1. The Praxis Mathematics Tests
Workshop PPT #2
Slide show created by Jolene M. Morris

2. ETS Praxis Website
Official website where you can find more information about the Praxis tests

3. NCTM Website
The National Council of Teachers of Mathematics (NCTM) is the national organization that sets the standards for what mathematics concepts should be taught and at what grade levels.

4. Jolene’s Website
Jolene’s website where you can find Praxis resources such as Jolene’s 400 Praxis flashcards in a format to import into an app such as Flashcards To Go or Anki or Quizlet

5. Agenda (Slide 5) Number Theory (hunter green) Fractions (black)
Decimals (light blue)
Ratios, Proportions, & Percentages (dk blue)
Integers (red)
Geometry & Measurement (burgundy)
Data, Statistics, & Probability (pink)
Algebraic Reasoning (green)

6. Number Theory

7. Our Number System

8. Place Value

9. Ordering Large Numbers
Write them above one another with the place values lined up.
Starting from the left, look for the largest value.
For example, if you are asked to order:
5, , , ,922

10. Rounding Numbers
Rounding a number requires that you understand place value.
Look at the digit to the right of the place being rounded.
If that digit on the right is 5 or higher, add 1 to the place being rounded; otherwise, leave the place being rounded as is.
Change all places to the right of the place being rounded to zeroes.

11. Comparison Symbols
< less than > greater than ≤ less than or equal to ≥ greater than or equal to = equal to Each of these symbols can also be negated by putting a slash mark through them, such as not equal to: ≠

12. Estimation

13. Squared & Cubed Numbers

14. Divisibility Rules
2 = Even numbers (ending in 0, 2, 4, 6, and 8) 3 = If repeated sums of the digits result in 3, 6, or 9 4 = If the last two digits are divisible by 4 5 = If the last digit is 0 or 5 6 = If the number is divisible by both 2 and 3 8 = If the last three digits are divisible by 8 9 = If repeated sums of the digits result in 9 10 = If the last digit is 0

15. Prime Numbers
Prime Numbers = Integers greater than 1 with exactly 2 factors or divisors; numbers that are evenly divisible by only 1 and themselves.
The number 2 is the first prime and it is the only even number that is prime. The number 1 is neither prime nor composite. Memorize the prime numbers 1-100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

16. Other Number Classifications
Prime vs. Composite
Odd vs. Even
Denominate numbers
Consecutive integers
Cardinal numbers
Ordinal numbers

17. Number Line
A number line is a straight line where each point of that line corresponds to a real number.
A line is made up of an infinite number of points and there are an infinite amount of real numbers.
Usually the line is marked off to show the integers, including zero.
A number line is generally written as a horizontal line.

18. Expanded Notation

19. Interpreting Remainders
Do you round up? Do you round down? Do you use the remainder as part of the answer (or as the entire answer)?
How many boxes can be filled? (use only the quotient; ignore the remainder)
How many cans are needed to paint the wall? (round the quotient to the next greater whole #)
How many in the last box that isn’t completely full? (use only the remainder)

20. Prime Factorization
Thus, the prime factorization of 120 = 2 x 2 x 2 x 3 x 5

21. GCF
12 = 2 × 2 × 3
18 = × 3 × 3
30 = × × 5
GCF = × 3
The factors common to all three numbers above are 2 and 3.
2 x 3 = 6 so 6 is the GCF of 12, 18, and 30.

22. LCM

23. Multiplying by Powers of 10

24. Dividing by Powers of 10

25. 4 Ways to Indicate Multiplication
Using a small “×”, such as 3 × 5. Note that the “×” is not used for multiplication in algebra because it might be confused with the variable “×”.
Using a small, raised dot, such as 3 • 5
Using parenthesis, such as (3)(5) or 3(5) or (3)5
Using no symbol, such as 3y meaning 3 times y

26. Properties of Operations

27. PEMDAS Please Excuse My Dear Aunt Sally
Order of Operations
Simplify inside parentheses or grouping symbols
Simplify any expressions with exponents
Perform multiplication & division from left to right
Perform addition & subtraction from left to right
PEMDAS Please Excuse My Dear Aunt Sally

28. Properties of Zero Division by zero is undefined
Zero is neither positive nor negative
Zero is the additive identity
Any number multiplied by zero equals zero
Zero is used as the universal place holder
Zero is neither prime nor composite
Zero has no multiplicative inverse
A number with an exponent of zero equals 1
Zero factorial equals 1

29. Properties of One One is the multiplicative identity
Any number multiplied by one equals the number
One is neither prime nor composite
A number with exponent of 1 equals the number
The number one raised to any power equals one
Any nonzero number divided by itself equals one

30. Identity Property

31. Number Theory
ANY QUESTIONS?

32. Fractions

33. What is a Fraction?
A fraction represents equal-sized parts of a whole.
The top number is called the numerator
The bottom number , the denominator, is a denominate number (measurement of size)
The line between numerator & denominator indicates division and is called a vinculum.

34. Proper & Improper Fractions
A proper fraction has a numerator smaller than the denominator and indicates a fraction less than one whole: 3/8 1/4 14/15 4/5
An improper fraction has a numerator larger than (or equal to) the denominator and indicates a fraction that is equal to one or more than one whole: 3/2 7/4 16/ / /5
An improper fraction can be changed into a whole number or a mixed number by dividing the denominator into the numerator.

35. Common Fraction

36. Mixed Number
A mixed number is a whole number and a proper fraction combined. Mixed numbers may also be called mixed fractions.
This graphic shows two whole pizzas and a fraction of 3 pieces out of 4  2 3/4

37. Greatest Common Factor
The GCF is the largest number that is a factor of all the numbers.
One way to find the GCF is to write the prime factorization of each of the numbers above each other. Then “bring down” those factors that are in common and multiply them:
Find the GCF of 12, 18, 30:
12 = 2 × 2 × 3
18 = × 3 × 3
30 = × × 5
GCF = × = 6

38. LCD
The Lowest Common Denominator is the smallest number that is a multiple of all the denominators.
One way to find the LCD is to count by each of the denominators and find the first number that is a multiple of all.
Another way to find the LCD is to write a prime factorization of each denominator, and then “bring down” one of each factor and multiply.

39. Equivalent Fractions
Fractions that simplify to the same simple fraction.
When two fractions are equivalent, their cross products are equal :
3 x 6 = 18 2 x 9 = 18

40. Simplify Fractions

41. Mixed Number
A mixed number indicates the whole amounts and the parts (a fraction).
To convert a mixed number to an improper fraction, multiply the whole number by how many parts are in a whole, and then add the remaining parts.
For example, 6 2/3 means there are 6 whole amounts of 3/3 (6 x 3 = 18) so there are 18/3.
Adding the remaining 2/3 results in a total of 20/3 in 6 2/3.

42. Improper Fractions
An improper fraction shows a total number of parts, but in those parts is at least one whole.
To convert an improper fraction to a mixed number, divide the numerator by how many parts are in a whole. The quotient becomes the whole number and the remainder becomes the numerator of the fraction part of the mixed number.
For example, 23/3 equals 23 divided by 3, which is 7 r 2 or 7 2/3

43. Why Find Common Denominators?
footers
footers
footers

44. Why Change to Common Denominators?

45. Why Change to Common Denominators?
3 apples
oranges
7 ??????

46. Add Fractions & Mixed Numbers
Write the two fractions/mixed numbers vertically above each other (lining up place value)
Change the fractions to a common denominator.
Add the numerators only.
Put that sum over the common denominator.
Simplify the answer.

47. Subtract Fractions & Mixed Numbers
Write the two fractions/mixed numbers vertically above each other (lining up place value)
Change the fractions to a common denominator.
Subtract the numerators only (careful to regroup one whole (2/2, 3/3, 4/4, etc.) if you need to borrow).
Put that difference over the common denominator.
Simplify the answer.

48. Multiply Fractions& Mixed Numbers

49. Reciprocal
A reciprocal is a fraction where the numerator and denominator have been switched.
Multiplying any fraction by its reciprocal results in an answer of 1. As such, a reciprocal is called a multiplicative inverse.
NOTE: Since there is no rule on how to divide fractions, but because multiplication is the inverse of division and a reciprocal is the inverse of a fraction, you can divide fractions by multiplying by the reciprocal of the divisor. Hence, the inverse of an inverse results in the same answer as if you had divided.

50. Divide Fractions& Mixed Numbers
Change any mixed numbers and any whole numbers to improper fractions with a “1” on the denominator.
Write the two fractions horizontally beside each other.
Write the reciprocal of the divisor (flip the second fraction upside down) and change the operation to multiplication.
Expand each numerator and each denominator into a prime factorization. “Cancel” any ones such as 3/3 or 5/5. Multiply what is left straight across.

51. Change Fractions to Decimals & Percentages
Change a fraction to a decimal by dividing the denominator into the numerator. Keep dividing until the decimal number repeats or terminates. Draw a line (vinculum) above the repeating portion.
Change a fraction to a percent by first changing it to a decimal as explained above, and then moving the decimal point two places to the right. Remember to append the percent symbol.

52. Comparing & Ordering Fractions
Cross Multiply (Means extremes property)
More than two fractions: change them to common denominators (or use reasoning, such as: are they more than half)

53. Fractions
ANY QUESTIONS?

54. Decimals

55. Decimal Point
A decimal point is a period that indicates the location of the one’s place – the decimal point always comes to the right of the one’s place. If there are no fractional decimal numbers to the right of the decimal point, the decimal point doesn’t have to be written. It is understood.

56. Comparing Decimal Numbers
Line up the decimal numbers according to place value (as though you were going to add them). Starting at the left-most place value, compare the numbers in each place value to find the largest, next largest, etc.

57. Addition
Decimal numbers are added exactly the same as whole numbers: line up the numbers by place value and add each place value from the right to the left.
When the decimal numbers are lined up by place value properly, the decimal points in each number are also lined up.
Any number without a decimal point is lined up so the ones place is right before the decimal point (there is an understood decimal point after the one’s place).
It may help to write zeros in empty places to facilitate addition.

58. Subtraction
Decimal numbers are subtracted exactly the same as whole numbers: line up the numbers by place value and subtract each place value from the right to the left.
When the decimal numbers are lined up by place value properly, the decimal points in each number are also lined up.
Any number without a decimal point is lined up so the ones place is right before the decimal point (there is an understood decimal point after the ones place). It may help to write zeros in empty places to facilitate subtraction.

59. Multiplication
Decimal numbers are multiplied by temporarily ignoring the decimal point. Multiply the two numbers as though they were whole numbers. In the final product, place the decimal point to signify the number of decimal places in both numbers of the original problem.
For example: 2.3 (one decimal place) x (three decimal places) is the same as 23 x 1456 with the answer having four decimal places (1 + 3 from the original problem)

60. Division
Division is not defined for decimal numbers. In order to divide by a decimal number, we change that divisor into a whole number: First multiply each number by powers of 10 – multiply by whatever is necessary to make the divisor a whole number. Then divide as you would with whole numbers. Wherever the decimal point is in the dividend, it floats directly up to that position in the quotient (answer).

61. Multiplying by Powers of 10

62. Dividing by Powers of 10

63. Scientific Notation 1234.5  1.2345 × 103
Scientific Notation is a way to write very large or very small numbers using powers of 10. To convert a number into scientific notation, move the decimal point so the resulting number is between 1 and 10. Then state the power of 10. Because we use a Base 10 number system, an easy way to know what power of 10 is needed, the exponent indicates the number of decimal places the decimal point was moved. The exponent is negative if the decimal point was moved to the right; the exponent is positive if the decimal point was moved to the left.
 × 103

64. Decimals
ANY QUESTIONS?

65. Ratios, Proportions, & Percentages

66. What is a Ratio?
A ratio is a comparison of two numbers using division.
Write a ratio using a fraction bar, a colon, or the word “to”
3:2
3/2
3 to 2

67. Fractions & Ratios A fraction compares PART to WHOLE
A ratio compares any two numbers
Convert a fraction into a ratio by changing denominator to the difference (D – N)
Example: 2/3 becomes 2:1

68. What is a Proportion?

69. How to Solve Proportions

70. (per = divided by; cent = 100)
What is a Percentage?
A percentage (or a percent) is a way of expressing a number, especially a ratio, as a fraction of 100.
(per = divided by; cent = 100)
The percent key on a calculator merely divides by 100. If your calculator doesn’t have a percent key, hit the divide key and then 100.

71. 3 Types of % Problems What number is 15% of 45?  x = (0.15) ∙ (45)
What percent of 45 is 15?  45 ∙ x = 15 or 45x = 15
15% of what number is 45?  (0.15) ∙ x = 45 or 0.15x=45

72. Convert % to Decimals & Fractions
Change a percent to a decimal by moving the decimal point two places to the left and removing the percent sign: 14% = 0.14
Change a percent to a fraction by writing the percent as a fraction over 100 and simplifying:
14% = 14/100, which simplifies to 7/50
Remember: per = divided by; cent = 100

73. Convert Decimals to % 0.14 = 14%
Change a decimal to a percentage by moving the decimal point two places to the right and appending the percent sign:
0.14 = 14%

74. Solve % Using Proportions

75. what number  variable (x) is  = of  multiply
Solve % Using Algebra
To solve percentages using algebra, write the problem as an algebraic statement where
what number  variable (x) is  = of  multiply
What number is 15% of 45?  x = (0.15) ∙ (45)
What percent of 45 is 15?  45 ∙ x = 15 or 45x = 15
15% of what number is 45?  (0.15) ∙ x = 45 or 0.15x=45

76. Unit Analysis Video: JoleneMorris.com, Math 115, Wk 2
Unit Analysis is the process of multiplying by successive conversion units (written in fraction form).
Unit Analysis Video: JoleneMorris.com, Math 115, Wk 2
Video explaining unit analysis for English units (JoleneMorris.com, Math 115, Wk6)
Video explaining unit analysis for Metric units (JoleneMorris.com, Math 115, Wk6)

77. Denominate Number
Specifies a quantity in terms of a number and a unit of measurement.
For example, 7 feet and 16 acres are denominate numbers.

78. Rate & Unit Rate
A rate is a ratio between two measurements with different units.
In addition to the three ways to write a ratio, rates may also use the word “per”.
Rates are usually simplified to a one in the denominator (unit rate).
13 miles per gallon
$4.59 per pound
12 inches per foot

79. Interest

80. Sales Tax
Sales Tax is written in percentages (which are converted to decimal to computer sales tax).
FP = MP + (ST × MP)
If you know two of those three amounts, you can use basic algebra to find the missing number. Remember to state the sales tax as a percentage in application problems.

81. Discounts Usually written as percentages.
Amount by which the purchase price is reduced.
Discount amount is the original sales price multiplied by the percentage of discount.
Discounted price is the difference of the original price and the discount amount.
OP – (OP * D) = DP where OP is the original price, D is the discount percentage written as a decimal number, and DP is the discounted price.

82. Percent of Change
The percent of change is also known as the percent of increase or the percent of decrease.
To calculate the percent of change,
Find the difference between the new amount and the original amount
Divide that difference by the original amount
Multiply by 100

83. Sales Commission
C = ar, where C = commission earned, a = amount of sale, and r = commission rate.
Example: Juana sells cars on a 3% commission rate. She just sold a car for $23,500. What was her commission?
C = ar
C = 23500(.03)
C = $705

84. Ratios, Proportions, & Percentages
ANY QUESTIONS?

85. Integers

86. What are Integers?
The counting numbers, their negatives, and zero (…, -3, -2, -1, 0, 1, 2, 3 …)
The symbol used for the set of integers is ℤ.

87. Number Line
A number line is a straight line where each point of that line corresponds to a real number.
A line is made up of an infinite number of points and there are an infinite amount of real numbers.
Usually the line is marked off to show the integers, including zero.
A number line is generally written as a horizontal line.

88. Absolute Value The value portion of a number without a sign.
Also described as the distance on a number line from 0.
Zero is the only number that is its own absolute value (because zero is neither positive nor negative).

89. Properties of Absolute Value

90. Ordering & Comparing Integers
Remember that negative numbers are always smaller than positive numbers.
It helps to place the numbers on a number line to compare them.
The larger the value of a positive number, the larger the number is.
The larger the absolute value of a negative number, the smaller the number.

91. Classroom Activity

92. Adding Integers
If the signs are the same, add the absolute values of the numbers and give the result their same sign.
If the signs are different, subtract the absolute values of the numbers and give them the same sign as the number with the larger absolute value.

93. Subtracting Integers
Because a negative number is the inverse of a positive number, and because subtraction is the inverse operation of addition, the RULE for subtracting integers is: Change the sign of the second number to its opposite and change the operation to addition. Then, follow the rules for adding integers.

94. Multiplying Integers
If the signs are the same, multiply the absolute values of the numbers and give the result a positive sign.
If the signs are different, multiply the absolute values of the numbers and give the result a negative sign.

95. Dividing Integers
If the signs are the same, divide the absolute values of the numbers and give the result a positive sign.
If the signs are different, divide the absolute values of the numbers and give the result a negative sign.

96. Review Properties in Number Theory Packet
Zero
One
Identity (additive & multiplicative)
Inverse (additive & multiplicative)

97. Integers
ANY QUESTIONS?

98. Geometry & Measurement

99. Geometry Symbols

100. Definitions Point Line Plane Line Segment Ray Angle Parallel
Perpendicular
Vertex
Arc
Coplanar
Collinear
Bisector
Chord
Diameter
Radius
Circumference
Symmetry
Surface Area
Perimeter
Area
Volume
Transversal
Polygon
Similar
Congruent
Protractor
Compass
Density
Unit Analysis
Golden Ratio

101. Angles An acute angle measures less than 90°
A straight angle measures 180°
A right angle measures 90°
An obtuse angle measures between 90°-180°
A reflex angle is measured in a clockwise direction as opposed to the normal counter-clockwise direction.

102. More About Angles
When two lines intersect, they form four angles. The two angles opposite each other are called vertical angles.
Complementary angles are two angles whose measure adds to 90°.
Supplementary angles are two angles whose measure adds to 180°.
Two angles are adjacent angles if they share a common vertex, they share a common side, AND they do not share any interior points.
An exterior angle is an angle on the outside of a polygon that is formed by extending the side of the polygon
When a transversal line crosses two other lines, it forms eight angles -- Corresponding angles are angles that are in the same position on each of the lines.

103. Triangles Classify triangles by their legs
Scalene triangle - all legs are of different length.
Equilateral triangle - all three legs (sides) are of equal measure.
Isosceles triangle - two of the three legs are of equal measure
Classify triangles by their angles
Right triangle - one angle is a right angle.
Acute triangle - all three angles are less than 90°
Obtuse triangle - one of the angles is obtuse.

104. Pythagorean Theorem

105. 3-Dimensional Solids Sphere Rectangular Solid (Cuboid) Cube Cylinder
Cone
Prism
Pyramid
Polyhedron
Euler’s Polyhedron Formula is V – E + F = 2

106. Measurement English System or Common System Metric System
Length = inch, foot, yard, rod, mile, etc.
Weight = ounce, pound, Ton, etc.
Volume = liquid ounces, cup, pint, quart, gallon, etc.
Metric System
Meter
Gram
Liter

107. Temperature

108. Converting Units of Measurement

109. Tessellations
A tessellation is a two-dimensional plane created by one or more polygon shapes fitted into each other so no “open space” remains.
Equilateral triangles, squares, and hexagons are the only regular polygons that tessellate.
Check out the tessellations of M. C. Escher:

110. Transformations
A transformation in geometry changes the position of a shape on the coordinate plane. There are four forms of transformation:
translation (slide)
rotation (turn)
dilation (scale)
reflection (flip)

111. Net (or network)
A net is a two-dimensional representation of a three-dimensional object. If a net is cut out, it can be put together to form the three-dimensional object it represents.

112. Geometry & Measurement
ANY QUESTIONS?

113. Data, Statistics, & Probability

114. Measures of Central Tendency
Data has a tendency to cluster or center on certain values. The term “average” is also used to indicate measures of central tendency.
Mean (evenly distributed / “average”)
Mode (bimodal distribution / most popular)
Median (outliers)
Range

115. Quartiles
Quartiles are three points that divide a set of ordered data into four equal groups.
The first quartile, also called the lower quartile, splits off the lower 25% of the data. It is denoted by Q1
The second quartile, also called the median, splits the data in half. It is denoted by Q2
The third quartile, also called the upper quartile, splits off the higher 25% of the data. It is denoted by Q3

116. Trends
A trend is the general direction data tends to move. From a line graph, a trend can be obvious when the line is going in an up or down pattern.
Example: In the stock market, when stocks are trending down, it is called a bear market. When stocks are trending up, it is called a bull market. (A mnemonic to remember which is which: A bear has claws that curve downward and a bull has horns which curve upward.)

117. Algorithm
An algorithm is a step-by-step process for solving a problem.
An example of an addition algorithm is:
Line up the numbers
Add each column starting on the right
Carry any tens-place digits to the next column
Place commas between periods in the answer
An algorithm is often written as a flowchart showing steps, branches, and decisions.

118. Probability
Ratio of how likely a specific event is to happen when compared to all possibilities of events that might happen.
Most often written as a fraction, but it may also be written as a decimal or a percentage.
Odds and probability are related concepts. With odds, you compare the number of favorable outcomes to the number of remaining (unfavorable) outcomes.

119. Probability of Multiple Events
Independent (OR) , mutually exclusive  add
Independent, non-exclusive  add then subtract the events in common
Dependent (AND)  multiply (Fundamental Counting Principle)
If dependent, be sure to use the conditional second probability

120. Odds Related to probability
Probability  favorable outcomes / possible outcomes
Odds  favorable outcomes / unfavorable outcomes
EXAMPLE: If you have a box with 2 red balls and 3 blue balls, the probability of randomly picking a red ball is 2 out of 5 or 2/5. The odds of randomly picking a red ball are 2 for and 3 against, or 2:3

121. 5 Rules of Probability

122. Tree Diagram
A tree diagram is a graphic organizer that lists all possibilities of a sequence of events in a systematic way.

123. Factorial

124. Sequences Series Arithmetic – a common number added
Geometric – a common number multiplied
Fibonacci – each term comprised of the sum of previous two terms
Series
Harmonic – sum of progressive unit fractions

125. Charts & Graphs (see study packet)
Bar
(broken scale)
Histogram
Line
Circle or Pie
Pictograph
Table
Scatterplot
Venn Diagram
Stem & Leaf Plot
Box & Whiskers Plot
Logic Diagram
Flow Chart

126. Deductive & Inductive
Deductive reasoning is a form of logic starting with statements of fact and drawing logical conclusions. If the laws of logic are followed from the statements of fact, the conclusions are true. It often helps to draw logic circles when working with deductive reasoning.
Inductive reasoning is making sufficient observations that conclusions can be formed.

127. Data, Statistics, & Probability
ANY QUESTIONS?

128. Algebraic Reasoning

129. Algebra & Algebraic Thinking
Algebra is the study of numbers, number patterns, and relationships among numbers.
Algebraic thinking is the study of our number system, patterns, representations, and mathematical reasoning.
A variable is used in algebra to represent a value that changes within the parameters of the problem. Lowercase letters of the alphabet are generally used to denote a variable.
The opposite of a variable is a constant.
The number multiplied to the variable is a coefficient.
-2x3

130. Real Number System

131. Properties of Operations

132. PEMDAS Please Excuse My Dear Aunt Sally
Order of Operations
Simplify inside parentheses or grouping symbols
Simplify any expressions with exponents
Perform multiplication & division from left to right
Perform addition & subtraction from left to right
PEMDAS Please Excuse My Dear Aunt Sally

133. Expressions & Equations
An expression is a collection of terms that have been added or subtracted.
An equation is a statement where an algebraic expression is equal to another algebraic expression or constant.
A literal equation is an equation made up of only known, measurable quantities. A literal equation is the same as a formula.
An inequality is similar to an equation, but the two sides are NOT equal.

134. Words that Signal + - × ÷
Addition: add, sum, increase, total, rise, plus, grow, added to, more than, increased by, gain…
Subtraction: subtract, subtracted from, minus, difference, take away less than, decreased by…
Multiplication: multiply, multiplied by, product, times, of, twice…
Division: divide, divided by, quotient, per, ratio, half, …

135. Function Relationship is simply a set of ordered pairs.
Function is where each input is related to exactly one output.
One-to-one Correspondence is a function where each output has exactly one input.
Domain (x)
Range (y)

136. Absolute Value
Absolute value is the value portion of a number without a sign. Absolute values are also described as the distance on a number line from 0. Zero is the only number that is its own absolute value (because zero is neither positive nor negative).
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137. Properties of Absolute Value

138. Literal Equation
A literal equation is an equation made up of only known, measurable quantities. A literal equation is the same as a formula.
With a literal equation, you are not solving for an unknown quantity that varies. Instead, you are manipulating the letters/variables in the equation to a different form to substitute values in it.

139. Solving Algebraic Word Problems (see the study packet)
Consecutive numbers
Rectangular area & perimeter
Triangles
Unit conversion
Mixtures
Investments with interest
Discounts & Commissions
Distance-Speed-Time & Uniform motion

140. Simultaneous Equations
Simultaneous Equations are two or more equations with multiple variables. These are often called systems of equations. There are many ways to solve a system of equations. Three ways discussed in beginning algebra are:
Elimination (sometimes called adding)
Substitution
Graphing

141. Polynomials
A polynomial is an algebraic expression with one or more terms. A polynomial cannot have a variable in the denominator (which is a negative exponent). A polynomial with one term is called a monomial; two terms, a binomial; and three terms, a trinomial. The term with the highest exponent (sum) determines the degree of the polynomial.
degree
type of polynomial 1
linear 2
quadratic 3
cubic

142. Classifying Polynomials

143. Factoring Polynomials
Factor out any common factors in all terms.
If the polynomial has four terms, factor it by grouping.
If it is a binomial, look for a difference of squares, a sum of cubes, or a difference of cubes. (Note that a sum of squares cannot be factored.)
If it is a trinomial and the coefficient of the x2 term = 1, un-FOIL to factor.
If it is a trinomial and the coefficient of the x2 term ≠ 1, use the AC method to factor.

144. Quadratic Equation
A quadratic equation is a second-degree polynomial equation (the exponent on the leading term is a 2). There are many ways to solve a quadratic equation, but the five most common ways are:
Factor and set each factor equal to 0
If there is no x-term, solve for x2 and apply the square root method.
Graph the equation (as a parabola) and determine the solutions where the parabola crosses the x-axis
Complete the square
Use the quadratic formula

145. Distance Formula
The distance formula is used to find the distance between two points. The distance formula can be obtained by creating a triangle and using the Pythagorean Theorem to find the length of the hypotenuse. The hypotenuse of the triangle will be the distance between the two points.

146. Coordinate Grid
A coordinate plane is a two-dimensional grid for locating points. There is an x-axis and a y-axis at 90-degree angles, which divide the grid into four quadrants that are numbered counter-clockwise using Roman numerals. The origin is where the two axes cross (0, 0). A coordinate pair is a pair of numbers indicating the location of a point (x, y). Sometimes called a Cartesian grid after the mathematician René Descartes ( )

147. Slope of a Line
The slope of a line is an algebraic concept used to graph linear equations. In the equation y = mx + b, the variable m represents the slope of the line. Slope is calculated by dividing the change in the y-coordinate (the rise) by the change in the x-coordinate (the run). Parallel lines have equivalent slopes. Perpendicular lines have slopes that are negative and reciprocal of each other. To graph a line when the slope and the y-intercept are known, plot the y-intercept and then use the slope to count UP and OVER to find another point on the line.

148. Graphing a Quadratic Equation

149. Rules for Exponents

150. Rules for Square Roots

151. Simplifying Square Roots

152. Approximating Square Roots
Approximating square roots means to find the approximate value of a number’s square root. We find approximate square roots by comparing the number to perfect square numbers where the square roots are known. For example, to find the approximate square root of 51, use the fact that 7 x 7 = 49 and 8 x 8 = 64. Since 51 is between the perfect squares of 49 and 64 (but closer to 49 than 64), the approximate square root of 51 is between 7 and 8 (but closer to 7 than 8). The approximate square root of 51 is 7.1 or 7.2

153. Algebraic Reasoning
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154. Praxis Workshop PPT #2
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