8th Grade Common Core

Mathematics...

...Simplified

Mrs. Ditrano’s Math Class

A function is a rule that assigns to each input exactly one output. These are graphs of functions: These are NOT the graphs of functions:

(every x-coordinate is different) (x-coordinates are the same for more than one point)

The VERTICAL LINE TEST can be used to determine if a relation is

a function. If a vertical line can be drawn that intersects the

relation more than once, then the relation is NOT a function.

If the vertical line only intersects the relation once, then the

relation IS a function.

These are tables of functions: These are NOT tables of functions:

(every x-coordinate is different) (x-coordinates are the same for more than one point)

The coordinates of a function: {(1, 6), (2, 3), (4, 3), (5, 7)} each x is only used once

NOT the coordinates of a function: {(1, 6), (2, 3), (1, 3), (5, 7)} one x is paired with two y’s

Use the same skill to write the

equation of a line from a graph:

Slope,

constant rate of change

(line climbs this way)

y-intercept,

initial value

(Start here)

To graph a line from the equation, start at the y-intercept,

then count up (or down if the slope is negative) and right,

according to the slope, to find another point on the line.

Connect the dots . Label the line.

EXAMPLE: y = 2

3x – 5

Start at (0, -5), go up 2, right 3, plot a point.

Linear Equations

do not include

exponents,

variables in the

denominator or

absolute value.

To write the equation of a line from a table, find the y-intercept by finding the

value of y when x = 0. Calculate the slope by determining the patterns in the y-

column and x-column and write them in fraction form 𝑐ℎ𝑎𝑛𝑔𝑒_𝑖𝑛_𝑦

𝑐ℎ𝑎𝑛𝑔𝑒_𝑖𝑛_𝑥.

y- intercept = 3

slope = 𝟑

𝟐

Two Way Tables Example Question Emma has collected information about the cats and dogs that children in her class

have as pets. For each pupil, there are four possible responses they could make:

The pupil has a cat and a dog.

The pupil has a cat but not a dog.

The pupil has a dog but not a cat.

The pupil does not have a cat or a dog.

If we were given the following data, the table would look like this:

8 pupils have a cat and a dog.

4 pupils have a cat but not a dog.

12 pupils have a dog but not a cat.

6 pupils do not have a cat or a dog.

To calculate the relative frequency, compare the amount in a category to the total

and convert that ratio into a percent.

Example: What is the relative frequency of the number of pupils who do not have

a cat or a dog (6) to the total (30)? 6

30 = 6 ÷ 30 x 100 = 20%

Values are not always compared to the grand total. Sometimes, the total is the

amount in a category. Example: Out of the pupils that have dogs (8 + 12 = 20),

what percent also have cats(8)? 8

20 = 8 ÷ 20 x 100 = 40%

6

Positive

Association

Negative

Association

No

Association

Positive Association: data points have an upward trend

(as one variable increases, so does the other one)

Negative Association: data points have a downward trend

(as one variable increases, the other one decreases)

No Association: data points have no trend

Outlier: numerically distant from

the rest of the data

Cluster: a group of data points Line of Best Fit: A straight

that are close together line drawn through the center

of a group of data points

𝐿𝑎𝑤𝑠 𝑜𝑓𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑠

When the base is the same, you can simply add the exponents

when multiplying.

𝟓𝟑 x 𝟓𝟔 = 𝟓𝟗

When the base is the same, you can simply subtract the

exponents when dividing.

𝟖𝟕÷ 𝟖𝟐 =𝟖𝟓

A negative exponent indicates that the value is the reciprocal

of that value with a positive exponent.

𝟕−𝟐 = 𝟏

𝟒𝟗

(34)5 = 320

Any number raised to the power of zero, except zero itself,

equals one. 𝟏𝟑𝟎 = 1

Numbers written in scientific notation must be in the form:

a x 10n a is greater than or equal to 1 but less than 10 n is an integer that represents the number of places the decimal point moves to write the number in standard form positive exponent: move the decimal point to the right negative exponent: move the decimal point to the left

Examples: Scientific Notation Standard Form

9.43 x 106 = 9,430,000 4.019 x 10-3 = 0.004019 When solving word problems, remember the phrase, “how many times” means to divide. Try this: The temperature at the surface of the sun is approximately 1.0 x 104 degrees Fahrenheit. The temperature at its center is approximately 2.7 x 107 degrees Fahrenheit. About how many times greater is the temperature at the center of the sun than at its surface?

Rota

tion

Dilation

Translation: figure moves according to a rule;

image is congruent to original figure

Right: Add to x-coordinate

Left: Subtract from x-coordinate

Up: add to y-coordinate

Down: subtract from y-coordinate

Reflection: mirror image over given line; each point on the figure

will be the same distance from the line of reflection, but on the

opposite side of the line; image is congruent to original figure

Rotation : the figure turns clockwise or counterclockwise; image is

congruent to original figure

90o = one turn

180o = two turns

270o = three turns

Dilation: figure shrinks or enlarges according to a scale factor (multiply ALL

coordinates by that scale factor); image is similar but NOT congruent to

original figure

Complementary angles Supplementary angles

add up to 90o. add up to 180o.

Vertical angles are congruent. Corresponding angles

are congruent.

Alternate interior angles Same side interior angles

are congruent. are supplementary.

Alternate exterior angles Same side exterior angles

are congruent. are supplementary.

*To solve algebraic pairs of angles problems, first determine the relationship between the

angles in question, then write an algebraic equation and solve for the variable.

Substitute your answer into the given expression to calculate the measure of the angle.

Have proportional side lengths

Have the same slope

Have congruent angles states

that if two angles in one triangle

are congruent to two angles in

another triangle, the triangles are

similar.

that the exterior angle of a triangle

equals the sum of the two

non-adjacent interior angles. states that the length of the

third side of a triangle must be

less than the sum and greater

than the difference of

the other two sides.

A = C + D

To eliminate fractions,

multiply the entire

equation by the LCM of

the denominators first.

This equation has one solution:

x = 4. If you end up with x = x,

there are infinite solutions to

that equation. If you end up

with two numbers that are not

equal, there are no solutions to

that equation.

(See Systems of Equations

for more information.)

Simplify (apply the distributive property, if possible, then

combine like terms).

Isolate the variable by using inverse operations.

Check your answer.

Example: 4(x + 1) – 3x + 18 = 9x – 10

4x + 4 – 3x + 18 = 9x – 10

x + 22 = 9x – 10

-9x – 9x

-8x + 22 = -10

-22 -22

-8x = -32

-8 -8

x = 4

Check: 4(4 + 1) – 3(4) + 18 = 9(4) – 10

4(5) – 3(4) + 18 = 9(4) – 10

20 – 12 + 18 = 36 – 10

8 + 18 = 36 – 10

26 = 26

Solve for x: 𝟐

𝟓x – 15 =

𝟓

𝟖x – 16

Systems of Equations The solution to a system of equations is the set of variables that make all equations

true. On a graph, this is the point where the lines intersect. Three ways to solve a

system of equations are:

Graphing: Substitution: replace one

variable with an equivalent value

Example: x = 3

y = 2x + 5

y = 2(3) + 5

y = 6 + 5

y = 11

Solution: (3, 11)

Elimination: combine the equations by adding or subtracting them to

eliminate one variable and solve for the other variable. 2x + 5y = 46 5x – 3y = 5 3x – 2y = 11

2x + 3y = 30 4x + y = 21 4x – 3y = 14

Subtracting eliminates ‘x’ Multiply the second Multiply both equations

2x + 5y = 46 equation by 3. to eliminate one variable.

- 2x + 3y = 30 3(4x – y = 11) 4(3x – 2y = 11)

2y = 16 12x + 3y = 63 3(4x – 3y = 14)

2 2 Now adding will

y = 8 eliminate ‘y’. 12x – 8y = 44

Substitute 8 for ‘y’ 5x – 3y = 5

(in either equation) + 12x + 3y = 63 Subtract to eliminate ‘x’.

to solve for ‘x’. 17x = 68 12x – 8y = 44

Solution: (3, 8) 17 17 - 12x – 9y = 42

x = 4 y = 2

Substitute 4 for ‘x’ Substitute 2 for ‘y’

(in either equation) (in either equation)

to solve for ‘y’. to solve for ‘x’.

Solution: (4, 5) Solution: (5, 2)

Try This: The math club and the

science club had fundraisers to buy

supplies for a hospice. The math club

spent $135 buying six cases of juice and

one case of bottled water. The science

club spent $110 buying four cases of

juice and two cases of bottled water.

How much did a case of juice cost? How

much did a case of bottled water cost?

This equation has one

solution: (3, 11). If you end up

with x = x, there are infinite

solutions to that equation

which means it is actually the

same line in each equation. If

you end up with two numbers

that are not equal, there are

no solutions to that equation.

This means that the lines are

parallel and will never

intersect on a graph.