 3 relationships:

– Part to Part

– Part to Whole

– Whole to Part

– “and” means order doesn’t matter

– Comparing cats and dogs can be 𝐶:𝐷 or 𝐷:𝐶

– “TO” means order does matter

– Comparing cats to dogs can only be 𝐶:𝐷

– Numbers do NOT have to be smaller to bigger.

A ratio is a comparison of 2

things using numbers.

A rate is a comparison of 2

things using numbers and

words.

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– The only difference between a unit rate and a unit price is a

unit price has money.

– No money = rate

– Money = price

– For a unit price, money always comes first!

– Per is the key word. If you use the word “Per” you don’t need

“for one” or “for each” or “for every”.

– Sometimes your unit rate/price is a decimal.

Unit rate is like a rate, but

the bottom number is always

1.

So you are finding the

amount or price for every 1

item, or for each, or per.

– Distance words:

– Miles

– Meters

– Yards

– Feet

– Rate words:

– ___ per ____

– Time words:

– Minutes

– Seconds

– Hours

– Days

– years

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– Three components:

– Part = IS

– Whole = OF

– Percent = %

– Equation:

– Part = Percent x whole

– 𝑃 =%

100• 𝑊

– Can use proportions

–𝑥1𝑦1=

𝑥2𝑦2

Percents are always out of

100.

Use % to show percent.

Can be written as a fraction,

decimal, or percent.

– Conversion sheet tells you customary to customary

– Conversion sheet tells you metric to metric

– You will be given conversion factor (what to multiply or divide by) if

going between systems

– Bigger to smaller unit Multiply

– Smaller to bigger unit Divide

Customary: US, Myanmar,

Liberia

No rhyme or reason

Metric: Everyone else:

Europe, Asia, South America,

Canada, Mexico, Caribbean

Base-ten system (powers of

10)

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Area

Surface Area/Nets

Volume

Graphing with Polygons

Formulas

Rectangle: 𝐴 = 𝑏ℎ or 𝐴 = 𝑙𝑤

Triangle: 𝐴 =1

2𝑏ℎ or 𝐴 =

𝑏ℎ

2

Parallelogram: 𝐴 = 𝑏ℎ

Trapezoid: 𝐴 =1

2ℎ(𝑏1+ 𝑏2) or 𝐴 =

𝑏1+𝑏2 ℎ

2or decompose

the figure

Composed Figure: decompose it

Area is represented by A.

Area is always squared,

meaning it has an exponent

of 2.

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– Nets are the 2-D unfolded figure of the 3-D figure

– Nets can help find surface area.

– Surface area is how much wrapping paper you need to go around

the 3-D shape.

SA if used to show surface area.

Surface area uses squared units

(exponent of 2).

– Formula: 𝑉 = 𝑙𝑤ℎ or 𝑉 = 𝐵ℎ

Length, width, height

Area of base times height

– 6th grade only has to find volume of right rectangular prisms

(including cubes).

– Fractional side lengths

Can leave answer as improper fraction.

Volume is represented by V.

Volume is cubed, or cubic units

(exponent of 3).

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– Corners are called vertices (vertex-singular) and labelled with letters.

– Four quadrants:

– To find side length or perimeter, count spaces between the points.

Coordinate plane has x-axis

(horizontal) and y-axis

(vertical), and meet at origin

(0,0).

Ordered pair is (x,y).

Statistical Questions

Data Displays

Measures of Center

Measures of Variability

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– A statistical question CAN have more than one answer.

– It doesn’t always have to have more than one answer, but the

possibility needs to be there.

– NOT Statistical:

– If it is asking you or just one person.

– If it is asking an opinion.

– If there is only one answer.

– If it is a yes or no question.

– Statistical:

– If it says per, for each, for every.

– If multiple people could answer and have difference responses.

Dot Plot

Has one dot to represent each data value

Good for smaller data sets and sets closer together

Allows you to see gaps, clusters, outliers

Can see the total number of data values

Histogram

Uses bars to represent a range, not a specific value

Can see frequencies of ranges but not frequencies of specific values

Can see the total number of data values

Good for values farther apart or many values

Allows you to see gaps, clusters, possibly outliers

Box Plot

Uses a box and whiskers (lines) to represent 5 data values of the data set

Cannot see specific data values

Allows you to see main data values as summaries (minimum, 1st/lower quartile, median, 3rd/upper quartile, maximum)

Good for big data sets and data sets with numbers far apart

Cannot see gaps, clusters

Can somewhat see outliers

Three types of data displays:

Dot plot

Histogram

Box plot

Shows data on a number line

and allows you to make

summaries and inferences

about the data.

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Dot Plots Histograms Box Plots

How represented? Dots – 1 for each datavalue

Bars – to represent ranges

Box and whiskers – to represent 5 boundary values

Can see specific data values?

Yes No No

Can see gaps,clusters, outliers?

Yes Yes Possibly outliers; no gaps, clusters

What kind of data sets?

Smaller, closer together

Bigger, spread apart Bigger, spread apart

Can see frequencies? Yes Somewhat – of ranges

No

Can see minimum, median, maximum?

Yes Somewhat – ranges Yes

– MoC are a one-number summary of the data set.

– You can find MoC using data displays or sets of data

values.

Mean – Average

Median – Middle

Mode – Most common

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– MoV tell you about data distribution

– How spread out the data are

– Distributions of the data:

– Gaps

– Clusters

– Outliers

– Skewed

– Symmetry

Range

Interquartile Range (IQR)

Mean Absolute Deviation

(MAD)

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