The Praxis Mathematics TestsWorkshop PPT #2

Slide show created by Jolene M. Morris

ETS Praxis WebsiteETS Praxis WebsiteOfficial website where you canfind more information about

the Praxis tests

www.ets.org/praxis/

NCTM WebsiteNCTM WebsiteThe 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.

www.nctm.org

Jolene’s WebsiteJolene’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

www.JoleneMorris.com

Agenda (Slide 5)Agenda (Slide 5)6.6. Number Theory (hunter green)Number Theory (hunter green)32.32.Fractions (black)Fractions (black)52.52.Decimals (light blue)Decimals (light blue)63.63.Ratios, Proportions, & Percentages (dk blue)Ratios, Proportions, & Percentages (dk blue)83.83.Integers (red)Integers (red)96.96.Geometry & Measurement (burgundy)Geometry & Measurement (burgundy)111.111.Data, Statistics, & Probability (pink)Data, Statistics, & Probability (pink)126.126. Algebraic Reasoning (green)Algebraic Reasoning (green)

Number TheoryNumber Theory

Our Number System

Place Value

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:

5,139 986,733 3,950 77,922

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.

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

Estimation

Squared & Cubed Numbers

Divisibility Rules2 = 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

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

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

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.

Expanded Notation

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)

Prime Factorization

Thus, the prime factorization of 120 = 2 x 2 x 2 x 3 x 5

GCF 12 = 2 × 2 × 3 18 = 2 × 3 × 3 30 = 2 × 3 × 5 GCF = 2 × 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.

LCM

Multiplying by Powers of 10

Dividing by Powers of 10

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

Properties of Operations

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

PEMDASPEMDASPlease Excuse My Dear Aunt Sally

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

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

Identity Property

Number Theory

ANY QUESTIONS?

FractionsFractions

What is a Fraction?• A fraction represents equal-sizedequal-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.

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 16/15 15/15 99/5

• An improper fraction can be changed into a whole number or a mixed number by dividing the denominator into the numerator.

Common Fraction

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

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 = 2 × 3 × 3

30 = 2 × 3 × 5

GCF = 2 × 3 = 6GCF = 2 × 3 = 6

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.

Equivalent Fractions

• Fractions that simplify to the same simple fraction.

• When two fractions are equivalent, their cross products are equal :

3 x 6 = 182 x 9 = 18

Simplify Fractions

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.

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

Why Find Common Denominators?

3 7-footers+ 2 6-footers

5 13-footers

Why Change to Common Denominators?

Why Change to Common Denominators?

3 apples+ 4 oranges

7 ??????

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.

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.

Multiply Fractions&Mixed Numbers

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 NOTE: Since there is no rule on how to divide fractions, but because multiplication is the inverse of division and a reciprocal is because multiplication is the inverse of division and a reciprocal is the inverse of a fraction, you can divide fractions by multiplying by the inverse of a fraction, you can divide fractions by multiplying by the reciprocal of the divisor. Hence, the inverse of an inverse the reciprocal of the divisor. Hence, the inverse of an inverse results in the same answer as if you had divided.results in the same answer as if you had divided.

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.

Change Fractions toDecimals & 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.

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)

Fractions

ANY QUESTIONS?ANY QUESTIONS?

DecimalsDecimals

Decimal PointA 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.

Comparing Decimal NumbersLine 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.

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.

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.

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 1.456 (three decimal places) is the same as 23 x 1456 with the answer having four decimal places (1 + 3 from the original problem)

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).

Multiplying by Powers of 10

Dividing by Powers of 10

Scientific NotationScientific 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.

1234.5 1.2345 × 103

Decimals

ANY QUESTIONS?ANY QUESTIONS?

Ratios, Proportions, & Ratios, Proportions, & PercentagesPercentages

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”

3:23/2

3 to 2

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

What is a Proportion?

How to Solve Proportions

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.

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

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

Convert Decimals to %• Change a decimal to a percentage by

moving the decimal point two places to the right and appending the percent sign:

0.14 = 14%

Solve % Using Proportions

Solve % Using AlgebraTo 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 = 1515% of what number is 45? (0.15) ∙ x = 45 or 0.15x=45

Unit AnalysisUnit 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)

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.

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 pound12 inches per foot

Interest

Sales TaxSales 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.

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 thediscount percentage written as a

decimal number, and DP is the discounted price.

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

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 = arC = 23500(.03)

C = $705

Ratios, Proportions, & Percentages

ANY QUESTIONS?ANY QUESTIONS?

IntegersIntegers

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

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.

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).

Properties of Absolute Value

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.

Classroom Activity

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.

Subtracting IntegersBecause 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.

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.

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.

Review Propertiesin Number Theory Packet

• Zero• One• Identity (additive & multiplicative)• Inverse (additive & multiplicative)

IntegersANY QUESTIONS?ANY QUESTIONS?

Geometry & MeasurementGeometry & Measurement

Geometry Symbols

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

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.

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.

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.

Pythagorean Theorem

3-Dimensional Solids• Sphere• Rectangular Solid (Cuboid)• Cube • Cylinder• Cone• Prism• Pyramid• Polyhedron

Euler’s Polyhedron Formula is V – E + F = 2

Measurement• English System or Common 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

Temperature

Converting Units of Measurement

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 tessellationsof M. C. Escher:

TransformationsA 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)

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.

Geometry & Measurement

ANY QUESTIONS?

Data, Statistics, & ProbabilityData, Statistics, & Probability

Measures of Central TendencyData 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

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

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.)

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.

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.

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

Odds• Related to probability• Probability favorable outcomes /

possible outcomes• Odds favorable outcomes / unfavorable

outcomesEXAMPLE: 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

5 Rules of Probability

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

Factorial

Sequences• 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

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

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.

Data, Statistics, & Probability

ANY QUESTIONS?ANY QUESTIONS?

Algebraic ReasoningAlgebraic Reasoning

Algebra & Algebraic ThinkingAlgebra & 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

Real Number SystemReal Number System

Properties of OperationsProperties of Operations

Order of OperationsOrder 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

PEMDASPEMDASPlease Excuse My Dear Aunt Sally

Expressions & EquationsExpressions & 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.

Words that Signal + - × ÷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, …

FunctionFunction• 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)

Absolute ValueAbsolute 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).

| |

Properties of Absolute ValueProperties of Absolute Value

Literal EquationLiteral 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.

Solving Algebraic Word ProblemsSolving Algebraic Word Problems(see the study packet)(see the study packet)

• Consecutive numbers• Rectangular area & perimeter• Triangles• Unit conversion• Mixtures• Investments with interest• Discounts & Commissions• Distance-Speed-Time & Uniform motion

Simultaneous EquationsSimultaneous 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

PolynomialsPolynomialsA 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

Classifying PolynomialsClassifying Polynomials

Factoring PolynomialsFactoring 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.

Quadratic EquationQuadratic 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

Distance FormulaDistance FormulaThe 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.

Coordinate GridCoordinate GridA 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 (1596-1650)

Slope of a LineSlope of a LineThe 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.

Graphing a Quadratic EquationGraphing a Quadratic Equation

Rules for ExponentsRules for Exponents

Rules for Square RootsRules for Square Roots

Simplifying Square RootsSimplifying Square Roots

Approximating Square RootsApproximating Square RootsApproximating 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

Algebraic ReasoningAlgebraic Reasoning

ANY QUESTIONS?ANY QUESTIONS?

Praxis Workshop PPT #2Praxis Workshop PPT #2ANY QUESTIONS?ANY QUESTIONS?