9Ordering Large Numbers Write them above one another with the place values lined up.Starting from the left, look for the largest value.For example, if you are asked to order:5, , , ,922
10Rounding NumbersRounding a number requires that you understand place value.Look at the digit to the right of the place being rounded.If that digit on the right is 5 or higher, add 1 to the place being rounded; otherwise, leave the place being rounded as is.Change all places to the right of the place being rounded to zeroes.
11Comparison 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: ≠
14Divisibility Rules2 = Even numbers (ending in 0, 2, 4, 6, and 8) 3 = If repeated sums of the digits result in 3, 6, or 9 4 = If the last two digits are divisible by 4 5 = If the last digit is 0 or 5 6 = If the number is divisible by both 2 and 3 8 = If the last three digits are divisible by 8 9 = If repeated sums of the digits result in 9 10 = If the last digit is 0
15Prime NumbersPrime Numbers = Integers greater than 1 with exactly 2 factors or divisors; numbers that are evenly divisible by only 1 and themselves.The number 2 is the first prime and it is the only even number that is prime. The number 1 is neither prime nor composite. Memorize the prime numbers 1-100:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
16Other Number Classifications Prime vs. CompositeOdd vs. EvenDenominate numbersConsecutive integersCardinal numbersOrdinal numbers
17Number LineA 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.
19Interpreting Remainders Do you round up? Do you round down? Do you use the remainder as part of the answer (or as the entire answer)?How many boxes can be filled? (use only the quotient; ignore the remainder)How many cans are needed to paint the wall? (round the quotient to the next greater whole #)How many in the last box that isn’t completely full? (use only the remainder)
20Prime FactorizationThus, the prime factorization of 120 = 2 x 2 x 2 x 3 x 5
21GCF12 = 2 × 2 × 318 = × 3 × 330 = × × 5GCF = × 3The factors common to all three numbers above are 2 and 3.2 x 3 = 6 so 6 is the GCF of 12, 18, and 30.
254 Ways to Indicate Multiplication Using a small “×”, such as 3 × 5. Note that the “×” is not used for multiplication in algebra because it might be confused with the variable “×”.Using a small, raised dot, such as 3 • 5Using parenthesis, such as (3)(5) or 3(5) or (3)5Using no symbol, such as 3y meaning 3 times y
27PEMDAS Please Excuse My Dear Aunt Sally Order of OperationsSimplify inside parentheses or grouping symbolsSimplify any expressions with exponentsPerform multiplication & division from left to rightPerform addition & subtraction from left to rightPEMDAS Please Excuse My Dear Aunt Sally
28Properties of Zero Division by zero is undefined Zero is neither positive nor negativeZero is the additive identityAny number multiplied by zero equals zeroZero is used as the universal place holderZero is neither prime nor compositeZero has no multiplicative inverseA number with an exponent of zero equals 1Zero factorial equals 1
29Properties of One One is the multiplicative identity Any number multiplied by one equals the numberOne is neither prime nor compositeA number with exponent of 1 equals the numberThe number one raised to any power equals oneAny nonzero number divided by itself equals one
33What is a Fraction?A fraction represents equal-sized parts of a whole.The top number is called the numeratorThe bottom number , the denominator, is a denominate number (measurement of size)The line between numerator & denominator indicates division and is called a vinculum.
34Proper & Improper Fractions A proper fraction has a numerator smaller than the denominator and indicates a fraction less than one whole: 3/8 1/4 14/15 4/5An improper fraction has a numerator larger than (or equal to) the denominator and indicates a fraction that is equal to one or more than one whole: 3/2 7/4 16/ / /5An improper fraction can be changed into a whole number or a mixed number by dividing the denominator into the numerator.
36Mixed NumberA mixed number is a whole number and a proper fraction combined. Mixed numbers may also be called mixed fractions.This graphic shows two whole pizzas and a fraction of 3 pieces out of 4 2 3/4
37Greatest Common Factor The GCF is the largest number that is a factor of all the numbers.One way to find the GCF is to write the prime factorization of each of the numbers above each other. Then “bring down” those factors that are in common and multiply them:Find the GCF of 12, 18, 30:12 = 2 × 2 × 318 = × 3 × 330 = × × 5GCF = × = 6
38LCDThe Lowest Common Denominator is the smallest number that is a multiple of all the denominators.One way to find the LCD is to count by each of the denominators and find the first number that is a multiple of all.Another way to find the LCD is to write a prime factorization of each denominator, and then “bring down” one of each factor and multiply.
39Equivalent FractionsFractions that simplify to the same simple fraction.When two fractions are equivalent, their cross products are equal :3 x 6 = 18 2 x 9 = 18
41Mixed NumberA mixed number indicates the whole amounts and the parts (a fraction).To convert a mixed number to an improper fraction, multiply the whole number by how many parts are in a whole, and then add the remaining parts.For example, 6 2/3 means there are 6 whole amounts of 3/3 (6 x 3 = 18) so there are 18/3.Adding the remaining 2/3 results in a total of 20/3 in 6 2/3.
42Improper FractionsAn improper fraction shows a total number of parts, but in those parts is at least one whole.To convert an improper fraction to a mixed number, divide the numerator by how many parts are in a whole. The quotient becomes the whole number and the remainder becomes the numerator of the fraction part of the mixed number.For example, 23/3 equals 23 divided by 3, which is 7 r 2 or 7 2/3
43Why Find Common Denominators? footersfootersfooters
45Why Change to Common Denominators? 3 applesoranges7 ??????
46Add Fractions & Mixed Numbers Write the two fractions/mixed numbers vertically above each other (lining up place value)Change the fractions to a common denominator.Add the numerators only.Put that sum over the common denominator.Simplify the answer.
47Subtract Fractions & Mixed Numbers Write the two fractions/mixed numbers vertically above each other (lining up place value)Change the fractions to a common denominator.Subtract the numerators only (careful to regroup one whole (2/2, 3/3, 4/4, etc.) if you need to borrow).Put that difference over the common denominator.Simplify the answer.
49ReciprocalA 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.
50Divide Fractions& Mixed Numbers Change any mixed numbers and any whole numbers to improper fractions with a “1” on the denominator.Write the two fractions horizontally beside each other.Write the reciprocal of the divisor (flip the second fraction upside down) and change the operation to multiplication.Expand each numerator and each denominator into a prime factorization. “Cancel” any ones such as 3/3 or 5/5. Multiply what is left straight across.
51Change Fractions to Decimals & Percentages Change a fraction to a decimal by dividing the denominator into the numerator. Keep dividing until the decimal number repeats or terminates. Draw a line (vinculum) above the repeating portion.Change a fraction to a percent by first changing it to a decimal as explained above, and then moving the decimal point two places to the right. Remember to append the percent symbol.
52Comparing & Ordering Fractions Cross Multiply (Means extremes property)More than two fractions: change them to common denominators (or use reasoning, such as: are they more than half)
55Decimal PointA decimal point is a period that indicates the location of the one’s place – the decimal point always comes to the right of the one’s place. If there are no fractional decimal numbers to the right of the decimal point, the decimal point doesn’t have to be written. It is understood.
56Comparing 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.
57AdditionDecimal 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.
58SubtractionDecimal 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.
59MultiplicationDecimal numbers are multiplied by temporarily ignoring the decimal point. Multiply the two numbers as though they were whole numbers. In the final product, place the decimal point to signify the number of decimal places in both numbers of the original problem.For example: 2.3 (one decimal place) x (three decimal places) is the same as 23 x 1456 with the answer having four decimal places (1 + 3 from the original problem)
60DivisionDivision 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).
63Scientific Notation 1234.5 1.2345 × 103 Scientific Notation is a way to write very large or very small numbers using powers of 10. To convert a number into scientific notation, move the decimal point so the resulting number is between 1 and 10. Then state the power of 10. Because we use a Base 10 number system, an easy way to know what power of 10 is needed, the exponent indicates the number of decimal places the decimal point was moved. The exponent is negative if the decimal point was moved to the right; the exponent is positive if the decimal point was moved to the left. × 103
70(per = divided by; cent = 100) What is a Percentage?A percentage (or a percent) is a way of expressing a number, especially a ratio, as a fraction of 100.(per = divided by; cent = 100)The percent key on a calculator merely divides by 100. If your calculator doesn’t have a percent key, hit the divide key and then 100.
713 Types of % Problems What number is 15% of 45? x = (0.15) ∙ (45) What percent of 45 is 15? 45 ∙ x = 15 or 45x = 1515% of what number is 45? (0.15) ∙ x = 45 or 0.15x=45
72Convert % 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% = 0.14Change a percent to a fraction by writing the percent as a fraction over 100 and simplifying:14% = 14/100, which simplifies to 7/50Remember: per = divided by; cent = 100
73Convert Decimals to % 0.14 = 14% Change a decimal to a percentage by moving the decimal point two places to the right and appending the percent sign:0.14 = 14%
75what number variable (x) is = of multiply Solve % Using AlgebraTo solve percentages using algebra, write the problem as an algebraic statement wherewhat number variable (x) is = of multiplyWhat number is 15% of 45? x = (0.15) ∙ (45)What percent of 45 is 15? 45 ∙ x = 15 or 45x = 1515% of what number is 45? (0.15) ∙ x = 45 or 0.15x=45
76Unit Analysis Video: JoleneMorris.com, Math 115, Wk 2 Unit Analysis is the process of multiplying by successive conversion units (written in fraction form).Unit Analysis Video: JoleneMorris.com, Math 115, Wk 2Video explaining unit analysis for English units (JoleneMorris.com, Math 115, Wk6)Video explaining unit analysis for Metric units (JoleneMorris.com, Math 115, Wk6)
77Denominate NumberSpecifies a quantity in terms of a number and a unit of measurement.For example, 7 feet and 16 acres are denominate numbers.
78Rate & Unit RateA rate is a ratio between two measurements with different units.In addition to the three ways to write a ratio, rates may also use the word “per”.Rates are usually simplified to a one in the denominator (unit rate).13 miles per gallon$4.59 per pound12 inches per foot
80Sales TaxSales Tax is written in percentages (which are converted to decimal to computer sales tax).FP = MP + (ST × MP)If you know two of those three amounts, you can use basic algebra to find the missing number. Remember to state the sales tax as a percentage in application problems.
81Discounts Usually written as percentages. Amount by which the purchase price is reduced.Discount amount is the original sales price multiplied by the percentage of discount.Discounted price is the difference of the original price and the discount amount.OP – (OP * D) = DP where OP is the original price, D is the discount percentage written as a decimal number, and DP is the discounted price.
82Percent of ChangeThe percent of change is also known as the percent of increase or the percent of decrease.To calculate the percent of change,Find the difference between the new amount and the original amountDivide that difference by the original amountMultiply by 100
83Sales CommissionC = ar, where C = commission earned, a = amount of sale, and r = commission rate.Example: Juana sells cars on a 3% commission rate. She just sold a car for $23,500. What was her commission?C = arC = 23500(.03)C = $705
84Ratios, Proportions, & Percentages ANY QUESTIONS?
86What are Integers?The counting numbers, their negatives, and zero (…, -3, -2, -1, 0, 1, 2, 3 …)The symbol used for the set of integers is ℤ.
87Number LineA 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.
88Absolute Value The value portion of a number without a sign. Also described as the distance on a number line from 0.Zero is the only number that is its own absolute value (because zero is neither positive nor negative).
90Ordering & Comparing Integers Remember that negative numbers are always smaller than positive numbers.It helps to place the numbers on a number line to compare them.The larger the value of a positive number, the larger the number is.The larger the absolute value of a negative number, the smaller the number.
92Adding IntegersIf 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.
93Subtracting IntegersBecause a negative number is the inverse of a positive number, and because subtraction is the inverse operation of addition, the RULE for subtracting integers is: Change the sign of the second number to its opposite and change the operation to addition. Then, follow the rules for adding integers.
94Multiplying IntegersIf 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.
95Dividing IntegersIf 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.
96Review Properties in Number Theory Packet ZeroOneIdentity (additive & multiplicative)Inverse (additive & multiplicative)
100Definitions Point Line Plane Line Segment Ray Angle Parallel PerpendicularVertexArcCoplanarCollinearBisectorChordDiameterRadiusCircumferenceSymmetrySurface AreaPerimeterAreaVolumeTransversalPolygonSimilarCongruentProtractorCompassDensityUnit AnalysisGolden Ratio
101Angles An acute angle measures less than 90° A straight angle measures 180°A right angle measures 90°An obtuse angle measures between 90°-180°A reflex angle is measured in a clockwise direction as opposed to the normal counter-clockwise direction.
102More About AnglesWhen two lines intersect, they form four angles. The two angles opposite each other are called vertical angles.Complementary angles are two angles whose measure adds to 90°.Supplementary angles are two angles whose measure adds to 180°.Two angles are adjacent angles if they share a common vertex, they share a common side, AND they do not share any interior points.An exterior angle is an angle on the outside of a polygon that is formed by extending the side of the polygonWhen a transversal line crosses two other lines, it forms eight angles -- Corresponding angles are angles that are in the same position on each of the lines.
103Triangles Classify triangles by their legs Scalene triangle - all legs are of different length.Equilateral triangle - all three legs (sides) are of equal measure.Isosceles triangle - two of the three legs are of equal measureClassify triangles by their anglesRight triangle - one angle is a right angle.Acute triangle - all three angles are less than 90°Obtuse triangle - one of the angles is obtuse.
1053-Dimensional Solids Sphere Rectangular Solid (Cuboid) Cube Cylinder ConePrismPyramidPolyhedronEuler’s Polyhedron Formula is V – E + F = 2
106Measurement English System or Common System Metric System Length = inch, foot, yard, rod, mile, etc.Weight = ounce, pound, Ton, etc.Volume = liquid ounces, cup, pint, quart, gallon, etc.Metric SystemMeterGramLiter
109TessellationsA tessellation is a two-dimensional plane created by one or more polygon shapes fitted into each other so no “open space” remains.Equilateral triangles, squares, and hexagons are the only regular polygons that tessellate.Check out the tessellations of M. C. Escher:
110TransformationsA transformation in geometry changes the position of a shape on the coordinate plane. There are four forms of transformation:translation (slide)rotation (turn)dilation (scale)reflection (flip)
111Net (or network)A net is a two-dimensional representation of a three-dimensional object. If a net is cut out, it can be put together to form the three-dimensional object it represents.
114Measures of Central Tendency Data has a tendency to cluster or center on certain values. The term “average” is also used to indicate measures of central tendency.Mean (evenly distributed / “average”)Mode (bimodal distribution / most popular)Median (outliers)Range
115QuartilesQuartiles 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 Q1The second quartile, also called the median, splits the data in half. It is denoted by Q2The third quartile, also called the upper quartile, splits off the higher 25% of the data. It is denoted by Q3
116TrendsA 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.)
117AlgorithmAn algorithm is a step-by-step process for solving a problem.An example of an addition algorithm is:Line up the numbersAdd each column starting on the rightCarry any tens-place digits to the next columnPlace commas between periods in the answerAn algorithm is often written as a flowchart showing steps, branches, and decisions.
118ProbabilityRatio of how likely a specific event is to happen when compared to all possibilities of events that might happen.Most often written as a fraction, but it may also be written as a decimal or a percentage.Odds and probability are related concepts. With odds, you compare the number of favorable outcomes to the number of remaining (unfavorable) outcomes.
119Probability of Multiple Events Independent (OR) , mutually exclusive addIndependent, non-exclusive add then subtract the events in commonDependent (AND) multiply (Fundamental Counting Principle)If dependent, be sure to use the conditional second probability
120Odds Related to probability Probability favorable outcomes / possible outcomesOdds favorable outcomes / unfavorable outcomesEXAMPLE: If you have a box with 2 red balls and 3 blue balls, the probability of randomly picking a red ball is 2 out of 5 or 2/5. The odds of randomly picking a red ball are 2 for and 3 against, or 2:3
124Sequences Series Arithmetic – a common number added Geometric – a common number multipliedFibonacci – each term comprised of the sum of previous two termsSeriesHarmonic – sum of progressive unit fractions
125Charts & Graphs (see study packet) Bar(broken scale)HistogramLineCircle or PiePictographTableScatterplotVenn DiagramStem & Leaf PlotBox & Whiskers PlotLogic DiagramFlow Chart
126Deductive & InductiveDeductive 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.
129Algebra & Algebraic Thinking Algebra is the study of numbers, number patterns, and relationships among numbers.Algebraic thinking is the study of our number system, patterns, representations, and mathematical reasoning.A variable is used in algebra to represent a value that changes within the parameters of the problem. Lowercase letters of the alphabet are generally used to denote a variable.The opposite of a variable is a constant.The number multiplied to the variable is a coefficient.-2x3
132PEMDAS Please Excuse My Dear Aunt Sally Order of OperationsSimplify inside parentheses or grouping symbolsSimplify any expressions with exponentsPerform multiplication & division from left to rightPerform addition & subtraction from left to rightPEMDAS Please Excuse My Dear Aunt Sally
133Expressions & Equations An expression is a collection of terms that have been added or subtracted.An equation is a statement where an algebraic expression is equal to another algebraic expression or constant.A literal equation is an equation made up of only known, measurable quantities. A literal equation is the same as a formula.An inequality is similar to an equation, but the two sides are NOT equal.
134Words that Signal + - × ÷Addition: add, sum, increase, total, rise, plus, grow, added to, more than, increased by, gain…Subtraction: subtract, subtracted from, minus, difference, take away less than, decreased by…Multiplication: multiply, multiplied by, product, times, of, twice…Division: divide, divided by, quotient, per, ratio, half, …
135Function Relationship is simply a set of ordered pairs. Function is where each input is related to exactly one output.One-to-one Correspondence is a function where each output has exactly one input.Domain (x)Range (y)
136Absolute ValueAbsolute 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).| |
138Literal EquationA 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.
139Solving Algebraic Word Problems (see the study packet) Consecutive numbersRectangular area & perimeterTrianglesUnit conversionMixturesInvestments with interestDiscounts & CommissionsDistance-Speed-Time & Uniform motion
140Simultaneous Equations Simultaneous Equations are two or more equations with multiple variables. These are often called systems of equations. There are many ways to solve a system of equations. Three ways discussed in beginning algebra are:Elimination (sometimes called adding)SubstitutionGraphing
141PolynomialsA 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.degreetype of polynomial1linear2quadratic3cubic
143Factoring Polynomials Factor out any common factors in all terms.If the polynomial has four terms, factor it by grouping.If it is a binomial, look for a difference of squares, a sum of cubes, or a difference of cubes. (Note that a sum of squares cannot be factored.)If it is a trinomial and the coefficient of the x2 term = 1, un-FOIL to factor.If it is a trinomial and the coefficient of the x2 term ≠ 1, use the AC method to factor.
144Quadratic EquationA quadratic equation is a second-degree polynomial equation (the exponent on the leading term is a 2). There are many ways to solve a quadratic equation, but the five most common ways are:Factor and set each factor equal to 0If there is no x-term, solve for x2 and apply the square root method.Graph the equation (as a parabola) and determine the solutions where the parabola crosses the x-axisComplete the squareUse the quadratic formula
145Distance FormulaThe distance formula is used to find the distance between two points. The distance formula can be obtained by creating a triangle and using the Pythagorean Theorem to find the length of the hypotenuse. The hypotenuse of the triangle will be the distance between the two points.
146Coordinate GridA coordinate plane is a two-dimensional grid for locating points. There is an x-axis and a y-axis at 90-degree angles, which divide the grid into four quadrants that are numbered counter-clockwise using Roman numerals. The origin is where the two axes cross (0, 0). A coordinate pair is a pair of numbers indicating the location of a point (x, y). Sometimes called a Cartesian grid after the mathematician René Descartes ( )
147Slope of a LineThe slope of a line is an algebraic concept used to graph linear equations. In the equation y = mx + b, the variable m represents the slope of the line. Slope is calculated by dividing the change in the y-coordinate (the rise) by the change in the x-coordinate (the run). Parallel lines have equivalent slopes. Perpendicular lines have slopes that are negative and reciprocal of each other. To graph a line when the slope and the y-intercept are known, plot the y-intercept and then use the slope to count UP and OVER to find another point on the line.
152Approximating Square Roots Approximating square roots means to find the approximate value of a number’s square root. We find approximate square roots by comparing the number to perfect square numbers where the square roots are known. For example, to find the approximate square root of 51, use the fact that 7 x 7 = 49 and 8 x 8 = 64. Since 51 is between the perfect squares of 49 and 64 (but closer to 49 than 64), the approximate square root of 51 is between 7 and 8 (but closer to 7 than 8). The approximate square root of 51 is 7.1 or 7.2