Combo Praxis 5001 Math

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divisor

the number you divide BY -the 2nd number in a division problem when it is across for example, 3 is the divisor in 2 ÷ 3 -It is the number outside the bracket in long division

Words that Signal Equals

will be is

Regular Polygon

An equiangular polygon is one where all angles of the polygon are the same measure. An equilateral polygon is one where all sides of the polygon are the same measure. If a polygon is both equiangular and equilateral, it is called a regular polygon.

Polyhedron (names)

Convex polyhedron are named according to the number of faces:<div>  4 = tetrahedron 5 = pentahedron 6 = hexahedron 7 = heptahedron 8 = octahedron 9 = nonahedron 10 = decahedron </div>

Integers

Integers = the counting numbers, their negatives, and zero (..., -3, -2, -1, 0, 1, 2, 3 ...). No fractions or decimals

Tip for solving comparison problems in Praxis

Make sure that all numbers are in the same format (all fractions or all decimals or all percentages).

Base 10

The Base 10 number system is the system we use - each place value is ten times the value of the place to the right of it. There are 10 digits in the Base 10 number system:  0-9. These 10 digits are all that is needed to make any number. Another name for the Base 10 number system is the decimal number system.

Binary Operations (name the four basic operations and any special relationships between them)

The four basic binary operations are: 1. Addition 2. Subtraction (the inverse of addition) 3. Multiplication (repeated addition) 4. Division (repeated subtraction and the inverse of multiplication)

When is the "mean" the best measure of central tendency?

When the data are consistently distributed and there are no outliers.

Properties of Number System Operations

"Properties of Number System Operations <img src=""pastehp_zce.png"" />"

Rates

"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 (second measurement) Examples: 13 miles per gallon $4.59 per pound 12 inches per foot

Cube (Definition, volume, and surface area)

"A cube is a three-dimensional solid where all angles are right angles and all faces are squares. A cube is also informally called a square box. <img src=""pastexwl_ol.png"" /><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">V = a3</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">SA = 6a2</div>"

Comparing Decimal Numbers

"Line up the decimal numbers according to place value (as though you were going to add them). Starting at the left-most place value, compare the numbers in each place value to find the largest, next largest, etc. <img src=""paste6lle1v.png"" />"

Official website where you can find more information about the Praxis tests

"<a href=""www.ets.org/praxis/"">www.ets.org/praxis/</a>"

Solving Percentages (using the Percent Proportion)

"To solve percentages using the percent proportion, use the means-extreme property of proportions (cross multiply). The percent proportion can be written as:    <img src=""pasteh2sohc.png"" /> "

The equation y = ax² + bx + c makes what shape when it is graphed?

"a parabola: <img src=""pastelupk8k.png"" />"

Inequality

An inequality is similar to an equation, but the two sides are NOT equal.

Denominate Numbers

A *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.

How can you tell when an expression is simplified?

An expression is simplified when • No parentheses appear • No powers are raised to powers • No more than one like term • No negative exponents appear

Whole Numbers

Whole numbers = the counting numbers and zero (0, 1, 2, 3, 4, ...). Positive numbers that have no fractions or decimals, including zero.

Average

arithmetic mean To average a group of numbers, add all the numbers together and divide by how many numbers there are. For example, the average of 5, 7, 12, and 8 is (5 + 7 + 12 + 8) / 4 = 8.

ordering fractions

convert fractions into decimals first, then put in order

When translating word problems, the word "is" means __________.

equals (=)

In the equation y = ax² + bx + c, a negative "a" makes the parabola _________.

face downward (frown)

The equation y = ax² + bx + c, a positive "a" makes the parabola _________.

face upward (smiley-face)

What are the parts of a multiplication problem?

factors; partial product; product

All whole numbers except for 1 and 0 are either _______ or ___________.

prime, composite

When translating word problems, the word "what" means __________.

the unknown - use a variable such as x, y, or n

Properties of Absolute Value

"Properties of Absolute Value:  For all real numbers a and b:<div>   <img src=""pastezyyzjg.png"" /> <img src=""pasterb6ojm.png"" /> <img src=""pastetefynd.png"" /> <img src=""paste8lrdxu.png"" /> <img src=""paste4ubvo0.png"" /> <img src=""pastekbhgom.png"" /></div>"

Box and Whiskers Plot

"A box and whiskers plot is a visual way to show the five statistical number summaries: Minimum, Q1, Q2, Q3, and Maximum <img src=""pastev61z4q.png"" />"

Exterior Angle

"An exterior angle is an angle on the outside of a polygon that is formed by extending the side of the polygon. In the diagram below, ∠d is an exterior angle: <img src=""pastese8bs7.png"" />"

In a right triangle with legs of 6 and 8 inches, the hypotenuse is _______.

10 inches<div>  This is the 3-4-5 triangle with a factor of ×2 </div>

Bob caught and tagged 54 Nene birds, then released them. A week later, he observes 38 Nene birds and notices that 15 of them are tagged. What is the total population of Nene birds?

137 birds (see picture for more detailed steps) x/54 = 38/15 15x =(38)*(54) x =136.8 Round up to 137 because cannot have 0.8 of a bird :)

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

There is a bag of candy with 8 chocolates, 11 peppermints, 13 gumdrops and 9 pieces of licorice inside. Amy takes one out randomly and gets a gumdrop. What is the probability that the next candy she takes out will be either a gumdrop or a peppermint?

23/40

On last night's lottery drawing, 8 of the last 13 numbers were even. What is the probability that the next number drawn will be even?

61% (because 8/13 =0.61...)

real-world problems adding fractions Chad, a pet store employee, wants to fit two fish tanks on one table. One fish tank is 2/5 of a foot wide and the other fish tank is 3/10 of a foot wide. When placed next to each other, what is the total width of the two fish tanks? At the neighborhood block party, Mario served 7/10 of a gallon of hot chocolate and 2/5 of a gallon of apple cider. How much more hot chocolate than apple cider did Mario serve?

7/10 ft

Which equation shows the commutative property of addition? 2+3 = 4+1 8+1 = 1+8 7+2 = 9

8+1 = 1+ 8 move numbers around and they mean the same thing *THINK COMMUTE = move The property that says that two or more numbers can be added or multiplied in any order without changing the result.

Even Numbers

Even numbers are integers that are evenly divisible by two (2). Zero is an even number.

Counterexample

A counterexample is one example used to prove a hypothesis false.

Whole Number Place Value

A number in standard form is marked into groups of three digits using commas. Each of these groups is called a period. Within each group, the place values are always the 100's place, the 10's place, and the 1's place (from left to right). Understanding place value is key to understanding our number system. Decimal numbers simply extend the place values to the right and use "ths" to identify the places (e.g. 100 millionths place).

Acceleration

Acceleration is the measure of how speed (velocity) changes over time. It can be expressed as the change in velocity divided by the change in time: α = ∆v/∆t

Equiangular Polygon

An equiangular polygon is one where all angles of the polygon are the same measure. If a polygon is both equiangular and equilateral, it is called a regular polygon.

Solving Algebraic Word Problems (involving consecutive numbers)

Numbers following each other in counting order are called consecutive. They can be denoted as x, x + 1, x + 2, x + 3, etc. Consecutive odd numbers are x, x + 2, x + 4, etc. (assuming x is odd), and consecutive even numbers are also x, x + 2, x + 4, etc. (assuming x is even). Example: Find three consecutive, even numbers whose sum is 90. x + (x + 2) + (x + 4) = 90 3x + 6 = 90 3x = 84 x = 28 so the numbers are 28, 30, 32

Slope of a Line

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

Solving Algebraic Word Problems (involving distance-speed-time)

The distance formula is D = st, where D is the distance traveled, s is the rate of speed, and t is the time. To calculate the rate of speed: s = D/t To calculate the time: t = D/s

supplementary

Two angles whose sum is 180 degrees

Rectangle

any quadrilateral with four right angles A = lw P = 2l + 2w

Pie Chart

"A pie chart (also known as a circle graph) is a circular graph where sections of the circle represent parts of the whole. A pie chart ALWAYS gives parts of a whole. The pie sections are usually labeled with percentages. <img src=""pastempdkul.png"" />"

Proper Fraction

*proper fraction* has a numerator smaller than the denominator and indicates a fraction less than one whole *Remember on test, if answers are in fraction form, correct answer will always be reduced*

Perpendicular Lines

"Perpendicular lines are lines that meet or intersect at 90° angles. The slope of one is the negative reciprocal of the other.     <img src=""pasteiktpd5.png"" />"

What measure of central tendency is used to track trends or popularity?

mode

When translating word problems, the word "of" means __________.

multiply

Complementary Angles

"Complementary angles are two angles whose measure adds to 90°.    <img src=""pastewt8zfx.png"" />"

Mass (Define & List Units of Measure)

"Mass (also known as weight on Earth) is the amount of matter in an object. Technically, weight is a measure of the force of gravity against an object, but on Earth, mass and weight can be thought of as the same thing. The metric measure of mass is the gram. In the customary or U. S. English system, refer to the table below:<div> </div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">16 ounces - 1 pound</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">2,000 pounds = 1 Ton</div>"

Plane Transformations (four types)

"Plane transformations: 1. Translation (move) 2. Rotation (turn) 3. Dilation (scale)  (enlarge/reduce) 4. Reflection (flip)     <img src=""pasteg5pge5.png"" />"

Primary Data

"Primary data is data obtained from an observation or experiment. It is raw data that has not been manipulated in any way. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Roman Numerals

"Roman numerals and the Roman number system are similar to the Arabic number system used in the United States. The Roman number system is based on 10 so it is decimal, but it does not have place value. Letters are used to represent various numbers (Roman number names). The rule with Roman numbers is to write the numbers in descending order (from greatest to smallest). The exception to this rule is if a smaller number comes before a larger number, we subtract that smaller number from the larger number.   I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000 "

Rules for Square Roots

"Rules for Square Roots: <img src=""paste6xwrtd.png"" /> <img src=""paste4a38gn.png"" /> <img src=""pastecmnozo.png"" /> <img src=""pastel_tjlb.png"" /> <img src=""pastegjgr9o.png"" /> "

Converting Measures of Time

"To convert measures of time, use Unit Analysis (multiply by the conversion factor in such a way that all units cancel out except the unit you want). For example, to change 16 hours to seconds: <img src=""pastegekrwh.png"" />"

Chord of a Circle

"A chord of a circle is a line segment whose endpoints are on the circle. The diameter is the largest possible chord in a circle. <img src=""pasteqjplvi.png"" />"

Combinations

"A combination is a way of selecting several things out of a larger group, where order does not matter. <img src=""paste16ysps.png"" /> n is the number of items selected k is the number of items in the larger group This term will probably not appear on the Praxis I test but may be on the Praxis II test."

Common Fraction

"A common fraction is also known as a simple fraction. It represents parts of a whole and is written as a division problem: <img src=""pasteiztjms.png"" />  or 1/4 The 1 in the example above is the numerator. The 4 in the example above is the denominator. <img src=""pastecx2anr.png"" /> "

Compass (geometry)

"A compass is a tool used in geometry to draw arcs and circles. These arcs may be used to bisect lines and angles. <img src=""pasteopobbi.png"" />"

Literal Equation

"A literal equation is an equation made up of only known, measurable quantities. A literal equation is the same as a formula. With a literal equation, you are not solving for an unknown quantity that varies. Instead, you are manipulating the letters/variables in the equation to a different form to substitute values in it. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics. "

Plane (geometry)

"A plane is a two-dimensional surface. It is like a sheet of paper that has no thickness, yet it extends in all directions for width and height. <img src=""pasteggeygg.png"" />"

Decimal System Place Value

"A number in standard form is marked into groups of three digits. Each of these groups is called a period. Within each group, the place values are always the 100's place, the 10's place, and the 1's place. Decimal numbers simply extend the place values to the right and use "ths" to identify the places (e.g. 100 millionths place). There is not a oneths place. <img src=""paste6wllx9.png"" />"

Inverse Property

"The inverse property defines what happens when you add or multiply inverse numbers. The additive inverse is the negative or opposite of a number. When you add a number and its opposite, the result is zero. The multiplicative inverse is the reciprocal of a number. When you multiply a number and its opposite, the result is one.<div> <img src=""pastew_tlj3.png"" /> <img src=""pasteq2uqjs.png"" /> </div>"

Standard Deviation

"The term standard deviation is used in statistics to describe how much the results deviate (differ) from the mean (average). A small standard deviation indicates that most of the data values are close to the mean. A large standard deviation indicates that the data have a large range of values. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics. "

Corresponding Angles

"When a transversal line crosses two other lines, it forms eight angles that are often used in geometrical problems. Corresponding angles are angles that are in the same position on each of the lines. In the figure below, Angle 2 corresponds to Angle 6. <img src=""pasteucky2t.png"" />"

Vertical Angles

"When two lines intersect, they form four angles. The two angles opposite each other are called vertical angles. Vertical angles are always the same measure. In the drawing below, Angle 1 and Angle 3 are vertical angles. Angle 2 and Angle 4 are vertical angles. <img src=""paste1g3y38.png"" />"

4 Ways to Indicate Multiplication

1. Using a small "×", such as 3 × 5. 2. Using a small, raised dot, such as 3 • 5 3. Using parenthesis, such as (3)(5) or 3(5) or (3)5 4. Using no symbol, such as 3y (which means 3 times y).

What number does MMCCXCIV represent in Roman numerals? explain

2,294 In Roman numerals, when a smaller value appears to the left of a higher value, flip those two numbers around and subtract. The number can only be one place value away though to do the flip thing. (ex: cannot do 2,300 - 6 as MMIIVCCC) MM = 2,000 CC = 200 XC = (100-10) =90 IV = (5 - 1) = 4 so, 2000+200+90+4 = 2,294

Absolute Value

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

Algebra

Algebra is the study of numbers, number patterns, and relationships among numbers. Algebra generalizes these numbers, number patterns, and relationships. It is often said that algebraic thinking is the study of number patterns.

Positive Integers

Positive integers are those integers greater than zero. Positive integers appear to the right of zero on a number line. The positive sign is understood if it isn't written. Negative integers are the opposite of the positive integers. The number zero is neither positive nor negative.

Regrouping

Regrouping is the modern term that should be used instead of "carrying" and "borrowing." Children are now taught to add and subtract by keeping place value in mind. When we need to carry or borrow, we now teach students to re-group units into 10s or 10s into units.

Zero (as an Integer)

Zero is an integer and divides a number line into  -|+ The number zero has some interesting properties: • 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

Function

A function is a relation between a set of inputs and a set of potential outputs with the property that each input is related to exactly one output.

dividend

A number that is divided INTO by another number. -the first number in a division problem when it is across for example, 2 is the dividend in 2 ÷ 3 -It is the number inside the bracket in long division

Converting Percentages 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* *On test, all fractions must be in most simplified version*

First 10 Cubed Numbers

First 10 Cubed Numbers: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

distributive property of multiplication

Multiplication outside parentheses distributing over either addition or subtraction inside parentheses does not affect the answer

Tip for comparing positive and negative numbers in Praxis

Remember that negative numbers are always smaller than positive numbers. Also remember that the further right on the number line, the larger the number.

Rounding Numbers

Rounding a number requires that you understand place value. Rounding a number is a type of estimation. Rounding is also called "rounding off." To round a number, 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.

Representing Addition or Subtraction of Rational Numbers using concrete models

See pictures for models These models (pictures) show 4/8 + 3/8 and 7/8 - 2/8

Solving Algebraic Word Problems (involving unit conversion)

Set up unit conversion problems as a proportion. Then cross multiply (Means- Extremes Property) and simplify. Example: Convert 16 yards to feet. Because there are 3 feet in 1 yard: yard 1 16 feet 3 x 1/3 = 16/x Cross multiply: 1x= 3(16) x = 48 feet (Unit analysis may also be used - see the flashcard on unit analysis.)

Additive Identity

The additive identity is zero. Any number added to zero results in a sum of that number. The additive identity does not change the number when it is added to it.

Median

The median is used in statistics to describe one of three measures of central tendency. To find the median, list all the data values in numerical order. The median is the middle data value. If there is an even number of values, find the mean (or average) of the middle two values. Medians are used instead of means when there are outliers in the list of values that would distort the average. A mnemonic used to remember this term is to think of the median in the center of a freeway. A statistical median is in the center of the data values.

Mode

The mode is used in statistics to describe one of three measures of central tendency. To find the mode, list all the data values in numerical order. The data value that appears the most in the list is the mode. A data set of values can have no modes, one mode, or more than one mode.  Mode is used to determine popularity or commonly recurring events. A mnemonic used to remember this term is to think of the word "most," which is similar to "mode." Mode is/are the values that appear the most.

Addition of Whole Numbers

The algorithm for whole number addition is:<div>    1. Line up the numbers vertically so place values are in the same column 2. Add beginning in the ones place 3. If the sum is greater than 9, write the tens place digit above the next column to the left 4. Put commas in the answer to separate the digits into periods </div>

Comparison Symbols

The following five symbols are called comparison symbols: <    less than >    greater than ≤    less than or equal to ≥    greater than or equal to =    equal to Each of these symbols can also be negated by putting a slash mark through them, such as "not equal to":   ≠

identity property of addition/multiplication

The property that states that the sum of 0 and any number is that number AND The property that states that the product of 1 and any number is that number ex: 3+0 = 3 ex2: 3*1 = 3

How do you convert a fraction such as 2/3 into a ratio?

To convert a fraction into a ratio, keep the numerator; the new denominator becomes the difference of the denominator and numerator. The denominator of a fraction is the WHOLE amount; the denominator of a ratio is the REMAINING part. Example: The fraction 2/3 is a ratio of 2 parts to 1 remaining part or 2:1

write numbers using base-10 numerals

Write numbers by showing how many of each place value it has: ex: 1,249 (1*1000) + (2*100) + (4*10) + (9*1) (***NOT 1,000 + 200+ 40 + 9, that is expanded form***)

How would you write 399 as a Roman numeral?

Write the number in expanded form and convert each term to Roman numerals: 300 + 90 + 9 = 399 CCC XC IX CCCXCIX represents 399.

What are the parts of an addition problem?

addends; sum

rational numbers

any number that can be shown as a fraction (decimals, fractions, whole numbers) the proper way to show rational numbers are as fractions (so integers should be written as being over 1)

In Roman numerals, when a smaller value appears to the left of a higher value, _______ _____ _________ _______ and then ________. Ex: XLI = ?

flip the numbers around, subtract M= 1000 D=500 C=100 L=50 X=10 V = 5 I = 1 XLI = XL + 1 = (50-10) + 1 = 41

Harmonic Series

An *harmonic series* is the sum of progressive unit fractions: 1/1 + 1/2 + 1/3 +1/4 + 1/5......

Improper Fraction & Mixed Numbers (Definition)

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 An improper fraction can be changed into a whole number or a mixed number by dividing the denominator into the numerator. A mixed number is the sum of a non-zero integer and a proper fraction. *Remember on test, if answers are in fraction form, correct answer will always be reduced* *Improper fractions: denominator might not be most simplified one - make into mixed number THEN SIMPLIFY FURTHER*

Algorithm

An algorithm is a step-by-step process for solving a problem. An example of an addition algorithm is: 1. Line up the numbers 2. Add each column starting on the right 3. Carry any tens-place digits to the next column 4. Place commas between periods in the answer An algorithm is often written as a flowchart showing steps, branches, and decisions.

Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides. A = bh P = 2(b1 + b2)

Interpreting Remainder Problems

On the Praxis exam, it is important to interpret answers that have remainders correctly. Do you round up? Do you round down? Do you use the remainder as part of the answer (or as the entire answer)? Do you ignore remainders?    1. How many boxes can be filled? (use only the quotient; ignore the remainder) 2. How many cans are needed to paint the wall? (round the quotient to the next greater whole #) 3. How many in the last box that isn't completely full? (use only the remainder)

Converting Decimals to Percentages & Fractions

*Change a decimal to a percent* by moving the decimal point two places to the right and appending a percent sign 0.14 = 14% *Change a decimal to a fraction* by writing the decimal over a power of 10 representing the right-most place value in the decimal, and then simplifying: 0.146 = 146/1000 = 73/500

The equation y = ax² + bx + c crosses the y axis at y = _____?

y = c

Symbol for "absolute value"?

| |

Solving Algebraic Word Problems (involving rectangular area & perimeter)

Problems involving area & perimeter require use of formulas: P = 2L + 2W, where P = Perimeter, L = length, W = width. A = LW, where A = area Example: The length of a rectangle is twice its width. If the perimeter of the rectangle is 60 in, find its area. L = 2W so P = 2(2W) + 2W 60 = 4W + 2W 60 = 6W W = 10 and since L = 2W L = 20 Knowing this, A = LW = 20(10) = 200 in²

solve multi-step mathematical and real-world problems using addition of rational numbers ADDING FRACTIONS

Step 1: Find common denominator -multiply each fraction (both top and bottom) by the denominator of the other fraction - we can do this because any fraction with the same number on the top and bottom is equal to 1, so we are really just multiplying by 1 (look at picture to understand better) Step 2: Add numerators (tops), put the answer over the denominator Step 3: Simplify the fraction (if needed) ex: ¼ + ¼= 2/4 ex2: 1/3 + 1/6 = (6/6) (1/3) + (3/3)(1/6) = (6/18) + (3/18) = 9/18 =1/2

Tip for Praxis problems containing a chart or graph

Study the chart or graph carefully before reading the question. Note whether the vertical axis is broken (see Flashcard #153).

Solving Algebraic Word Problems (involving discounts)

Problems involving discounts require use of the discount formula: S = 1r - rd, where S is the sale price, r is the retail price, and d is the rate of discount. Example: A coat is on sale for $125. The coat was discounted 20%. What was the original retail price? 125 = 1r - (0.2)r 125 = (0.8)r then multiply both sides by 10 to clear decimal 1250 = 8r r = 1250/8 r = $156.25

Solving Algebraic Word Problems (involving investments)

Problems involving investments require use of the interest formula: i=Prt, where i = interest earned, P = principal (original amount), r = annual rate of interest, and t = time in years. Example: An investment is made at 5% simple interest for 12 years. It earned $420 interest. How much was originally invested? 420 = P(.05)(12) 420 = 0.6P 4200 = 6P P = 4200/6 P = 700

Solving Algebraic Word Problems (involving mixtures)

Problems involving mixtures use M₁V₁ + M₂V₂ = M₃V₃, where M is the percentage of each mixture, and V is the volume or amount of each mixture. Example: How much of a 16% solution is needed to combine with 34 ml of a 12% solution to make 50 ml of a 15% solution? 0.16x + 0.12(34) = 0.15(50) 16x + 12(34) = 15(50) 16x + 408 = 750 16x = 342 *x = 21.375 ml*

Real Numbers

Real Numbers are numbers that can be located on the number line, numbers that can exist in the real world - could be whole numbers, fractions, percents, etc. The opposite of real numbers are imaginary numbers. The symbol used for the set of real numbers is R. Sets in mathematics include the set of integers (Z), rational numbers (Q), primes (P), real numbers (R), natural numbers (N), whole numbers (W), etc. *Remember on test, if answers are in fraction form, correct answer will always be reduced*

Solving Algebraic Word Problems (involving uniform motion)

Solving Algebraic Word Problems (involving uniform motion): In problems where you are given data about an object traveling with and then against a moving object, use a table (see picture), and then set the SAME quantities equal to each other & solve. The distance formula is D = st (Distance = speed * time) Example: A boat can travel 12 mi/hr in still water. If the boat can travel 5 mi downstream in the same time it takes to travel 3 mi upstream, what is the rate of the river's current? (see picture for table and solution)

Rectangular Arrays

Visual model that shows students how to use dots to count Doesn't matter how dots are aligned, as long as it's clear that each of the sets of dots are their own individual digit in a problem ex: 2 + 4 can be shown as °°+°°°° ex2: 2 * 4 can be shown as ₀₀₀₀ ⁰⁰⁰⁰

When is the "median" the best measure of central tendency?

When the data contains outliers

Solving Algebraic Word Problems (involving commissions)

When we think of sales commissions, we often think of car sales. Thus, it is appropriate that the formula for commission is C = ar, where C = commission earned, a = amount of sale, and r = commission rate. Example: Juana sells cars on a 3% commission rate. She just sold a car for $23,500. What was her commission? C = ar C = 23500(.03) C = $705

*Test Tip* Fraction Number Lines (Ruler)

When working with fractional number line, like a ruler, where you have to measure between point A and point B, don't do the complicated calculations, just count the number of 1/8ths or 1/16th between the points. Count the number of increments in each inch, that number is the denominator. Then count how many of those increments is between points A and B. That number is your numerator. Simplify if you can. Ex: 11/16th Ex2: 2/8ths = 1/4

Which property of addition is shown? 5 + (7 + 1) = (5 + 7) + 1

associative Changing the grouping of numbers will NOT change the value. For example: (7 + 4) + 8 = 7 + (4 + 8) also works with multiplication ex2: as multiplication: (4 × 2) × 1 = 4 × (2 × 1) Grouping numbers together *THINK "associates" means group members/friends

When translating word problems, the word "percent" means _________.

divided by 100

Algebraic Symbol Manipulation

"Algebraic Symbol Manipulation means to solve for a variable in a formula means to find an equivalent equation in which the desired variable is isolated. Follow the same general strategies as solving any equation.       Example: P = 2L + 2W, solve for W    P = 2L + 2W -2L   -2L               Addition Property P - 2L   =  2W P - 2L = 2W    2           2             Multiplication Property <img src=""paste2kdom8.png"" /> "

Measuring Angles

"Angles are measured in degrees using a protractor. Place one ray of the angle along the zero edge of the protractor. The other ray of the angle points to the number of degrees that are in the angle. <img src=""pastev1xnd2.png"" /> "

Area (Definition)

"Area is the two-dimensional measure of how many square units can fit inside the interior of an object <img src=""paste7hksh1.png"" /><div> </div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">1 acre = 43,560 ft2</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">640 acres = 1 mile2</div><div> </div>"

Congruent Shapes

"Congruent shapes are two shapes of exactly the same size and shape. The two shapes may be rotated or flipped. The common way to mark the matching sides and angles of congruent shapes is with hash marks as shown below: <img src=""pastepp88mc.png"" />"

Sales Tax

"Sales Tax is written in percentages (which are converted to decimal to computer sales tax). The final purchase price of an item = marked price + (sales tax times marked price). Using FP for final price, MP for marked price, and ST for sales tax, the algebraic equations is: 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. "

Scientific Notation

"Scientific Notation is a way to write very large or very small numbers using powers of 10. To convert a number into scientific notation, move the decimal point so the resulting number is between 1 and 10. Then state the power of 10. Because we use a Base 10 number system, an easy way to know what power of 10 is needed, the exponent indicates the number of decimal places the decimal point was moved. The exponent is negative if the decimal point was moved to the right; the exponent is positive if the decimal point was moved to the left. 1234.5 <img src=""pastexurkr0.jpg"" /> 1.2345 × 103"

Secondary data

"Secondary data is data obtained from someone else other than the user. Secondary data can be thought of as second-hand data; nevertheless, secondary data can be extremely useful depending on how the data was original obtained and manipulated.   This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Similar Shapes

"Similar shapes are shapes that have the same angles but the size of the sides is different; they are the same shape but not the same size. The similar shape may be flipped or rotated, but it is still similar if the two shapes are merely dilations of each other. <img src=""paster6ixgb.png"" />   <img src=""paste1br9if.png"" />"

Speed  (measurement)

"Speed is a measurement that tells how fast an object is moving. The rate of speed is usually expressed as a ratio of distance over time. Distance = rate of speed × time  (D = rt) <img src=""pastexupyh8.png"" /> <img src=""pasteb9ccsr.png"" />"

Supplementary Angles

"Supplementary angles are two angles whose measure adds to 180°.     <img src=""pasteskh7hx.png"" />"

Time Measurement

"Time can be formatted in using a 12-hour clock with a.m. and p.m. or using a 24-hour clock (military time). In most places of the world, time is adjusted twice a year by one hour for Daylight Saving Time (note that there is no "S" on Saving -- it is not Daylight Savings Time). The common units of time are: 60 seconds = 1 minute ; 60 minutes = 1 hour ; 24 hours = 1 day ; 7 days = 1 week ; 28-31 days = 1 month ; 12 months = 1 year 365 days = 1 common year  (366 days = 1 leap year) "

Converting Measures of Area

"To convert measures of area, use Unit Analysis (multiply by the conversion factor in such a way that all units cancel out except the unit you want). For example, to change 16 square feet to square inches: <img src=""paste7skkfe.jpg"" /> "

Variance

"Variance is a statistical measure of how far the data are spread out from the mean (average). This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics. "

Symbol for "summation"?

"<div style=""font-size: 1.5em; font-family:cambria;"">∑</div>"

Symbol for "infinity"?

"<div style=""font-size: 1.5em; font-family:cambria;"">∞</div>"

Symbol for "congruent"?

"<div style=""font-size: 1.5em; font-family:cambria;"">≅</div>"

Mixed Number

"A mixed number is a whole number and a proper fraction combined. Mixed numbers may also be called mixed fractions. <img src=""paste65g730.jpg"" /> This graphic shows two whole pizzas and a fraction of 3 pieces out of 4  <img src=""pasteey5lnj.jpg"" /> 2 3/4 "

Isosceles Triangle

"An isosceles triangle is one where two of the three legs (sides) are of equal measure, which means two of the angles are of equal measure. In diagrams representing triangles (and other geometric figures), "tick" marks along the sides are used to denote sides of equal lengths: <img src=""paste8nmnux.png"" />"

Obtuse Angle

"An obtuse angle is an angle that measures between 90°-180°     <img src=""pastehdxi8l.png"" />"

Adjacent Angles

"Two angles are adjacent angles if they share a common vertex, they share a common side, AND they do not share any interior points. In other words two angles that are side-by-side are adjacent. <img src=""pastequlmnb.png"" />"

Representing Multiplication or Division of Rational Numbers using concrete models

*See pictures for models* These models (pictures) show: 8/9 * 3/4 and 5/9 ÷ 1/4 Remember: To *multiply fractions*, top x top, bottom x bottom, reduce if you can. To *divide fractions*, flip (invert) the 2nd fraction and multiply top x top, bottom x bottom, simplify if you can.

Negative and Positive Addition/Subtraction

+ plus - = - - plus - = -

Negative and Positive Multiplication/Division Rules

+ times + = + Ex: 2 * 3 = 6 + times - = - Ex: 2 * -3 = -6 - times - = - Ex: -2 * -3 = 6 *Remember by: If a good thing happens to a good person, that's good If a good thing happens to a bad person, thats bad If a bad thing happens to a bad person, that's good

Tip for working with Praxis measurement problems

1. Determine if it's customary or metric units 2. Determine if it's length, volume, mass, time, or temperature 3. Change all measurements to the same unit

Tip for solving Praxis problems of area, perimeter, and volume

1. Determine if it's customary or metric units 2. Determine if it's length, volume, or mass 3. Change all measurements to the same unit 4. Use the appropriate formula

*Test Tips: Translating Word Problems into Equations*

1.) Last sentence of word problem = what the question is asking you to find out/what you actually need to solve (Begin with the end in mind) 2.) What do I wish I knew? What would you want to ask if you could ask the test makers to clarify? This could be a variable, like "x" you need to find. 3.) How can I work backwards to find this info? *Approximating can sometimes save time on multiple choice questions.* *Ex:* Tom is at a plant nursery where he spends a total of $41.92. Ferns are $5.24, avocado trees are $3.56, gardenias are $7.13 and succulents are $4.05. He buys a total of 8 of one type of plant. What plant does he buy? Can estimate to guess: Ferns: 8 x $5 = $40 Succulents: 8 x $4=$32 ^^ Already know the answer is ferns because others would be either more than or less than $40. *Ex2:* How much do 11 ferns cost? (fill in the blank) Do TEN x (price), then add one more, because it is much faster to do that mental math. 10 x $5.24 = $52.40 + $5.24 = $57.64

Patterns

A pattern is a type of theme of recurring events or objects, sometimes referred to as elements of a set of objects. It is said that algebra is a study of patterns.

Algebraic Thinking

Algebraic thinking is the mathematics we teach and learn to prepare us to understand algebra. In elementary schools, algebraic thinking is the study of our number system, patterns, representations, and mathematical reasoning.

Approximating Square Roots

Approximating square roots means to find the approximate value of a number's square root. We find approximate square roots by comparing the number to perfect square numbers where the square roots are known.<div> </div><div>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</div>

Multiples

A *multiple* is the product of the number and the counting numbers (1-9) any quantity and an integer. Multiples may be found by counting by the number. A calculator may also be used to find multiples - enter the quantity, hit the plus sign, enter the quantity again, and then hit the equal symbol. Then keep hitting the equal symbol to see successive multiples. Example: Find Multiples of 23: 23 + 23 = 46, = 69, = 92, = 115, etc. 23 x 1, 23 x 2, 23 x 3, 23 x 4, 23 x 5, etc.

Prime Factorization

A *prime factorization* of a number is a list of all prime factors that multiply together to make that number. A factor tree is often used to find the prime factorization of a number. The prime factorization can be written as the product of individual factors or exponents can be used to write the product of repeated prime factors. Ex: 120 = 2 x 2 x 2 x 3 x 5 This can also be written as 120 = 23 x 3 x 5

Ratios (Definition & three ways to write)

A *ratio* is a FIXED relationship between 2 quantities, or in other words, are a comparison of 2 numbers using division. The most common ratios we see are fractions. Write a ratio using a fraction bar, a colon, or the word "to": 3:2 3/2 3 to 2 When we *set two ratios equal to each other,* we say these 2 ratios/fractions are *proportionate.* The comparison between the numerators and the denominators is *the same fixed relationship*. ex: 1/2 = 2/4 "One half is equal to two fourths." ="One half is proportionate to two fourths."

Convert Mixed Number to Improper Fraction

A mixed number indicates the whole amounts and the parts (a fraction). For each whole number, there are a complete number of parts (for example, if the parts are measured in thirds, each whole has three thirds). 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.

Rectangular Solid (Definition, volume, and surface area)

A rectangular solid (also known as a cuboid) is a three-dimensional solid where *all angles are right angles* and *opposite faces are equal.* A rectangular solid is also informally called a rectangular box. V = Lwh SA = 2wh + 2hL + 2wL

diameter

A straight line passing from side to side through the center of a circle or sphere. It is radius * 2 d=2r

Trapezoid

A trapezoid is a convex quadrilateral with at least one pair of parallel sides. The parallel sides are called bases and the other two sides are called legs. A = 1/2h(b1 + b2) P = add all four sides

Solution  (algebra)

An algebraic solution is the answer to an equation. The solution will give a value or multiple values for the variables in the equation. A solution is also called a root of the equation. For more than one variable, the solution will be an ordered pair, an ordered triple, etc.

Convert Improper Fraction to Mixed Number

An improper fraction shows a total number of parts, but contained in those parts is at least one whole. If the parts are measured in thirds, each three in the numerator of the improper fraction makes 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

Tip for Praxis problems with both fractions and decimals

Change all numbers to fractions or change all numbers to decimals before beginning your calculations.

Classify Numbers in Real Number System

Classify Numbers in Real Number System Sets in mathematics include the set of integers (Z), rational numbers (Q), primes (P), real numbers (R), natural numbers (N), whole numbers (W), etc.

Imaginary Numbers

Imaginary Numbers are numbers that contain the imaginary number "i", which is the square root of negative one: Note that if the discriminant portion of the quadratic equation is negative, the function or quadratic equation has no real solutions. The symbol used for the set of complex numbers is C. ***This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics.***

Perfect Squares (numbers)

Numbers that have a whole number square root. The first ten perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. Is zero a perfect square? There is a debate about this in the mathematics community - some believe zero is a perfect square because 0 times 0 = 0; some disagree because they say the definition of a perfect square is "numbers that have a POSITIVE integer square root" and zero is not positive. ....so the debate continues.

Percentage of a population

Numerator: # of thing you are interested in Denominator: Total population Ex: There is an orchard with 30 apple trees, 25 orange trees, 15 fig trees and 5 plum trees. What percentage of the trees are fig trees? Numerator: # of fig trees Denominator: Total # of trees 15/(30+25+15+5) = 15/75 Can just divide 15 by 75, or can simplify fraction: 15/75 = (3*5)/(3*25)= (5)/(25) = 1/5 = 20% 20% of the trees are fig trees

Tip about Praxis problem words in ALL CAPITAL LETTERS

One of the four question types on the Praxis test is an exception question where you need to find the answer that does NOT fit the pattern or does NOT answer the question. The signal words that this is an exception question will be written in ALL CAPITAL LETTERS, such as EXCEPT, NOT, and LEAST.

Parentheses

Parentheses are a way to group numbers. Other grouping symbols are: braces { }, square brackets and the vinculum or fraction bar. To remove parentheses, we distribute the number immediately outside the parenthesis (with its sign). We distribute by multiplying by the number. 3(3x +1) = 9x + 3 -(2x - 5) = -2x + 5

Ordering Integers

To order or compare 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 is (remember negatives act in an opposite way from positive numbers).

Comparing Integers

Simplify Algebraic Expressions

To simplify algebraic expressions, use the distributive property and combine like terms. This can also be stated in step-by-step fashion:<div>   1. Clear the parenthesis (by following the distributive property or the rules of exponents) 2. Add the coefficients of like terms 3. Add the constant terms </div>

Solving Linear Equations

To solve a linear equation, isolate the variable. Apply the addition principle of equality (add the same number to both sides of the equation), and then apply the multiplication principle of equality (multiply the same number to both sides of the equation). It is often helpful to clear parentheses and clear fractions first.

Addition of Fractions & Mixed Numbers

To add fractions & mixed numbers: 1. Write the two fractions/mixed numbers vertically above each other (lining up place value) 2. Change the fractions to a common denominator. 3. Add the numerators only. 4. Put that sum over the common denominator. 5. Simplify the answer. *Remember on test, if answers are in fraction form, correct answer will always be reduced*

Area of Irregular Shapes

To find the area of irregular shapes, divide the shape into regular two-dimensional shapes or picture a regular shape that has been removed:

Square

a regular polygon made up of four equal sides and four equal angles of 90 degrees each if s = side A = s^2 P = s*4

Area Models

Visual model that helps students understand how quantities can be used to describe 2D objects, like 3 shaded squares in a set of 4 squares is 3/4

Translating English (Word Problems) into Math: Percentages

What = x percent = /100 of = * is = equals "What percent of 8 is 2?" = [x/100 * 8 = 2]

When translating word problems, the word "per" means _________.

division

What are the parts of a division problem?

divisor; dividend; quotient; remainder

real-world problems diving fractions : ex1: Kevin has 8/10 cups of water. He has to split this water up between dogs' water dishes. He poured 2/5 cups of water into each water dish. How many water dishes can he fill with the water he has? ex2: Cameron has 5/6 yards of string. She needs 1/3 yard of string to make a bracelet. How many bracelets can she make?

ex1: 8/10 c ÷ 2/5 c per bowl = number of bowls This is asking "How many groups of 2/5 fit in 8/10?" 8/10 ÷ 2/5 = ? INVERT the 2nd Fraction: =8/10 * 5/2 = ? Multiply across: (8*5)/(10*2) = ? (40) / (20) = ? 4/2 =? =2 He can fill 2 water dishes with this much water ex2: 5/6 ÷ 1/3 = ? using inverse fraction and multiplying: 5/6 * 3/1 =(5*3)/(6*1) =15/6 both can be divided by 3, to simplify down to =5/2 =2 and 1/2 or dividing across: 5/6 ÷ 1/3 =(5÷1)/(6÷3) =5/2 =2 and 1/2

Factor

numbers that divide evenly into other numbers - without a remainder. For example, 5 divides into 40 evenly so 5 is a factor of 40. We often create a factor tree or a prime factorization of numbers to help us recognize the factors of a number See pic for factor tree which shows that the prime factorization of 120 = 2 x 2 x 2 x 3 x 5

Triangle

polygon with three angles/vertices and three sides made up of line segments -can be named by its three vertices: ΔABC P = a + b + c A=1/2(bh)

circumference

the distance around the outside of a circle C = (π)d = 2(π)r

Decimal Number 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). <img src=""pastex0nanh.png"" />"

Cylinder (Definition, volume, and surface area)

"A cylinder is a three-dimensional solid where the top and bottom are circles. <img src=""pastebdvock.png"" /><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">V = πr2h</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">SA = 2πr2 + 2πrh</div>"

First 10 Square Numbers

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Decimal Number 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) Technically, decimal numbers are multiplied by powers of 10 in order to make them whole numbers before multiplication. Then the answer (product) is divided by that same power of 10.

Proportions (Definition)

A true statement that two ratios or fractions are equal. If have a variable, cross multiply to solve. (Each side will now be a linear equation, no longer in fraction form. See example) ex: x/9 = 12/54 54x= (9)(12) 54x= 108 (54x)÷54 = 108÷54 x=2 Every proportion is made up of 4 numbers. When we *set two ratios equal to each other,* we say these 2 ratios/fractions are *proportionate.* The comparison between the numerators and the denominators is *the same fixed relationship*. ex: 1/2 = 2/4 "One half is equal to two fourths." ="One half is proportionate to two fourths."

Irrational Numbers

All real numbers except the rational numbers; -numbers that are square roots of non-perfect square numbers -non-terminating, non-repeating decimal s (like pi - just keeps going no end) -most well-known irrational number is pi (π) which is approximately equal to 22/7 or 3.14 The opposite of Irrational numbers are the rational numbers - ALL real numbers are either rational or irrational.

Odd vs. Even Numbers

An even number is an integer that is evenly divisible by 2 (without a remainder). Note that the number zero is an even number. An odd number is an integer that is NOT evenly divisible by 2.

Geometry Symbols (13 of them)

"Geometry Symbols (13 of them) <img src=""pastenutarn.png"" />"

Binomial

"A binomial is an algebraic expression with exactly 2 terms Example:  3x - 2y"

Circle Graph

"A circle graph (also known as a pie chart) is a circular graph where sections of the circle represent parts of the whole. A pie chart ALWAYS gives parts of a whole. The pie sections are usually labeled with percentages. <img src=""pastempdkul.png"" />"

Closure

"Closure is a property of operations. Number sets are closed under addition and multiplication, but not under subtraction or division. This means that you can add any two whole numbers and the result will be a whole number. But you can't subtract any two whole numbers and be guaranteed that the result will be a whole number (it might be a negative number, which is not a whole number). This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Concave

"Concave describes an object with a hollowed out or cut out portion—a part of the object has been "caved" in. The opposite of a concave polygon is a convex polygon. <img src=""pastem9enkn.png"" />"

Continuous Data

"Continuous variables can assume an infinite number of values between any two specific values. They are obtained by measuring. They often include fractions and decimals. <img src=""pasteo0hvqs.png"" />"

Convex

"Convex describes an object such as a polygon that is not concave. All vertices of a convex polygon are less than 180-degrees in measure: <img src=""pastecrexir.png"" /> The opposite of convex is concave."

Correlation

"Correlation is a statistical relationship between dependent variables. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics. "

Data Collection

"Data collection is the process of collecting and preparing data for statistical purposes. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Deductive Reasoning

"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. A trick to remember the difference: Deductive reasoning = Drawing logical conclusions from factual statements Inductive reasoning = I-witness (eye-witness) observations    "

Density

"Density is the measure of how compact the material is inside an object. Density is measured in terms of the mass per unit volume. We have all heard the story of Archimedes who discovered the concept of density when he saw how his body displaced the water in his bathtub. He cried, "Eureka! Eureka!" <img src=""pastevurmyl.png"" /><div>  This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics.</div>"

Dependent (Data)

"Dependent data is the opposite of independent data. Data that is dependent means that it changes depending on how other data changes. Independent data is independent of changes in other data. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Discounts

"Discounts are usually written as percentages. A discount is an amount by which the purchase price is reduced. Be careful when working with discount problems because the answer may be the discount percentage, an amount of money discounted, the sales price after the discount, or the original sales price. Discount amount is the original sales price multiplied by the percentage of discount. The discounted price is the difference of the original price and the discount amount. <img src=""pastez1nm88.jpg"" />"

Discrete Data

"Discrete variables can be assigned values such as 0, 1, 2, and 3 are said to be countable. Examples of discrete variables are the number of children in a family, the number of students in a classroom, and the number of calls received by a switchboard operator each day for a month. <img src=""pasteo0hvqs.png"" /> This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Division of Whole Numbers

"Division of whole numbers is the inverse of multiply and is repeated subtraction. We divide to find how many groups of a number (the divisor) can be created from another number (the dividend). The symbol ÷ is used to signify division; it means "divided by." The algorithm for division Is: Divide the divisor into the digit in the largest place value of the dividend, multiply, subtract, bring down the next digit, and divide again. <img src=""pastexskq85.png"" />"

Euler's Polyhedron Formula

"Euler's Polyhedron Formula is V - E + F = 2 which means the number of vertices subtract the number of edges add the number of faces in any polyhedron always equals 2. <img src=""pasteoshytu.png"" /> This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Expanded Notation

"Expanded Notation is writing a number to show each digit's place value. Example: Write 123,456 in expanded notation<div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">(1 × 100,000) + (2 × 10,000) + (3 × 1,000) + (4 × 100) + (5 × 10) + (6 × 1)</div><div> or this number can be written using exponents:</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">(1 × 105) + (2 × 104) + (3 × 103) + (4 × 102) + (5 × 101) + (6 × 100) </div>"

Factorial

"Factorial is a unary operation. The exclamation point is the symbol used to denote factorial. Factorials are most commonly used in permutations and combinations. To find the factorial of a number n, multiply all the numbers from 1 to the number n. Example: Find 6!<div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">6! = 6 × 5 × 4 × 3 × 2 × 1 = 720</div><div> By convention, 0! = 1 This term will probably not appear on the Praxis I test but may appear on the Praxis II test. </div>"

Inductive Reasoning

"Inductive reasoning is making sufficient observations that conclusions can be formed. 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 conclusion is true. It often helps to draw logic circles when working with deductive reasoning. A trick to remember the difference: Deductive reasoning = Drawing logical conclusions from factual statements Inductive reasoning = I-witness (eye-witness) observations "

Length (Define & List Units of Measure)

"Length is a measure of distance. The metric measure of length is the meter. In the customary or U.S. English system, refer to the table below:<div> </div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">12 inches = 1 foot</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">3 feet = 1 yard</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">5280 feet = 1 mile</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">1760 yards = 1 mile</div><div> </div>"

Normal Distribution

"Normal distribution is used in statistics to describe the normal patterns of data. When graphed, data that is a normal distribution is shaped like a bell curve. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Parallel Lines

"Parallel lines are lines in a plane that do not intersect or touch at any point. The lines are equidistant from each other.    <img src=""pasteldxdrc.jpg"" />"

Probability

"Probability is the ratio of how likely a specific event is to happen when compared to all possibilities of events that might happen. Probability is most often written as a fraction, but it may also be written as a decimal or a percentage. The numerator of the fraction tells how many possibilities of a specific event, the denominator tells how many total possibilities. For example, in a deck of face cards, the probability of drawing a heart is 13 out of 52 (13 hearts in a deck of 52 face cards). That is written as 13/52 but simplified to 1/4.    The three types of probabilities are classical, empirical, and subjective. "

Qualitative Data

"Qualitative variables are variables that can be placed into distinct categories, according to some characteristic or attribute. For example, if subjects are classified according to gender (male or female), the variable gender is qualitative. Other examples of qualitative variables are religious preference and geographic locations. <img src=""pasteo0hvqs.png"" /> This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Quantitative Data

"Quantitative variables are numerical and can be ordered or ranked. For example, the variable age is numerical, and people can be ranked in order according to the value of their ages. Other examples of quantitative variables are heights, weights, and body temperatures. <img src=""pasteo0hvqs.png"" /> This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Reduce a Fraction

"Reducing a fraction is the same as simplifying a fraction. The term "reduce" is seldom used in mathematics today. To simplify a fraction, expand the numerator and denominator into prime factorizations. "Cancel" any ones such as 3/3 or 5/5. Then multiply straight across those numbers that are left. True story: When I did my student teaching 40 years ago, I was teaching how to add fractions and I used the term "reduce" a fraction. After the lesson, my cooperating teacher took me aside and told me to use "simplify" instead -- he said, "Jolene, only women reduce. Fractions simplify." "

Rules for Exponents

"Rules for Exponents: <img src=""pastepclu9m.png"" /> <img src=""pastew7lolv.png"" /> <img src=""pastebcyy2v.png"" /> <img src=""pastegsz6at.png"" /> <img src=""pastebs4gbp.png"" /> <img src=""pasteyyuzdq.png"" /> <img src=""paste7tne0g.png"" /> <img src=""pastezmrr27.png"" /> <img src=""pastezr82vh.png"" /> "

Symmetry

"Symmetry (also known as reflection symmetry) is when both halves of an object are exact copies of each other. The line down the middle between the two halves is the line of symmetry. <img src=""pastejzerao.png"" />"

Converting Measures of Capacity/Volume

"To convert measures of capacity in the Metric System, merely move the decimal point to the "place value" of the new unit of measure:   kilo-  hecto-  deka-   l   deci-  centi-  milli- To convert measures of volume in the U. S. Customary System, use Unit Analysis (multiply by the conversion factor in such a way that all units cancel out except the unit you want). For example, to change 5 quarts to ounces: <img src=""pastehrkfkk.png"" /> "

Converting Measures of Length

"To convert measures of length in the Metric System, merely move the decimal point to the "place value" of the new unit of measure:   kilo-  hecto-  deka-   m   deci-  centi-  milli- To convert measures of length in the U. S. Customary System, use Unit Analysis (multiply by the conversion factor in such a way that all units cancel out except the unit you want). For example, to change 16 feet to inches: <img src=""pastefyusyu.png"" />"

Bell Curve

"A bell curve is a line graph where the data values peak in the middle and fall off drastically in both the positive and negative direction. It is named a bell curve because the line resembles the shape of a bell. Another name for a bell curve is a Gaussian function. A bell curve is often used in grading—employing the belief that the majority of the students are C-students and a small minority are A-grade or Failing students. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics. "

Converting Measures of Mass/Weight

"To convert measures of mass in the Metric System, merely move the decimal point to the "place value" of the new unit of measure:   kilo-  hecto-  deka-   g   deci-  centi-  milli- To convert measures of weight in the U. S. Customary System, use Unit Analysis (multiply by the conversion factor in such a way that all units cancel out except the unit you want). For example, to change 5 pounds to tons: <img src=""pasteysfwlh.png"" /> "

Converting Units of Measurement

"To convert from larger units of measurement to smaller units, multiply. To convert from smaller units of measurement to larger units, divide. To convert denominate numbers, use unit analysis. <a href=""http://www.jolenemorris.com/mathematics/Math115/Wk6/convertDenomNum/convertDenomNum.html"">Video explaining unit analysis for English units</a> <a href=""http://www.jolenemorris.com/mathematics/Math115/Wk6/convertMetricDen/convertMetricDen.html"">Video explaining unit analysis for Metric units</a> (both videos are at JoleneMorris.com, Math 115, Week 6) "

Unit Analysis

"Unit Analysis is the process of multiplying by successive conversion units (written in fraction form). unit analysis video: <a href=""http://www.jolenemorris.com/mathematics/Math115/Wk2/2.8prob28/2.8prob28.html"">JoleneMorris.com, Math 115, Wk 2</a> <a href=""http://www.jolenemorris.com/mathematics/Math115/Wk6/convertDenomNum/convertDenomNum.html"">Video explaining unit analysis for English units</a> <a href=""http://www.jolenemorris.com/mathematics/Math115/Wk6/convertMetricDen/convertMetricDen.html"">Video explaining unit analysis for Metric units</a> (both videos are at JoleneMorris.com, Math 115, Week 6) "

Symbol for "approximately"?

"<div style=""font-size: 1.5em; font-family:cambria;"">≈</div>"

Symbol for "therefore"?

"<div style=""font-size: 2.0em; font-family:cambria;"">∴</div> Three dots in a triangle"

What are the three "special right triangles"?

"<div style=""font-size:1.3em; font-family:cambria; font-style:italic;"">30° - 60° - 90° 45° - 45° - 90° 3n° - 4n°  -5n°</div>"

The ratios of side lengths in a 45-45-90 triangle are ______.

"<img src=""pastedi_qox.jpg"" />"

The ratios of side lengths in a 30-60-90 triangle are ______.

"<img src=""pasteqwcsqx.jpg"" />"

The total degrees of measure inside every n-sided shape

"180°(n - 2) Subtract 2 from the number of sides and multiply by 180 degrees."

Using algebra, how can you express the sum of 3 consecutive numbers?

"x + (x + 1) + (x + 2)"

Using algebra, how can you express the sum of 4 consecutive odd numbers?

"x + (x + 2) + (x + 4) + (x + 6)"

Pythagorean Triple

"A Pythagorean Triple is a set of any three integers (a, b, c) such that   a2 + b2 = c2 The three numbers of a Pythagorean Triple describe the length of the three sides of a right triangle. Perhaps the most well-known Pythagorean Triple is 3-4-5. The other common Pythagorean Triple you will see on the Praxis exam is 5-12-13. There are 16 Pythagorean Triples with c < 100: <img src=""pastemfl7al.jpg"" />"

Sample Space

"A Sample Space is the set list all possible outcomes of a probability experiment. Two ways to list sample spaces when there are two outcomes done in sequence are using tree diagrams and tables. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Venn Diagram

"A Venn diagram is a diagram using circles to show logic statements. Most Venn diagrams display overlapping circles. Venn diagrams are named after John Venn who developed the concept of Venn diagrams about 1880. By shading portions of the overlapping circles, set theory concepts such as UNION and INTERSECTION can be shown. <img src=""pasteuvbfyv.png"" />"

Bar Graph

"A bar graph is used to demonstrate relative values in a data set. The primary purpose of a bar graph is to compare values: The height (or length) of the bars shows how the values compare to other values in the data set. <img src=""pastev4gmvn.png"" />"

Binomial Distribution

"A binomial distribution is a number indicating the number of results in a two-way experiment. The experiment tests only two outcomes (YES/NO, TRUE/FALSE, etc.). In the simplest explanation, the binomial distribution indicates the probability of the outcomes. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics. "

Coefficient (algebra)

"A coefficient is the number part of a term in an algebraic expression. For example, negative two is the coefficient of the following expression: -2x3 The coefficient is a factor of the term. The coefficient is multiplied and is, therefore, a multiplicative factor. "

Cone (definition and volume)

"A cone is a three-dimensional shape with a circular base. A cone can be formed by spinning a triangle in three-dimensional space.<div> </div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">V = 1/3πr2h <img src=""pastexppscd.png"" /></div>"

Quadrants (Coordinate Grid)

"A coordinate grid 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). <img src=""pastepfryy3.png"" />"

Coordinate Grid

"A coordinate grid 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). <img src=""pastepfryy3.png"" />"

Origin (coordinate grid)

"A coordinate plane is a two-dimensional grid for locating points. There is an x-axis and a y-axis at 90-degree angles, which divide the grid into four quadrants that are numbered counter-clockwise using Roman numerals. The origin is where the two axes cross (0, 0). A coordinate pair is a pair of numbers indicating the location of a point (x, y).  Sometimes called a Cartesian grid after the mathematician René Descartes (1596-1650) <img src=""pastepfryy3.png"" />"

Coordinate Plane (Definition to include five terms)

"A coordinate plane is a two-dimensional grid for locating points. There is an x-axis and a y-axis at 90-degree angles, which divide the grid into four quadrants that are numbered counter-clockwise using Roman numerals. The origin is where the two axes cross (0, 0). A coordinate pair is a pair of numbers indicating the location of a point (x, y).  Sometimes called a Cartesian grid after the mathematician René Descartes (1596-1650) <img src=""pastepfryy3.png"" />"

Correlation Coefficient

"A correlation coefficient is a statistical measure of the relationship between dependent variables. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics. "

Decimal Point

"A decimal point is a period that indicates the location of the one's place - the decimal point always comes to the right of the one's place. If there are no fractional decimal numbers to the right of the decimal point, the decimal point doesn't have to be written. It is understood. <img src=""pasteays8m1.png"" /> "

Function Machine

"A function machine is a visual device to help young students understand the concept of a function. Each function machine has a rule it applies to numbers that are put into the machine (the inputs). After the machine applies the rule, it outputs the result. <img src=""pastes6cmhn.png"" />"

Geometric Sequence

"A geometric sequence is an ordered list of numbers where each number is formed by multiplying a constant number to the previous number. An example of a geometric sequence is: 3, 9, 27, 81, 243, etc. where each number is formed by multiplying by 3. An arithmetic sequence may be written algebraically as <div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">an = kan-1</div><div> Do not confuse this with a geometric series where numbers are not is a list but form an addition problem. An example of a geometric series is 3 + 9 + 27 + 81 + ... </div>"

Histogram

"A histogram is a bar graph where the bars are vertical; the bars represent continuous groups of numerical data; and the bars touch. The term comes from the late 1890's meaning an historical diagram. Example: children's ages at the movie: A bar graph would have a bar showing how many children of each age attended the movie, i.e. 6-year olds, 7-year olds, 8-year olds, etc. A histogram would have a bar showing how many of each age GROUP attended the movie, i.e., 6-9-year-olds, 10-13-year-olds, 14-17-year-olds. "

Line Graph

"A line graph is a statistical graph showing the data points connected by line segments. A line graph is used to visualize trends in the data. The graphic below is a line graph showing two trend lines. <img src=""pastet1aa0g.png"" />"

Line Segment

"A line segment is a portion of a line with two endpoints. The line segment is named by the two endpoints. The symbol for a line segment is a line segment placed above the two endpoint letters: <img src=""pastewodbmg.png"" />"

Line (geometry) -- How is a line named?

"A line is an infinite collection of collinear points. A line has no width or depth—it has only one dimension: length. A line is named with a lowercase letter or by two points on the line. The symbol for a line is a double headed arrow. Thus, the line below is line l,  or <img src=""pasterfgrge.png"" /> <img src=""paste9jxvzp.jpg"" /> "

Logic Diagram

"A logic diagram is a visual way to determine the truth or logic of statements. A truth table may also be used. With a logic diagram, use circles to show relationships. For example:  (1) ALL cats have tails. (2) SOME cats are black. (3) Goldy is a cat. ALL means the circle is completely inside another circle. SOME means the circle is partially inside another circle. NONE means the circles are completely separate. <img src=""pastefezadz.png"" /> From the logic diagram above, we see that Goldy definitely has a tail but may or may not be black. "

Monomial

"A monomial is a polynomial with a single term. A term contains a coefficient with possibly one or more variables all multiplied. Neither a term nor a monomial has any addition or subtraction. Here are samples of four monomials:<div> </div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">3        3x        3x2       x5 </div>"

Net or Network (geometry)

"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. <img src=""pastega8pp2.png"" /> "

Number Cubed

"A number cubed is the same as the number times itself times itself again. The exponent of 3 is often called cubed because the volume of a cube is the side times itself times itself.<div> </div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">n3 = n cubed = n × n × n</div>"

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. <img src=""pasteoetkwm.png"" />"

Number Squared

"A number squared is the same as the number times itself. The exponent of 2 is often called squared because the area of a square is the side times itself. n2 = n squared = n × n"

Permutations

"A permutation is a way of selecting several things out of a larger group, where order does matter. (permute = changing order) <img src=""pasten_w36g.png"" /> n is the number of items selected k is the number of items in the larger group This term will probably not appear on the Praxis I test but may appear on the Praxis II exam. "

Pictograph

"A pictograph is a graph or chart where pictures are used to indicate a specific number of objects. <img src=""pastewhwbe7.png"" />"

Point

"A point is a non-dimensional location on a plane. A point is usually labeled with a capital letter of the alphabet. <img src=""pastefmrxti.png"" />"

Polygon (definition and names)

"A polygon is a two-dimensional shape drawn on a plane. A regular polygon is where all sides and all angles of the polygon have the same measure. <img src=""pastem4ndua.png"" />"

Polynomial (terms; degrees; types)

"A polynomial is an algebraic expression with one or more terms. A polynomial cannot have a variable in the denominator (which is a negative exponent). A polynomial with one term is called a monomial; two terms, a binomial; and three terms, a trinomial. The term with the highest exponent (sum) determines the degree of the polynomial. <img src=""pasteerhr6y.jpg"" />"

Prism

"A prism is a three-dimensional object with two bases of the same figure. The side faces of a prism are quadrilaterals. Prisms are named according to their bases. As such, if the two bases are triangles, it is a triangular prism. If the two bases are hexagons, it is a hexagonal prism. <div> </div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">V = area of base × height</div>"

Proportions (How to Solve)

"A proportion is when two ratios are equivalent: <img src=""paste4snc75.png"" /> When two ratios are equivalent, the cross products are equal. Thus, to solve a proportion, cross multiply the two numbers that are diagonal from each other, and then divide by the number diagonal from the unknown. <img src=""pastewj9gxm.png"" /> "

Protractor

"A protractor is a geometry tool used to measure angles. Place one ray of the angle along the zero edge of the protractor. The other ray of the angle points to the number of degrees that are in the angle. <img src=""pastev1xnd2.png"" />"

Pyramid (geometry)

"A pyramid (in geometry) is a three-dimensional object. The base of the pyramid is a polygon. Line segments connect the base of the pyramid to a single point, called the apex. Each base edge and the apex form a triangle -- thus all faces of a pyramid (except the base) are triangular. <img src=""pastexr4d5j.png"" />   <img src=""pastei5rbce.png"" />"

Quadratic Equation (definition & five ways to solve)

"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: 1. Factor and set each factor equal to 0 2. If there is no x-term, solve for x2 and apply the square root method. 3. Graph the equation (as a parabola) and determine the solutions where the parabola crosses the x-axis 4. Complete the square 5. Use the quadratic formula This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics. "

Words that Signal Subtraction

"• subtract • subtracted from • minus • difference • take away • less than • decreased by <img src=""pasteanxscw.jpg"" /> "

Quadrilateral

"A quadrilateral is a polygon with four sides (and four vertices). Other names for a quadrilateral are a quadrangle and a tetragon. The interior angles of a quadrilateral add to 360◦. An excellent graphic showing the Euler diagram of quadrilateral types can be found on Wikipedia: <a href=""http://en.wikipedia.org/wiki/File:Euler_diagram_of_quadrilateral_types.svg"">http://en.wikipedia.org/wiki/File:Euler_diagram_of_quadrilateral_types.svg</a> "

Radius

"A radius is a line segment that goes from the center of a circle to any point on the circumference of the circle. The measure of the radius is half the diameter. Often, the term radius is also used to denote the measure of the radius line segment. <img src=""pasteqjplvi.png"" />"

Ray (geometry)

"A ray is often mistakenly called a half line. A ray is part of a line that has an endpoint but goes infinitely in a straight line from that endpoint. Two rays that have the same endpoint form an angle. A ray is named by its endpoint and any other point on the ray. The symbol for a ray is a small ray (single headed arrow). <img src=""pasteqch7ya.png"" />"

Reciprocal

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

Reflex Angle

"A reflex angle is an angle measured in a clockwise direction as opposed to the normal counter-clockwise direction. <img src=""pastehdxi8l.png"" />"

Relationships (data pairs)

"A relationship is simply a set of ordered pairs. If the set of ordered pairs has only one y-value for any x-value, this relationship is a function. If the set of ordered pairs has only one y-value for any one x-value, it is a function with a one-to-one correspondence. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Rhombus (Definition, perimeter, & area)

"A rhombus is a two-dimensional quadrilateral where all four sides are the same length. Thus, a square is a specialized rhombus.    <img src=""pastewevixu.png"" /><div> </div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">P = 4s</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">A = bh </div>"

Right Angle

"A right angle is an angle that measures 90°    <img src=""pastepowczn.png"" />"

Right Triangle

"A right triangle is a triangle in which one angle is a right angle. The right angle is usually marked with a small square: <img src=""paste4dlk9j.png"" />"

Scalene Triangle

"A scalene triangle is a triangle in which all three sides are of different length. In diagrams representing triangles (and other geometric figures), "tick" marks along the sides are used to denote sides of equal lengths: <img src=""pastemmbztr.png"" />"

Scatter Plot

"A scatter plot is a graph showing a collection of two-coordinate points. The points are not connected with line segments, but the points may demonstrate a trend. A technique called the "line of best fit" determines a line through the points where about half of the points are above the line and about half the points are below the line. This line of best fit visually demonstrates a trend. <img src=""pastenavbm4.png"" />"

Sets

"A set is a collection of objects. The objects in a set can be numbers, expressions, and other mathematical objects. Georg(e) Cantor developed set theory in the late 1800's. Common operations on sets include intersection, union, complements, and Cartesian products. Other concepts include the Universal set, Null set, members or elements, and sub-sets. Sets in mathematics include the set of integers (Z), rational numbers (Q), primes (P), real numbers (R), natural numbers (N), etc. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics. "

Simulation

"A simulation is a game, computer program, or device that approximates a real-life event. Simulations are used when actual events are dangerous or impractical to duplicate. Examples of simulations are driving teaching machines, dissection of frogs, and chemical experiments. Simulations are used in statistics to approximate results from a real-life event. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Sphere (definition, volume & surface area)

"A sphere is a three-dimensional, perfectly round shape. Sphere is from the Greek word for "ball." <img src=""pasteenjykf.png"" /> <img src=""paste6krvua.png"" /> Technically, in mathematics, a sphere only includes the "surface" and not the interior. "

Straight Angle

"A straight angle is an angle that measures 180° <img src=""pastehdxi8l.png"" />"

Table (of Data)

"A table is used to display the information in an organized manner. A table usually has column headings -- each record is in a row of the data table. <img src=""paste9ycib5.png"" /> <a href=""http://earthquake.usgs.gov/regional/neic/"">http://earthquake.usgs.gov/regional/neic/</a>"

Transformations

"A transformation in geometry changes the position of a shape on the coordinate plane. There are four forms of transformation: 1. translation (slide) 2. rotation (turn) 3. dilation (scale) 4. reflection (flip) <img src=""pasteg5pge5.png"" />"

Transversal

"A transversal is a line that crosses two or more other lines. A transversal of two lines forms eight angles that are often used in geometrical problems (see: vertical angles, adjacent angles, corresponding angles, interior angles, and exterior angles). <img src=""pastepuu9z8.png"" />"

What is a Tree Diagram? What is it used for?

"A tree diagram is a graphic organizer that lists all possibilities of a sequence of events in a systematic way. A tree diagram is used in determining probability - it is a way to calculate the total possible outcomes and view each possible scenario. <img src=""pastebn9e6n.png"" />"

Trend

"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.) <img src=""pasteaiankc.png"" />"

Trinomial

"A trinomial is an algebraic expression with exactly 3 terms Example:  3x2 + 2x - 1"

Vertex

"A vertex (plural: vertices) is a point that describes the corners or intersections of geometric shapes.    <img src=""pasteunnzhs.png"" /> <img src=""paste_p3_wg.png"" />"

Dependent Variable

"A dependent variable is one that depends on another variable (independent variable) for its value. In an equation with two variables, you can choose any value you want for one of them (the independent variable), but once that value is chosen, the dependent variable has limited values in order for the equation to be true. Dependent and independent variables are often interchangeable - in an equation with an x-variable and a y-variable; it doesn't matter which variable you choose a value for, but that sets the value of the other variable. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Common Factors

"All counting numbers (except 1) have at least two factors. Common factors are those factors that are in common with two or more numbers. For example,    The factors of 6 are 1, 2, 3, and 6    The factors of 9 are 1, 3, and 9    The common factors of 6 and 9 are 1 and 3 "

Common Multiples

"All counting numbers have an infinite number of multiples. Common multiples are those multiples that are in common with two or more numbers. For example,     The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, etc.     The multiples of 3 are 3, 6, 9, 12, 15, 18, etc.     The common multiples of 2 and 3 are 6, 12, 18, etc. "

Acute Angle

"An acute angle is an angle that measures less than 90°    <img src=""pastehdxi8l.png"" /><div> </div>"

Acute Triangle

"An acute triangle is any triangle where all three angles are less than 90°:   <img src=""pastep3fl4r.png"" /><div> </div>"

Angle Bisector

"An angle bisector is a ray (or line or line segment) that divides one angle into two angles of equal measure. In other words, an angle bisector cuts an angle in half. <img src=""paste3zxp_e.png"" />"

Arc

"An arc is a section of a circle. It is a set of points all equidistant from a center point. Arcs of the same size cut equal measured central angles in the circle. <img src=""paste1uhjoc.jpg"" />"

Arithmetic Sequence

"An arithmetic sequence is an ordered list of numbers where each number is formed by adding a constant number to the previous number. An example of an arithmetic sequence is: 3, 6, 9, 12, 15, 18, etc. where each number is formed by adding 3 to the previous number. An arithmetic sequence may be written algebraically as  an = an-1 + k Do not confuse this with an arithmetic series where numbers are not is a list but form an addition problem. An example of an arithmetic series is 3 + 6 + 9 + 12 + ... "

Equilateral Triangle

"An equilateral triangle is one where all three legs (sides) are of equal measure. An equilateral triangle is also equiangular, which all three angles are of equal measure (60°). In diagrams representing triangles (and other geometric figures), "tick" marks along the sides are used to denote sides of equivalent lengths: <img src=""pastezot6cz.png"" />"

Experiments

"An experiment is an attempt to prove or disprove an hypothesis. For the experiment to be valid, it must be repeatable and verifiable. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Exponent

"An exponent is a symbol to indicate a shortcut of multiplication by the same number. The exponent signifies the base is multiplied by itself (not by the exponent) the exponent number of times. Example:<div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">63 = 6 × 6 × 6 = 216</div><div> </div><div>Careful:</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">63 ≠ 6 × 3 </div>"

Obtuse Triangle

"An obtuse triangle is a triangle where one of the angles is obtuse (greater than 90-degrees). <img src=""pastew_b9rw.png"" />"

Oblique Triangle

"Any triangle that is NOT a right triangle is an oblique triangle. As such, acute triangles and obtuse triangles are in the category of oblique triangles. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Approximate Conversions (English/Customary units to/from Metric)

"Approximate Conversions (English/Customary units to/from Metric):<div> </div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">1 inch ≈ 2.54 cm</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">1 yard ≈ 0.91</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">1 mile ≈ 1.61 km</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">1 ounce ≈  28 g</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">1 pound ≈  2.2 kg</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">1 quart ≈ 0.94 l</div><div> </div><div><div>A meter is a little more than a yard. A gram is about the weight of a paper clip. A liter is a little more than a quart. </div></div>"

Algebraic Constant

"As opposed to a variable, an algebraic constant is a known number that does not vary. <div>In the trinomial below, the 5 is a constant.<div>2x2 - 3x + 5. </div><div> </div><div> A constant term is a term in a polynomial without a variable in it. </div></div>"

Scale on Bar or Line Graph

"Bar graphs and line graphs will often have a symbol on the vertical axis to indicate that some numbers have been left out. Be aware of these "broken scales" when analyzing graphs. The break in the scale is used when one or more of the bars/lines are significantly out of range of the other bars/lines. These broken scales can visually give a wrong impression. In the bar graph below, the red bar looks about twice the size of the green bar; whereas, the red bar is actually six or seven times the size of the green bar! <img src=""pastevduqsu.png"" />"

Dividing by Powers of 10

"Because we use a Base 10 number system, each place value is 10 times the place value to its right. As such, dividing by powers of 10 is simply a matter of moving the decimal point to the left (one place for each power of 10).<div> </div><div>Example:  123.456 ÷ 102 = 123.456 ÷ 100 = 1.23456 In this example, we were dividing by 10 to the power of 2, which is a 1 followed by 2 zeroes, so we merely move the decimal point 2 places to the left. </div>"

Multiplying by Powers of 10

"Because we use a Base 10 number system, each place value is 10 times the place value to its right. As such, multiplying by powers of 10 is simply a matter of moving the decimal point to the right (one place for each power of 10).   Example:   123.456 × 102 = 123.456 × 100 = 12345.6 In this example, we were multiplying by 10 to the power of 2, which is a 1 followed by 2 zeroes, so we merely move the decimal point 2 places to the right. "

Independent (Data)

"Dependent data is the opposite of independent data. Data that is dependent means that it changes depending on how other data changes. Independent data is independent of changes in other data. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Sum of Interior Angles

"In a polygon, the sum of the interior angles is equal to the number of sides, subtract 2, and then multiply by 180°: <div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">180(n - 2)°</div>"

Quartiles

"In statistics, 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 This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics. "

Sample (statistics)

"In statistics, often the population is too large to test everything in the population. Instead, a subset of the population is tested. This subset is called a sample. An entire college class is offered at most large universities to explain the best way to sample a representative subset of the population without introducing bias. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Fibonacci Sequence

"The Fibonacci Sequence is formed by starting with 0 and 1, adding those two terms to obtain the third term, adding the second and third terms to obtain the fourth term, and continuing by adding the last two terms to find the next term. The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci. <div> </div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...</div><div> The Fibonacci numbers appear in biological settings and are used in computer programming. </div>"

Fundamental Counting Principle

"The Fundamental Counting Principle: If there are m ways for one event to occur and n ways for another event to occur, there are m x n ways for both to occur. These events (in a sample space) are listed using a tree diagram or a table. <img src=""pastek07rva.png"" />"

Golden Ratio

"The Golden Ratio is a common ratio found in nature, arts, and mathematics. It is also known as the Golden Section or Divine Proportion. The Greek letter phi (Ф) is used for the Golden Ratio. A Golden Ratio exists between two numbers a and b (with a being the larger value) if <img src=""pastezffbgl.png"" />     Leonardo Da Vinci used the Golden Ratio in creating his statues and paintings. The Golden Ratio drawing of Da Vinci (shown to the right) is famous. <img src=""pasteluupkc.png"" /> This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

LCD

"The Lowest Common Denominator (LCD) of two or more fractions 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. <img src=""pastepukqul.png"" />    Another way to find the LCD is to write a prime factorization of each denominator, lined up by factors. Then "bring down" one of each factor and multiply. <img src=""pastemx82fz.png"" />"

Metric System of Measurement

"The Metric System is an international system of measurement based on the decimal system. The prefixes of the Metric System are shown in the table to the right: length <img src=""pasteey5lnj.jpg"" /> meter capacity (volume) <img src=""pasteey5lnj.jpg"" /> liter mass (weight) <img src=""pasteey5lnj.jpg"" /> gram <img src=""pastepaakoa.png"" />"

National Council of Teachers of Mathematics

"The National Council of Teachers of Mathematics (NCTM) is the national organization that sets the standards for what mathematics concepts should be taught and at what grade levels. States have the option of accepting the NCTM standards as is, modifying them for state use, or writing their own standards. Most states have accepted the NCTM with some limited modifications. <a href=""www.nctm.org"">www.nctm.org</a> <img src=""pasteylvtdj.png"" />"

Pythagorean Theorem

"The Pythagorean Theorem is used to determine the measure of an unknown leg or hypotenuse of a right triangle. <div> </div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">a2 + b2 = c2</div><div>    <img src=""pasteo6twj3.png"" /></div>"

Additive Inverse

"The additive inverse is a number such that when a number and its additive inverse are added, the resulting sum is zero (the additive identity). The additive inverse is the negative of a number and the opposite of the value of a variable:<div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">(n) + (-n) = 0</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">(6) + (-6) = 0 </div>"

Associative Property

"The associative property says that three numbers may be added/multiplied using the first two numbers first and then the third -or- they may be added/multiplied using the last two numbers first and then the first number. The sum/product will be the same regardless:<div> </div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">(a + b) + c = a + (b + c)</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">(a × b) × c = a × (b × c) </div>"

Commutative Property

"The commutative property says that two numbers can be added/multiplied in any order and the sum/product will be the same regardless:<div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">a + b = b + a</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">a × b = b × a </div>"

Diameter of a Circle

"The diameter of a circle is the longest chord in a circle. It is a line segment that passes through the center of the circle and whose endpoints are on the circle. The symbol for the diameter is Φ, which is made on a Windows computer by ALT+8960. <img src=""pasteqjplvi.png"" />"

Distance Formula

"The distance formula is used to find the distance between two points. The distance formula can be obtained by creating a triangle and using the Pythagorean Theorem to find the length of the hypotenuse. The hypotenuse of the triangle will be the distance between the two points.     <img src=""pastejog92i.png"" /> The distance formula can be derived from the Pythagorean Theorem. As such, you do not need to memorize the distance formula if you understand the Pythagorean Theorem."

Distributive Property

"The distributive property involves both addition and multiplication. If two numbers are added inside parenthesis but multiplied by a third number outside the parenthesis, that third number may be distributed to each of the numbers inside parenthesis and then added. The distributive property is one case where multiplication comes before parenthesis in the order of operations:<div> </div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">a(b + c) = ab + ac</div>"

Domain & Range of a Function

"The domain of a function is the set of all possible x values. The range of a function is the set of all possible y values. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics."

Rules of Probability

"The five rules of classical probability theory: 1. The probability of any event will always be a number from zero to one ( ). 2. When an event cannot occur, the probability will be 0. 3. When an event is certain to occur, the probability will be 1. 4. The sum of the probability of all outcomes in the sample space is 1. 5. The probability that an event will not occur is equal to 1 minute the probability that it will occur. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics. "

Hypotenuse

"The hypotenuse is the longest side of a right triangle, often labeled "c".     <img src=""paste_75kns.png"" />"

Identity Property

"The identity property defines what happens when you add or multiply by the identity numbers. The additive identity is zero. When you add zero and a number, the result is that number. The multiplicative identity is one. When you multiply one and a number, the result is that number.<div> </div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">n + 0 = n</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">n × 1 = n </div>"

Multiplicative Identity

"The multiplicative identity is the number one (1). We can multiply any number by one and the result is the original number. Multiplying by the multiplicative identity does not change the value of the number. We use the multiplicative identity when we simplify fractions. If the numerator has a factor of 2, and the denominator has a factor of 2, that is the fraction 2/2, which is the whole number 1. Since multiplying by 1 does not change the value of the original fraction, we can "cancel" that 1 to simplify the fraction: <img src=""pastexoxefs.png"" />"

Multiplicative Inverse

"The multiplicative inverse of a number is the reciprocal of that number. We can multiply any number by its multiplicative inverse and the result is the number one. <img src=""pasteno7s61.png"" />"

Probability of Multiple Events

"The probability of multiple events has different calculations depending on whether the events are independent (OR) or dependent (AND) and whether the events are mutually exclusive (have possibilities in common) • Independent, mutually exclusive <img src=""pasteey5lnj.jpg"" /> add • Independent, non-exclusive <img src=""pasteey5lnj.jpg"" /> add then subtract the events they have in common • Dependent <img src=""pasteey5lnj.jpg"" /> multiply (first event doesn't affect probability of the second event) • Dependent <img src=""pasteey5lnj.jpg"" /> multiply first probability and the conditional second probability "

Square Root of a Number

"The square root of a number is the value when multiplied by itself makes the number. A number's square root is always smaller than half of the number. The symbol for square root is called a radical sign. There is an index of 2 on the radical sign, but an index of 2 is rarely written -- it is understood to be 2 if not written. Finding a square root is the inverse of squaring a number. Although -6 times -6 also makes 36, we always use the positive square root (also called the principal square root). <img src=""pastejysnds.png"" />"

Stem & Leaf Plot

"The stem & leaf plot is a table used to show all data values, but in an organized manner. The table has two columns. The right column gives the ones place and the left column displays the tens place values: <img src=""pastezzgijr.jpg"" /> The stem & leaf plot above displays the numbers 1, 5, 8, 8, 13, 16, 16, 16, 17, 31, 32, and 35."

Vertex of a Parabola

"The vertex of a parabola is a single point where the parabola changes direction from upward to downward (or downward to upward).  The x-coordinate of a parabola's vertex is found by <div> <img width=""50px"" src=""paste1joj3i.png"" /> <img src=""pastew3fxjr.png"" /></div>"

Square Units of Measurement

"The area of an object is the number of square units that can fit inside the object. For example, if the object is measured in feet, the square unit of measurement for the area of that object is a one-foot by one-foot square. Be careful when finding area of objects when the answer is wanted in a different unit of measure:<div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">1 yd2 = 9 ft2</div><div style=""font-style:italic; font-size: 1.3em; font-family:cambria;"">1 ft2 = 144 in2</div>"

Random Variable

"The term random variable is used in statistics and probability to indicate a variable where the value of the variable may change according to the probability. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics. "

Regression

"The term regression is used in statistics to identify a technique or algorithm for estimating relationships between variables. This term will probably not appear on the Praxis test but is included here for completeness of mathematical topics. "

Legs of a Triangle

"The legs of a triangle are the sides of the triangle. In a right triangle, the legs are usually labeled "a" and "b". The legs are the two shorter sides of a right triangle. The legs of a right triangle are perpendicular to each other.     <img src=""pasted1v_n8.png"" />"

Percentages, the three types of problems

"There are three types of percentage problems depending on what value is missing in the equation: What number is 15% of 45? <img src=""pastexurkr0.jpg"" />   x = (0.15) ∙ (45) What percent of 45 is 15?     <img src=""pastexurkr0.jpg"" />  45 ∙ x = 15  or 45x = 15 15% of what number is 45?  <img src=""pastexurkr0.jpg"" />  (0.15) ∙ x = 45 or 0.15x=45 "

Interest

"There are basically two kinds of interest: simple and compound. Simple interest is paid on the principal amount only. Compound interest is paid on the principal amount plus accrued interest. The formula to find simple interest is I = prt where p is the principal, r is the interest rate, and t is the time period.    The formula to find compound interest is A = p(1 + r)n where p is the principal, r is the interest rate, and n is the number of interest periods. "

Temperature

"There are two scales used to measure temperature. The majority of the world uses the Celsius scale (formerly called Centigrade). In the United States, we commonly use the Fahrenheit scale. Water freezes at 0°C and at 32°F Water boils at 100°C and at 212°F To convert temperatures between the two scales: <img src=""pastec0ifzm.png"" /> <img src=""pastew9c1i3.png"" /> "

Percentages, Solving

"There are two ways to solve a percentage problem. To solve percentages using the percent proportion, use the means-extreme property of proportions (cross multiply). The percent proportion can be written as: <img src=""pasteh2sohc.png"" /> The second way to solve a percentage problem is with simple algebra: Write the percentage as an algebraic equation where "what number" → variable (x),   is → =,   and of → multiply. Then solve the equation. "

Evaluate Algebraic Expressions

"To evaluate an algebraic expression, substitute the given values for each variable into the expression, and then follow the order of operations (PEMDAS) to simplify the expression. 1. Perform the operations inside a parenthesis first 2. Then follow rules for exponents 3. Then multiplication and division, from left to right 4. Then addition and subtraction, from left to right"

Simplifying Square Roots

"To simplify square roots, take out the square root of any perfect squares that are factors inside the radicand. Perhaps the easiest way to do this is the factor the number inside the radicand so it is obvious which factors can be taken out. Example: Simplify <img src=""pastenr9prq.png"" /> <img src=""paste2w8pyb.png"" />"

Solving Percentages (using the algebraic equations)

"To solve percentages using algebra, write the problem as an algebraic statement where what number <img src=""pasteeimgot.jpg"" /> variable (x) is <img src=""paste_ubim_.jpg"" /> = of <img src=""pasteiboybt.jpg"" /> multiply What number is 15% of 45? <img src=""paste_vyywq.jpg"" />   x = (0.15) ∙ (45) What percent of 45 is 15?     <img src=""pasteey5lnj.jpg"" />  45 • x = 15  or 45x = 15 15% of what number is 45?  <img src=""pasteuwfwqh.jpg"" />  (0.15) • x = 45  or 0.15x=45 "

Writing Algebraic Expressions (from English statements)

"To write an algebraic expression from an English statement, we convert each word or phrase to the equivalent algebraic symbol. For example, "of" means to multiply, "per" means to divide, "total" means addition, and "is" means equals. Other flashcards contain ALL common words that can be converted to algebra. Here's an example:  The sum of two numbers is 16. One of the numbers is twice the other number.    x + y = 16     x = 2y"

Intersecting Lines

"Two lines on the same plane that share a single point are said to be intersecting lines.   <img src=""pastevfekmv.jpg"" />"

Bimodal Distribution

"When a list of numbers has TWO numbers that appear the most, this distribution of numbers in the list is called a bimodal distribution. For example, this list is a bimodal distribution with 3 and 7 as the two modes: <img src=""pastem841zv.jpg"" /> "

Words that Signal Addition

"• add • sum • increase • total • rise • plus • grow • added to • more than • increased by • gain <img src=""pasteanxscw.jpg"" />"

Words that Signal Division

"• divide • divided by • quotient • per • ratio • half <img src=""pasteanxscw.jpg"" />"

Words that Signal Multiplication

"• multiply • multiplied by • product • times • of • twice <img src=""pasteanxscw.jpg"" />"

Converting Fractions to Decimals & Percentages

*Change a fraction to a decimal* by dividing the denominator into the numerator. Keep dividing until the decimal number repeats or terminates. Draw a line (vinculum) above the repeating portion. (can just divide on calculator or can do long division, see pic) *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 add the percent symbol.

Solving Algebraic Word Problems (involving number relationships)

*Solving Algebraic Word Problems (involving number relationships):* Translate verbal statements into English words using an equal symbol (=) for the word ""is"" or using comparison symbols for statements of inequality.Then, solve the equation or inequality by isolating the variable.

Solving Algebraic Word Problems (involving triangles) Problems involving triangles require you to know facts about triangles, such as:

-Two angles that form a straight line (called supplementary angles) add up to 180°. -All the angles inside a triangle add up to 180°. Example: Find the unknown angle (x):

Units of Capacity (Define & List Units of Measure)

Capacity (also known as volume) is the amount of space inside an object.  The measurement of capacity/volume in the metric system is the liter. In the customary or U. S. English system, refer to the table below: 1 pint (pt) = 2 cups (c) 1 quart (qt) = 2 pints (pt) 1 quart (qt) = 4 cups (c) 1 gallon (gal) = 4 quarts (qt) 1 gallon (gal) = 8 pints (pt) 1 gallon (gal) = 16 cups (c)

Cardinal Numbers

Cardinal Numbers are numbers used to indicate quantity. The Cardinal Numbers are the same as the Natural Numbers (for the purposes of elementary school students' understanding).

Circumference (definition)

Circumference is the distance around a circle. (circum = around, fer = carry)

Collinear

Collinear describes two or more points that are on the same line (they are in a straight row or lined up).

Common Decimal and Percentage Equivalents (1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/8, 3/8, 5/8, 7/8, 1, 2, 3 1/2)

Common Equivalents 1/2 = 0.5 = 50% 1/3 = 0.33 1/3 = 33 1/3% 2/3 = 0.66 2/3 = 66 2/3% 1/4 = 0.25 = 25% 3/4 = 0.75 = 75% 1/5 = 0.2 = 20% 2/5 = 0.4 = 40% 3/5 = 0.6 = 60% 4/5 = 0.8 = 80% 1/6 = 0.16 2/3 = 16 2/3% 5/6 = 0.83 1/3 = 83 1/3% 1/8 = 0.125 = 12.5% 3/8 = 0.375 = 37.5% 5/8 = 0.625 = 62.5% 7/8 = 0.875 = 87.5% 1 = 1.0 = 100% 2 = 2.0 = 200% 3 1/2 = 3.5 = 350%

Consecutive Integers

Consecutive integers are integers that differ by 1. Example:  -3 and -2 are consecutive integers Example:  x and x+1 are consecutive integers

Coplanar

Coplanar describes two-dimensional figures that are on the same plane.

Estimation

Estimation is a mathematical process of finding an approximate value of a variable, expression, or operation. The most common way to find an estimate is to round off the numbers in the problem to numbers that are easy to calculate and then find the answer to the rounded problem. The symbol for an estimated or approximate answer is ≈. For example:   287 + 321 + 878 ≈ 300 + 300 + 900 = 1,500

Factoring Polynomials

Factoring Polynomials 1. Factor out any common factors in all terms. 2. If the polynomial has four terms, factor it by grouping. 3. 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.) 4. If it is a trinomial and the coefficient of the x² term = 1, un-FOIL to factor. 5. If it is a trinomial and the coefficient of the x² term is not 1, use the AC method to factor.

Central Tendency Measures

Measures of Central Tendency are statistical measures such as mean, median, and mode. Data has a tendency to cluster or center on certain values. The term "average" is also used to indicate measures of central tendency; as such, the mean, median, and mode are ALL averages.

Multiple of an Integer n

Multiples of integers are formed when the integer, n, is successively multiplied by the whole numbers. As such, the multiples of the integer 3 are 0, 3, 6, 9, 12, 15, etc. The way the "multiple of an integer n" is written in algebra is kn where "k" is a whole number.

Negative Integers

Negative integers are those integers less than zero. Negative integers appear to the left of zero on a number line. The negative sign is always written. Negative integers are the opposite of the positive integers. The number zero is neither positive nor negative.

Odds (Statistical)

Odds and probability are related concepts. With probability, you compare the number of favorable outcomes to the total possible number of outcomes. With odds, you compare the number of favorable outcomes to the number of remaining (unfavorable) outcomes. 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

Ordinal Numbers

Ordinal numbers, unlike cardinal numbers that indicate a quantity, are numbers that indicate order or rank. Ordinal numbers are 1st, 2nd, 3rd, 4th, 5th, etc., unlike cardinal numbers that indicate a quantity, are numbers that indicate order or rank. Ordinal numbers are 1st, 2nd, 3rd, 4th, 5th, etc.

Perimeter (Definition)

Perimeter is the measure of the distance around a polygon     (peri = around, meter = measure)

Simultaneous Equations (definition and 3 ways to solve)

Simultaneous Equations are two or more equations with multiple variables. These are often called systems of equations. A solution gives values for the variables that are true for all equations in the system. There are many ways to solve a system of equations. Three ways discussed in beginning algebra are:<div>  1. Elimination (sometimes called adding) 2. Substitution 3. Graphing Another method presented in intermediate/advance algebra is the use of matrices.</div>

Steps to Solve Praxis Problems

Steps to Solve Praxis Problems: 1. Read the question carefully, circling, underlining, and/or writing down what you are looking for. 2. Pull out important information. 3. Draw, sketch, or mark in diagrams or on scratch paper. 4. If you know a simple method or formula, work the problem out as simply and quickly as possible. 5. If you don't know a simple method or formula... a. Try eliminating some unreasonable choices b. Work backwards from the answers c. Substitute in numbers—work a simpler problem d. Try approximating to clarify your thinking 6. Check to be sure your answer is reasonable.

Surface Area

Surface area is the total area of the faces and curved surfaces of a solid figure.

Properties of One

The number one has some interesting properties: • One is the multiplicative identity • Any number multiplied by one equals the number • One is neither prime nor composite • A number with an exponent of 1 equals the number • The number one raised to any power equals one • Any nonzero number divided by itself equals one

Properties of Zero

The number zero has some interesting properties: • 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

Triangles (two ways to classify)

Triangles can be classified in two ways: 1. By the angles in the triangle: acute, obtuse, and right. 2. By the sides in the triangle:  equilateral, isosceles, and scalene.

Velocity

Velocity is a measure of the speed of an object and the direction in which it is going. For example, the wind is blowing 22 mph in a NE direction.

Volume (Definition)

Volume is the measure of the number of cubic units that can fit inside an object. Volume is also known as capacity.

Flow Charts

A flow chart is a diagram used to visually describe an algorithm. A flow chart shows a step-by-step path for the algorithm.    Small circles are used to show the start and end points. Diamonds are used for decisions -- to ask questions and branch the flow chart depending on the answer to the question. A rectangle is used to show a process or action step. Input and output are represented by parallelograms. Other symbols are used in more complex flow charts.

What is the difference between a fraction and a ratio?

A fraction compares PART of something to its whole. A ratio compares two different things - neither thing is always the whole or sum of the two.

What is a fraction?

A fraction is a numeral showing a part of a group or a part of a set expressed as division. The top number is called the numerator; the numerator indicates the part. The bottom number is called the denominator; the denominator indicates the total in the group or set.

Frequency Distribution

A frequency distribution is an organized way to display the results from an experiment. Frequency distribution tables are often used in probability for situations that do not use sample spaces (a set listing ALL possible outcomes in an experiment).  The three types of probabilities are classical, empirical, and subjective. The probability for situations that use a frequency distribution is called Empirical Probability or Relative Frequency Probability. Classical probability uses a list of ALL possible outcomes. Subjective probability is based on a person's knowledge of the situation (an educated guess).

Polyhedron (definition only)

A polyhedron is any three-dimensional solid with faces, edges, and vertices. Euler's formula describes an interesting property of convex polyhedron:  V - E + F = 2

Sequence

A sequence is a set of members where order is important. For example, the sequence of letters ABC is entirely difference from the sequence of letters ACB - although we are using the same three letters, they are in different order.

Tessellation

A tessellation is a two-dimensional plane created by one or more polygon shapes fitted into each other so no "open space" remains. Kepler first discussed tessellations in the early 1600's. Equilateral triangles, squares, and hexagons are the only regular polygons that tessellate. There exists an entire branch of geometry about tessellations, begun by Russian scientist Fyodorov in the late 1800's. Tessellations for 3+ dimensional spaces are also defined.

Variable

A variable is used in algebra to represent a value that changes within the parameters of the problem. The opposite of a variable is a constant. Lowercase letters of the alphabet are generally used to denote a variable. There are two types of variables: dependent and independent.

proportion

A true statement that two ratios or fractions are equal. Every proportion is made up of 4 numbers. (2 numerators and 2 denominators) If have a variable, cross multiply to solve. ex: x/9 = 12/54 54x= (9)(12) 54x= 108 (54x)÷54 = 108÷54 x=2

Item Analysis (Praxis study tip)

After taking a practice test, do an item analysis of each problem you missed: 1. What competency/domain is it assessing? 2. What specific concept or skills it is assessing? 3. What is the reason you missed the problem? 4. What were the possible stem distractors that caused the error? 5. What vocabulary should you learn? Add this concept to your study plan as an area of focus.

Questions to ask yourself about your Praxis answer

Always ask yourself if your answer is reasonable. If you have time left over at the end of the test, go back through each answer to be sure it is reasonable - but do not change your answer unless it is clearly wrong.

Equation

An equation is a statement where an algebraic expression is equal to another algebraic expression or constant. Equations and expressions are often confused. The equation has an equal symbol; whereas, the expression does not have a comparison symbol (merely a collection of terms). We solve an equation but we evaluate or simplify an expression.

Equilateral Polygon

An equilateral polygon is one where all sides of the polygon are the same measure. If a polygon is both equiangular and equilateral, it is called a regular polygon.

Expression

An expression is a collection of terms that have been added or subtracted. Equations and expressions are often confused. The equation has an equal symbol; whereas, the expression does not have a comparison symbol (merely a collection of terms). We solve an equation but we evaluate or simplify an expression.

Ordered Pair

An ordered pair is a pair of numbers indicating the location of a point. The first number, called the first coordinate, tells how far the point is right or left on the horizontal x-axis. The second number, called the second coordinate, tells how far the point is up or down on the vertical y-axis. The actual point is the intersection of those two coordinates.

Angle

An angle is the figure formed by two rays, called the sides of the angle, which share a common endpoint, called the *vertex* of the angle. An angle can be named by its vertex or by naming a point on each leg with the vertex point in the middle. The angle below is Angle B or Angle ABC, also written as ∠ABC.

working with unit fractions

Any fraction where there is a 1 in the numerator They make up all fractions, for example: 5/7 = 1/7 + 1/7 + 1/7 +1/7 + 1/7

Subtraction of Integers

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

Tip for Praxis problem with fractions of unlike denominators

Change all fractions to common denominators before beginning your calculations.

Rule of Threes *to solve Ratio/Proportion Problems*

Every proportion is made up of *4 numbers* (2 numerators and 2 denominators) If word problem gives you 3 out of the 4 numbers in a proportion problem, you can cross multiply and solve for the 4th (missing) number. (When you cross multiply a fraction/ration, equation should be linear, no longer fractions - see example) Ex: Margery is pouring orange juice for the kids in her class. She pours 3 cups every 2 minutes (Margery is super slow at pouring juice). How long will it take her to fill up 24 cups of juice? 3 cups/2 mins = 24 cups/x minutes *cross multiply to get:* 3x = 24*2 3x=48 x=16 It will take Margery 16 minutes to pour 24 cups of juice. (Get it together, Margery!) *TRICKY QUESTION!!*: If the question said: "Margary *poured* (as opposed to pourS) 3 cups *in* 2 mins....how long will it take for her to *finish* filling all 24 cups for her class?" then you would have to subtract out the 3 cups she ALREADY filled!! 24 cups total - 2 cups already filled = 21 cups left to fill 3 cups/2 mins = 21 cups/x minutes *cross multiply to get:* 3x = 21*2 3x=42 x=14 *BE VERY CAREFUL ABOUT THIS TRICK!*

For making educated guesses on the Praxis test, what is good to know about the choices?

Except in cases where the problem asks you to compare or order numbers, all answer choices are listed in numerical order. If you must guess, start with the middle answer choice and adjust up or down from there until you find the correct answer.

Percent Change

The *percent of change* is also known as the percent of increase or the percent of decrease. To calculate the percent of change, 1. Find the difference between the new amount and the original amount, that *difference*= numerator 2. Divide that *difference* (not the new amount!, the difference) by the original amount, original amount = denominator 3. Multiply by 100 ex: Population was 25,000 in 2014 Population was 28,000 in 2015 What is the percent change in the population? 28,000 - 25,000 = 3,000 3,000/25,000 = 3/25 = 12/100 = 12%

Customary System of Measurement

The Customary System of Measurement is also called the English System of Measurement (or the Common System of Measurement). This is the primary measurement system used in the United States. Length = inch, foot, yard, rod, mile, etc. Weight = ounce, pound, Ton, etc. Volume = liquid ounces, cup, pint, quart, gallon, etc.

English System of Measurement

The English System of Measurement is also called the Customary System of Measurement (or the Common System of Measurement). This is the primary measurement system used in the United States. Length = inch, foot, yard, rod, mile, etc. Weight = ounce, pound, Ton, etc. Volume = liquid ounces, cup, pint, quart, gallon, etc.

Sieve of Eratosthenes

The Sieve of Eratosthenes is a technique to teach young children about prime numbers. A paper is numbered from 2-100. Circle 2 because it is prime. Then count every 2 numbers and cross them off (cross off the multiples of 2). The next number after 2 that is not crossed off is 3, circle it. Count every 3 numbers (multiples of 3) and cross them off. The next number after 3 that is not crossed off is 5, circle it. Count the multiples of 5 and cross them off.... This process continues for the entire chart. All crossed-off numbers are composite; the prime numbers are circled.

Mean

The mean is used in statistics to describe one of three measures of central tendency. Find the mean by adding the data values and dividing by the number of data values. (Also known informally as average.)

Range (of Data)

The range is used in statistics to describe the difference between the smallest value and the largest value in a data set. To find the range, find the smallest value (the minimum or min) and find the largest value (the maximum or max). The range is the difference when you subtract the min from the max.

Addition of Integers

The RULE for adding integers is:<div>    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. </div>

Division of Integers

The RULE for dividing integers is: 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.

Multiplication of Integers

The RULE for multiplying integers is: 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.

Multiplication of Whole Numbers

The algorithm for whole number multiplication is: 1. Line up the numbers vertically so place values are in the same column 2. Multiply beginning in the ones place -- multiply each digit in the multiplicand. If the product is greater than 9, write the tens place digit above the next column to the left (regrouping or carrying) 3. Next, multiply by the tens place -- to signify that you are in the tens place, put one zero in the partial product. Multiply each digit in the multiplicand. 4. Add the partial products, and put commas in the answer to separate the digits into periods

Subtraction of Whole Numbers

The algorithm for whole number subtraction is: 1. Line up the numbers vertically so place values are in the same column 2. Subtract beginning in the one's place 3. Use regrouping (formerly called borrowing) if the top number is too small to allow subtraction 4. Put commas in the answer to separate the digits into periods

vinculum

The line that separates the numerator from the denominator

The number π is a mathematical constant that is the ratio of a circle's circumference to its diameter. The constant, sometimes written pi, is an irrational number approximately equal to 3.14159 or 22/7. (i.e., a circle's diameter can be wrapped around its circumference pi times - 3 times and a little bit more.)

Operation Properties (Three of them)

There are three primary properties of operations: • commutative • associative • distributive Other operation properties are closure, zero, inverse, and identity.

Plotting a Point on a Coordinate Grid

To plot a point with the coordinates of (x, y), we follow along the x-axis until we get to the first coordinate, and then we follow along the y-axis until we reach the second coordinate. We mark a small dot at the location where these two coordinates intersect.

Order of Magnitude

a measure of powers of 10 (basically the number of zeroes to the right of the number if positive ~ or number of zeroes to the left of the number if the ~ is negative.) This concept is often used in working with scientific notation. A number that has been multiplied by 10 has increased its _____________ by 1. A number that has been multiplied by 1000 has increased its _____________ by 3. 1 has a ~ of 0. 10 has a ~ of 1 100 has a ~ of 2 0.1 has a ~ of -1 0.01 has a ~ of -2 0.001 has a ~ of -3

What are the parts of a subtraction problem?

subtrahend; minuend; difference

Percentages (Definition)

a way of expressing a number, especially a ratio, as a fraction of 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. *Remember: per = divided by; cent = 100*

Counting Numbers

same thing as natural numbers whole numbers 1 and up **DOES NOT INCLUDE 0**

radius

The distance from the center of a circle to any point on the circle. It is half the diameter. r= d/2

Ordering Whole Numbers

"When you are asked to order whole numbers, write them above one another with the place values lined up. Then, 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 Write them above each other with the place values lined up as you would if you were going to add the numbers. Looking at the place values from left to right, the largest number is 986,733. The next largest number is 77,922. Both the first and third numbers start in the same place value but 5 is larger than 3 so 5,139 is larger than 3,950

probability equation

#possibilities of what I want -------------------- total # possible outcomes

Division of Fractions & Mixed Numbers

*Division of Fractions & Mixed Numbers: is not defined.* As such, we use the properties of our number system to *change the division problem into a multiplication problem*, which *is* defined. (see pic) -Change mixed numbers to improper fractions. -Change 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. A reciprocal is the inverse of a fraction. Multiplication is the inverse of division. The inverse of an inverse is the same as the original problem. As such, multiplication of a reciprocal is the same as dividing by the original divisor. *Remember on test, if answers are in fraction form, correct answer will always be reduced* *Do NOT need common denominators* *Answer should be MORE than the original factors* (ex: ½ ÷ ¼= 4/2 = 2) <--2 is more than 1/2 or 1/4*

Equivalent Fractions

*Equivalent fractions* are fractions which simplify to the same simple fraction When two fractions are equivalent, their cross products are equal (see picture) If have a variable, cross multiply to solve. (Each side will now be a linear equation, no longer in fraction form. See example) ex: x/9 = 12/54 54x= (9)(12) 54x= 108 (54x)÷54 = 108÷54 x=2 *Remember on test, if answers are in fraction form, correct answer will always be reduced*

LCM

*Lowest Common Multiple (LCM)* of two or more numbers is the smallest number that is a multiple of all the numbers. One way to find the LCM is to count by each of the numbers and find the first number that is a multiple of all. For example, find the LCM of 9, 12, and 18 Then make a Venn Diagram with all the factors, the shared factors in the center. Lastly, multiply all the numbers together. This is the LCM. (see pic) example LCM word problem: One day, Edward and his friends had lunch while sitting at tables of 15. Another day, they had lunch at tables of 10. What is the *smallest number of people that could be in the group?* (30 people)

Simplifying Fractions

*Simplifying fractions* is sometimes (incorrectly) called reducing a fraction. Expand the numerator and denominator into prime factorizations (see pic). "Cancel" any ones such as 3/3 or 5/5. Then multiply straight across those numbers that are left. The resulting (simplified) fraction is equivalent to the original fraction but is in simplified form. *Remember on test, if answers are in fraction form, correct answer will always be reduced*

solve multi-step mathematical and real-world problems using multiplication of rational numbers MULTIPLYING FRACTIONS

*When we multiply fractions, we are saying "(fraction 1) OF (fraction 2)* (Ex: 1/2 * 2/3 means 1/2 OF 2/3) https://learnzillion.com/lesson_plans/7906 Just multiply across, then simplify

Rate Conversion

1 Euro =$1.25 Clock costs 55 Euros. How much would that cost in dollars? (1.25 USD/1 Euro) * Price $1.25/Euro *55 Euros = $68.75

Divisibility Rules

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

Decimal Number 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.

Decimal Number 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.

Fractions (Definition, Meaning, and Parts)

A *fraction* represents equal-sized parts of a whole. The top number of a fraction is called the *numerator* because it is the number of parts. The *denominator* is a denominate number (measurement of size) telling how many of the equal-sized parts are in the whole. The line between the numerator and denominator indicates division and is called a *vinculum* Note that the parts MUST be equal-sized. The parts are usually indicated by coloring them or shading them *Remember on test, if answers are in fraction form, correct answer will always be reduced*

Circle

A circle is a closed figure made up of all points that are equidistant from another point (the center). The distance from the center point to the edge of the circle is called the radius. A = (π)r^2 C = (π)d = 2(π)r

Factor Tree

A factor tree is a method used to find a number's prime factorization. Start by splitting the number into any two factors that multiply to make that number. Then split each of those two factors into two factors each. Keep splitting each "branch" of the factor tree until you reach a prime number, which cannot be split into factors. The final prime numbers at the end of each "branch" of the factor tree are the prime factorization of the number. See pic for factor tree which shows that the prime factorization of 120 = 2 x 2 x 2 x 3 x 5

Rational Numbers

Any number that can be represented as the RATIO of two INTEGERS (integers are the counting numbers, their negatives, and zero (..., -3, -2, -1, 0, 1, 2, 3 ...).) -integers (because every integer is the ratio of the integer over 1. ex: 2 = 2/1) -fractions -decimals that end (like 12.5) or repeat the same numbers over and over (like 23.666666) -percentages **Square roots of non-perfect squares can NOT be expressed as the fraction of two integers, so they IRrational numbers* All real numbers are either rational or irrational. Sets in mathematics include the set of integers (Z), rational numbers (Q), primes (P), real numbers (R), natural numbers (N), whole numbers (W), etc.

solve multi-step mathematical and real-world problems using division of rational numbers DIVIDING FRACTIONS

Flip the 2nd fraction upside down, then multiply straight across, then simplify answer *If numerator and denominator are both divisible by the 2nd number, can just divide across* ex: (2/3) ÷ (3/4) = (2/3) * (4/3) = (2*4) / (3*3) = (8/9) cannot divide 2 by 3, so cannot divide across in this problem ex2: 2/3 ÷ 5 Make the whole number into fraction: = 2/3 ÷ 5/1 INVERT the 2nd fraction: =2/3 * 1/5 MULTIPLY ACROSS: =(2*1)/(3*5) =2/15 cannot divide 2 by 5, so cannot divide across in this problem ex3: 3 ÷ 1/4 INVERT the 2nd Fraction: = 3/1 ÷ 1/4 Multiply across: =3/1 * 4/1 = (3*4)/(1*1) =12/1 SIMPLIFY = 12 ex4: 8/10 ÷ 2/5 = ? INVERT the 2nd Fraction: =8/10 * 5/2 = ? Multiply across: (8*5)/(10*2) = ? Simplify (40) / (20) = ? 4/2 =2 On this one, the numerator and denominator are both divisible by the 2nd number, so could also just divide across 8/10 ÷ 2/5 = (8÷2)/(10÷5) = 4/2 = 2 (*When you flip the 2nd fraction upside down, it is called the RECIPROCAL of the 2nd fraction, or INVERTING the 2nd fraction) ♫ "Dividing fractions, as easy as pie, Flip the second fraction, then multiply. And don't forget to simplify!" ♫

Multiplying Fractions & Mixed Numbers

Multiplying Fractions & Mixed Numbers: Two ways to do it: 1.) multiply top times top, multiply bottom times bottom simplify if you can 2.) (see pic) -Change mixed numbers to improper fractions. -Change whole numbers to improper fractions with a "1" as the denominator. -Write the two fractions to be multiplied horizontally beside each other. -Expand each numerator and each denominator into a prime factorization (see pic). -"Cancel" any ones such as 3/3 or 5/5. Multiply what is left straight across *Remember on test, if answers are in fraction form, correct answer will always be reduced* *Do NOT need common denominators* *Answer should be LESS than the original factors* (ex: ½ * ¼= 1/8) <--1/8 is smaller than both 1/2 and 1/4*

Composite Numbers

Numbers that are not prime numbers. numbers that have more than two factors or divisors

Order of Operations

PREMDAS 1. Simplify inside parentheses/brackets/etc. 2. Simplify any expressions with roots 3. Simplify any expressions with exponents 3. Perform multiplication / division from left to right 4. Perform addition / subtraction from left to right PREMDAS or Please really excuse my dear Aunt Sally

Unit Rates

Price per item Total Cost divided by number of items Ex: Roses are $24/dozen $24/12 = $2 each

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,* because a prime number is a number with only 2 factors one AND itself, one is itself. 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

GCF

The *Greatest Common Factor* (GCF) of two or more numbers 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. Then make a Venn Diagram with all the factors, the shared factors in the center. Lastly, multiply all the numbers in the middle of the Venn diagram together. (see pic) example GCF word problem: Mei is a dental sales representative who wants to distribute 16 brochures and 10 pamphlets to local dental offices. She wants to deliver *the same combination* of brochures and pamphlets *to each* office, *without having any materials left over.* What is the *greatest number* of dental offices Mei can distribute materials to? (2 offices)

Subtraction of Fractions & Mixed Numbers

To subtract fractions & mixed numbers: 1. Write the two fractions/mixed numbers vertically above each other (lining up place value) 2. Change the fractions to a common denominator. 3. Subtract the numerators only *be careful to regroup one whole (2/2, 3/3, 4/4, etc.) if you need to borrow* 4. Put that difference over the common denominator. 5. Simplify the answer. *Remember on test, if answers are in fraction form, correct answer will always be reduced*

How would you write 99 as a Roman numeral? explain why

XCIX In Roman numerals, when a smaller value appears to the left of a higher value, flip those two numbers around and subtract. The number can only be one place value away though to do the flip thing. (ex: canNOT do IC as 99 (which would be 100 -1), can only do (XC = 100-10=90) + (IX= 10-1 = 9) ) Write the number in expanded form and convert each term to Roman numerals: 90 + 9 = 99 XC IX XCIX represents 99.

reciprocal

a fraction inverted (flipped upside down) *if you multiply 2 fractions that are reciprocals of each other, the answer is always 1 (ex: 2/3 * 3/2 = 6/6 = 1)

Natural Numbers

the counting numbers (1, 2, 3, 4 ...) whole numbers 1 and up **DOES NOT INCLUDE 0**

When the dividend is GREATER than the divisor, the answer should be _________ than one.

greater ex: (3/4) ÷ (5/8) =? 3/4 is GREATER than 5/8, so the answer should be GREATER than one the answer is 1 and 1/5, so this is correct

When the dividend is LESS than the divisor, the answer should be _________ than one.

less ex: (1/2) ÷ (5/8) = ? 1/2 is LESS than 5/8, so the answer should be LESS than one. the answer is 4/5, so this is correct

round multi-digit numbers to any place value

round 847,256 to the 10,000s place: 850,000 round 847,256 to the thousands place: 847,000 round 847,256 to the hundreds place: 847,300

sample space

what are all the possible outcomes? what are all the things you could pick (in a probability experiment)? The set of all possible outcomes of a probability experiment (all the possibilities NOT the number of possibilities) Usually shown in a tree map, table or chart Lets you see how many possible outcomes there are, so you can figure out probability more easily

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