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# Math Review

Before reading this section, make sure you have read the appropriate description of the mathematics section test (computerized or paper) to understand what is expected of you in the mathematics section of the Praxis I.

There are many hints and strategies presented below to help get you started on studying and to give you an idea of what to expect on the test.  You should consult a test prep book to get a comprehensive and detailed description of topics and problems with thorough explanations.

### What to Expect

For the computer test, you will have 75 minutes for 46 multiple-choice questions focusing on key mathematics concepts, problem solving, and quantitative reasoning. For the paper test, you will have 60 minutes for 40 multiple-choice questions focusing on key mathematics concepts, problem solving, and quantitative reasoning.

Conceptual Knowledge and Procedural Knowledge
Conceptual Knowledge

• Number Sense and Operations Sense with whole numbers, fractions, and decimals
• Order and Equivalence among whole numbers, fractions, and decimals
• Numeration and Place Value
• Whole Number Properties
• Operation Properties of addition, subtraction, multiplication, and division
• Order of Operations

Procedural Knowledge

• Computation, including identifying needed information and/or operations to solve a problem
• Estimation
• Ratio, Proportion, and Percent
• Probability
• Equations and Inequalities
• Algorithmic thinking, including following algorithms, and analyzing and finding patterns in procedures

Representations of Quantitative Information

• Interpretation: interpret various types of graphs, plots, charts, flow charts, and diagrams; recognize relationships in data, find and interpret basic statistical measures, such as mean, median, mode, range
• Detect and interpret trends
• Make inferences and draw conclusions from data
• Identify patterns in data, such as variation
• Make connections between various representations of data, such as tables; graphs; formulas and rules; symbols, numbers, and words

Measurement and Informal Geometry, Formal Mathematical Reasoning
Measurement and Informal Geometry

• Systems of Measurement, including conversions and use of appropriate units in U.S. customary and metric systems
• Measurement, including solving problems by using geometric concepts, formulas, estimation, indirect measurement, visual comparison, scaling, proportional reasoning, and nonstandard units
• Geometric Properties

Formal Mathematical Reasoning

• Logical Connectives and Quantifiers
• Validity of Arguments, based on deductive reasoning
• Generalization

As you read this list, if you could not identify or define a term or two, write those down. Then make sure to read about these topics before your test.

### Resources

Through the ETS website, you can purchase an actual full-length PPST Mathematics test (Practice Test: PPST Mathematics) that has been used in previous test administrations.  ETS recommends you use this in conjunction with their Study Guide.

*Praxis Made Easy, by Lynn Gardner.  Available from http://www.mgmtutoring.com/  You can purchase a hard copy of the book or get same day access by purchasing the electronic copy.  This is a great resource!  There are no long explanations and complicated theory, just short rules and hints for each topic.  And there are tons of problems, each with detailed answers and explanations.  Check out the test-taking tips on the website, too!

Any of the general Praxis test prep books will help you prepare for the mathematics section.  *Barron's How to Prepare for the PPST and Computerized PPST, by Robert Postman, has a good mathematics section.  You may want to take the Mathematics Review Quiz first.  The answers to the quiz identify the specific mathematics topic involved in each question, so you can go directly to the topic that you need to review.

*available for check-out in 221 White Hall

### Study Strategies

There are many concepts, formulas, and rules you will need to know.  There are two preparation strategies you should consider.

1. Go through a review book section by section.
• If you do not feel comfortable at all with mathematics and you feel like you need as much help as you can get then you should develop a comprehensive and structured review schedule for yourself.
• Find a prep book that you like and study a specific section or set of topics everyday.
• Test yourself on those concepts and move on after you feel confident with these ideas.
1. Review specific concepts/topics.
• If you feel more comfortable with mathematics in general or with specific topics in mathematics, then take a practice test first.
• Identify the areas you need to work on.  For example, take the Mathematics Review Quiz in the prep book, Barron's How to Prepare for the PPST and Computerized PPST, by Robert Postman. The answers are organized by review section, so you can go directly to the topic that you need to review.
• Study the particular topics you need help on by reviewing the specific sections in your test prep book, a math book, or an on-line resource.
• There are some great on-line math resources that are arranged by topic.  Try purplemath.com or math.com.

When taking practice tests, simulate the test-taking environment as much as possible so that you become comfortable with the constraints of the actual test-taking situation.

• Work on problems in a timed environment.
• Use scratch paper, not a calculator.

When checking answers to practice tests, read through the explanations for all the problems, even the ones you answered correctly, and make note of the topics you need to read up on.

Calculators are prohibited!  So practice problems in a timed environment.  You will want to be able to complete your calculations quickly so that you can focus on your problem-solving strategies.

Yes, calculators are prohibited!  So practice your estimation skills.  It is often not necessary to solve the entire problem to find the appropriate response.  If you look at the answers first, you may not have to complete the problem, or you may be able to work with round numbers to make the computations easier.

Remember, calculators are prohibited!  So know your times tables!  This will save you from wasting valuable time.

Did I mention calculators are prohibited?  So practice adding, subtracting, multiplying and dividing by hand.  Practice with whole numbers, decimals, and fractions!

Practice times tables.

• You can practice filling in a chart.
 X 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
• Buy a set of flashcards for young learners.

Review divisibility rules.

Review scientific notation.

• A number greater than or equal to 1 but less than 10 multiplied by a power of 10 is written in scientific notation.

• Example: 2,350,000 written in scientific notation is 2.35 x 106.

• To write out a number given in scientific notation, use the following rules:

• If the power of 10 is a positive number, add zeroes to the right of the base number and move the decimal place to the right the number of the power.

• Example: To expand 1.68 x 104, write 1.680000 and move the decimal place to the right four times to get 16800.

• If the power of 10 is a negative number, add zeroes to the left of the base number and move the decimal place to the left the number of the power.

• Example: To expand 4.12 x 10-5, write 000004.12 and move the decimal place to the left five times to get 0.0000412.

Practice converting between decimals, percents, and fractions.

• To convert a percent to a decimal, move the decimal point to the left two spaces.

• Example: 85.5% is 0.855.

• To convert a decimal to a percent, move the decimal point to the right two spaces.

• Examples: 0.60 is 60% and 0.04 is 4%.

• To convert a decimal to a fraction, place the number to the right of the decimal point in the numerator (on top) and place a 1 followed by as many zeroes as there are digits to the right of the decimal place in the denominator (on the bottom).

• Example: 0.416 = .

• To convert a fraction to a decimal, divide the denominator into the numerator.

• Example: .

Review metric measurement units.

Review basic geometry principles.

• Know properties of basic shapes, such as quadrilaterals (including rectangles, squares, and trapezoids), circles, triangles.
• Learn the formulas for area, perimeter, and volume for basic two-dimensional and three-dimensional shapes.
• Review the Pythagorean theorem.  Learn about the Pythagorean Theorem at math.com

Practice data analysis.

• Look at tables, charts, graphs, etc. in newspapers, magazines, and on-line resources of your interest.  What does the table, chart, or graph tell you?
• Practice interpreting data by writing a sentence or two explaining the results in the table, chart, or graph.

Review basic probability rules.

• Probability is represented by a number between 0 and 1.
• The probability of a specific outcome is represented by

.

• Example: There are 3 red marbles and 2 blue marbles in a bag.  The probability that I pick a blue marble is 2/5, or 0.4.

### Problem Solving Strategies and Hints

First, identify what area of mathematics is involved.
Examples

• Do they ask which numbers are larger, smaller, or equivalent?  You will need to use your number sense.
• Do they use words like area, perimeter, angle, side, triangle, rectangle, etc.  You will need geometry.
• Do they ask questions about data shown in a chart, graph, or pictograph?  You will need to use your data analysis skills.
• Do they use words like ratio or ask questions like, “If it takes 4 hours to sew 3 dresses, how long will it take to sew 7 dresses?”  You will need to use your ratio and proportion skills.

Identify knowns/unknowns.

Example:
The sum of 6 and 3 times a number is 30.  What is the number?

Known: the sum of two numbers is 30; one of the numbers is 6

Unknown: the other number

Example:
A square and a rectangle have the same area.  The rectangle has length 9 and width 4.  What is the perimeter of the square?

Known: length and width of the rectangle, area of the rectangle and the square (because area of a rectangle is length x width)

Unknown: the length of the side of the square; the perimeter of the square

Translate word problems into mathematical expressions.
Example: The sum of 6 and 3 times a number is 30.  What is the number?

Start with “3 times a number”.  Use n to represent the number that you do not know yet.  Then “3 times a number” is 3 x n or 3n.

Next, look at “the sum of 6 and 3n”.  You can rewrite that as 6 + 3n.

Then, 6 + 3n “is 30” may be written as 6 + 3n = 30.

Then solve for n (see below).

Maintain equalities and inequalities.

• Whatever you do on one side, you must do on the other.

Example:
6 + 3n = 30
6 – 6 + 3n = 30 – 6
3n = 24
3n  3 = 24  3
n = 8

• If you multiple or divide both sides of an inequality by a negative number, you must switch the direction of the inequality

Example:
-7 < -4
(-1) (-7) > (-1)(-4)
7 > 4

Example:
-2x + 6 > 14
-2x + 6 - 6 > 14 – 6
-2x > 8
-2x  (-2) < 8  (-2)
x < -4

Draw pictures, graphs, visual representations.
Examples

• Use a number line to determine the order of numbers.
• Draw pictures of geometric shapes.

There are some questions where you need to use the answer choices to find the correct response.

• Don’t try to solve the problem independently of the answer choices.
• Guess and check each answer choice until you find the correct answer.

Example:
If 2x + 4y = 8, which of the following answer choices are possible values for x and y?
(A) x = 2 and y = 6
(B) x = 10 and y = -3
(C) x = 10 and y = 3
(D) x = 4 and y = 2
(E) x = -2 and y = 6

Try answer choice (A): 2(2) + 4(6) = 4 + 24 = 28 ¹ 8
Try answer choice (B): 2(10) + 4(-3) = 20 + (-12) = 8.
The correct answer is (B).  You do not need to check the remaining answer choices.

### Test Taking Strategies

Skip questions that may take longer.  You may return to them later.  You may want to make a reasonable guess (you have a 20% chance!) so that you have an answer recorded in case you run out of time.

Read the possible answers along with the question.  You may be able to eliminate one or more possible responses right away.

Rephrase the question into a real life situation.

Try testing patterns or relationships with smaller, easy-to-work-with numbers before returning to the given numbers or symbols.

Try to keep your written calculations organized (near the problem in the paper-based test booklet or on the computerized test scrap paper) so that you can easily find them again if you go back to a question.

Good luck!

Do not worry about your difficulties in Mathematics, I can assure you mine are still greater  – Albert Einstein