Applied Probability and Statistics
Module 1: Basic Numeracy and Calculations
1.2 Number Systems
1) Whole Numbers – a whole number is 0,1,2,3,4,5,6,7,8,9, …
a) Whole numbers are numbers whose values are “whole”, such as 1 or 2. Fractions or
decimals, on the other hand, can be “parts of a whole”, such as “one half.”
Whole numbers can be represented without fractional or decimal component and aren’t
negative.
2) Integers – like whole numbers, are numerical figures that don’t contain a fractional or
decimal component. Integers, unlike whole numbers can be either a positive number, a
negative number, or zero. Ex. (Negative)-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, (Positive) 1, 2, 3, 4,
5, 6, 7, 8, 9. See number line below:
a) Positive integers have a value that is greater than zero (meaning to the right of 0 on the
number line, while negative integers are less than zero (or to the left of 0 on the number
line. It’s important to remember that zero is neither positive or nor negative. (However,
zero is considered an integer).
3) Rational numbers – are numbers that can be expressed as a fraction. This class of numbers
4
includes all integers since all integer can be expressed as a fraction: 4= 1
a) Rational numbers are also decimal numbers that have expansions that end or continue
to repeat forever, rather than continuing forever without repeating.
−3
For Ex 101 In decimal form is -0.02970297…, which we can also write as 0.0297 (With bar
above 0297) means the 0297 continues repeating forever.
b) Rational numbers include all the integers plus fractions and decimals that end or repeat.
3
1) Ex. One and three-fourths (1 4 )is located between one and two. Negative fractions
and decimals are placed between negative integers. For ex. -3.5 is between
negative 3 and negative 4.
4) Real Numbers – A real number is any number that can be placed on the number line,
whether that be negative or positive, fraction or decimal. Real numbers also include
decimals that don’t end and can’t be written as a fraction. For ex. Pi is approximated to
3.14, but the decimals don’t end and don’t repeat.
5) Everything included on the number line below is considered a real number.
1.3 Negative Numbers – Negative numbers are less than zero. One common way people try to
wrap their heads around positive and negative number is to imagine a number line. When
comparing any two quantities, the number to the right is always greater than the number to the
left. In the number line below, for example, 5 is greater than 3, 0 is greater than -3, and -3 is
greater than -5. Whenever comparing two numbers, you can imagine where they are on the
number line to determine which is greater.
1.4 Ordering
1) Real Numbers – are the numbers that can be placed on a number line. One important
component of numeracy is the ability to understand the value of a number at a glance.
A) For ex. -3 is greater than -4, but less than 0.
B) Whole Numbers of 16, 1, 3024, and 5 can be placed in numeric order from least to greatest:
1, 5, 16, 3024. The numbers increase as you go to the right on the number line.
C) You can use a similar strategy for placing integers on the number line. Remember, however,
that for negative integers, the value is smaller the farther from zero. Notice that -3 is to the
left of 2 on the number line. Therefore, we know that -3 is less than 2.
D) Placing a rational number on the number line requires recognizing that any partial amount
in a fraction or decimal makes the number greater than the whole number part of the
fraction or decimal. For ex. 4.3 should be placed on the number line between 4 and 5, not
3
between 4 and 3. Likewise, 1 4 Is greater than 1 and therefore between integers 1 and 2.
E) Infinite ∞, Repeating Decimal is considered a Rational number.
F) Negative fractions and decimals can be placed between negative integers the same way,
though remember that the lesser value integer will be to the left of the greater integer. For
negative numbers, farther away from zero, the smaller the value.
1.5 Intervals and Sets
1) Set – A collection of numbers. In Statistics, a collection of numbers is often referred to as
Data. These groups of numbers and data can be categorized as discrete or continuous.
A) A collection of numbers is discrete if its values are distinct, separate and unconnected.
B) If the values within the set are connected, without gaps, the collection is considered to
be continuous.
C) An interval is a set of numbers between two specified values. An interval can be
visualized as a segment of the number line. The segment of the number line above that
falls between 1 and 2 is called an interval.
D) The set of real numbers between 1 and 2, highlighted green in the above number line, is
a continuous set. This set contains all numbers within the interval; there are no gaps in
value. The values within the set are connected, without gaps.
E) A set of data is continuous if it can hold any value within the set. An example of
continuous data might be age. Possible to be 22.67 years of age, real numbers are
considered continuous.
F) Discrete - Can only have certain, distinct values, is "counted", contains unconnected
points, in mathematics, whole numbers, integers, and even integers are all examples of
discrete sets. These sets contain unconnected elements, with gaps between each value.
G) In statistics, some data sets will be discrete. Examples of discrete data sets are the
number of adults in a household, the results of rolling two dice, and number of
machines in operation, as these are distinct groups.
H) Continuous - Can have any value within an interval, is "measured", does not have clear
boundaries between elements or data points.
I) In mathematics, the set of real numbers is an example of a continuous set. This sets
contains continuous elements, with no discernible gaps between each element.
Remember that the number line is a visual representation of the set of real numbers.
Just as the number line is continuous with no gaps, so is the set of real numbers.
J) In statistics, some data sets will be continuous. Examples of continuous data sets are
temperature, distance, and time, as the set of possible values within these groups is
continuous. An element in these groups can hold any real number within a certain
interval, dependent upon the scale used.
K) An entire set of data can also be categorized as discrete or continuous, if a distribution
turns out to have a defined number of outcomes, the distribution is considered discrete.
If the distribution results in any number of outcomes in a interval, its considered
continuous.
1.6 Notation: Intervals and Sets
1) Inequalities become very important in defining interval sets. Let’s consider the interval
between the real numbers 1 and 2. We use a variable such as x to represent numbers
found in the interval. We know that x is greater than 1 but less than 2, so it can be
expressed as 1 < x < 2 <—> This expression would read “1 is less than x which is less
than 2.
A) This type of expression can come in many different forms. Say, interval , a included
all the values between 1 and 2, as well as 1 and 2.
B) The number line above displays 1 ≤ a ≤ 2, meaning 1 is less than or equal to a which
is less than or equal to 2.
2) Interval Notation – There’s a number of different ways intervals can be written and
interpreted. The first row on the chart displays the ways less than and greater than
inequalities appear both in writing and drawn on a number line. Second row shows how
less than or equal to and greater than or equal to intervals appear.
A) A Parenthesis, ( , in writing denotes that the value alongside it is not to be included
as in < and >).
B) A Bracket, ], means the value alongside it is included (as in ≤ and ≥). Use a comma
between the two value to represent the interval.
C) Example – Write 4 < x ≤ 9, with x being the interval. Place the lower boundary on
the left and the upper boundary on the right: 4, 9. Use an open parenthesis before
the 4 to represent greater than, and a closed bracket after the 9 to represent less
than or equal to: (4, 9]
D) On a number line: An open circle denotes that the value is not to be included.
(as in < and >). A closed circle means that the value is included (as ≤ and ≥).
Connected the plotted values on the number line to represent the interval.
E) Example: Display −1 ≤ x ≤ 3 using interval notation. [-1, 3]
F) Example: Display −5 < x < −2 using interval notation. (-5, -2)
G) Example: Display -5 < x ≤ -2 using interval notation. (-5, -2]
H) When the interval is infinite – as in rather than being between two values, its
compared only to one – use an infinity symbol, ∞, in writing and an arrow on the
number line to display the continuous interval.
I) Example: Display x > -2 using interval notation (-2, ∞)
J) Example: Display x ≤ 8 using interval notation (- ∞, 8]
K) Discrete Sets – often use a different notation. A discrete set is often denoted with a
pair of closed braces { }. Integers between 5 and 16 inclusive: Set = {5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16}
1.7 Arithmetic
1) Mathematical Expressions – Expressions are a group of symbols, such as numbers and
operators, that has mathematical validity. Most basic expressions are arithmetic
expressions, such as 3+3, 6÷2. “Evaluate an expression”, is a synonymous with the
statement “solve the problem.”
a) Addition of Positive Numbers – denoted by the + sign. Sum is when multiple
numbers are added to one another. Sum can be found by moving that many places
to the right long the number line.
1) Identities – Addition is cumulative – the order in which the numbers appear in
the sum can be reversed.
2) Identity property – adding 0 to any number doesn’t change the original
number. 0.924 + 0 = 0.924
3) Additive inverses – Special relationship between values that are equally far
from 0 on opposite side of the number line, such as -3 and 3. The sum of any
number and its inverse is 0.
b) Subtract of Positive Numbers – denoted by – (minus) sign. Difference is when
multiple numbers are subtracted from one another. A difference can be found by
moving that many places to the left along the number line.
c) Subtraction - is not cumulative
1.8 Muti-Step Expressions
1) All addition and subtraction operations are performed from left to right. Multi-Step
expressions should be solved from left to right.
1.9 Multiplication and Division
1) Multiplication is the act of adding the same number multiple times. Multiplication can
be denoted by a few different signs, dot-sign, ∙, known officially as an interpunct.
2) Product - is when numbers are multiplied.
3) Identities: Cumulative, meaning the order in which you multiply the numbers doesn’t
matter. Multiplicative inverse of a number x is the number you must multiply x by to
1
get 1. The numbers 5 and Multiplicative inverses.
5
1.10
Negative Integers: Addition and Subtraction
1)
Addition of Negative Numbers – are represented with a minus sign directly in front of
the number – Ex. (-5) When adding a negative number, you will see an addition (plus)
sign followed by a negative (minus) sign – Ex. (5+ - 6). Adding a negative number is the
same as subtraction. To understand how this works, think of the negative sign as an
“opposite” sign. The addition sign is followed immediately by a negative (opposite) sign,
it turns the addition operation into its opposite: subtraction. Ex. Our sum becomes a
difference.
2) Subtraction of Negative Numbers – negative numbers and subtraction are both
denoted by the minus sign, - ,. When subtracting a negative number, you’ll see a
subtraction (minus) sign followed by a negative (minus) sign. Parentheses are often
added around the negative number 3 – (-2). This difference is read “three minus
negative two”. Subtracting a negative number is the same as addition. Think of the
negative sign as an “opposite” sign. The subtraction sign is followed immediately by a
negative (opposite) sign, it turns the subtraction operations into its opposite: addition.
3)
Examples: 1) -56 + -8 = -64, 2) 7 – 11 + -5 = (Problem Solved): 7-11+ -5= -4 + -5 =-9
40 + (-30) – 5 = (Problem Solved): 40 + (-30) – 5 = 10 – 5 = 5),
4)
7 + 6 + (-25) = (Problem Solved): 7 + 6 (-25) = 13 + (-25) = 13 – 25 = -12,
5)
40 – (-2) + (-8) = (Problem Solved): 40 – (-2) + (-8) = 42 + (– 8) = 42 – 8 = 34
6)
-603 + 69 + (-25) = (Problem Solved): -603 + 69 + (-25) = (-534) + (-25) = -559
7)
410 – (-261) + (-86) = (Problem Solved): 410 – (-261) + (-86) = 671 + (-86) = 671- 86 =
585.
3)
1.11
Negative Integers: Multiplication and Division
1) Multiplying a Positive Number by a Negative Number
a) When multiplying a positive number by a negative number, the product will always
be negative. This type of operation involves unlike signs, a positive sign (+) and a
negative sign (-). Ex. −8 ∙ 4 ∙ 2 = −64. In this example, multiplying any number of
positive numbers by a negative number changes the sign.
2) Multiplying a Negative Number by a Negative Number
a) When multiplying a negative number by a negative number results in a positive
product. Multiplying products of two negatives will always equal a positive
(negative x negative = positive) Ex’s. -12 x -4 = 48, -1 x -32 = 32, 1- x -2 x -1 = -2
(Problem Solution): 1 x -2 = 2 x -1 = -2
b) Even numbers of negative numbers will yield a positive result; while an odd number
of negative numbers will yield a negative result.
3) Dividing a Positive Number by a Negative Number
a) When dividing a positive number by a negative number (or a negative number by a
positive number), the quotient will always be negative. A positive number divided
by a negative number must equal a negative quotient.
b) Dividing a negative number by a negative number equals a positive number. Two
negative numbers equal a positive number.
1.12
Zero: Multiplication and Division
1) Multiplying by Zero - The product of multiplication with zero will always be zero;
whether multiplying by a positive or negative, any integer multiplied by zero will result
in zero as the product.
a) Ex. 4 x 0 = 0; 0 x 25 = 0; 100 x 0 = 0
b) The only way for a product to equal 0 is if one of the numbers is 0. So, in the
expression a x b = 0, either a or b (or both) must be equal to 0.
c) When multiplying a negative number by zero, the above rule still stands; the
product will equal zero.
d) Ex. 0 x -8 = 0; 0 x -32 = 0
2) Dividing Zero by any Number – Dividing 0 by any nonzero number yields 0.
a) Ex. 0 ÷ 4 = 0; 0 ÷ 0 x 16 = 0; 0 ÷12 = 0
3) Dividing any Number by Zero
a) Dividing an integer by zero doesn’t equal that integer (because any integer divided
by 1 equals that integer, so dividing by zero can’t follow the same principle.)
b) So, when we divide an integer by zero, the answer is always “undefined.”; whether
positive or negative, an integer divided by zero will deliver an “undefined” result.
c) Ex. -6 ÷ 0 = undefined; -64 ÷ 0 = undefined; -28 ÷ 0 = undefined.
1.13
Exponents – are a representation of multiplication. The base number represents the
number being multiplied by itself, and the exponent represents how many times that number is
to be multiplied.
a) Ex. 63 (6 x 6 x 6 = 1,296); 35 (3 x 3 x 3 x 3 x = 243); (-2)2 2 means -2 will appear –
(negative) or be multiplied –(negative) twice (-2 x -2 = 4)
b) Any non-zero number with an exponent of 0 (or raised to the zero power) equals 1. Ex.
80 = 1
c) Any base with an exponent of 1 equals the base. Ex. 51 = 5. 1 means the 5 will only
appear once.
1.14
1)
2)
3)
4)
Application of Exponents
The repeated multiplication associated with exponents often represents repeated
events in life, such as receiving interest from a bank or determining probabilities
associated with sampling. Exponents are also used to represent large numbers that
occur in the real world.
When looking at the parameters of a test, the base number exhibits the number of
possible outcomes of one attempt or trial, and the exponent illustrates how many
times that trial or attempt is performed.
The Probability is determined by calculating the exponents, which can tell us what the
chances are of something occurring.
Applied Example: Say you flip a coin 6 times, there are 2 possible outcomes for every
flip – heads or tails.
1.15
Introduction to Order of Operations
1) Order of Operations – is a set of rules that defines the order in which mathematical
operations should be performed. When expressions have more than one operations
(meaning more than one calculation using addition, subtraction, multiplication or
division), we have to follow rules for the Order of Operations.
A) PEMDAS – Convenient way to remember the order of operations.
1) (P)Parentheses – First - do all operations that lie inside parentheses.
2) (E)Exponents – Second – Do any work with exponents or roots.
3) (M)Multiplication
Third – working from left to right, do all multiplication and division.
4) (D)Division
5) (A)Addition
Finally – working from left to right, do all addition and subtraction.
6) (S)Subtraction
B) Identities
a) Addition in multi-step expressions is associative – the placement of the
parentheses can be changed without affecting the result under addition.
b) Subtraction in multi-step expression is not associative. The placement of the
parentheses matters.
c) Associative property holds again if you add a negative number.
d) Applying Order of Operations
Evaluating Using PEMDAS To evaluate the expression correctly, follow Order of
Operations. The two operations in this expression are subtraction and division.
Using Order of Operations, we know to perform the division operation before
the subtraction operation: Correct: 8 - 4 ÷ 2 = 6. Problem Solution: 4 ÷ 2 = 2,
then subtract 8 – 2 = 6
e) Ex. 2 x 9 – 3(6 -1) + 1 = Problem Solution: 2 x 9 – 3(5) + 1 = 4
f)
Focus on the parentheses first, then the exponents and roots, then
multiplication and division. The final step is addition and subtraction from left to
right.
1.16
Approximating a Mathematical Solution
1) Estimation is a calculation that uses approximations instead of exact numbers in order
to arrive at a result that is close to, but distinct from, the real solution to a problem.
2) Advantages of Estimation:
a) Estimations can provide quicker calculations.
b) Can offer a more global picture of the situation.
c) Can get results in the absence of information.
3) Disadvantages of Estimation:
a) Estimations aren’t accurate. Can’t be used for calculations that need to be precise.
b) Can be further off than the estimator realizes, leading to wrong conclusions.
c) Sometimes the exact numbers are just as easy as the calculations.
1.17
Factors and Divisibility
a) Divisibility helps break down integers to their smallest parts.
b) Factors are integers that evenly divide the initial integer. When multiplied together, the
product of these factors give you the initial integer. Ex. 6 x 5 = 30; 30 ÷ 5 = 6. Because 5
divides 30 without a remainder, 5 is a factor of 30, and 6 is likewise a factor of 30.
c) Divisibility and factors are elements of both positive and negative numbers.
d) Ex. What are the factors of 28? Write 1 and 28, with space for additional factors. Check
to see if 28 is evenly divisible by 2. It’s 28 ÷ 2 = 14. Write 2 and 14 in the list of factors.
Check if 28 is evenly divided by 3, 4, 5, 6, 7 and add the numbers in the list of factors.
e) Ex. What are the factors of 16? Write 1 and 16, with space for additional factors. Check
if 16 is evenly divisible by 2. It’s 16 ÷ 2 = 8. Write 2 and 8 in the list of factors. Check if
16 is divided by 3, 4, 5, 6, 7, 8 and add the numbers in the factor list.
1.18
Prime and Composite Numbers
1) Prime number is a positive integer with exactly two positive factors, 1 and itself; it can’t
be divided evenly by any other two integers. For example, the only positive numbers
that divide 3 are 1 and 3. Therefore, 3 is a prime number.
a) Examples of Prime Members:
1) Positive Factors of 2: 1, 2
2) Positive Factors of 7: 1, 7
3) Positive Factors of 11: 1, 11
4) Positive Factors of 23: 1, 23
2) Composite number has at least one positive factor other than 1 and itself. For example,
10 can be divided evenly by 1, 2, 5, 10.
a) Examples of Composite Numbers:
1) Positive Factors of 6: 1, 2, 3, 6
2) Positive Factors of 12: 1, 2, 3, 4, 6, 12
3) Positive Factors of 25: 1, 5, 25
1.19
Prime Factorization and the Greatest Common Factor
1) Fundamental Theorem of Arithmetic states that all composite numbers can be
represented as the product of prime numbers. This theory simply states that any
integer greater than 1 is either prime or is the product of prime numbers. Prime
numbers can be considered the building blocks of composite numbers.
2) Prime Factorization is the act of identifying which prime factors make up a particular
composite number.
3) GCF – Greatest Common Factor of any two integers a and b is the greatest number that
is both a factor of a and a factor of b.
4) Factorization of Composite Numbers:
a) Positive factors of 6: 1, 2, 3, 6 – The greatest common factor 6 and 12 is 6. It’s the
greatest factor each number has in common.
b) Positive factors of 12: 1, 2, 3, 4, 6, 12
c) Positive factors of 25: 1, 5, 25 – greatest common factor of 12 and 25 is 1.
d) Positive factors of 32: 1, 2, 4, 8, 16, 32 – greatest common factor of 12 and 32 is 4.
1.20
Introduction to Square Roots
1) Square Root is a number that produces a specified number when its multiplied by
itself. Square roots are like exponents in that a number is multiplied by itself to
produce a particular product.
A) Perfect Square Root – Ex. 42 = 4 x 4 = 16; 62 = 6 x 6 = 36; (-82) = -8 x -8 = 64
B) Principal Square root is a positive square root
C) Negative Square root is a negative square root.
Examples - 42 = 16, so 4 is the principal square root of 16; 62 = 36, so 6 is the
principal square root of 36; (-8)2 = 64, so -8 is the negative square root of 64
2) Radical Sign √ - is a radical sign.
3) Radicand is the number under the radical sign. It’s the number within the radical
sign whose square root is to be taken. We use the radical sign as symbol for the
principal square root √16 = 4. The negative square root a negative sign must be
placed in front of the radical sign -√49 = -7
4) Square Roots in Equations – Square root in expressions should be treated as
exponents. Complete square roots, first before multiplication and division. Treat
the square root as a positive integer, unless specified with a minus sign outside the
radical sign.
5) Ex. ((√25) + 10
= 5 + 10 = 15
6) Ex. √81 𝑥 2 − 30
= 9 x 2 – 30
= 18 x 30 = -12
7) Ex. −√144 ÷ 12 + 5
= - 12 ÷ 12 + 5
= -1 + 5 = 4
1.21
Flashcards
Module 1
1) Radicand – The number within the radical sign whose square root is to be taken.
2) Superscript – The symbol (such as a number or letter) written above and
immediately to the left or right of another character.
3) Discrete – A collection of numbers whose values are distinct, separate, and
unconnected.
4) Perfect Square – The product of any integer with itself yields a perfect square. So, a
number is a perfect square if it can be written as the square of an integer. Ex. 9 is a
perfect square because 3*3 = 9
5) Subtraction – Taking one or more values away from another
6) Identity Property – The property that 0 can be added to any number without
changing the value of the number. Likewise, 1 can be multiplied by any number
without changing the value of that number.
7) Multi-Step Expression – An expression or equation with more than two values and
two or more operators that requires multiple steps to be solved.
8) Radical Sign – The symbol which indicates to take the square root of the number
that follows
9) Multiplication – the act of adding the same number multiple times. Often denoted
by the dot sign, ∙.
10) Parentheses – are used to separate operations within an expression. Any
operations within the parentheses should be performed first.
11) Multiplicative inverse – Multiplicative inverse of a number x is the number you
must multiply x by to get 1. For Ex. 5 and ⅕ are multiplicative inverses.
12) Prime Factorization – Determining the set of prime numbers whose product is the
original integer.
13) Factor Tree – A graphical method used to identify the prime factorization of an
integer
14) Square Root – A Numbers that produces a specified number when its multiplied by
itself
15) Data – A set of values of qualitative or quantitative variables; pieces of data are
individual pieces of information.
16) Principal Square Root – The positive square root of a number. For example, the
principal square root of 36 is 6.
17) Negative Square Root – The negative square root of a perfect square. For example,
-6 is the negative square root of 36
18) Negative Number – The number whose value is less than zero. On the number line,
negative numbers are to the left of zero.
19) Operators – A word or symbol (such as + or - ) that indicates an operation between
values.
20) Fundamental Theorem of Arithmetic – A concept which states that any integer
greater than 1 is either prime or is the product of a unique set of prime numbers.
21) Sum – The result of multiple numbers being added together
22) Continuous – A collection of numbers whose values are not dividable into distinct
units.
23) Operation – is a procedure which generates a new value from one or more
operands or mathematical values
24) Factor – An integer that divides another integer. We say an integer, x, is a factor of
another integer, y, if the quotient y/x is also equal to an integer.
25) Addition – Finding the total of two or more values
26) Associative Property – The associative property holds that under certain operations
in a multi-step expression, the computations may be done in any order. Commonly
represented as (a + b) + c = a + (b + c). Addition and multiplication are associative.
27) Whole Number – A number whose value is 0 or greater (negative numbers are not
considered whole numbers) and can be represented without a fractional or a
decimal component.
28) Prime Number – A number with only two factors; one and itself
29) Numbers – A word or symbol (such as five or 16) that represents a specific amount
or quantity.
30) Product – The result of multiplying values
31) Difference – The result of one number being subtracted from another number.
32) Positive Number – A number whose value is greater than zero. On the number line,
positive numbers are to the right of zero.
33) Base Number – The number multiplied by itself when paired with an exponent. For
example, in 8 to the third power, 8 would be the base number.
34) Expressions – A string of terms that are connected by division, addition, and
subtraction operations.
35) Additive Inverse – Two numbers equidistant from 0 on a number line whose sum is
0. For example, 3 and -3 are additive inverses.
36) Estimation – Approximating a value for a calculation
37) Commutative – The property that the order of the numbers under the operations
does not change the result. Addition and multiplication are commutative: a + b = b +
a and ab = ba.
38) Order of Operations – A set of rules that defines the order in which mathematical
operations should be performed.
39) Factorization – The process of determining the prime factors of a composite
number.
40) Composite Number – A number with more factors than just one and itself.
41) Exponents – Sometimes called a power, it’s a quantity that represents repeated
multiplication.
42) Division – Splitting values into equal parts or groups
43) Set – A collection of numbers
44) Integer – A number (positive, negative, or zero), that can be represented without
fractional or a decimal component.
45) Quotient – The result of a division expression.
46) GCF (Greatest Common Factor) – The greatest common factor of any two integers a
and b is the greatest number that is both a factor of a and a factor of b.
47) Rational number – A rational number is a number that can be written as a ratio of
integers, which means it can be written as a fraction.
48) Interval – A set of numbers between specified values
49) Real Number – Any numbers on the number line. Real numbers include zero,
negative and positive integers, fractions, and decimals.
Module 2: Fractions, Decimals, and Percentages
2.2) Introduction to Fractions
1) Fractions – Fractions are a means of expressing numbers which are part of a whole.
A) Fractions contains two integers, separated by a slash, /, or a fraction bar, - ,
B) Numerator – The number that is written before the slash, or above the fraction bar. The
numerator represents the number of equal parts.
C) Denominator – The second Numbers written after the slash, or below the fraction bar.
The denominator indicates how many of these equal parts make up one whole unit.
2) Proper Fractions
A) The numerator is less than the denominator, therefore the value is less than 1 (Ex:
⅞, ½, ⅘
3) Improper Fractions
1) The numerator is greater than the denominator and therefore value is greater than 1
7
(except if the fraction is negative) For Ex.
4
4) Mixed Number consists of a whole number and a proper fraction. Negative sign in front
1
1
2
applies to both parts of the mixed number. For Ex. 14 2 , 1 3 , −8 5
5) Changing a mixed number to an improper fraction.
6) Changing an improper fraction to a mixed number
a) Write the fraction as a division problem: 9 divided by 5.
b) Solve by dividing numerator 9 by 5. 5 goes into 9 one time with a remainder of 4
c) Write the answer using the quotient, 1, followed by a fraction whose numerator is the
remainder, 4, and whose denominator is the documentation from the original fraction,
4
5. 1 5
d) Ex. Changing Improper Fractions to a Mixed Number
e) Ex. Changing Improper Fractions to a Mixed Number
7
52
9
=
52
÷
9
=
5.
𝐴𝑛𝑠𝑤𝑒𝑟
5
9
30
2
= 30 ÷ 7 = 4. 𝐴𝑛𝑠𝑤𝑒𝑟 4 7
7
f)
4
Ex. Changing Mixed Numbers to Improper Fractions 5 5 = 5 × 5 = 25. 25 + 4 =
29. 𝐴𝑛𝑠𝑤𝑒𝑟
29
5
3
4
g) Ex. Changing Mixed Numbers to Improper Fractions 2 = 2 𝑥 4 = 8. 8 + 3 =
11. 𝐴𝑛𝑠𝑤𝑒𝑟
11
4
2.3) Identify Equivalent Fractions
1) Equivalent Fractions are fractions that have the same value, such as one-half and two1
2
fourths. In numerical terms 2 = 4
2) Ex. How many 24ths are equal to 5/6 – The denominator 24 is 4 times the denominator 6,
so the numerator of the equivalent fraction must be 4 times the numerator 5.
Therefore 20 24ths are equivalent to 5/6.
3) Ex. How many 27ths are equal to 4/9 a) Step: 1 Write out the expression 4/9 = ?/27.
b) Step 2: Calculate how many times 9 goes into 27. 27 ÷ 9 = 3.
c) Step 3: Calculate the numerator. The denominator 27 is 3 times larger than the
denominator 9. Therefore, the numerator must also be 3 times larger than 4. 3 x 4 =
12.
d) Step 4: Calculate the equivalent fraction. 4/9 = (4×3)(9×3) = 12/27. These two
fractions are equivalent fractions 4/9 = 12/27.
4) Ex. An inch is 1/12 of a foot. How many inches are in 2/3 of a foot.
a) Step 1: Write out the expression. 2/3 = ?/12
b) Step 2: Calculate how many times 3 goes into 12. 12 ÷ 3 = 4
c) Step 3: Calculate the numerator. The denominator 12 is 4 times larger than the
denominator 3. Therefore, the numerator must also be 4 times larger than 2. 4 x 2 = 8
d) Step 4: Calculate the equivalent fraction.
8
.
12
2
3
(2×4)
= (3×4) =
𝑇ℎ𝑒𝑠𝑒 𝑡𝑤𝑜 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠 𝑎𝑟𝑒 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠.
2
e) Step 5: State the answer to the original question. There are 8 inches in 3 𝑂𝑓 𝑎 𝑓𝑜𝑜𝑡.
5) What fraction with a denominator of 40 is equivalent to ¼.
a) The denominator 40 is 10 times the denominator 4, so the numerator of the
equivalent fraction must be 10 times the numerator 1. Therefore, ¼ is equivalent to
10/40.
2.4) Reducing Fractions
1) Reducing Fractions – When we found equivalent fractions, we multiplied the numerator
and the denominator by a common number. Reducing a fraction is the opposite process
of finding equivalent fractions. Reducing requires cancelling common factors. A fraction
is reduced or simplified to its lowest term when its numerator and denominator have no
common factors. It’s much easier to multiply, divide, add, and subtract, when fractions
are reduced to lowest terms.
2) Reducing Fractions Using Prime Factorization
a) List the prime factors to both the numerator and denominator
b) Cancel the factors that are common to both numerator and denominator
c) Multiply across the numerator and denominator.
1) Step 1: List the prime factors of both the numerator and denominator.
2) Step 2: Cancel, or divide by, all the factors that are common to both the
numerator and denominator. In this example, the common factor is 2
2×3
6/ 8 =
𝑟𝑒𝑚𝑜𝑣𝑒 𝑡ℎ𝑒 𝑐𝑜𝑚𝑚𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟 2.
2𝑥2×2
3) Step 3: The numerator is 3. The denominator is found by multiplying the
×3
3
remaining factors ×2×2 = 4
D) Ex. Reducing the following fraction:
−5
1) 20
2) Step 1: List the prime factors of both the numerator and denominator
−5
2×2×5
−5
20
=
3) Step 2: Cancel, or divide by, the factor that is common to both the numerator
−5
−
and denominator. The common factor is 5.
=
20
2×2𝑥
4) Step 3: When all of the factors in the numerator have been canceled, you put a 1
−1
in the numerator. The denominator is 2 x 2 = 4. =
4
4) Ex. Reduce the following fraction using the dividing by a common factor method.
a) Step 1: Notice that the numerator and denominator are both divisible by 7.
−28 −28÷7
−4
1)
=
42 42÷7
6
b) Step 2: Notice that
−4
𝑆𝑡𝑖𝑙𝑙 ℎ𝑎𝑣𝑒 𝑎 𝑐𝑜𝑚𝑚𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟. 𝑁𝑒𝑒𝑑 𝑡𝑜 𝑟𝑒𝑑𝑢𝑐𝑒 𝑓𝑢𝑟𝑡ℎ𝑒𝑟.
6
−4
−4÷2
−2
1) 6 = 6÷2 = 3
5) Ex. Reduce the following fraction with a negative sign in the numerator and
−27
denominator, using the method of dividing by common factors.
−81
A) Step 1: The first simplification should be to cancel the negative signs in the
−27
27
27
numerator and denominator.
= . 3 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑏𝑜𝑡ℎ 27 𝑎𝑛𝑑 81.
=
(27÷3)
(81÷3)
=
9
27
B) Step 2: 3 still divides both
C) Step 3: 3 still divides both
−81
81
81
(9÷3)
9
3
= (27÷3) =
27
9
(3÷3)
3
1
= (9÷3) = 3
9
2.5) Butterfly Method: Identify Equivalent Fractions
1) Butterfly Method is a way to cross-multiply two fractions to determine whether they
are equal.
2) Write the fractions as if they are equal with a question mark over the equal sign.
3) Now draw "butterfly wings" around the opposite numerators and denominators.
4) Multiply the numbers in each of the "butterfly wings." If the products are equal, then
the fractions are equivalent.
5) Multiply the numbers in each of the butterfly wings
6 x 101 = 606
17 x 36 = 612
Since the products aren’t equal (606 ≠ 612), the fractions are not equal.
2.6)
Multiples and the Least Common Multiple
1) Multiples are the opposite of factors: integers that are created by multiplying the
number times another integer. Multiples are needed to add and subtract fractions, and
factors are needed to reduce them. Ex. Multiples of 7: 7 x 1 = 7; 7 x 2 = 14; 7 x 3 = 21;
7 x 4 = 28. Multiples of 11: 11 x 1 = 1; 11 x 2 = 22; 11 x 3 = 36; 11 x 4 = 44
2) LCM (Least Common Multiple) the common multiple whose positive value is smallest.
1) How to find the least common multiple ?
a) Situation 1: If the two numbers do NOT share a common factor, multiply them
together to find the least common multiple. For Ex. 2 & 3 have no common
factors, so the LCM is 6. Another Ex. Least Common Multiples of 7 and 8
Multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56, 63 … Multiples of 8 are: 8, 16,
24, 32, 40, 48, 56, 64, 72 …. Notice that since 7 and 8 have no common factors,
LCM of 7 and 8 is 7 x 8 = 56.
b) Situation 2: If the two numbers do have a common factor, you can list out the
multiples of the greater number and divide the smaller number into each
multiple. The first multiple of the greater number that the smaller number
evenly divides is the LCM. Ex. Another Ex. Least Common Multiples of 6 and 8.
6 and 8 have a common factor of 2, so don’t multiply them by one another.
Begin with the first multiple of 8, which is 8. Does 6 divide 8 evenly?, No. Next
take the second multiple of 8, 8 x 2 = 16. Does 6 divide 16 evenly ?, No. Take
the next multiple of 8: 8 x 3 = 24. Does 6 divide 24 evenly ?, Yes. You have
found the LCM. LCM of 6 and 8 is 24.
c) Situation 3: Sometimes the smaller of the two numbers will divide the larger.
The larger number then is the LCM for both numbers. For Ex, 12 is a multiple of
6 because 6 x 2 = 12. Therefore, 12 is the LCM of 12 and 6. Ex. 9 and 27 have a
common factor of 9, so don’t multiply them by one another. Begin with the
first multiple of 27, which is 27. Does 9 divide 27 evenly ?, Yes. You have found
the LCM. The LCM of 9 and 27 is 27.
2.7) Least Common Denominators
1) To make the denominators of two fractions with different denominators the same,
we need to to first find the least common multiple of those different denominators.
A) This is called the least common denominator. 1/3 and -2/7.
Does 3 and 7 have a factor in common?, No.
B) This is Situation 1: Multiply 3 x 7 = 21. 21 is the least common multiple of the
numbers 3 and 7, therefore it is the least common denominator for 1/3 and -2/7. 21 is
the LCD for 1/3 and -2/7.
C) Determine the least common denominator of ¾ and 1/6.
D) Does 4 and 6 have a common factor?, YES
E) This is Situation 2: Take the larger number, 6, and divide 4 into multiples of it. 4
doesn’t divide 6, but it divides 6 x 2 = 12, so 12 is the least common multiple.
12 is the LCD for ¾ and 1/6.
F) Transforming Fractions – Step 1: Divide the least common denominator by the
current denominator. Step 2: Multiply both the numerator and the denominator by
this integer.
G) Transform 1/3 and -2/7 each into their equivalent fractions that share a common
denominator of 21. First, convert 1/3 to an equivalent fraction with a denominator
of 21. Step1: Divide the LCD (new denominator) by the current denominator 21 ÷ 3
(|𝑥7)
7
= 7. Step 2: Multiply the numerator and denominators by 7. (3𝑥7) = 21
−2
1
3
7
= 21
H) Transform
𝐼𝑛𝑡𝑜 𝑡ℎ𝑒 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑐𝑜𝑚𝑚𝑜𝑛 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟.
7
I) Step 1: Divide the LCD (new denominator) by the current denominator. 21 ÷ 3 = 7
J) Step 2:
(−2×3)
(7×3)
=
−6
21
7
The two new fractions with a common denominator are: 21 𝐴𝑛𝑑
−6
21
2.8) Fractions: Addition and Subtraction
1) Adding & Subtracting Fractions with the Same Denominator
To add fractions with the same denominator, follow these two steps:
Same Denominators: Step 1: Add or subtract the numerators of all the fractions in the
expression. Step 2: Keep the same denominator! (Don’t add the two denominators). Step 3:
Reduce the answer, if necessary.
(4÷2)
(6÷2)
2
=3
2) Adding & Subtracting Fractions with Different Denominators
To add or subtract fractions with different denominators, you need to find the least common
denominator. This is the smallest number that can be divided by both denominators.
Step 1: Find the least common denominator (LCD). Step 2: Use the LCD to find the equivalent
fractions and rewrite the expression. Step 3: Add or subtract the numerators of all the fractions
in the expression. Step 4: Keep the denominator the same – do not add them.
Step 5: If necessary, reduce the answer.
3) Add the following fractions:
Step 1: Find the least common multiple (LCM) of both 4 and 3. The LCM of 4 and 3 is 12.
Step 2: Convert fractions to have like denominators: Multiplying both the numerator and
denominator by 3 is equivalent to dividing each of the old parts into 3 parts. Multiplying the
numerator and denominator by 4 is equivalent to dividing each of the old parts into 4 parts. The
amount of each fraction (shaded region of the pies) remains the same, but we have the thinner
slices in each pie. The point is we have the same thinner slices for each of the pies.
Step 3: Add the like Fractions:
1
4
1×3
4𝑥3
1
+3
1𝑥4
+ 3𝑥4
1
4
1
3
+ =
3
12
7
12
+
4
12
=
7
12
2.9) Multiplying Fractions and Mixed Numbers
1) Multiply the numerators to obtain a new numerator.
Multiply the denominators to obtain a new denominator.
Write the answer in fraction form and reduce it to the lowest terms, if necessary.
Change any improper fractions to mixed numbers.
2) Multiplying Fractions: Example #1
A) To answer the question we would multiply the fractions:
3 1
𝑥
5 5
3𝑥1
3
=
5𝑥5 25
3) Multiplying Fractions: Example#2
a) Multiply the Following Fractions:
4
−1
𝑥 3
13
A) Step 1: Multiply the numerators to obtain a new numerator.
4𝑥 − 1 =−4
B) Step 2: Multiply the denominators to obtain a new denominator
−4
13×3
C) Step 3: Write the answer in fraction form and reduce, if necessary
−4
39
=
−4
39
4) Multiplying Mixed Numbers
1) Change any mixed numbers to improper fractions.
Multiply the numerators to obtain a new numerator.
Multiply the denominators to obtain a new denominator and write the answer in fraction
form. Change the improper fraction back to a mixed number. Reduce the mixed number to
the lowest terms, if necessary.
2) Multiplying Mixed Numbers: Example#1
2 1
a) Multiplying the following mixed number and fraction: 6 3 𝑥 2
2
b) Step 1: Change any mixed numbers to improper fractions
c) Step 2: Multiply the numerators
20×1=20
d) Step 3: Multiply the denominators
20
3×2
=
63
6𝑥3 +2=
𝑇ℎ𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑖𝑠 𝑛𝑜𝑤
20
6
e) Step 4: Convert the improper fraction to a mixed number
2
1
20
6
20
1
𝑋 2
3
20
3×2
2
= 36
f) Step 5: Reduce the fraction to lowest terms 3 = 3
6
3
3) Multiplying Mixed Numbers: Example#2
2
1
a) Multiply the following mixed number and fraction: 3 𝑥 3 3
b) Step 1: Change any mixed numbers to improper fraction.
1
10
2 10
3 =
𝑇ℎ𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑖𝑠 𝑛𝑜𝑤 𝑥
3
3
c) Step 2: Multiply the numerators
d) Step 3: Multiply the denominator
3
2×10
20
3×3
3
=
=
20
𝑇ℎ𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑖𝑠 𝑛𝑜𝑤
20
3×3
20
9
e) Step 4: Convert the improper fraction to a mixed number
20
9
2
= 29
2
f)
Step 5: Reduce the fraction to lowest terms. The fraction 29 Is in lowest terms.
4) Canceling
A) Canceling is a way of reducing fraction before multiplying them together. Canceling is
dividing the numerator and denominator by the same number.
7 1
1 1
1
B) 9 𝑥 14 = 9 𝑥 2 = 18
2.10) Dividing Fractions & Mixed Numbers
1) To Divide Fractions, follow these steps:
A) Change the division sign (÷) to a multiplication sign (x).
B) Write the reciprocal (the reciprocal of any number is just the numerator and
denominator reversed) of the second fraction.
C) Multiply the numerators
D) Multiply the denominators
E) Write the answer in the form of a fraction
F) Reduce the fraction to the lowest terms, if necessary.
1
2
2) Example#1 – Divide the following fractions: 2 ÷ 3
A) Step 1: Change the division sign (÷) to a multiplication sign (X):
1
𝑥
2
1 3
𝑥
2 2
B) Step 2: Write the reciprocal of the divisor, or second fraction:
Notice you can’t cancel the 2s which are both in the denominator
1×3=3
C) Step 3: Multiply the numerators
D) Step 4: Multiply the denominators
3
2×2
=
3
4
3
E) Step 5: Write the answer in fraction form 4
F) Step 6: Reduce the fraction to the lowest terms, if necessary.
2
3
3) Example #2 – Divide the following mixed number and fraction: −2 ÷
7
2
A) Step 1: Convert all mixed numbers to improper fractions −2 7 =
B) Step 2: Change the division sign (÷)to a multiplication sign (x)
C) Step 3: Write the reciprocal of the second fraction:
D) Step 4: Multiply the reciprocal of the divisor
−16𝑥7
7×3
−16 7
𝑥
7
3
=
7
−2𝑥7+2
7
=
−16
𝑥
7
=
−112
21
E) Step 5: Convert the improper fraction to a mixed number
−112
21
7
= −5 21
7
1
F) Step 6: Reduce the fraction to the lowest terms, if necessary −5 21 = −5 3
−16
7
2.11) Decimals
1) Decimals are denoted by a decimal point, which will fall on the right side of a whole
number. The part of the whole is determined by how many numbers fall to the right of the
decimal point.
5
2) Example below, 5 is in the tenths place, which translates to "five tenths" or 10 (fractions!).
We read the decimal point as and, so this number would be read "one and five tenths." In
the example below, we expand the decimal further to the right. Two places to the right of
the decimal point is known as the hundredths place.
In the example above, we have 5 in
the tenths place or
5
;
10
𝟓𝟑 𝒊𝒏 𝒕𝒉𝒆 𝒉𝒖𝒏𝒅𝒓𝒆𝒅𝒕𝒉𝒔 𝒑𝒍𝒂𝒄𝒆 𝒐𝒓
538
1000
53
.
100
Finally, we have 538 in
the thousandths place or
Therefore 1.538 would be read as "one and five hundred
thirty-eight thousandths."
3) Understanding Place Value – tells what part of the whole the decimal represents.
2.12) Rounding
1) Rounding is used with both terminal decimals and repeating decimals.
A) Terminal Decimal – a decimal number that has digits that don’t go on forever.
B) Repeating Decimal – a decimal that has a repeating sequence of digits that go forever
2) Traditional Rounding Procedures
A) Rounding a decimal can reduce the digits in the number while still keeping the value
Similar. Numbers can be rounded up or down, depending on the digit following.
B) Rounding Rules of Thumb:
1) If the digit in the place value to the right of the digit being rounded is 4 or less the digit
being rounded stays the same.
2) If the digit in the place value to the right of the digit being rounded is 5 or greater, add 1
to the digit being rounded.
C) Rounding Procedures
1) Identify the digit in the place value to which you’re rounding
2) Evaluate the digit according to the Rounding Rules of Thumb.
3) Rewrite the number from left to right, until you reach the digit being rounded. If this
digit is to the left of the decimal, fill in the digits to the right with zeros until you reach
the decimal point.
4) Example#1 Round 138.791 to the nearest tens.
A) Step 1: Identify the digit in the place value
1) In this example, the digit in the tens' place value is 3
B) Step 2: Evaluate the digit according to the Rounding Rules of Thumb
1) In this example, we would round the tens digit, 3, up because the digit that
follows, 8, is greater than 5. Therefore, we would add 1 to 3. 1 + 3 = 4
C) Step 3: To complete the whole number, fill in the remaining places with
zero(s) until you reach the decimal point. Therefore, the answer is: 140
5) Example#2 Round 138.791 to the nearest hundredths.
a) Step 1: Identify the digit in the place value.
1) In this example, the digit in the hundredths' place value is 9.
b) Step 2: Evaluate the digit according to the Rounding Rules of Thumb.
1) In this example, the thousandths' digit, 1, is less than 5. Therefore, the 9 in the
hundredth’s place stays the same.
c) Step 3: To complete the answer, the digits after the hundredths place are dropped.
Therefore, the answer is: 138.79
2.13) Decimals: Addition and Subtraction
1) Adding or Subtracting Decimals
A) Write the problem vertically, lining up the decimal points.
B) Fill in the missing decimal places with zeros as placeholders.
C) Add or subtract as you would whole numbers
D) Bring down the decimal point
E) Write the answer, dropping any trailing zeros which follow the decimal point (if
any)
2) Zero as a Placeholder – zeros can be used as a placeholder when adding or subtracting
decimals that don’t have the same number of decimal places. This is done to keep the
decimals the same length, keeping all of the digits in the same column.
3) Addition Example Below:
4) Example: Addition, carrying values when sum is greater than 10
5) Example: Subtraction
2.14) Decimals: Multiplication & Division
1) Multiplication with Decimals – The decimal point in your product or quotient, will reflect the
total number of decimal places from within the problem. Note: in multiplication, don’t add
placeholder 0s.
2) For ex. For example, if you are multiplying a number with a decimal in the tenths place by a
number with a decimal in the hundredths place, your answer will contain a decimal in the
thousandths place ( 10×100=1,000)
3) Multiplying Decimals
a) Write the problem vertically, writing the decimal with the most digits on the top.
b) Multiply the two decimals as you would if you were multiplying whole numbers
c) Determine the number of decimal places in both factors
d) Working from the far right to left count the number of places that correspond to the
number of decimal places needed and put in a decimal point.
e) Write the answer, dropping any trailing zeros which follow the decimal point (if any)
4) Trailing Zeros – zeros that follow the last nonzero digit in the decimal part of a number.
5) Example#1. Decimal number 425.67000, has three trailing zeros (three zeros after the
seven). These zeros have no value, and should be dropped from the answer
6) Example#2
7) Division with Decimals
a) Write the problem so the dividend is under the division symbol and the divisor is to
the left of the symbol. If number in the dividend is an integer, put a decimal point
next to the units place.
b) Convert the divisor to a whole number by moving the decimal point to the right as
many places as are needed. Move the decimal point in the dividend the same number
of places. Add zeros if necessary.
c) Place a decimal point for the answer (quotient) directly above the decimal point of
dividend, above the division bar.
d) Divide as you would for whole numbers.
e) Example#1 – 1.05 ÷ 0.02
f)
Example#2 – 0.064 ÷ 1.6
2.15) Percentages denoted by sign, %, percentages are also part of whole – but that whole is 100.
1) Adding and subtracting percentages is similar to adding and subtracting whole numbers and
decimals. Below example of Adding percentages
2) Subtracting Percentages 94% - 33.45% = 60.55%
3) Multiplying and Dividing Percentages require that the percentages be written in decimal form.
Once the percentage values are converted to their decimal form, the rest of the process is like
multiplying and dividing decimals. A percentage less than 100% is only part of the whole, so the
percentage needs to be divided by 100, and after multiplication, the answer will be less than the
original amount.
A) Convert a Percentage to a Decimal. To find the decimal equivalent of a percentage, you
simply move the percentage two places to the left, that is you’re dividing by 100. Any
number divided by 100 will have a decimal point moved 2 places to the left.
B) Percentages have to be converted to decimals for division problems as well. If you are dividing
by 50%, you will be dividing by 0.5 (or 50 ÷ 100). Convert 75% of a medication (“of” almost
always signals multiplication), we can multiply 0.75 x 5 = 3.75ml.
C) Example#1
D) Division with Percentages / Example#2
E) Percentage “Of” – is to determine some percentage of another quantity.
F) For example: 7% of a population to have type O negative blood and we’re collecting blood
from 200 volunteers, how many people could we estimate to have type O negative blood?
For these types of problems, we convert the percentage to a decimal and then multiply by the
7
quantity. Converting 7% to a decimal 100 = 0.07 𝑎𝑛𝑑 𝑡ℎ𝑒𝑛 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡ℎ𝑎𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑏𝑦 200 −
0.07 𝑥 200 = 14. 𝑊𝑒 𝑤𝑜𝑢𝑙𝑑 𝑒𝑥𝑝𝑒𝑐𝑡 𝑎𝑏𝑜𝑢𝑡 14 𝑝𝑒𝑜𝑝𝑙𝑒 𝑡𝑜 ℎ𝑎𝑣𝑒 𝑡𝑦𝑝𝑒 𝑂 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑏𝑙𝑜𝑜𝑑.
G) Another Example: How many mL of a drug is contained in 100mL of a 1% solution?
1
First convert the percentage to a decimal
= 0.01. 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑡ℎ𝑎𝑡 𝑏𝑦 100 𝑚𝐿 −
100
0.01 𝑥 100 = 1. 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 1𝑚𝐿 𝑜𝑓 𝑚𝑒𝑑𝑖𝑐𝑎𝑡𝑖𝑜𝑛.
2.16) Proportions and Unknown Quantity
1) Ratio – is a comparison of two numbers. This comparison can be written in a variety of different
ways, each expressing that for every x amount of this you have y amount of that.
2) Rate – is a ratio that compares two quantities having different units of measure.
3) Proportion – is a true statement in which two ratios are equal to each other. Proportions can
be used when comparing two ratios, frequently concluding that they are equal or unequal.
4) Butterfly Method – Multiplying across an equality sign is also the method used to establish if
4
20
two proportions are the same 5 𝐼𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 25
5) Conditional Proportion is when one part of a proportion is a variable or unknown quantity.
1) Solving Conditional Proportions
A) Determine which part of the fraction, the numerator or denominator, has the unknown
values. Work with the other part of the fraction, the one with 2 unknown values.
B) Divide the larger numerator or denominator by the smaller numerator or denominator
(the conversion number)
C) Multiply this conversion number by both the numerator and denominator of the smaller
fraction to establish an equivalent fraction.
D) Match the numerators of the fractions and the denominators of the fractions to solve the
unknown part.
E) Example 2:
6) Percentage Proportion is a way of converting a percentage into a proportion. This is
needed in order to find the percentages of quantities. To setup the percent proportion
𝟕
you express the percentage over 100. So 70% would be
. The proportion can be set
𝟏𝟎𝟎
equal to an amount x amount over y. Here x and y would represent the ratio you’re
given, and the percent is the part of 100 (variable) you’re trying to solve for.
7) An example of this type of problem is the question "What percentage is 3 of 4? Here the
𝑥
ratio 3 over 4 is set equal to
.
100
You solve these problems either through the butterfly method of cross-multiplication
after you have setup the proportion for the unknown or by finding an equivalent fraction,
whichever process is easiest.
8) Another example: what number is 36% of 20?
9) Convert from a Proper Fraction or an improper fraction to a decimal.
a)
b) Convert from a mixed number to a decimal
a) Keep the whole number to one side. Only work with the fraction part.
Then, divide the numerator by the denominator as explained on the previous
page.
b) Convert from a fraction to a percentage. Convert to a decimal first.
c) Convert from a decimal to a proper fraction:
1) Identify the place value for the farthest right number. Ex. in the decimal
.2078, the digit farthest to the right is an 8. The 8 occupies the ten
thousandths place. Place the denominator as the whole number associated
with the place value name. Remove the decimal point from the decimal and
place as the numerator. Reduce the fraction to lowest terms:
2) Example 2: Convert 10.39
Step 1: The place value occupied by the 9 is the hundredth's place.
Step 2: We need a denominator of 100
Step 3: Remove the decimal point and put the 39 over the denominator of
100
Step 4: The fraction does not need to be reduced.
d) Convert from a percentage to a fraction directly.
1) Remove the percentage sign.
2) Put the percentage over 100
3) If you have a decimal within the percentage, remove it by multiplying the
numerator and denominator by as many 10s as are needed.
4) Reduce the fraction to lowest terms
2.17) Unit Conversion
1) Steps to Calculate a Unit Conversation
a) List starting units (Unit conversion is performed when we’re given a measurement in one unit
system, but we need to convert to a different measurement)
b) List desired units (represents a value in different units)
c) Determine unit conversion rates (when converting between different units, you need to
know the conversion rates
d) Set up fractions so that certain units cancel (Unit conversions are executed by unit
cancellation. If the units divide one another, they cancel each other out. Units cancels if there’s
an instance of a unit in the numerator and the same unit in the denominator of a fraction. The
final measurement should be cancel. Only want the desired units that don’t cancel out. To
accomplish this, create a special fraction – Unit Multiplier Fraction (A fraction that has two
equivalent quantities with different units as its numerator and denominator) and multiply it by
the original fraction.
e) Perform calculations and cancel units
2) Multiple Unit Multipliers
A) Remember that a unit cancels when it is both in a numerator and in a denominator.
Two unit multipliers—gallons to quarts, and quarts to pints—are used to obtain the desired ending
Units.
3) Convert Units in both numerator and denominator
2.18) Temperature Conversion
1) Measuring temperature, Fahrenheit (F) and Celsius (C) – Both measure in degrees.
2) An equation is used to convert from one temperature scale to the other. One for Fahrenheit
and the other for Celsius.
9
3) Convert to Fahrenheit: Temperature in degrees Fahrenheit = Celsius × 5 + 32
9
4) Convert to Celsius: Temperature in degrees Celsius = (Fahrenheit − 32) × 5
5) Example#1 – Convert Celsius to Farenheit
a) Step 1: Find the correct formula
We will use the first formula because we want our answer in Fahrenheit. Temperature in
𝟗
degrees Fahrenheit = Celsius × + 32
𝟓
b) Step 2: Substitute the known value
9
Temperature in degrees Fahrenheit = 30 × + 32
5
c) Step 3: Simplify the equation to find a solution
30×9
Temperature in degrees Fahrenheit = 5 + 32
270
Temperature in degrees Fahrenheit = 5 + 32
Temperature in degrees Fahrenheit = 54 + 32
Temperature in degrees Fahrenheit = 86 ( 86°F )
6)Example#2 – Convert Fahrenheit to Celsius - adult body temperature is 98.6°
A) Step 1: Find the correct formula
𝟓
Temperature in degrees Celsius = (Fahrenheit − 32) × 𝟗
B)Step 2: Substitute the known value
5
Temperature in degrees Celsius = (98.6 − 32) ×
9
C)Step 3: Simplify the equation to find a solution
5
Temperature in degrees Celsius = 66.6 x
66.6×5
Temperature in degrees Celsius = 9
Temperature in degrees Celsius = 37
9
Module 3 Algebraic Conventions and Notation
3.2.0) Arithmetic Terminology
1) Operations – An operation is a mathematical procedure which can generate a new value
(simplest and most common operations: addition, subtraction, multiplication, and division)
2) Equation – Two mathematical expressions separated by the equals sign, =. The two expressions
on either side of the equals sign hold the same value as one another.
3) Constant – is a number with a fixed value. All real numbers are constants, regardless of if they
are prime, composite, odd, or even, rational or irrational.
4) Arithmetic Expression – is a string of numbers connected by elementary operations (addition,
subtraction, multiplication, and division).
5) Exponent – sometimes called a power, is a quantity that represents repeated multiplication.
To the right, the base is 6 and the exponent 3. 63 is equal to 6 x 6 x 6.
3.2.1) Understanding Variables
1) Variables – A symbol that represents a mathematical value or holds the place of a numerical
value. A variable will often be a letter from the Western alphabet (a, b, c, …) or Greek alphabet
(α, β, γ, …) The actual letter or symbol being used as a variable isn’t important, the numerical
value represented by the symbol is what gives the variable its importance.
2) Value – A number or numerical worth.
3) Example#1 Variable Value
a
1
Variable, a, represents the numerical value of 1. Therefore, the statement (a + 5) is equivalent
to (1 + 5). Because the variable a represents the numerical value of 1, variable a can be replaced
with the number 1.
4) Example#2. Variable
Value
b
1
C
3
d
2
Using the table above, we can substitute any variable with the numerical value it represents.
3.2.2) Algebraic Expressions
1) Algebraic Expressions – are useful whenever you want to perform a calculation multiple times.
2) Creating Terms: Multiplication. Multiplication is the only elementary operation that can be
denoted by variables and constants simply being written next to one another: 3 ∙ x can be
written as 3x. Similarly, -6 ∙ a can be written as -6a. Addition, subtraction, and division all
require some type of sign (+, -, ÷) to denote that they’re being performed. Not multiplication.
3) Term – An individual number, a variable, or numbers and variables multiplied together. They
are separated by addition, +, or subtraction, -, in an expression.
4) Exponents are used for repeated multiplication of a variable, the same as with repeated
multiplication of a constant (ex. a ∙ a = a2)
A) Example#1: d ∙ d ∙ d = d3 ; x ∙ y ∙ y = xy2 ; h ∙ h ∙ g = h2g
5) Coefficients – is a number by which a variable is being multiplied. Coefficients are written in
front of variables. So, in 16x, 16 is the coefficient and x is the variable. If a variable is without a
number in front of it, the coefficient is 1. Though not written, there is essentially an invisible 1
in front of any variable without a numerical coefficient.
6) Terms – can be many things: a single constant, such as 5, a single variable, such as x, or a term
can also be any number of constants and variables multiplied together, such 7ab.
A) Terms can include:
 Multiplication
 Constants, which can be coefficients
 Exponents
 Variables
7) Algebraic Expression – is a string of terms connected by division, addition, and subtraction.
a) Examples (10÷2) − 22
2x2 + 9
6a2b − π
4w3z – wz + 8
3.2.3) Coefficients and Exponents –
1) Exponents represent how many times a number or variable is to be multiplied by itself – as
opposed to added to itself
2) Coefficient is a constant written in front of a variable. Coefficient is written next to, specifically
in front of, a variable because its being multiplied by that variable.
3.3) Inverse Operations –
1) Inverse Operations - Two operations that undo one another
2) Example: Each of the elementary operations have an inverse operation:
 Subtraction is the inverse of addition.
 Addition is the inverse of subtraction.
 Division is the inverse of multiplication.
 Multiplication is the inverse of division.
3.4.0) Introduction to Combining Like Terms
1) Combining terms that have the same variable component. Combining like terms helps to
consolidate when solving an equation.
2) Example: (jar)+2(jar): one jar + two jars = three jars (can be written: (jar) + 2 (jar) = 3(jar)
or j + 2j = 3j)
3.4.1) Identifying Like Terms
1)
To combine like terms, we first need to identify which terms qualify as “like terms”
a) Like Terms: always have the same variable(s) with the same exponent(s).
b) Examples of Like Terms: a and 2a (are like terms because they share the same variable a);
-4 and 7 (are like terms because they are both constants); 3D and 6D2 (are like terms
because they share the same variable (D) raised to the same power (2)
2) Unlike Terms: a and b (the terms have two different variables: a and b); 7x and 7y (the terms
have 2 different variables once coefficients are removed: x and y. The variables don’t match.
3m and 7m2 (the terms have the same variable, m, but with different exponents
3) Degree – the largest exponent in an expression
A) Expression Classification – Expressions are often classified by their degree.
B) Names of Expressions Based on Their Degree
 0. An expression of degree 0 is known as a constant
 1. An expression of degree 1 is known as linear
 2. An expression of degree 2 is known as quadratic
 3. An expression of degree 3 is known as cubic
3.4.2) Combining Like Terms with Addition:
a) Example#1: 3gh + 13gh = 16gh
b) Example#2: -4m3 + 25m3 = 21m3
2) Combining Like Terms with Subtraction:
a) Example#1: 5d2 – 14d2 = -9d2
b) Example#2: 6w – 18w = -12w
3.4.3) Combining Terms in Longer Expressions:
1) Step 1: Identify Like Terms
2) Step 2: Move the Like Terms Next to One Another
3) Step 3: Add or Subtract Coefficients
A) Move like terms -8d and -3d next to one another.
B) Always keep the sign with the proper term. The subtraction sign in front of 8d should stay in
front of 8d, and the subtraction sign in front of 3d should stay in front of 3d.
C) Add or subtract coefficient. The other two terms (10d2 and 5e) aren’t like terms, so they can’t
be simplified. Final Answer: 10d2 – 11d + 5e
D) Example#1:
 Identify all the sets of terms: 2a2 – 6C + 8a2 – 14b – 18C
 Move the like terms next to one another: 2a2 + 8a2 – 6C – 18C – 14b
 Combine the like terms: 10a2 – 24C – 14b
3.5.0) Introduction to the Distributive Property
1) Distributive Property – a mathematical principle used to multiply one term by multiple terms.
This principle is employed when there are parentheses around multiple terms, which are in turn
multiplied by a single term. The distributive property uses a process known as distribution.
Right column evaluates the expression without using the distributive property. In the left
column, we distribute the term that is outside of the parentheses (2) to each of the terms that
are inside of the parentheses (4 and 3), using multiplication. Multiply 2 by both 4 and 3,
individually
A) Example1: Term (Term + Term + ∙∙∙ + Term)
B) Example2: 2(4+3) –
3.5.1) The Distributive Property in Algebraic Expressions
1)
2) Examples: S ∙ (5s + x3); (−2a + 11b) ∙−8
3) Distribution on Negative Numbers - When in doubt, change all subtraction operations to the
addition of negative numbers.
4) Example1: 3x (5y − 2))
5) Example2: −6a (2b − 4)
With that multiplication, the resulting expression is −12ab +24a. Since −12ab are not like terms,
they cannot be combined, and so the simplified expression is −12ab+24a.
3.5.2) Distributive Property: Longer Expressions
1) Example#1: 5(−3x + y) − 15. 5 is to be distributed to the -3x and the y inside the parentheses.
Because -15 isn’t inside the parentheses, its not multiplied by 5. = (5 ∙ -3x) + (5 ∙ y) – 15
= Final Answer -15x + 5y – 15
A) Example#2: 4 (b + 3) - 5 (2 – 6b)
4𝑥+2
)
12
2) Distributive Property: Fractions: (
A) Dividing our numerator (4x+2) by 12 is the same as multiplying (4x + 2) by
1
1
12
1
The simplified expression is equal to (3) 𝑥 + (6)
1𝑥
Through the regular methods of multiplication, ( 3 ) 𝑊𝑜𝑢𝑙𝑑 𝑎𝑙𝑠𝑜 𝑏𝑒 𝑎𝑐𝑐𝑒𝑝𝑡𝑎𝑏𝑙𝑒.
B) Example: 3(2x + 2) – (y + 6) = The answer is 6x−y
There are two instances in which the distributive property should be used. For 3(2x+2)
distribute the 3 to the terms inside the parenthesis to get 6x+6. For −(y+6) imagine an
invisible −1 outside the parentheses. Remember that if an expression inside a set of
parenthesis is being multiplied by −1, you need to apply that negative sign to the entire
encased expression, so distributing that to the terms inside the parentheses gets us −y−6.
With all relevant terms distributed, simplify the expression by combining like terms.
Therefore, 6x + 6 – y – 6 =6x – y.
C) Example:
D) Remember to be Careful to
Distribute negative (-) signs along with the number or variable
if the negative (−) sign is attached to a term outside the parentheses
Follow negative rules for products:
 Negative ∙ negative = positive
 Negative ∙ positive = negative
 Positive ∙ Negative = Negative
 Simplify strings of terms by increasing the exponent on like variables: X ∙ X equals x2
3.2.6) Simplifying Longer Expressions
1) Example#1: An exponent is created in the term 8x2 because we’re multiplying the x in 2x by the
16−8
4
x in 4x. 2x ∙ 4x = 2 ∙ 4 ∙ x ∙ x = 8x2
8
=4=2
2) Example#2: −4g(5a−3a)
A) Answer =−4g(2a)
3.2.7) Substitution Method
1)
Substitution Method - When a variable is substituted by its known value in an algebraic
expression or equation: replacing x with its value. When substituting a value for a variable, it
is a good idea to enclose the value within parentheses. Using parentheses is a good visual way
to keep track of values that have replaced variables.
6
6
6
2) Example1: a + 𝑎−5 = 7 + 7−5 = 7 + 2 = 7 + 3 = 10
16−2𝑦
16−2(4)
3) Example2:
=
=
4
4
3.8) Substitution for Multiple Variables
16−8
4
8
=4=2
1) Example1: If x=2 and y = 3 x-y; Final Answer: =2−3 = −1
3.9) Algebraic Expressions
1) Equation – An equation is denoted by the equals sign, = , represents a mathematical
relationship. Following its name, equations denote that entities are equal to one another.
An equation is two terms or expressions with an equals sign in between them.
2) Algebraic equation - is any equation that contains variables, constants, or mathematical
operations.
A) Linear Equations – an algebraic express that consists of constants and “simple variables”.
A linear equation is an algebraic equation with degree of 1. This means that none of the
terms in the equation have an exponent larger than 1.
𝑑
Linear Equation Examples: b = 2 + x; b = 2 ; 7 = 8𝐷
3.10) Linear Equations
1) Principle of Equality – if you perform equivalent operations to both sides of an equation, the
result will always be an equivalent equation.
A) Example: 4 = 4; 4 - 2 = 4 – 2; 2 = 2. (we want to manipulate the four on the left side of the
equation above, we need to subtract 2 on the left side and perform the same operation on
the right side. So, subtracting 2 from the right and left side the expression now
becomes 2 = 2.
B) Example: If b = x + 3; b - 3 = x + 3 – 3
2) Elementary Operations – (Addition, subtraction, multiplication, and division) and they are used
to solve equations.
A) Addition Principle of Equality – Add the same number to both sides of an equation, and the
result will be an equivalent equation. Example: x + 5 = 9; Subtract -5 on both sides; x + 5 –
5=9–5 =x4
B) Subtraction Principle of Equality – Subtract the same number from both sides of an
equation, and the result will be an equivalent equation.
3.12) Manipulating Equations with Addition and Subtraction
1) Addition and Subtraction are inverse operations, they undo one another.
A) Multi-Step Expressions - Manipulating equations with addition and subtraction can feature
multi-step expressions, as well. Using the Order of Operations, we perform (in order):
parentheses, exponents and square roots, multiplication and division, and then addition and
subtraction. Take a look at the problem below.
B) Step 1: Example. (7 ∙ 2) – 6 + y = 18 (From the Left Side) First evaluate the expressions
containing known values to find the value of y.
(7 ∙ 2) − 6 + y = 18
(14) − 6 + y = 18
8 + y = 18
C) Step 2: 8 + y = 18
−8
−8
y = 18 – 8
y = 10
D) Combining Like Terms to Solve Equations. 4x and -3x are a set of like terms
a. Step 1: 4 x + 8 − 3x = 2
b. Step 2: 4x + 8 −3x = 2
c. Step 3:
x+8=2
d. Step 4:
−8 −8
e. Step 5:
X = −6
f. After combining the like terms, subtract 8
Note: from the left and right sides of the equation. The final equation reads.
X = −6 (2 – 8 = −6)
E) Example 2: (Addition Principle of Equality). If 26 = a− 42, then what does 26 = a − 42
equal?
a) Step 1: Undo the -42 by using the inverse of subtraction, which is addition. Need to add
42 to both sides of the equal in order to isolate a
b) Step 2: 26 = a – 42
c) Step 3: + 42 + 42
d) Step 4: 26 + 42 = a
e) Step 5: 68 = a
F) Example 3: (Subtraction Principle of Equality) If x + 4 = 10, then what does x equal ?
(Subtract 4 from both sides of equation)
a) Step 1: x + 4 = 10
b) Step 2: - 4 - 4
c) Step 3: x = 10 – 4
d) Step 4: x = 6
3.13) Manipulating Equations with Multiplication and Division
1) Multiplication Principle of Equality – If both sides of an equation are multiplied by the same
non-zero number, the result is an equivalent equation.
a) Example – Multiplication Principle of Equality – a ÷ 4 = −3. Solve for a.
A ÷ 4 = -3
∙4 ∙4
A = -3 ∙ 4
A = -12
2) Division Principle of Equality – If both sides of an equation are divided by the same non-zero
number, the result is an equivalent equation.
a) Example – Division Principle of Equality – 2x = 10 - On the left side of the equation is a
multiplication operation: 2 ∙ x. Therefore, we need to undo the multiplication operation ∙ x
Multiplication and division are inverse operations, so we can use division to undo the
multiplication here and isolate x. By the division principle of equality, we divide both sides
of the equation by 2: 2x = 10
÷2 ÷2
2x ÷ 2 = 10 ÷ 2 = x = 5
3) Steps for Solving an Equation With Complex Expressions
A) Substitute any variable's known value for the variable itself
B)




Simplify expressions on either side of the equation following order of operations:
Distribute
Combine Like terms
Add and subtract constants
Complete any other process that serves to simplify the expression
C) Move terms across the equation, using the Addition and Subtraction Principles of Equality:
 Move all constants to one side of the equation
 Get all terms with the variable to be solved on the opposite side of the equation
D) Simplify the expressions on either side of the equation:
 Combine like terms on one side, if necessary
 Add and subtract constants on the other side, if necessary
E) Isolate the lone variable on one side of the equation, using the Multiplication and Division
Principles of Equality:
 The variable will be across from its value
F) Check your answer:
 Plug in your solution to the original equation. Perform the arithmetic on both sides of the
equation. If the two sides of the equation are equal, you have successfully solved the equation!
4) Solve for x: 3(x − 1) = −1 + 3x − 2x + 8
a) Step 1: Substitute any variable's known value for the variable itself.
Since we do not have any known values, we can skip this step.
b) Step 2: Simplify both sides of the equations
3(x−1) = −1 + 3x − 2x + 8. Distribute the 3
3x – 3 = −1 + 3x − 2x + 8. Combine like terms of 3x and -2x
3x – 3 = −1 + x + 8. Combine the −1 and 8
3x – 3 = x + 7
C) Step 3: Move terms across the equation, using the Addition and Subtraction Principles of
Equality.
3x – 3 = x + 7. Subtract x from both sides to get all the terms with x on one side.
-x
-x
2x – 3 = 7
2x – 3 = 7. Add 3 to both sides to get all the constants on the other side.
+3 +3
2x = 10
D) Step 4: Simplify the expressions on either side of the equation (if necessary)
We are finished with this step because we have all the X 's on one side and a number on the
other side of the equation.
E) Step 5: Isolate the lone variable on one side of the equation, using the Multiplication and
2𝑥
10
= 2
Division Principles of Equality. 2
𝑥=5
F) Step 6: Check your answer by plugging it in
3 (5−1) ≟ −1 + 3 (5) – 2 (5) + 8
3 (4) ≟ −1 + 15 – 10 + 8
12 ≟ 14 – 10 + 8
12 ≟ 4 + 8
12 = 12
This confirms that x = 5.
G) Example: 3 + x ÷ 6 = 1 – 3 (4 − 7) Simplify each side of the equation:
3 + x ÷ 6 = 1 – 3 (−3)
3+x÷6=1+9
3 + x ÷ 6 = 10
Bring all constants to one side of the equation:
3 – 3 + x ÷ 6 = 10 − 3
X÷6=7
Multiply both sides by 6:
X = 42
H) Example: y − 2y + 1 = 7y − 4y – 7. Simplify each side of the equation by combining like terms.
4y + 1 = 3y – 7
4y + 1 = 3y – 7
Now, isolate y by getting all the numbers on one side and all the y
y's on the other.
4y + 1 – 1 = 3y – 7 – 1
4y = 3y – 8
We still have a y on both sides. Subtract 3y from each side to get them on one side.
4y − 3y = y – y – 8
Y = −8
3.14) Solving Longer Equations
3.15) The Butterfly Method – Cross-multiply two fractions to determine whether they are equal.
1) Multiply the numbers in each of the butterfly wings:
Write the products as an equation. 299j = 525. Divide both sides by 299 to isolate j
299𝑗 525
=
299 299
𝑗=1
226
299
3.16) Graphing on the Coordinate Plane
1) Graphs are another way to organize and display data. A graph or coordinate plane, consists of
an x-axis and y-axis; the x-axis is the horizontal line that passes through the point where y = 0,
while the y-axis is the vertical line that passes through the point where x = 0.
2) Coordinate plane – A tool used for graphing that is a display of a two dimensional plane.
It consists of an x-axis and a y-axis. The x-axis being a horizontal number line that passes
through the origin of the coordinate plane, the y-axis is a vertical number line that passes
through the origin of the coordinate plane, and the axes meet at the origin.
3) Horizontal Line – A line that runs left to right on the coordinate plane, parallel to the x-axis.
It has no “rise” or change in y-value
4) Vertical Line – A line that runs up and down on the coordinate plane, parallel to the y-axis.
It has no “run” or change in x- value.
5) Points plotted on a graph are located via coordinates. The location on the x-axis is written first,
and the location on the y-axis second – the x-coordinate followed by the y-coordinate.
These two coordinates – also referred to as ordered pairs
6) Ordered Pairs - Two numbers written in the form (x,y), where x represent the x-value or xcoordinate, and y represents the y-value or y-coordinate, tells us the locations of each point
and in which quadrant
7) Quadrant - the four sections of the coordinate plane, separated and defined by the x-axis and
y-axis
8) Origin – The point (0,0) on a coordinate plane, which is where the x-axis and y-axis intersect.
9) Plot the point F (3, -3). To plot this point, count 3 units to the right of the origin and then 3 units
down
10) Plot the point C (-2, 1) To plot this point, count 2 Units to the left of the origin and 1 point up
3.17) Slope-Intercept Form
1) The most basic to graph are those that generate a straight line – also known as linear equations.
2) Linear Equation – is an equation that has a degree of 1 (An algebraic equation that consists of
constants and “simple variables”.) When drawn on a graph, a linear equation creates a straight
line (linear). Any equation that does not contain square roots (which a number that produces a
specified number when it is multiplied by itself) or exponents (Sometimes called power, it’s a
number that shows how many times the base is multiplied by itself), greater than 1 is a linear
equation. Each graph above includes a continuous, straight line. Below are examples of linear
equations that will result in a straight line.
A)
✔
✔
✔
✔
Examples of Linear Equations
x + 9 = 2y
y = 4x + 10
−6 + y = −2x
𝑦
=5
3
3) Slope-Intercept Form – A common format that a linear equation can take that is helpful for
graphing purposes. Its of the form y = mx + b, where m is the slope of the line and b is the yintercept. Ex. Y = mx + b. Where m is the slope of the line and b is the y-intercept.
A) Slope-Intercept form is made up of two unique parts: The slope and y-intercept. The slope,
or m, represents the steepness of the line drawn on a graph. The y-intercept, or b,
determines where on the y-axis the line crosses.
B) Example: Getting the Equation into Slope-Intercept Form
a) First, make sure the y term is on the left side of the equation.
b) Put all x’s and constants on the right side of the equation.
c) Multiply or divide to make the coefficient of y be 1
3.17.1) Slope
1) Calculating Slope - How is slope actually calculated? The slope is a proportion. It is a consistent
ratio of distance along the y-axis to distance along the x-axis. Consider the equation y = 5x. The
coefficient 5 in front of the x (the “m” in y = mx + b), tells us that for every additional 1 of xvalue, the y-value will increase by 5. When x = 1, y = 5. When x = 2, y = 10. The difference
between initial y-value of 5 and the second y-value of 10 is 5. As x increases by 1, y increases by
5. We can see that graphically as well: See graph below.
2) Example: Find the slope of the line that contains the points (1,5) and (0, -3)
A) Use the formula
𝑅𝑖𝑠𝑒: (𝑦2 −𝑦1 )
𝑅𝑢𝑛: (𝑥2 −𝑥1 )
B) Using the two points we are given, we substitute them into the formula
−3−5
0−1
−8
C) Slope equals −1 𝑜𝑟 8
3.18) Graphing Linear Equations
1)
3.19) Linear Inequalities
1) Direction of Inequality – Simplify any term or expression on either side of the inequality. Add or
subtract from both sides / multiply or divide both side by a positive number.
2) Multiplying or dividing both sides by a negative number will cause the direction of an inequality
to reverse.