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HINTS, TIPS AND FACTS*
*Check out this helpful website: www.ixl.com and go to the 8th grade page.
Numbers
Natural/Counting numbers: 1, 2, 3, 4…
Whole numbers: 0, 1, 2, 3…
Integers: …-3, -2, -1, 0, 1, 2, 3 …
Rational numbers: Quotient of two integers; Repeating decimals; Terminating decimals
Irrational Numbers: Non-repeating, non-terminating decimals; p , 3
Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
(1 is NOT prime; 2 is the only even prime number)
Perfect squares: 1, 4, 9, 16, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Factor: 3 is a factor of 12
Multiple: 12 is a multiple of 3
Order of Operations
Please Excuse My Dear Aunt Sally Þ
Parentheses Exponents Multiply Divide Add Subtract
Properties
Commutative: a + b = b + a; ab = ba
Associative: (a + b) + c = a + (b + c); (ab)c = a(bc)
Distributive: a(b + c) = ab + bc
Additive Inverse: a + -a = 0
Additive Identity: a + 0 = a
1
Multiplicative Inverse: a ·
=a
Multiplicative Identity: a · 1 = a
a
Absolute Value
Definition: The distance from zero
Equations:
Translate words into math symbols:
+
–
x

Add
Plus
Sum
More
Increase
Total
Subtract
Minus
Difference
Less
Decrease
Fewer
Multiply
Times
Product
Twice (x 2)
Triple (x 3)
Of
Divide
Quotient
Share equally
“percent of” Þ multiply
“fraction of” Þ multiply
“square” of a number Þ x2
“square root” of a number Þ
x
Consecutive Integers: x, x + 1, x + 2, x + 3
Consecutive Odd/Even Integers: x, x + 2, x + 4, x + 6
To solve equations with fractions Þ Multiply each term by the least common denominator.
x x
+ =5
3 2
æ x xö
6 ·ç + ÷ = 6 · 5
è 3 2ø
Ex. 2x + 3x = 30
5x = 30
x =6
Inequalities:
If you multiply or divide by a negative number, you reverse the inequality sign.
2 < x < 5 means that x is between 2 and 5
Formulas & Conversions:
Distance = Rate · Time
d=r ·t
Simple Interest = Principal · Rate · Time
I = prt
9
Fahrenheit to Celsius Conversion: F = C + 32
5
5
Celsius to Fahrenheit Conversion: C = (F – 32)
9
Ratios & Proportions:
To solve proportion problems: Ratio = Ratio Þ Cross Multiply!
To set up proportion problems be sure to compare the correct categories:
in in
=
mi mi
If you’re given 2 categories and asked to compare them to a total, add the ratio categories.
Ex: In a class the ratio of boys to girls is 2 to 3. The total number of students is 30.
2+3=5
2
x
To find the number of boys:
=
. So, 60 = 5x. Then x = 12
5 30
To find the number of girls: Either subtract 12 from 30; or solve
Percents
Word problems: ______Part_______
Total or original
=
3
x
=
5 30
%
100
For long “wordy” problems, try “translating” each word into a math symbol.
Average
Median is the middle number (when they are listed from smallest to largest).
Mode is the number that occurs most often.
sum
Mean is the average =
# terms
Exponent Rules
(xa)(xb) = xa + b
(xa)b = xab
x0 = 1
x–a =
xa
= xa – b
b
x
1
xa
Polynomials
Add polynomials: (3x2 + 4x + 5) + (2x2 + 6x + 7) = (5x2 + 10x + 12)
Subtract polynomials: (3x2 + 4x + 5) - (2x2 + 6x + 7) = (x2 – 2x – 2)
Multiply binomials (FOIL): (x + 2)(2x + 3) = 2x2 + 3x + 4x + 6 = 2x2 + 7x + 6
Divide polynomials by a monomial:
8x 3 + 6x 2 + 2x
= 4x2 + 3x + 1
2x
Factoring:
GCF (Greatest common factor)
ax2 + axy = ax (x + y)
5x2 + 10x = 5x(x + 2)
IAM (I add & multiply)
x2 + bx + c = (x + m)(x + n)
where m · n = c; m + n = b
x2 + 5x + 6 = (x + 3)(x + 2)
DOTS (Difference of two squares)
x2 – y2 = (x + y)(x – y)
x2 – 25 = (x + 5)(x – 5)
Slope & Lines
y -y
rise
slope = m =
= 2 1
x 2 - x1
run
Equation of a line Þ y = mx + b
m = slope; b = y-intercept
y = a number Þ horizontal line
x = a number Þ vertical line
(zero slope)
(undefined slope)
Positive slopes rise upward to the right.
Negative slopes rise upward to the left.
Parallel lines have equal slopes.
Perpendicular lines have slopes that are negative reciprocals.
GEOMETRY*
Triangles:


*Remember to review all vocabulary on Schoolwires.
The longest side is across from the largest angle.
The shortest side is across from the smallest angle.
5 in

12 in
25
120
35
4 in
A missing side is always between the sum and the difference of the two given sides.
8
8–3< x <8+3
3
5 < x < 11
x

An exterior angle is equal to the sum of the two non-adjacent interior angles.
x = 115 + 40
115
x
Polygons:
25
40
x = 155
(n = number of sides in a polygon)

Polygon: A simple, closed figure made up of line segments.

Regular Polygon: Polygon with all sides congruent and all angles congruent.

Sum of the interior angles of a polygon: (n – 2) 180

One interior angle of a regular polygon:

Sum of the exterior angles of any polygon = 360

One exterior angle of a regular polygon:

Diagonals from one vertex: (n – 3); Diagonals from all vertices:
(n - 2)180
n
360
n
(n)(n - 3)
2
Quadrilaterals: