Download Introduction, Math study habits, Review of Prealgebra

PREALGEBRA REVIEW DEFINITIONS
Commutative Laws
The "Commutative Laws" say you can swap numbers over and still get the same answer….
when you add:
a + b = b + a,
or when you multiply:
a×b = b×a
Associative Laws
The "Associative Laws" say that it doesn't matter how you group the numbers (i.e. which you calculate first) ...
when you add:
(a + b) + c = a + (b + c)
... or when you multiply:
(a × b) × c = a × (b × c)
Distributive Law
The “Distributive Law’ says you get the same answer when you:
 multiply a number by a group of numbers added together, or
 do each multiply separately then add them
a × (b + c) = a × b + a × c
NOTE!!! These laws are to do with adding or multiplying, not dividing or subtracting.
4 - 3 = 1, but 3 – 4 = -1
The Commutative Law does not work for subtraction division:
12 / 3 = 4, but 3 / 12 = ¼
The Associative Law does not work for subtraction or division:
(9 – 4) – 3 = 5 – 3 = 2, but 9 – (4 – 3) = 9 – 1 = 8
(20  10)  5 
The Distributive Law does not work for subtraction or division:
2
but 20  (10  5)  10
5
6 – (3-1) = 4 but (6 - 3) – 1 = 2
24 / (4 + 8) = 24 / 12 = 2, but 24 / 4 + 24 / 8 = 6 + 3 = 9
PERCENT (%) - “parts of 100.”
63% means 63 parts of 100, or 63/100.
To convert to a percent, just multiply by 100. If the number is a decimal, multiply by 100 (just move the decimal point 2
places to the left) and attach the % sign
.02 = .02 X 100 = 2%
To convert from a percent to a decimal, just drop the % sign and divide by 100 (move the decimal point 2 places to the rght).
2% = 2/100 = .02
Every statement of percent involves three numbers. For example,8 is 50% of 16.
8 is called the Amount. 50% is the Percent. 16 is called the Base. The Base always follows "of." In any percent problem, we are given
two of those numbers, and we are asked to find the third.
Amount = % x Base, where the % is put in decimal or fraction format.
8 = 50% of 16 can be translated into 8 = .50 x 16 or 8 = ½ x 16
PREALGEBRA REVIEW DEFINITIONS
ORDER OF OPERATIONS (PEMDAS)
Please: do all operations within parentheses and other grouping symbols (such as [ ], or operations in numerators and denominators of
fractions) from innermost outward.
Excuse: calculate exponents
My Dear: do all multiplications and divisions as they occur from left to right
Aunt Sally: do all additions and subtractions as they occur from left to right.
Example:
20 – 2 + 3(8 - 6)2 Expression in parentheses gets calculated first
= 20 – 2 + 3(2)2 Next comes all items with exponents
=
20 – 2 + 3(4) Next in order comes multiplication. Multiplication and Division always come before addition or division, even if to the
right.
= 20 – 2 + 12 Now when choosing between when to do addition and when to do subtraction, always go from left to right, so do 20-2
first, because the subtraction is to the left of the addition.
= 18 + 12 Now finally we can do the addition.
= 30
The Test of Reasonableness
In general applying a test of reasonableness to an answer means looking at it in relation to the numbers operated upon to determine if
it’s “in the ballpark.” Put in simple terms, you look at the answer to see if it makes sense. For example, if you determine 10% of $75
to $750, you should immediately notice something is very wrong, because 10% of something is much smaller than the original amount.
The test of reasonableness comes very much into play in word problems. For example, questions asking for length, dollars, or time
should never give negative answers because those things would not make sense if they were negative (what is -3 feet?). Also, think
about what the answer should be. If I invest $400 dollars in a bank at 8% interest for 3 years, I would expect the balance to be larger
than the original amount that I invested (that’s why we put money in the bank!), so an answer that is smaller than $400 is obviously
wrong.
VOCABULARY DEFINITIONS
Factor - One of two or more quantities that divides a given quantity without a remainder. For example, 2 and 3 are factors of 6; a and
b are factors of ab.
Factor X Factor = Product
Product - The number or quantity obtained by multiplying two or more numbers together.
Factor X Factor = Product
Dividend - A quantity to be divided.
Divisor - The quantity by which another quantity, the dividend, is to be divided.
Quotient - The number obtained by dividing one quantity by another. In 45 ÷ 3 = 15, 15 is the quotient.
Dividend/ Divisor = Quotient
or Dividend ÷ Divisor = Quotient
quotient
divisor dividend
Remainder - The number left over when one integer is divided by another: The remainder plus the product of the quotient times the
divisor equals the dividend.
If there is a remainder, then
Dividend = Quotient X Divisor + Remainder
PREALGEBRA REVIEW DEFINITIONS
Least Common Multiple (LCD) - The smallest quantity that is divisible by two or more given quantities
without a remainder: 12 is the least common multiple of 2, 3, 4, and 6. Also called lowest common multiple.
Greatest Common Factor (GCF) - The largest number that divides evenly into each of a given set of numbers.
The greatest common divisor is useful for reducing a fraction into lowest terms. Consider for instance
where we cancelled 14, the greatest common factor of 42 and 56.
Prime number is a whole number that has exactly two factors: itself and 1.
Composite number is a whole number that has more than two factors.
The prime factorization of a whole number is the number written as the product of its prime factors.
EXAMPLE Write the prime factorization of 72.
Solution
We start building a factor tree for 72 by dividing 72 by the smallest prime, 2.
Because 72 is 2 · 36, we write both 2 and 36 underneath the 72. Then we circle the 2 because it is prime.
Next we divide 36 by 2, writing both 2 and 18, and circling 2 because it is prime. Below the 18, we write 2 and 9, again circling the 2.
Because 9 is not divisible by 2, we divide it by the next smallest prime, 3. We continue this process until all the factors in the bottom
row are prime.
The prime factorization of 72 is the product of the circled factors.
72 = 2 × 2 × 2 × 3 × 3
We can also write this prime factorization as
.
PREALGEBRA REVIEW DEFINITIONS
A set is a collection of objects. The objects in the set are called the elements of the set. The roster method of
writing sets encloses a list of the elements in braces.
Example: The set of even natural numbers less than 10 can be written like this: {2,4,6,8}.
The set of natural numbers is {1,2,3,4,5,6,7,….}. These are basically the “counting numbers.”
The set of whole numbers is the set of natural numbers and the number, 0. {0, 1, 2, 3, 4, 5, ….}
The set of integers is the set of whole numbers and their opposites. {…. -4,-3,-2,-1,0,1,2,3,4,….}
The number 0 is an integer, but it is neither negative nor positive. For any two different places on the number
line, the integer on the right is greater (>) than the integer on the left.
The absolute value (using the | | symbol) of a number is its distance from zero on the number line. The
absolute value of a number is ALWAYS POSITIVE (or 0).
| 5| = 5,
|-5| = 5,
|3-8| = |-5| = 5, |3-3| = |0| = 0
|-4|=4
-4 is 4 units away
from 0
|4|=4
+4 is 4units away
from 0
PREALGEBRA REVIEW DEFINITIONS
INTEGERS
Adding and Subtracting Integers
When adding two integers with the same sign, just ignore the signs, then attach them on the answer.
-3 + -5 = - (3 + 5) = -8
When adding two integers with different signs, take the absolute values (make both numbers positive), and subtract the smaller one
from the larger one. The sign of the integer with the larger absolute value will be the sign of your answer.
7 + (-8)
The absolute value of 7 is |7| = 7
The absolute value of -8 is |-8| = 8 larger
8–7=1
Remember in the original problem, the integer whose absolute value was 8 was -8, so our answer is negative.
7 + (-8) = -1
Subtracting a negative integer is the same as ________ a _______ integer.
3 – (-5) = 3 + 5 = 8
-2 – (-3) = -2 + 3 = 1
Subtracting a positive integer is the same as _____ a _____ integer.
5 – 3 = 5 + (-3) = 2
-2 – 3 = -2 + (-3) = 1
Multiplying and Dividing Integers
(positive integer) X (positive integer) = (_____ integer)
(positive integer) ÷ (positive integer) = (______ integer)
(negative integer) X (negative integer ) = (______ integer)
(negative integer) ÷ (negative integer ) = (_______ integer)
(positive integer) X (negative integer) = (______ integer)
(positive integer) ÷ (negative integer) = (_______ integer)
(negative integer) X (positive integer ) = (negative integer)
(negative integer) ÷ (positive integer ) = (negative integer)
PREALGEBRA REVIEW DEFINITIONS
Memory Tip:
When something bad happens to a good person, that’s bad.
When something good happens to a bad person, that’s bad.
When something good happens to a good person, that’s good.
When something bad happens to a bad person, that’s good.
When a negative integer is raised to an even power, the result is _____.
(-2)4= (-2)(-2)(-2)(-2) = 16
When a negative integer is raised to an even power, the result is ______.
(-2)5= (-2)(-2)(-2)(-2)(-2) = -32
Exponents are only applied to the number directly diagonally left of it. If a negative integer is raised to a power, it must be in
parentheses () in order for the exponent to apply to the negative number.
(-2)2 ≠ - 22
(-2)2 = (-2)(-2) = 4
- 22 = - (2)(2) = -4
EXPONENTS
Exponent - A number or symbol, as 3 in (x + y)3, placed to the right of and above another number, variable, or expression (called the
base), denoting the power to which the base is to be raised. Also called power.
In this example: 82 = 8 × 8 = 64
The exponent (or power) tells how many times the base is to be multiplied by itself.
Example 1:
(x + y)3 = (x + y)(x+y)(x+y)
Example 2:
(-3)4 = (-3)(-3)(-3)(-3) = 81
PREALGEBRA REVIEW DEFINITIONS
FRACTIONS
A fraction is just a division problem.
Numerator
= Numerator ÷ Denominator
Denominator
A fraction is in LOWEST TERMS when the numerator and denominator have no common factors.
6
3
is not in lowest terms,
is in lowest terms
10
5
3
Mixed Number - A number, such as 6 , consisting of an integer and a fraction. A mixed number is just the sum of a whole
5
number and a fraction.
Improper Fraction - A fraction in which the numerator is larger than the denominator. Converting from mixed number to
improper fraction:
3 6  5  3 33
6 

5
5
5
Mixed numbers must be converted to improper fractions or decimals before doing ANY MULTIPLICATION OR DIVISION
OPERATIONS on them.
A fraction can be converted into a decimal by dividing: Numerator ÷ Denominator
denominator numerator
When ADDING or SUBTRACTING FRACTIONS, they must have the same denominator, then you just add the numerators
and leave the denominator the same.
1 3
 
6 8
The denominators, 6 and 8, are not the same, so we must find the LEAST COMMON DENOMINATOR to convert these fractions into
equivalent ones with the same denominator. The LCM of 6 and 8 is the SMALLEST NUMBER THAT BOTH 6 and 8 can
go into. Choose the larger denominator (which is 8 in this case) and start taking multiples until 6 can go into it.
Does 6 go into 8? NO
Does 6 go into 8x2? NO
Does 6 go into 8x3? Yes, 6 goes into 24. Therefore 24 is the LCM.
Multiply the numerator and denominator of each fraction by whatever it takes to get the LCM as the new denominator.
Then once you have the same denominators in each fraction, just add the numerators and leave the denominator the same.
1  4 33 4
9 13


   
6  4  8  3  24 24 24
MULTIPLYING FRACTIONS
Multiplying Fractions uses a different rule. When multiplying fractions, you multiplying the numerators AND the denominators.
 1  3  1  3 3

   
 6  8  6  8 48
This fraction can simplified by canceling out common factors in the numerators and denominators (tops and bottoms)before
multiplying across. 6 can be rewritten as 2x3, and the 3’s can be cancelled out.
1 3
1
 1  3  1  3


   
 6  8  6  8 2  3  8 16
DIVIDING FRACTIONS
The rule for dividing fractions is simple. Just take the RECIPROCAL(flip the top and bottom) of the DIVISOR (the fraction to the
right of the ÷ symbol) multiply
 1   3   1   8  1 8 1 2  4 4


        
 6   8   6   3  6  3 2  3 3 9
PREALGEBRA REVIEW DEFINITIONS
GEOMETRY
straight angle = 180 degrees.
Adjacent angles that form a straight angle add up to 180 degrees.
A
B
A + B = 180°
right angle = 90 degrees. Complementary angles area adjacent angles that form a right angle and add up to 90 degrees.
acute angle = less than 90 degrees
obtuse angle = greater than 90 degees.
Vertical angles (opposite angles formed from intersecting lines) are congruent (the same).
a2+b2=c2
Pythagorean Theorem:
The sum of the squares of the legs of a
right triangle is equal to the square of the
hypotenuse.
c
a
b
POLYGONS
QUADRILATERALS (4-sided polygons):
Parallelogram
Rhombus
Rectangle
(opposite sides parallel) (all sides same size)
Square Trapezoid (only 2 sides are parallel)
(each angle 90°)
(each angle 90°
and all sides same size)
Triangles
Scalene Triangle
(no sides the same)
Isosceles Triangle
Equilateral Triangle
Right Triangle
(2 sides the same)
(3 sides the same)
(one angle is 90°)
Angles of a triangle add up to 180 degrees.
Perimeter – distance around the edges of an object
Perimeter of a square = 4s, where s = length of one side
Perimeter of a rectangle = 2W + 2L, where W = width and L = length
Perimeter of a triangle = side 1 + side 2 + side 3
Area – amount of surface covered by an object.
Area of a square = s2
Area of a rectangle = L*W , where L = length, W = width
Area of a triangle = ½ bh, where b = length of base and h = height
Area of a parallelogram = bh, where b = length of base and h = height
Volume – capacity, or the amount of 3-dimensional space an object occupies.
Volume of a cube = s3
Volume of a box (rectangular prism) = L*W*H, where L=length, W = width, H = height
Volume of a cylinder = πr2*h
Volume of a cone = ⅓ πr2*h
Volume of a sphere = 4 r 3
3
See next page for explanations of circles and definitions of π and r.
CIRCLES:
A circle is the set of all points that are a fixed distance from the center.
The fixed distance is called the radius, often abbreviated by the variable, r.
The diameter starts at one side of the circle, goes through the center and ends on the other side.
So the Diameter is twice the Radius:
Diameter = 2 × Radius = 2r
Half a circle
is called a
Semicircle.
Center
Given this property, you can make a formula for
the circumference by multiplying both sides of this equation
by the diameter.
Circumfere nce
  ( Diameter )
Diameter
Circumfere nce   (Diameter )
Circumfere nce   (2r )  2r
(Diameter)
Area of a Circle:
If you make a parallelogram out of a circle by piecing together the
slices of pie, you will use half the circumference for one base and
half the circumference for the other base (so base = ½(2πr)=πr).
The height of the parallelogram will be the radius, r.
Now we now the area of a parallelogram is base X height.
So the area of a circle is (πr)(r).
Area of a Circle = πr2
VARIABLE EXPRESSIONS AND LIKE TERMS
Expressions (contain no “=” sign) :
An expression is one or numbers or variables having some mathematical operations done on them.
Numerical Expressions:
3+5
3(4)
6/2
5-1
4
Expressions can be evaluated or simplified: 3 + 5 can be simplified to 8
This just means, “Whatever you see, do”
Algebraic Expressions:
x+5
3x  in Algebra, it is implied that 3x means 3 times x. The “proper” way to write the product of a number and a variable
is to always write the number to the left of the variable. x times 5 = 5x
When numbers are multiplied by variables, they are given a special name, “coefficient”.
5 is the coefficient of 5x.
Algebraic expressions can be simplified by using the associative and distributive properties.
-4m(-5n) can be simplified by rearranging the terms (we can do this when the only operation is multiplication) so that all
the constants are grouped together and all the variables are grouped together(alphabetically))
-4m(-5n) = (-4)(-5)mn = 20mn
2(-4z)(6y)=2(-4)(6)yz = -48yz
3(s+7) can be simplified by using the distributive property
3s + 3(7) = 3x + 21
-6(-3x -6y + 8) can also be simplified. Change any subtraction to adding a negative.
-6(-3x + -6y + 8)
Now distribute the -6 and make sure to glue the negative sign on that -6 wherever you distribute it!
-6(-3x) + -6(-6y) + -6(8) = 18x + 36y + -48 = 18x + 36y + 48.
Like terms are terms with exactly the same variables raised to the exactly the same powers.
expression are considered like terms. Terms that are not like terms are called unlike terms.
Like Terms
Unlike Terms
2x, 3x, -4x
Same variables,
each with a power
of 1.
2x, 2x
Different powers
3, 5, -1
Constants
3, 3x, 3x
2
2
5x , -x
Same variables and same
powers
2
2
Different powers
2
2
5x , 5y
Different variables
Any constants in an
Only Like Terms can be combined!
Combining like terms is to add or subtract like terms. Combine like terms containing variables by combining their
coefficients and keeping the same variables with the same exponents.
Example: 3x2 – 8x2 = (3 - 8)x2 = -5x2
Example: 5x + 2x = (5 + 2)x = 7x
Example:
3x2 + 5x + 2 + x2 – x - 3
=3x2 + x2 + 5x-x + 2-3
= 4x2 + 4x – 1
Algebraic expressions can only be simplified if they have “like terms,” which are terms with the same variable and same
exponent.
x+5+2
Only 5 + 2 can be simplified. x + 5 + 2 becomes x + 7
3x + x + 2
can be simplified to 4x + 2
x2 + x
cannot be simplified because they don’t have the same exponents and are therefore not like terms.
Algebraic expressions can be EVALUATED if you are given the value for the variables.
 x  15
for x = 3. Substitute 3 for x in the expression.
6
 (3)  15  3  15  3  (15)  18



 3
6
6
6
6
Example: Evaluate
Example:
2
If a=-2, evaluate -3a + 4a
2
-3(-2) + 4(-2) Do exponents first
= -3(-2) + 4(4) Multiplication left to right
= 6 + 4(4)
= 6 + 16
Addition
= 22
17. Translating Verbal Expressions into Mathematical Expressions
Verbal Expressions Examples
Addition
added to
more than
the sum of
increased by
the total of
6 added to y
8 more than x
the sum of x and z
t increased by 9
the total of 5 and y
Math
Translation
6+y
8+x
x+z
t+9
5+y
Subtraction
minus
less than
subtracted from
decreased by
the difference
between
x minus 2
7 less than t
5 subtracted from 8
m decreased by 3
the difference between y and 4
x-2
t-7
8-5
m-3
y-4
10 times 2
one half of 6
the product of 4 and 3
10 X 2
(1/2) X 6
4X3
multiplied by
y multiplied by 11
11y
divided by
the quotient of
x divided by 12
the quotient of y and z
x/12
y/z
the ratio of
the ratio of t to 9
t/9
Power
the square of
the cube of
squared
the square of x
the cube of z
y squared
x
3
z
2
y
Equivalency
equals
is
is the same as
1+2 equals 3
2 is half of 4
½ is the same as 2/4
1+2 = 3
2 = (½)X4
yields
represents
3+1 yields 4
y represents x+1
3+1 = 4
y=x+1
greater than
less than
greater than or equal
to
at least
no less than
less than or equal to
at most
no more than
-3 is greater than -5
-5 is less then -3
x is greater than or equal to 5
-3 > -5
-5 < -3
x≥5
x is at least 80
x is no less than 70
x is less then or equal to -6
y is at most 23
y is no more than 21
x ≥ 80
x ≥ 70
x ≤ -6
y ≤ 23
y ≤ 21
Multiplication times
of
the product of
Division
2
1
2

2
4
Comparison
Solving Application Problems
Problem-Solving Strategy:
•
Analyze the problem. What are you trying to find? What’s the given info?
•
Work out a plan before starting. Draw a sketch if possible. Look for indicator words (e.g.
gained, lost, times, per) to know which operations (+,-, x,÷) to use.
•
Estimate a reasonable answer.
•
Solve the problem.
•
Check your work. If the answer is not reasonable, start over.
Example:
Each home in a subdivision requires 180 ft of fencing. Find the number of homes that can be fenced
with 5760 ft of fencing material.
•
What are we trying to find? Total number of homes that can be fenced.
Given info: 5760 ft of material, 180 ft of fencing per home.
•
Work out a plan. We want to know how many homes can be fenced with 5760 ft of fencing. If
we divide 180 feet per home into 5760 feet we will get the total number of homes ( The feet
units cancel out).
•
Estimate a reasonable answer.
Round 5760 to 6000
and 180 to 200.
6000÷200 = 30
4) Solve the problem.
5) Check your work.
Check division. Does 32x18=5760 ? Yes.
Is 32 close to our estimate, 30? Yes.
So our answer is 32 homes. (Make sure the units “homes” is in the answer!)
5760 ft 
1 home 5760

 32 homes
180 ft
180
32
180 5760
- 540
360
- 360
0