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“Factorials” are written with an
exclamation point (!), and mean that
you should multiply that number by
every counting number less than it.
For example:
3! = 3 × 2 × 1 = 6
If you are dividing two factorials,
you can save time by canceling first.
For example:
5! 5 × 4 × 3 × 2 × 1
=
= 5 × 4 = 20
3!
3×2×1
“Exponents” are written as a small
number to the upper right of a
number (the “base”), and mean that
you should multiply the regularsized number by itself the small
number of times. E.g.
25 = 2 × 2 × 2 × 2 × 2 = 8 × 4 = 32
If you are dividing two exponents
with the same base, you can save
time by canceling first. E.g.
25 2 × 2 × 2 × 2 × 2
=
=2×2=4
3
2
2×2×2
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑠𝑝𝑒𝑒𝑑 × 𝑡𝑖𝑚𝑒
If you bicycle at ten miles per hour for
three hours, your total mileage will be
10 × 3 = 30 miles.
You can also rearrange the equation as
𝑠𝑝𝑒𝑒𝑑 =
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑡𝑖𝑚𝑒
or
𝑡𝑖𝑚𝑒 =
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑠𝑝𝑒𝑒𝑑
So if you go 100 miles in four 4, you must
100
have gone an average speed of
= 25
4
miles per hour. If you went 60 miles at 5
60
miles per hour, it would take = 12
5
hours.
In a right triangle, the two sides next
to the right angle are called “legs”,
and the side opposite the right angle
is called the “hypotenuse”. If we call
the lengths of the legs “a” and “b”,
and the length of the hypotenuse
“c”, the equation
a
leg
𝑎2 + 𝑏 2 = 𝑐 2 is true.
hypotenuse
c
E.g., if the legs are 3 and 4, 𝑐 2 =
32 + 42 = 9 + 16 = 25 = 𝑐 2 , so
the hypotenuse is 5.
leg
b
The “triangle inequality” says that
any two sides of a triangle have to
add up to more than the other side.
For example, if two sides of a
triangle are 3 m and 7m, the other
side can’t be 1 m, because 1 + 3 is
not more than 7. The other side
also can’t be 20 m, because 3 + 7 is
not more than 20.
We normally write numbers in “base
10”, where the rightmost digit is
100 = 1, the next one is 101 = 10,
then 102 = 100, etc. Any counting
number can be a base, however.
For example, computers think in
“base 2”, where the rightmost digit
is 20 = 1, the next one is 21 = 2,
2
then 2 = 4, etc. If a computer is
thinking about the number 1012 ,
0
that number is the same as 1 × 2 +
0 × 21 + 1 × 22 = 1 × 1 + 0 + 1 ×
4 = 1 + 4 = 5 in base 10.
Instead of adding over and over, it is
frequently easier to multiply, so long
as you can turn the addition
problem into addition of the same
number.
For example, if you’re trying to add
the numbers from 1 to 10, you
might notice that 1 + 10 = 11, 2 +
9 = 11, etc. You might also realize
that there were 10 numbers to start
with, so there must be 5 pairs that
each add to 11, for a total of 5 ×
11 = 55.
In a data set, the mode is the
number that appears the most
often.
The median is the number that
appears in the center if the numbers
are arranged in increasing order.
The mean is the average of the
numbers, which you get by adding
all the numbers and dividing by how
many numbers there are.
The range is the difference between
the largest and smallest numbers in
the set.
When a sequence is not obvious,
looking at the differences can help.
If you don’t see a pattern there, try
it again, etc...
For example, in the sequence 9, 12,
17, 24, 33, 44, … the differences are
3, 5, 7, 9, 11, … If you still don’t see
it, the differences of the differences
are 2, 2, 2, 2, … so the next double
difference is probably 2, the next
difference is probably 13, and the
next number in the sequence is
probably 57.
For many problems, making an
organized list can often help if you
don’t know the “best” way to do it.
For example, if you want to find all
of the ways to add three different
single-digit numbers and get 15,
start with 1 & 2. What can be added
to 1 & 2 to get 15? 12, which isn’t a
single-digit number. 1 & 3? No. 1 &
4? No. 1 & 5? Yes, 9 works, so we
write 159, then 168, but not 177.
Then we try 2 & 3 (no), then 2 & 4,
which works, so we write 249, 258,
& 267, then 348, 357, & 456.
A prime number is a number with
exactly two factors, 1 and itself.
1 is NOT prime, because it has only
one factor, 1.
2 is prime, because it has exactly
two factors, 1 and 2.
3 is prime, because it has exactly
two factors, 1 and 3.
4 is NOT prime, because it has three
factors: 1, 2, and 4.
To turn in a circle, you must turn
360° (degrees).
If you turn only halfway around, you
turn 180°, which is a “straight
angle”.
When two lines intersect at a vertex,
angles that are opposite one
another are equal, and are called
“vertical angles”.
When a line intersects two parallel
lines, corresponding angles at each
intersection will be equal.
Any convex n-gon (n-sided polygon)
can be divided into 𝑛 − 2 triangles
by drawing all of the diagonals from
one vertex.
For example, a pentagon (a 5-gon)
can be divided into 5 − 2 = 3
triangles.
Every triangle has angles that add up
to 180°.
So every n-gon has angles that add
up to (𝑛 − 2) × 180°.
The perimeter of a 2D figure is the
distance around its edge.
The area of a 2D figure is the
number of 1x1 squares it would take
to fill the figure, even if they might
need to be chopped up.
The volume of a 3D solid is the
number of 1x1x1 cubes it would
take to fill the space (perhaps with
chopping).
The surface area of a 3D solid is the
sum of the areas of its surfaces.
Generally, “sum” means “add”,
“difference” means “subtract”,
“product” means “multiply”, and
“quotient” means “divide”.
When discussing fractions, “of”
often means “multiply”.
When discussing probability, “and”
frequently means “multiply” and
“or” often means “add”.
In the metric system, there is a set
of prefixes that indicate how many
of something you’re talking about.
“kilo” means 1000, “hecta” means
100, “deka” means 10, “deci” means
1
1
, “centi” means , and “milli”
10
1
100
means
, but there are many
1000
others.
For example, a kilometer is 1000
1
meters, and a deciliter is of a liter.
10
When naming things,
mathematicians frequently use
prefixes to indicate numbers. Some
common ones are “uni” for 1, “duo”
for 2, “tri” for 3, “tetra” for four,
“pent” for five, “hex” for six, “sept”
for seven, “oct” for eight, “non” for
nine, “dec” for ten, but there are
many more.
For example, a tetrahedron is a solid
with four faces, and a hexagon is a
two-dimensional figure with six
sides.
To convert a repeating decimal to a
fraction, you must find a way to
eliminate the repeating part of the
decimal.
̅̅̅, we will
For example, to convert . ̅63
consider that 100 of that number
̅̅̅. If we subtracted
would be 63. ̅63
them, we’d get 99 of that number,
which would be 63. Thus, one of
63
21
7
that number would be = = .
99
33
11
Whenever you’re doing complicated
arithmetic, remember the made-up
word PEMDAS, which is supposed to
help you remember to do anything
in parentheses first, then any
exponents, then all multiplication &
division (which are sort-of the same
thing) from left to right, then all
addition & subtraction (also sort-of
the same) from left to right.
For example,
4 + 5 × 6 = 4 + 30 = 34 and
3 × (7 − 1)2 ÷ 4 = 3 × 36 ÷ 4 = 27.
In algebra, a letter or other symbol
might be used to represent a
number we don’t know. For
example, 2𝑤 + 3 = 45 means if
double some number (math people
avoid writing × if they can) and add
3, we get 45. To figure out what 𝑤
is, we need to work backwards; 2𝑤
must be 3 less than 45, so we write
2𝑤 = 45 − 3 = 42. Then 𝑤 must
be half of 42, so we write 𝑤 = 42 ÷
2 = 21.
Guess and check is a totally valid
solution strategy, but should
probably be your last resort, and is
better if you have some reasons for
guessing what you’re guessing (e.g.
“My guess gave me an answer that
was WAY to big; I should guess a lot
less this time.”).
Every counting number other than 1
can be written as the product of
prime numbers in exactly one way;
this is called the “prime
factorization” of the number.
For example,
12 = 2 × 3 × 3 = 21 × 32 and
120 = 2 ⋅ 5 ⋅ 2 ⋅ 3 ⋅ 3 = 22 ⋅ 32 ⋅ 51 .
One way to find prime factorizations
is to find numbers that multiply to
the number, e.g. 120 = 10 × 12,
then do the same for those numbers
until you only have prime numbers.
Probability is a measure of how
likely something is. If something is
certain, it has a probability of 1. If it
is impossible, it has a probability of
0. In between, probabilities are
fractions, and represent the number
of ways the thing could happen
divided by the total number of ways
𝑔𝑜𝑜𝑑
anything could happen (
𝑡𝑜𝑡𝑎𝑙
).
For example, if a bag has three red
marbles and two blue marbles, the
probability of drawing a red marble
𝑔𝑜𝑜𝑑
𝑟𝑒𝑑
3
is
=
= .
𝑡𝑜𝑡𝑎𝑙
𝑎𝑙𝑙
5
The “counting principle” says that if
you have two (or more) independent
sets of choices, you can multiply the
numbers of choices to figure out the
overall number of choices.
If you have three pairs of pants and
five shirts, you have 3 × 5 = 15
outfits.
If you are flipping four coins, and
each of them could land two
different ways, there are 2 × 2 ×
2 × 2 = 16 total ways they could
flip.
An “arithmetic” sequence is one
where the same number is added
(or subtracted) between terms, such
as 1, 3, 5, 7, … or 100, 93, 86, 79, …
A “geometric” sequence is one
where the same number is
multiplied (or divided) between
terms, such as 1, 3, 9, 27, … or 3, 6,
12, 18, …
Sometimes two sequences are
interspersed, such as 1, 1, 3, 3, 5, 9,
7, 27, …
Any number can be written in
“scientific notation”, which is a
number between 1 and 10
multiplied by a power of 10. For
example, 0.098 = 9.8 × 10−2 and
76543 = 7.6543 × 104 . The
exponent of the 10 is the number of
decimal places you moved the
decimal point to the left. It will be a
negative number if you move the
decimal point to the right.
When two angles add up to a
“straight angle” (180°), they are
“supplementary”, and each is the
“supplement” of the other.
When two angles add up to a “right
angle” (90°) they are
“complementary”, and each is the
“complement” of the other.
A “permutation” is when you are
picking items from a group and the
order in which you pick them
matters. For example, if five people
run a race and you’ve got a firstplace trophy, a second-place
certificate, and a third-place
handshake, the order matters.
There are 5 × 4 × 3 = 20 × 3 = 60
ways you could give the awards.
A “combination” is when you’re
picking items from a group, but
order doesn’t matter. For example,
if you’re electing three kids from
your five-person math club to be
part of the student council, the
order doesn’t matter. The math
works out just like a “permutation”,
except that you divide by the
“factorial” of the number of things
you’re picking (in this case 3). So,
5×4×3
5×4×3
there are
=
=5×2=
3!
3×2×1
10 ways to pick them.
In a circle, the diameter (𝑑 ) is the
distance across the circle through
the center.
The radius (𝑟) is the distance from
the center to the edge, so it is half
the diameter (𝑑 = 2𝑟).
The circumference (𝐶 ) is the
perimeter, the distance around the
circle, and is equal to the diameter
times 𝜋 (pi) (𝐶 = 𝜋𝑑 ).
The area (𝐴) is equal to 𝜋 times the
square of the radius (𝐴 = 𝜋𝑟 2 ).
𝜋 (pi) is a really awesome number
that comes up in circle problems,
and in a lot of other places as you
learn more math.
𝜋 is an irrational number, which
means that it is not equal to any
fraction or mixed number, and that
its decimal representation goes on
forever without repeating.
𝜋 is commonly approximated as
1
22
3.14, 3 , or , but you should only
7
7
use these if a problem tells you to.
A palindrome is a number that reads
the same forward and backward,
such as 11, 202, 9999, or 87678.
Polygons are two-dimensional
figures who sides are all line
segments. Their corners are called
vertices.
Polyhedrons are three-dimensional
solids whose faces are all polygons.
Their corners are called vertices,
their “corner sides” are called edges,
and their “flat sides” are called
faces.
Factors are numbers that can divide
into another number. For example,
2 and 8 are both factors of 24.
Multiples are the opposite of
factors. For example, 24 and 38 are
both multiples of 2.
The greatest common factor of two
numbers is the largest number that
is a factor of both of them.
The least common multiple of two
numbers is the smallest number that
is a multiple of all of them.