THEA Math Test Prep

General Testing Information

THEA-IBT Guidelines

THEA-IBT IS FOR NEW STUDENTS ONLY!

THEA-IBT consists of 3 sections: Reading, Writing and Math. You may take in any order.

The Reading section consists of approximately 40 multiple-choice questions matched to about seven reading selections of 300-750 words each.

The Math section consists of approximately 50 multiple-choice questions covering four general areas: fundamental mathematics, algebra, geometry, and problem solving.

The Writing portion consists of 2 subsections: approximately 40 multiple-choice questions and a writing sample. For the writing sample you will be asked to write a 300-600 word essay in response to a prompt. Your essay may NOT be more than 1,000 words. You must complete both before moving on to the next section.

You cannot attempt another section of the THEA until you have completely finished the section on which you are working.

You may work the sections in any order.

If you " run out" of time to complete the THEA-IBT you must wait 14 days to take the remaining portions. You must state that you " ran out" of time. DO NOT CLICK END TEST IF YOU RAN OUT OF TIME! This will give you a score for that portion of the test. You will be administered a new and different test and you will have the option to take only the sections that you need.

You can start with which ever section that you choose.

Time Management: You are allowed 4 hours for the ENTIRE test. This includes reading the directions and the writing sample of the writing portion of the test.

Additional information can be found at http://www.thea.nesinc.com/index.asp

To view your scores, go to the counselors 48 hours after test or go to www.thea.starttest.com .

Strategies for effective Test performance before the test:

Good night's sleep and eat breakfast.

Wear comfortable clothes (layer).

Allow plenty of time to get ready and get to test site.

Follow directions carefully and read individual test questions.

Guess wisely (don't skip questions) and check answers.

Pace yourself- plan to stay the entire time.

Read passages with care.

Estimate in Math: formulas, 4 function calculator.

Plan your writing sample.

Study and gain self-confidence.

Get help.

 

Formula Chart

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Mathematics Fundamentals

Exponent Rules

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Examples:

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Watch the Video: Math Help Exponents 1: Definitions by Pat McKeague

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Watch the Video: Math Help Exponents 3: Multiplication by Pat McKeague

 

Operations of Signed numbers (rules with add/sub/mult/div)

Addition & Subtraction

  1. Same (like) signs ADD and KEEP that sign.
  2. Different (unlike) signs SUBTRACT and KEEP sign of the larger absolute value.

Examples:

Same sign. Add and keep the sign.

Different signs. Subtract and keep the sign of the larger absolute value.

Different signs. Subtract and keep the sign of the larger absolute value.

Same signs. Add keep the sign.

Multiplication and division of Integers

  1. Multiply or divide, if the signs are same (like) the sign of the product or quotient will be positive.
  2. Multiply or divide, if the signs are different (unlike) the sign of the product or quotient will be negative.

 Examples:

Same signs. The sign is positive.

Same signs. The sign is positive.

 

Different signs. The sign is negative

Same signs. The sign is positive.

 

Different signs. The sign is negative

 

Same signs. The sign is positive.

 

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Watch the Video: Adding/Subtracting negative numbers by Kahn Academy

 

 

Operations with Fractions and Decimals Add/Sub/Mult/Div

Multiplication rules

  1. Put all MIXED NUMBERS in IMPROPER FORM.
  2. Reduce all fractions before multiplying
  3. Multiply the numerator times the numerator and the denominator times the denominator.

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 To Multiply decimals numbers:

Multiply the numbers, then count the number of decimal places.

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Watch the Video: Arithmetic Basics: Multiplying Decimals by Patrick JMT

 

Division rules

  1. CHANGE division sign to a multiplication sign.
  2. Write the RECIPROCAL of the divisor (2nd number).
  3. Follow the multiplication rules.

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 To DIVIDE decimal numbers:

1. Move the decimal point in the divisor so that it becomes a whole number.

2. Move the decimal point in the dividend the same number of places to the right

3. Divide as if working with whole numbers. Write the decimal point in the answer directly above the decimal point in the dividend.

 

 Toggle open/close quiz question

 

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Watch the Video: Arithmetic Basics: Dividing Decimals by PatrickJMT

 

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Watch the Video: Fractions- Multiplying and Dividing by PatrickJMT

 

 

Addition and subtraction rules

  1. Must have a common denominator
  2. Add or subtract numerators only
  3. Denominator remains the same
  4. Reduce to lowest terms
  5. When adding or subtracting mixed numbers

a. Add or subtract whole numbers

b. With addition and subtraction extra step (carrying or borrowing) may be needed

c. Answers must always be in lowest terms

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Examples:

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Examples:

Add the following numbers: 2.046, 0.658 and 1.39.                    

Line up the decimals, fill in any needed zero,. and add the columns.

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 Subtract the following numbers: 10.8 - 3.52

LIne up the decimals, fill in any needed zeros and subtract the columns.

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Watch the Video: Fractions 1: Reducing by Pat McKeague

 

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Watch the Video: Fractions 2: Multiplication by Pat McKeague 

 

 

Calculations using Scientific Notation

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Check Yourself: Click on the activity.

 Toggle open/close quiz question

 

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Watch the Video Converting between Scientific notation and decimal notation by PatrickJMT

 

Order of Operations, calculations using Percents

1. Always operate within grouping symbols-- parentheses, brackets, braces, division bar.

2. Exponets and roots.

3. Multiplication/ Division in order from LEFT to RIGHT.

4. Addition/Subtraction in order from LEFT to RIGHT.

 Example:

Add inside the parentheses.

Do the exponents.

Multiply

Subtract

48

Answer

 

 Toggle open/close quiz question

 

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Watch the Video:Introduction to Order of Operations by Khan Academy

 

Percent word Problems

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Word Problems with Integers, Decimals and Units of Measure

Steps to Solving Word Problems (strategies)

1. Read and Understand the problem.

2. Develop a mathematical plan for solving the problem.

3. Carry out the plan accurately.

4. Check your answer to make sure it is reasonable.

 

 

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Problems involving ratios and proportions

  1. Use the two related numbers as your first ratio.
  2. Write the second ratio by matching units and using x as the unknown.
  3. Cross multiply and solve for x.
  4. Check your answer.
  5. Make sure that you have answered the question.

Examples:

Step 1: Use the two related numbers as your first ratio.

 

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Step 2: Write the second ratio by matching and using x as the unknown.

Step 3: Cross multiply & solve.

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Step 4: Check

Step 5: Did you answer the question?

Step 1: Use the two related numbers as your first ratio.

 

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Step 2: Write the second ratio by matching and using x as the unknown.

Step 3: Cross multiply & solve.

 

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Step 1: Use the two related numbers as your first ratio.

 

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Step 2: Write the second ratio by matching and using x as the unknown.

Step 3: Cross multiply & solve.

 

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Step 4: Check

Step 5: Did you answer the question?

 

 

 

Measures of Central Tendency

Mean & Median

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Mode

Mode: The number that occurs most often. If there are 2 numbers that occur the same amount of times, then the set has 2 modes. If none of the numbers repeat more than once, then the set is said to have no mode.

Example: 5, 9, 99, 3, 2, 8, 73, 1, 4,16

Solution: The set has no mode, none of the numbers in the set repeat.

Example: 20, 43, 46, 43, 49, 43, 49

Solution: The mode is 43.

Example:1.1, 0.7, 0.9, 1.1, 0.5, 1.3, 0.5, 1.4, 1.8

Solution: The set has 2 modes, 1.1 and 0.5.

Variability

Variability: The spread of the data around the mean.

Example: The mean for set A is 90 and the mean for set b is 70. What is the variability?

Variability = 90 - 70 = 20

 

  

 

Graphs of Number Relationships

Cartesian Plane

Cartesian coordinate system or Rectangular Coordinate System – a grid system used to draw graphs.

 

Coordinate Terms:

The ordered pair (x, y) represents one point on a graph. The ordered pair of numbers is called its "coordinates".

 

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Plotting points

To plot the point (3, 2), start at the origin (0, 0), and count right 3 and up 2.

Example:

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x & y-Intercepts

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Graphing a line using x & y Intercepts

Step 1: Let x = 0 and solve for y

Step 2: Let y = 0 and solve for x.

Step 3: Plot the points on the graph. (WARNING: the line extends past the points)

Example:

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Watch this video: Graphing Linear Functions by Finding X,Y Intercept by PatrickJMT

 

 

Graph a linear equations by plotting points:

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Graphing a Line Using the Slope-Intercept Method: 

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Find the slope when given two points

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Use the graph to determine the slope

 

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Writing Equations of lines

Write the equation when given the slope and y-intercept.

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Horizontal and Vertical lines

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Slope and y-intercept from an equation:

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Linear Inequalities in One Variable

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Solving Linear Inequalites in One variable

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Watch the video: Algebra Help: Inequalities by Pat McKeague

 

 

Graphing Linear Inequalities in Two Variables: 

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Watch this video: Graphing Systems of Linear Inequalities - Example 1 by PatrickJMT

 

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Click on the applet: Families of Functions Applet

 

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Click on the applet: Functions 1

 

 

Direct and Inverse Variation: 

Forumlas:

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To solve a variation problem:

1.      Translate each problem into an equation.

2.      Use the given information to find k.

3.      Rewrite the equation, using the value you found for k.

4.      Solve the equation in #3 to answer the question.

 

Example:

The distance (d) a spring will stretch varies directly as the force (f) applied to the spring.  If the force of 5 pounds is required to stretch a spring 2 inches, what force is required to stretch the spring 5 inches?

Step 1: Translate

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Step 2: Find k.

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Step 3: Re-write the equation.

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Step 4: Solve

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Example:

The speed (v) of a gear varies inversely as the number of teeth (t).  If a gear that has 48 teeth makes 20 revolutions per minute, how many revolutions per minute will a gear make that has 30 teeth?

Step 1: Translate

 

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Step 2: Find k.

 

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Step 3: Re-write the equation.

 

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Step 4: Solve

 

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Watch the video: Algebra Word Problem Variation by Pat McKeague

 

One & Two Variables Equations

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Solving Linear Equations in One Variable

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Solving Linear Equations in Two Variables for a Specified Variable

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Systems of Linear Equations

A System of Linear Equations in Two Variables is two linear equations in one or two variables considered together.

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The solution to a system of linear equations in two variables is an ordered pair (value for x and y) that makes each equation true. The solution to the example system is (2,3) .

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Solving Systems of Linear Equations in Two Variables Using the Addition/Elimination Method:

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Watch the Video: Algebra: Solving a System of Equations by Pat McKeague

 

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Watch the Video: Solving Systems of Equations Using Elimination By Addition by PatrickJMT

 

Solving Systems of Linear Equations in Two Variables Using the Substitution Method:

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Watch the video: Solving Linear Systems of Equations Using Substitution by PatrickJMT

 

 

Systems with an Infinite Number of Solutions and with No Solution 

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Absolute values

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Solving an Equation Involving Absolute Value:

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Watch the video: Solving Absolute Value Equations - Example 1 by PatrickJMT

 

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Watch the video: Absolute Value and Evaluating Numbers by PatrickJMT

 

 

Graph Absolute Value Equations in Two Variables

  Plot points that satisfy the equation (as described earlier).  The graph is a V-shape so you must use enough ordered pairs so that you can find the lowest point of the V .  The graph will be symmetrical

  about the vertical line that passes through that point.

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Watch the Video: Solving Linear Absolute Value Equations and Inequalities by PatrickJMT

 

 

 

Quadratic Equations

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Solve Nonlinear Systems of Equations in Two Variables by Graphing

Graph each equation (as described earlier) on one set of axes. The point(s) of intersection of the graphs have coordinates that will satisfy the system of equations.

 

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Solve Nonlinear Systems of Equations in Two Variables by Substitution

Use the substitution method described earlier for solving systems of linear equations in two variables.

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These solutions also represent the points of intersection of the graphs of the equations.

 

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Watch the video: Graphing Quadratic Functions - Example 1 by PatrickJMT

 

Word Problems with Variables

Translating Words into Algebraic Terms

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Examples: Translate the following expressions.

a number added to four

x + 4

the product of two and a number subtracted from 5

5 - 2x

7 times a number reduced by eight

7x - 8

 

Word Problems involving one variable

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Example: Jerri has 3 children: Jen, Joe and Jill. Jen's age is 2 years more than 14 times Joe's age. Jill's age is 1 year less than twice Joe's age. Find each child's age if the sum of the ages is 35.

Step 1: identify the variables.

Step 2: Translate

 

Joe = x

Jen = 2 + 14 *Joe= 2 + 14x

Jill = 2*Joe -1= 2x -1

Step 3: Solve for the variable.

Step 4 & Step 5: check and re-read to determine if you have answered the question.

Joe = 2 years

Jen = 2 + 14*2 = 30 years

Jill =2*2 - 1 = 3 years

 

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Watch the Video: Word Problem: Finding Consecutive Numbers That Satisfy a Given Requirement - Ex 1 by PatrickJMT

 

 

Solving Typical Word Problems

Distance Rate time

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Example: Two boats leave port at the same time, one heading north at 35 knots and the other south at 47 knots. How long will it take to be 738 nautical miles apart?

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Watch Video: Algebra Help: Distance, Rate, and Time by Pat McKeague

 

Simple Interest 

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Substitue in the given information and solve for the unknown variable.

Example: Find the interest earned on $1000 for 1 year at rate of 6%

 Substitute in the given information.

 

 

Rates of Interest

Example: Maria has $2500 to invest for 1 year, CD's are paying 5% simple interest and her savings is paying 8% simple interest. How much did she invest in CD's and her savings account if she earned $180.50?

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Watch the Video: Understanding Simple Interest and Compound Interest by PatrickJMT

 

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Watch the Video: Solving an Investment Problem by PatrickJMT

 

 

Mixture Solutions

Example: A jeweler needs to mix an alloy with a 16% gold content and an alloy with a 28% gold content to obtain 32 ounces of a new alloy with a 25% gold content. How many ounces of each of the original alloys must be used?

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Algebraic Expressions and Functional Notation

Simplifying Polynomial Expressions

Examples: 6x & 7x are like terms

Combine the like terms:

Combine the like terms of 4x & -7x

Answer

 

Combine the like terms of -6t, 2t & 5 , -9

Answer

 

Multiplying Polynomial Expressions

A monomial times a polynomial

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Do the distributive property. Combine like terms if needed.

multiply

 

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Watch the Video: Multiplying Polynomials - Slightly Harder Examples #1 by PatrickJMT

 

 

Multiplying two binomials (F.O.I.L.)

Examples: Multiply the following binomials

Multiply the first term of each binomial.

Multiply the outside term of each binomial.

multiply the inside of each binomial.

Multiply the last term of each binomial.

Combine like terms if needed.

 

The Square of a binomial

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Watch the Video: Review of Video Foil by Nelson Carter

 

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Watch the Video: Math Help: Distributive Property by Pat McKeague

 

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Watch the Video: Algebra Help: Function Notation 1 by Pat McKeague

 

 

Factoring

Find the GCF of two or more numbers (terms)

Steps for finding the GCF:

1. Write each number (term) as a product of prime factors.

2. Determine the prime factors common to all the terms.

3. Multiply the common factors found in step 2.

4. The product is the GCF.

 

Example 1: Find the GCF of 20 and 24.

Step 1: Write the factors of 20 & 24

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Step 2: Determine the common factors.

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Step 3: Multiply the common factors.

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Step 4: The product of the common factors = the GCF.

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Example 2: Find the GCF of 15, 30 and 45

Step 1: Write the factors of 15, 30 and 45.

 

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Step 2: Determine the common factors.

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Step 3: Multiply the common factors.

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Step 4: The product of the common factors = the GCF.

GCF = 15

 Example 3: Find the GCF of

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Step 1: Write the factors of

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Step 2: Determine the common factors.

Step 3: Multiply the common factors.

3xy

Step 4: The product of the common factors = the GCF.

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Watch this Video: Factoring a Number by PatrickJMT

 

 

 

Check yourself: Drop and Drag Activity.

  

 

Factor a GCF from two terms

 Steps for factoring common monomial from two terms (GCF):

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1. Find the numerical factors that are common to the coefficients of all terms.

2. Find the variable factors common to all terms (lowest exponent of common factors)

3. The GCF is the product of the numerical factors from step 1 and the variable factors from step 2.

4. Then write the polynomial as the product of the GCF and the factor that remains when each term is divided by the GCF.

 

Example 1: Factor the GCF from each term in the expression.

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Step 1:Find the numerical common factors of 10 & 5.

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GCF = 5

Step 2:Find the variable factors common to all terms.

none

Step 3: The GCF of the numerical expression and GCF of the variable.

GCF = 5 

Step 4: Write the polynomial as the product of the GCF and the factor expression that remains.

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Example 2: Factor the GCF from each term in the expression.

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Step 1:Find the numerical common factors of 16, 12, 24.

 

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Step 2:Find the variable factors common to all terms.

 

xxx

xx

x

GCF = x

Step 3: The GCF of the numerical expression and GCF of the variable

GCF = 4x

Step 4: Write the polynomial as the product of the GCF and the factor expression that remains

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Example 3: Factor the GCF from each term in the expression.

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Step 1:Find the numerical common factors of 14 & 6.

 

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GCF = 2

Step 2:Find the variable factors common to all terms.

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Step 3: The GCF of the numerical expression and GCF of the variable

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Step 4: Write the polynomial as the product of the GCF and the factor expression that remains.

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Watch this Video: Factoring Using the Greatest Common Factor, GCF EXAMPLE 2 by PatrickJMT

 

 

Check Yourself: Click on Activity

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Factor by Grouping

How to factoring 4 term polynomials     Pattern: GCF(     )   ±   GCF(     )     =     (     ) (GCF )

1. Determine if all four terms have a common factor, if so, factor out the GCF.

2. Group the terms in pairs such that each pair has a GCF.

3. Factor out the GCF from each pair. (You should now have a common binomial  factor.) If you do not have a common binomial factor and you have factored correctly, try grouping the terms differently.

4. Factor out the common binomial factor.

5. Write the expression as a product of factors.

 

Example 1: Factor completely.

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Step 1: Check for common GCF

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Step 2: Group the terms.(Divide in two)

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Step 3: Factor out the GCF of each pair.

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Step 4: Factor out the binomial factor.

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Example 2: Factor completely.

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Step 1: Check for common GCF

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Step 2: Group the terms.(Divide in two)

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Step 3: Factor out the GCF fo each pair.

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Step 4: Factor out the binomial factor.

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Example 3: Factor completely.

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Step 1: Check for common GCF

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Step 2: Group the terms.(Divide in two)

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Step 3: Factor out the GCF fo each pair.

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Step 4: Factor out the binomial factor.

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Watch this Video: Factor by Grouping -EX 1 by PatrickJMT

 

 

 Check Yourself: Click the Self Check Activity.

 Toggle open/close quiz question

 

Factoring Special Binomials:

Difference of Squares

The difference of two perfect square terms, factors as two binomials (conjugate pair) so that each first term is the square root of the original first term and each second term is the square root of the original second term.

1. Factor out the GCF, if necessary.

2. Determine the pattern a =____ b = ______

3. Write the expression as a product of factors.

Hints: Does it fit the pattern (something square minus something squared)

 

Example: Factor

1. No GCF

2. Determine the pattern

a = x b =9

(x +9) (x - 9)

3. Write the expression as a product of factors.

 

(x +9) (x - 9)

 

Example: Factor

1. GCF = 2

2. Determine the pattern.

3. Write the expression as a product of factors.

 

2(x + 6)(x - 6)

 

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Watch the Video: Difference of Squares EX-1 by PatrickJMT

 

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Watch the Video: Difference of Squares EX-2 by PatrickJMT

 

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Watch the Video: Difference of Squares EX-3 by PatrickJMT

 

   

Check Yourself: Click on Activity

 Toggle open/close quiz question

 

Perfect Square trinomial

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Chart of Squares & Cubes

Learn these perfect squares and perfect cubes!!!!  

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Factoring Special Binomials: Difference of Cubes & Sum of Cubes

Difference fo cubes: Pattern

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Sum of Cubes:

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The difference or sum of two perfect cube terms have factors of a binomial times a trinomial. 

Step 1: Factor out the GCF, if necessary.

Step 2:Write each term as a perfect cube. 

Step 3: Identify the given variables.

Step 4:The terms of the binomial are the cube roots of the terms of the original polynomial. 

The first term of the trinomial is the first term of the binomial squared. The second term of the trinomial is the opposite sign of the product of the two binomial terms. The last term of the trinomial is the last term of the binomial squared.

Hint to remember the signs of the factors: S.O.A.P. (Same sign, Opposite sign, Always Plus)

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Example 3: Factor completely.

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Step 1: Factor out the GCF, if necessary.

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Step 2:Write each term as a perfect cube. 

Step 3: Identify the given variables.

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Step 4:The terms of the binomial are the cube roots of the terms of the original polynomial. 

 

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Watch the Video: Factoring Sum and Difference of Cubes by PatrickJMT

 

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Watch Video: Factoring Sum and Difference of Cubes EX 3 by PatrickJMT

 

 

  

Check Yourself: Click on Activity

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Factoring Trinomials where a = 1

 Trinomials     quad.gif      =(binomial) (binomial)        Hint:You want the trinomial to be in descending order with the leading coefficient positive. 

Steps for Factoring where a = 1

Step 1: Write the ( ) and determine the signs of the factors.

Step 2: Determine the factors (make a t-chart)

If the sign of the last term is positive, you want the factors of the last term whose sum is the coefficient of the middle term. The signs will be the same sign as the middle term. (++ or - -)

If the sign of the last term is negative, you want the factors of the last term whose difference is the coefficient of the middle term. The signs will be different (+ - or - +) and the sign of the middle term will go on the larger factor.

Step 3: Write the factors of the given expression.

Example: Factor

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Example: Factor

Step 1: Write the ( ) and determine the signs of the factors.

(minus)(minus)

Step 2: Determine the factors (make a t-chart)

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Step 3: Write the factors of the given expression.

(x - 2)(x - 3)

 

Example: Factor

Step 1: Write the ( ) and determine the signs of the factors.

(plus)(minus)

Step 2: Determine the factors (make a t-chart)

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Step 3: Write the factors of the given expression.

(x - 6)(x + 1)

 

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Watch the video:Factoring Tricks by Nelson Carter

 

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Watch this video: Factoring Trinomials where a = 1 by Nelson Carter

 

 

Check Yourself: Click on Activity

 

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Factoring Trinomials where a > 1 Using Trial &Error

You need the first terms to multiply and give the first term of the trinomial and you need the last terms to multiply and give the last term of the trinomial. The middle term is obtained by combining the inner and outer terms. If the last term is positive, the signs will be alike (++ or - -) and they will be the sign of the middle term. If the last term is negative, the signs will be different (+ - or - +).

If the factors you choose for the last term do not work, trade places with them; if that doesn't work, use different factors. If when you combine the inner and outer terms you get the right number but the wrong sign, you swap the signs.

Example 1: Factor

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Step 1: Write the ( ) and determine the signs of the factors.

(plus)(plus)

Step 2: Determine the factors (make a t-chart)

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Step 3: Write the factors of the given expression.

(2x +1)(x + 3)

 

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Watch the video: Factoring Trinomials When a Does Not Equal 1 by Nelson Carter

 

 

Check Yourself: Click on Activity

 

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Factoring Trinomials where a>1 Other Factoring Methods

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Watch the video: Another Method for Factoring Trinomials When a is Not Equal to 1 by Nelson Carter

 

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Watch this video: Factor Grouping V 1. by PatrickJMT

 

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Watch this video: Factor by Grouping by PatrickJMT 

 

 

Factoring Hints 

Ask yourself the following questions to help you factor:

1)   Is there a GCF?     GCF(               )

2)   Are there 2 terms? If so, is it the difference of squares?     (       ) (       )    Or is it the difference or sum of cubes?     (       ) (     )

3)   Are there 3 terms? if so, is  a=1     ?   Tricks       (       ) (       )  If so, is a>1      ?  Trial and Error (      ) (      ) or AC Method ( ) ( ) or grouping ( ) ( )

4)   Are there 4 terms?       Grouping           (         ) (       )

 

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Watch the video:Factoring Tricks by Nelson Carter

 

 

Polynomial Long Division

Example:

long.gif

1. Set up long division problem.

2. Estimate

3. Put the estimate on the top and multiply.

4. Subtract

5. Bring down the next term

long1.gif

Answer is

long 2.gif

 

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Watch the video: Long Division of Polynomials by Patrick JMT

 

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Watch this Video: Polynomial Division by Khan Academy

 

 

Simplifying Rational Expressions

rational 1.png

Example:

rational2.png

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Watch this video:Simplifying Rational Expressions by Kahn Academy

 

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Watch Video: Adding and Subtracting Rational Expressions by PatrickJMT

 

Radical Expressions

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Examples:

rad1.gif

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Watch the video: Simplifying Radical Expressions by PatrickJMT

 

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Watch the video: Algebra Help: Simplifying Radicals 1 by Pat McKeague

 

Multiplying Radical Expressions

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Examples

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Addition and Subtraction of Radical Expressions Like radicals

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 Examples:

radexp.gif

Exponential Notation for nth roots

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Watch the video: Algebra Help: Square Roots by Pat McKeague

 

 

Solving Quadratic Equations

Terminology

1. A Quadratic equations is an equation that contains a second-degree term and no term of a higher degree.

2. The standard form of a quadratic equation is ,quad1.gif where a, b & c are real numbers and .

 

Steps for Solving Quadratic Equations by Factoring

1. Write the equation in standard form:

2. Factor completely.

3. Apply the Zero Product Rule , by setting each factor containing a variable to zero.  If ab = 0, then a = 0 or b = 0.                                     

4. Solve the linear equations in step 3.

5. Check.

Note: Most quadratic equations have 2 solutions . The 2 solutions correspond to  the x-intercepts of the graph of a quadratic function.

 quad2.gif

quad3.gif

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Watch the Video: Solving Quadratic Equations by Factoring Basic Examples by PatrickJMT

 

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Watch the Video: Solving Quadratic Equations by Factoring another Example by PatrickJMT

 

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Watch the Video: Math Help Quadratics: Solve by Factoring by Pat McKeague

 

 

Check Yourself: Click on Activity

 Toggle open/close quiz question

 

Steps for solving Quadratic application problems:

1. Draw and label a picture if necessary.

2. Define all of the variables.

3. Determine if there is a special formula needed. Substitute the given information into the equation.

4. Write the equation in standard form.

5. Factor.

6. Set each factor equal to 0. And solve the linear equation. Eliminate any unreasonable answers. (Hint: We can't have -5 ft. of carpet.)

7. Check your answers.

 

 Area of a rectangle and Landscaping/border/frame problems . 

Example 1:A vacant rectangular lot is being turned into a community vegetable garden measuring 8 meters by 12 meters. A path of uniform width is to surround garden. If the area of the lot is 140 square meters, find the width of the path surrounding the garden.

Step 1:Draw and label a picture if necessary.

 E1.gif

Step 2:Define all of the variables.

e2.gif  

Step 3:Determine if there is a special formula needed. Substitute the given information to the equation.

e3.gif  

Step 4:Write the equation in standard form.

e4.gif  

Step 5:Factor.

 e5.gif

 

Step 6:Set each factor equal to 0. And solve the linear equation. Eliminate any unreasonable answers.

e6.gif  

Step 7:Check your answers.

ck.png  

 

Example 2:Each side of a square is lengthened by 7 inches. The area of this new larger square is 81 square inches. Find the length of a side of the original square.

Step 1:Draw and label a picture if necessary.

 101.png

Step 2:Define all of the variables.

 39.gif

Step 3:Determine if there is a special formula needed. Substitute the given information to the equation.

40.gif  

Step 4:Write the equation in standard form.

41.gif  

Step 5:Factor.

 100.png

Step 6:Set each factor equal to 0. And solve the linear equation. Eliminate any unreasonable answers.

43.gif  

Step 7:Check your answers.

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Pythagorean Theorem Problems: 

Example 3:A guy wire is attached to a tree to help it grow straight. The length of the wire is 2 feet greater than the distance from the base of the tree to the stake. The height of the wooden part of the tree is 1 foot greater than the distance from the base of the tree to the stake.

Step 1:Draw and label a picture if necessary.

 

 45.gif

Step 2:Define all of the variables.

 46.gif

Step 3:Determine if there is a special formula needed. Substitute the given information to the equation.

47.gif  

Step 4:Write the equation in standard form.

102.png  

Step 5:Factor.

 49.gif

Step 6:Set each factor equal to 0. And solve the linear equation. Eliminate any unreasonable answers.

103.png  

Step 7:Check your answers.

 51.gif

 

Example 5:A piece of wire measuring 20 feet is attached to a telephone pole as a guy wire. The distance along the ground from the bottom of the pole to the end of the wire is 4 feet greater than the height where the wire is attached to the pole. How far up the pole does the guy wire reach?

Step 1:Draw and label a picture if necessary.

 52.gif

Step 2:Define all of the variables.

53.gif  

Step 3:Determine if there is a special formula needed. Substitute the given information to the equation.

54.gif  

Step 4:Write the equation in standard form.

55.gif  

Step 5:Factor.

56.gif  

Step 6:Set each factor equal to 0. And solve the linear equation. Eliminate any unreasonable answers.

104.png  

Step 7:Check your answers.

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Motion Problems using the formula

Example 4:You throw a ball straight up from a rooftop 384 feet high with an initial speed of 3 feet per second. The function

59.gif

describes the height of the ball above the ground, s (t), in feet, t seconds after you threw it. The ball misses the rooftop on its way down and eventually strikes the ground. How long will it take for the ball to hit the ground?

Step 1:Draw and label a picture if necessary.

 60.gif

Step 2:Define all of the variables.

 t = time, s(t) = height

Step 3:Determine if there is a special formula needed. Substitute the given information to the equation.

The formula that was given. 

Step 4:Write the equation in standard form.

61.gif  

Step 5:Factor.

62.gif  

Step 6:Set each factor equal to 0. And solve the linear equation. Eliminate any unreasonable answers.

105.png  

 

Step 7:Check your answers.

63.gif  

 

Example 5:Use the same function

59.gif

to determine when the height of the ball is 336 feet. 

Step 1:Draw and label a picture if necessary.

 factor5.jpg

Step 2:Define all of the variables.

 t = time, s(t) = height

Step 3:Determine if there is a special formula needed. Substitute the given information to the equation.

 The formula that was given

Step 4:Write the equation in standard form.

106.png  

Step 5:Factor.

66.gif  

Step 6:Set each factor equal to 0. And solve the linear equation. Eliminate any unreasonable answers.

 107.png

 

Step 7:Check your answers.

67.gif  

  

Solve a Quadratic Equation by COMPLETING THE SQUARE .

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Watch The Video: Solving Quadratic Equations by Completing the Square by Patrick JMT

 

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Watch the Video: Quadratics: Completing the Square by Pat McKeague

 

Solving Quadratic Equations using the Quadratic Formula

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Example:

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Watch the Video: Solving Quadratic Equations using the Quadratic Formula-Example 3 by Patrick JMT

 

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Watch the Video: Math Help Quadratics: The Quadratic Formula by Pat McKeague

 

 

Graphing QuadraticFunctions

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Watch the Video: Graphing Quadratic Functions by Patrick JMT

 

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Click on this applet: Quadratic Function Calculator

 

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Click on the this applet: Quadratic Functions(General form)

 

Graphing Quadratic Inequalities

 208.png

 

 

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Problems with Geometric Figures

Find Perimeter: Rectangle, Square, Triangles, and Circumference of a circle

perimeter1.gif

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Watch this Video: Area and Perimeter by Kahn Academy

 

 

Finding Area: Rectangles and Squares, Triangles, Circles, Cylinders, and Rectangular Solids

area1.gif

 

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Watch this video: Circles: Radius, Diameter and Circumference by Kahn Academy

 

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Watch this video: Area of a circle by Kahn Academy

 

 

 Finding Volume: Rectangular Solids and Cylinders

volume1.gif

Using the Pythagorean Theorem to solve Right Triangle problems

Step 1: Draw and label the picture.

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Step 2: Substitute the given values into the formula.

Step 3: Solve for the unknown variable.

 

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Watch this video: Pythagorean Theorem by Kahn Academy

 

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Watch this video: Word Problems Using the Pythagorean Theorem - Example 1 by Patrick JMT

 

Problems with Geometric Concepts

POINT   A point identifies a position in space. It is the building block of geometric figures. A point is represented by a dot which is labeled with a capital letter.    

point.gif      

LINE   A line is made of an infinite number of points and extends indefinitely (infinitely) in both directions. It is labeled by two points on the line.

 

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PLANE   A plane is a flat surface that extends infinitely in both directions.

 

LINE SEGMENT   A segment is part of a line which consists of two points on a line and all the points between. The two points are called the ENDPOINTS. It is labeled by its endpoints.

line seg.gif

RAY   A ray is part of a line which has a fixed point at one end (endpoint) and continues infinitely in the other direction. It is labeled by the endpoint (first) and another point on the ray.
ray.gif

ANGLE   An angle is formed when two rays share a common endpoint.

 

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RIGHT ANGLE   A right angle measures exactly 90°.

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COMPLEMENTARY ANGLES   Two angles are complementary when the sum of the measures is 90°.   Each angle is called the complement of the other.

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SUPPLEMENTARY ANGLES   Two angles are supplementary when the sum of their measures is 180°. Each angle is called a supplement of the other.

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ADJACENT ANGLES   Two angles that have a common vertex and a common side are adjacent.

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CONGRUENT   means the same size and the same shape. The symbol for congruency is. Two angles that are congruent have the same measure.

 

VERTICAL ANGLES   Two intersecting lines form two pairs of vertical angles. These vertical angle pairs are congruent.

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STRAIGHT ANGLE   A straight angle measures exactly 180° and forms a straight line.

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PERPENDICULAR LINES   If two lines intersect to form right angles, then the two lines are perpendicular.   The sign for perpendicular is.

perp.gif

PARALLEL LINES   Two lines in the same plane that will never intersect are called parallel lines. The symbol for parallel is.

parallel.gif

 

TRANSVERSAL   A line that intersects two or more coplanar lines in different points.

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ALTERNATE INTERIOR ANGLES   

alt inter.gif

ALTERNATE EXTERIOR ANGLES

alt inter.gif

CORRESPONDING ANGLES

 

alt inter.gif

  

 

Congruent triangles   Triangles with the same size and the same shape are congruent triangles. If two triangles are congruent, then their corresponding parts (sides and angles) are congruent. 

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SSS (side-side-side)   If three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent.

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ASA (angle-side-angle)   If two angles and the side between them in one triangle are congruent to two corresponding angles and the included side of a second triangle, then the triangles are congruent.


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SAS (side-angle-side)   If two sides and the included angle in one triangle are congruent to the two corresponding sides and included angle of a second triangle, then the triangles are congruent.

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SIMILAR TRIANGLES Triangles with the same shape aresimilar.The sign for similarity is ~.

sim tri.gif

SIMILAR TRIANGLES

AAA (angle/angle/angle)   If 3 angles of one triangle are congruent to the corresponding angles of a second triangle, then the two triangles are similar.

 

SSS (Side/Side/Side) If the three sets of corresponding sides of two triangles are proportional, then the two triangles are similar.       

 

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Watch this Video: Similar Triangles by Kahn Academy

 

 

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Reasoning Skills

Inductive Reasoning:

 

Deductive Reasoning:

 

Read each conclusion and highlight the general statement. Specific information to general is inductive reasoning. General information to Specific is deductive reasoning. Decide whether inductive or deductive reasoning would be used for each conclusion.

Examples:

1. Rodney has $425 and must pay thefollowing bills: $200, $125, $85, and $50. He does not have enough to pay his bills.

If we have general information to specific information we use Deductive reasoning.

2. Tina tried to stop her car but her brakes did not work. She tried the brakes again and they still did not work. Her brakes were not going to work until repaired.

If we have specific information to general information we use Inductive reasoning

 

Find the next number in the pattern.

1, 3, 5, 7, 9 ,_____

Answer: Notice that we are adding 2 to each number to find the next number, so the next number will be 11.

 

Find the next number in the pattern:

25, 20, 15, 10, _____

Answer: We start at 25 and subtract 5 to find the next number, so the next number will be 5.

 

Find the next shape in the pattern.

indrea.gif

The answer is:

 

indans.gif

  

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Watch this Video: Deductive Reasoning:1 by Khan Academy

 

Other web sites and links:

Take the Practice test.

http://www.southtexascollege.edu/dev-math/THEA/Version 1/       

Other Web sites for reference & Tutorials

THEA home page

http://www.thea.nesinc.com/index.asp  

THEA Quick Reference Guide

http://www.thea.nesinc.com/PDFs/THEA_QuickRef.pdf  

THEA Practice Test (THEA Web site)

http://www.thea.nesinc.com/Practice.htm    

Practice Test version 2

http://www.southtexascollege.edu/dev-math/THEA/Version2/  

A & M tutorials

http://www.wtamu.edu/academic/anns/mps/math/mathlab/thea/math_help.htm#bat3  

videos topics from prealgebra to Calculus and beyond

 www.PatrickJMT.com

Mathtv.com Videos by topic

http://www.mathtv.com/videos_by_topic

 Free Math tutorials, problems and worksheets with applets:

http://www.analyzemath.com/

3-d Functions that spin

http://www.houseof3d.com/pete/applets/graph/index.html 

 Math.com

http://www.math.com/students/tools.html

Transformation of Functions:

http://www.calculusapplets.com/transform.html

Oscilloscope & Function Generator - Analysis of Signals in Time Domain

http://www.ece.ncsu.edu/virtuallab/JAVA/applets/osc.html