AHSGE Review for Math :AHSGE Review for Math From Passing the Alabama Graduation Exam in Mathematics from The American Book Company


Statistics and Range :Statistics and Range Statistics is a branch of mathematics that is used to organize data into forms that are easily understandable The difference between the largest number and the smallest number, in statistics, is called the range Ex. The range of the following numbers is 58 16, 73, 26, 15, and 35 73(largest) -15(smallest) = 58 so the range is 58


Mean :Mean The mean is the same as the average It is a way to find the middle number of the data To find the mean of a list of numbers, fist add together all the numbers in the list and divided by the number of items in the list Ex. Find the mean of : 38, 72, 110, 548 38+72+110+548 = 768 768/4 = 192 The mean is 192


Median :Median The median is another way to find the middle number in a group of numbers To find the median you will arrange the numbers in numerical order and if there is an odd number of items the median is the middle number; is there is an even number of items in the list the median is the average of the two middle numbers Ex. Find the median of: 42, 35, 45, 37, and 41 35 37 41 42 45 The median is 41


Mode :Mode Mode is the number that occurs most frequently in a list of numbers Ex. Find the mode of the list of grades: 70, 88, 92, 85, 99, 85, 70, 85, 99, 100, 88, 70, 99, 88, 88, 99, 88, 92, 85, 88 70 – 3 times 88 – 6 times The mode is 88 92 – 2 times 85 – 4 times 99 – 4 times 100 – 1 time


Tally Charts and Frequency Tables :Tally Charts and Frequency Tables Tally charts show each individual time that something occurs In tally charts you would use tick marks to show each time Frequency tables have the times in a list and record the frequency that each occurs In frequency tables you would just put the number of times each thing happened


Histograms :Histograms A histogram is a bar graph of data in a frequency table All you do is take the data that you are given and put it into bar graph form to show a visual of the frequency


Ratio :Ratio Ratio is used to compare things and may be in a linear form or a fraction form The numbers of a ratio must be written in the order that they are requested Ex. In a recipe it calls for 8 cups of sugar for every 6 cups of strawberries. What is the ratio of strawberries to sugar? 6/8 or 6:8 is the correct ratio


Probability :Probability Probability is the chance something will happen and it is most often expressed as a fraction Ex. Billy had 3 red marbles, 5 white marbles, and 4 blue marbles on the floor. If one marble rolls under a chair, what is the probability it was a red marble? total # of red on top of fraction 3 total # of marbles is on bottom 12 The answer may be expressed in lowest terms 3/12 = 1/4 1 out or every 4 is red


Solving Proportions :Solving Proportions Two ratios, or fractions, that are equal to each other are called proportions Ex. 1/4 = 2/8 To find a missing number from a proportion you will cross multiply and solve for the unknown Ex. 15 * 8 = 120 5 * x = 5x 5x = 120 x = 24


Integers and Order of Operations :Integers and Order of Operations Integers are all negative and positive whole numbers plus zero Ex. The absolute value of a number is the positive of the number Ex.


Adding Integers :Adding Integers When adding integers with the same sign you will add the numbers together and give the answer the same sign Ex. -4 + -7 = -11 When adding integers with opposite signs you will ignore the signs and find the difference and give the answer the sign of the larger number Ex. 3 + -7 = -4


Subtracting Integers :Subtracting Integers The easiest way to subtract integers is to change the problem to an addition problem First you would change the subtraction sign to addition and then you will change the sign of the second number to the opposite sign Ex. -6 – (-2) = Change the subtraction sign to addition and -2 to 2 -6 + 2 = -4


Multiplying and Dividing Integers :Multiplying and Dividing Integers If the numbers in multiplication or division have the same sign, then the answer is positive Ex. 6 x 8 = 48 -4 * -3 = 12 If the numbers in multiplication or division have different signs, the answer is negative Ex. 6 x -8 = -48 4 * -3 = -12


Understanding Exponents :Understanding Exponents When it is necessary to multiply a number by itself it is written as an exponent The first number is the base and the raised number is called the exponent Ex. 63 = 6x6x6 When using a negative base, an even exponent will give a positive answer and an odd exponent will give a negative answer Ex. (-2)2 = (-2) x (-2) = 4 (-2)3 = (-2) x (-2) x (-2) = -8 Any base number raised to the exponent of 1 equals the base number Any base number raised to the exponent of 0 equals 1


Square Root :Square Root To find a square root you need to know what number multiplied by itself equals the number you want to find the square root of Ex. Because 8 x 8 = 64 If a and b are two positive real numbers, then it is always true that


Order of Operations :Order of Operations An easy way to remember the correct sequence to work from in math is to remember the sentence “Please Excuse My Dear Aunt Sally” P is for parentheses E is for exponents M is for multiplication and D is for division (just start on the left of the equation and perform in the order they appear) A is for Addition and S is for subtraction (just start on the left of the equation and perform in the order they appear)


Substituting Numbers for Variables :Substituting Numbers for Variables In these problems, all you have to do is replace a known number with an unknown variable and solve for the correct answer Ex. Substitute 10 for the variable a in the following a + 1 becomes 10 + 1 17 – a becomes 17 – 10 9a becomes 9 x 10 a3 becomes 103


Understanding Algebra Word Problems :Understanding Algebra Word Problems Words that indicate that addition needs to be done are “and, increased, more, more than, plus, sum, total” Ex. 7 more than 4 is (7+4=11) Words that indicate that subtraction needs to be done are “decreased, difference, less, less than, left, lower than, minus” Ex. 8 less than 23 is (23-8=15) Words indicating that multiplication needs to be done are “double, half, product, triple, twice” Ex. The number is 3 times 4 (3x4=12) Words that indicate that division needs to be done are “ divide into, divide by, divide among, quotient Ex. The group of 70 divided into 10 teams (70/10=7)


Setting up Word Problems :Setting up Word Problems Remember that all problems have to have an equal signal Use a variable to represent the unknown or unknowns the problem is looking for The words used to mean equal are such as “is, are, or equals” Read the problems from the beginning and write the problem as you read it through Ex. Mike is twice as old as Steven. How old is Mike if Steven is 13? Mike’s age (M) = 2 x Steven’s age (s) M = 2 x 13 Mike is 26


One-Step Algebra Problems with Addition and Subtraction :One-Step Algebra Problems with Addition and Subtraction The goal in any algebra problem is to move all the numbers to one side of the equal sign and have the letter (variable) on the other side To move numbers from one side of a problem to the other you have to do the opposite to what it is asking Ex. 5 + x = 25 To move the 5 to the right you have to subtract it from both sides x = 25-5 x = 20


One-Step Algebra Problems with Multiplication and Division :One-Step Algebra Problems with Multiplication and Division To move numbers from one side of a problem to the other, using multiplication and division, is just as easy as it was with addition and subtraction You just have to remember to perform the opposite function to move the number Ex. 4x = 20 Divide both sides by 4 x = 20/4 x = 5


Multiplying and Dividing with Negative Numbers :Multiplying and Dividing with Negative Numbers The answer to an algebra problem should not have a negative sign in front of the variable Ex. -x = 5 is not completely solved. You will need to divide both sides of the equation by -1 to move the negative sign over x = 5/-1 so x = -5 A negative fraction can be written several different ways Ex. Ex. -3x = 15 Divide both side by -3 x = 15 / -3 x = -5


Solving Inequalities by Addition and Subtraction :Solving Inequalities by Addition and Subtraction If you add or subtract the same number to both sides of an inequality, the inequality remains the same (just like an equation) Ex. Solve the following inequality Add 2 to both sides and you get the answer


Solving Inequalities by Multiplication and Division :Solving Inequalities by Multiplication and Division If you multiply or divide both sides of an inequality by a positive number, the inequality symbol stays the same Ex. 4x > 20 divide both sides by 4 and get x > 5 If you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol Ex. 6 > -x/3 multiply both sides by -3 and reverse the inequality symbol to get -18 < x


Two-Step Algebra Problems :Two-Step Algebra Problems In two-step algebra problems, addition and subtractions are performed first and then division Ex. -4x + 7 = 31 Subtract 7 from both sides - 7 -7 -4x = 24 Divide both sides by -4 x = 6


Combining Like Terms :Combining Like Terms Terms having the same variable can be combined to simplify the expression Ex. 5x – 4 – 3x + 7 if you combine the like variables you get 5x - 3x = 2x and -4 + 7 = 3 so after combing you get 2x + 3


Removing Parentheses :Removing Parentheses To remove parentheses you will have to use the distributive principle Ex. 2(a + 6) multiply through by 2 2a + 12 Ex. 7(2b - 5) multiply through by 7 14b - 35 Ex. 4(-5c + 2) multiply through by 4 -20c + 8 Ex. -2(b – 4) multiply through by -2 -2b + 8


Multi-Step Algebra Problems :Multi-Step Algebra Problems You can use all that has been shown to you to now solve multi-step algebra problems and inequalities Ex. 3(x + 6) = 5x – 2 3x + 18 = 5x – 2 Use distributive property to remove parentheses -5x -5x Subtract 5x from both sides -2x + 18 = -2 - 18 - 18 Subtract 18 from both sides -2x = -20 -2 -2 Divide both sides by -2 to solve for x x = 10


Uniform Motion Problems :Uniform Motion Problems The formula used in Uniform Motion is d = rt d = distance r = rate of travel (speed) t = time spent traveling Ex. A train leaves a station traveling at 50mi/hr. Two hours later, a second train leaves on a parallel track traveling the same direction at 60mi/hr. In how many hours will the second train catch up to the first train? (Assume there are no stops and both trains travel at constant speeds) Rate (r) x Time (t) = Distance (d) 1st train 50mi/hr x t + 2 = 50 (t + 2) 2nd train 60mi/hr x t = 60t since the distance is the same set the two distances equal and solve for t 50 (t + 2) = 60t become 50t + 100 = 60t becomes 100 = 10t becomes t = 10 hrs


Consecutive Integer Problems :Consecutive Integer Problems Consecutive integers follow each other in order Ex. 1,2,3,4 n, n +1, n +2, n + 3 Consecutive even integers follow evenly Ex. 2,4,6,8 n, n + 2, n + 4, n + 6 Consecutive odd integers follow evenly Ex. 3,5,7,9 n, n + 2, n + 4, n + 6


Graphing Linear Equations :Graphing Linear Equations You can use linear equations to find ordered pairs to plot on a graph Ex. To find coordinates for y = 2x – 5 substitute some numbers for either x or y and then solve for the other x y 0 -5 1 -3 2 -1 3 1 A graph of a linear equation will always be a straight line


Understanding Slope :Understanding Slope The slope of a line refers to how steep a line is Slope is often represented by the letter m The formula for the slope of a line is: you could also do the rise/run method Ex. What is the slope of the following line that passes through the ordered pairs (-4,-3) and (1,3)? The slope is 6/5


Understanding Slope cont. :Understanding Slope cont. Slope > 0 Slope < 0 Slope = 0 Slope is undefined


Finding Distance Between Points :Finding Distance Between Points To find the distance between two points on a Cartesian plane you will use the following formula Ex. Find the distance between (-2,1) and (3,-4)


Midpoint of Line Segment :Midpoint of Line Segment You can use the coordinates of the endpoints of a line segment to find the coordinates of the midpoint of the line segment by using the formula below Ex. Find the midpoint of the line segment having endpoints at (-3,-1) and (4,3).


Slope-Intercept Form of a Line :Slope-Intercept Form of a Line To put a linear equation in slope-intercept form, solve the equation for y Slope-intercept form follows the pattern of y = mx + b (“m” represents slope and “b” represents the y-intercept) Ex. What is the slope and y-intercept of 2x + 3y = 15 2x + 3y = 15 -2x -2x 3y = -2x + 15 slope-intercept form: 3 3 3 The slope is & y-intercept is 5


Solving Systems of Equations by Substitution :Solving Systems of Equations by Substitution You can solve systems of equations algebraically by using the substitution method Ex. Find the point of intersection of the following 2 equations: Equation 1: x – y = 3 Equation 2: 2x + y = 9 1st solve one equation for x or y and then substitute that equation into the other equation for that variable Eq. 1: x = y + 3 and plug y + 3 in where x is in Eq. 2 Eq. 2: 2(y + 3) + y = 9 and solve for y 2y + 6 + y = 9 3y + 6 = 9 3y = 3 y = 1 Now take the answer for y and plug it into the 1st equation and solve for x Eq. 1: x = 1 + 3 x = 4


Polynomials :Polynomials Polynomials are algebraic expressions which include monomials containing one term, binomials which contain two terms, and trinomials, which contain three terms Terms are separated by plus and minus signs Monomials Binomials Trinomials Polynomials 4f 4t + 9 x2 + 2x + 3 x3 – 3x2 + 3x – 9 3x3 9 – 7g 5x2 – 6x – 1 p4+2p3+p2-5p+9 4g2 5x2 + 7x y4+15y2+100 2 6x3 – 8x


Adding and Subtracting Monomials :Adding and Subtracting Monomials Two monomials can be added or subtracted as long as the variable and its exponent are the same (called combining like terms) Ex. 4x + 5x = 9x 2x2 – 9x2 = -7x2 6y3 – 5y3 = y3 5y + 2y = 7y Remember that when the integer in front of the variable is “1”, it is usually not written


Adding Polynomials :Adding Polynomials When adding polynomials, make sure the exponents and variables are the same on the terms you are combining Ex. Add 3x2 + 14 and 5x2 + 2x 3x2 + 5x2 + 2x + 14 = 8x2 + 2x + 14 Ex. (4x3 – 2x) + (x3 – 4) = 4x3 – 2x + x3 – 4 = 5x3 – 2x -4


Subtracting Polynomials :Subtracting Polynomials When subtracting polynomials, remember to change all the signs in the subtracted polynomial and then add Ex. (4y2 + 8y + 9) – (2y2 + 6y – 4) = 4y2 + 8y + 9 – 2y2 – 6y + 4 = 2y2 + 2y + 13


Multiplying Monomials :Multiplying Monomials When two monomials have the same variable they can be multiplied The exponents are added together If there is no exponent, it is understood to be one Add Ex. 4x4 x 3x2 = 12x6 Multiply


Multiplying Monomials with Different Variables :Multiplying Monomials with Different Variables You Cannot Add The Exponents of Variables That Are Different Ex. (-4wx)(6w3x2) -4 * 6 = -24 w * w3 = w4 x * x2 = x3 Put the answer together to get -24w4x3


Multiplying Monomials by Polynomials :Multiplying Monomials by Polynomials Works by multiplying the terms inside the parentheses by the term outside the parentheses Ex. -5t(2t2 – 7t + 9) -10t3 + 35t2 – 45t


Removing Parentheses and Simplifying :Removing Parentheses and Simplifying First you will multiply the monomials and polynomials and then combine like terms Ex. 8x(2x2 – 5 + 7) – 3x(4x2 + 3x – 8) 16x3 – 40x2 + 56x – 12x3 – 9x2 + 24x 4x3 – 49x2 + 80x


Multiplying Two Binomials :Multiplying Two Binomials When multiplying two binomials you must multiply each term in the first binomial by each term in the second binomial The easiest way to multiply two binomials is to use the FOIL Method F stands for first O stands for outside I stands for Inside L stands for Last


Multiplying Two Binomials cont. :Multiplying Two Binomials cont. F O I L First Outside Inside Last Ex. (x + 6)(x – 5) (x + 6)(x – 5) (x + 6)(x – 5) (x + 6)(x – 5) x * x = x2 x * -5 = -5x 6 * x = 6x 6 * -5 = -30 x2 + -5x + 6x + -30 Combine like terms and write your answer x2 + x – 30


Factoring :Factoring In a multiplication problem, the numbers multiplied together are called factors and the answer to a multiplication problem is called the product Ex. Find the greatest common factor of 2y3 + 6y2 Look at the whole numbers and decide what the greatest common factor of 2 and 6 is and factor that number out of both 2(y3 + 3y2) Look at the remaining terms and find the common factor and factor it out of both y2(y + 3) Then you write you answer: 2y2(y + 3)


Factoring Trinomials :Factoring Trinomials Ex. Factor x2 + 6x + 8 Step 1: When the trinomial is in descending order as in the example above, you need to find a pair of numbers in which the sum of the two numbers equals the number in the second term, while the product of the two numbers equals the third term ____ + ____ = 6 and ____ x ____ = 8 The pair of numbers that satisfy both equations is 4 and 2 Step 2: Use the pair of numbers in the binomials x2 + 6x + 8 are (x + 4)(x + 2) Use the FOIL Method to check your answer


Checklist for Factoring Polynomials :Checklist for Factoring Polynomials First, Check to see if it has a greatest common factor that can be factored out first Second, see if it is the difference of two squares. Remember that the sum of two squares cannot be factored Third, if a polynomial has more than 3 terms, try factoring by grouping Fourth, if none of the above are possibilities, factor by trial and error


Quadratic Equations :Quadratic Equations Any equation that can be put in the form below is a quadratic equation ax2 + bx + c = 0 (a, b, and c are real numbers) The formula is: Ex. Solve y2 – 4y – 5 = 0 5 and 1 Substitute the numbers in the equation to make sure it works


Relations :Relations A relation is a set of ordered pairs The set of the first members (x values) of each ordered pair is called the domain of the relation The set of the second members (y values) of each ordered pair is called the range Ex. State the domain and range of the following {(2,4), (3,7), (4,9),(6,11)} Domain: {2,3,4,6} Range: {4,7,9,11}


Functions :Functions Some relations are also functions A relation is a function is for every element in the domain there is exactly one element in the range Basically for each value of x there is only one unique value of y Ex. {(2,4), (2,5), (3,4)} is not a function because the there are two x values with different y values


Determining Function :Determining Function If you have a graph and want to know if it is a function or not it is simple to do by using the vertical line test If any vertical line intersects a graph of a relation in more than one point, then the relation graphed is not a function


Types of angles :Types of angles Acute angle is less than 90o Right angle is 90o Obtuse angle is greater than 90o Straight angle is 180o


Adjacent Angles :Adjacent Angles Adjacent angles are two angles that have the same vertex and share one ray, but they do not share space inside the angles Ex. B In this diagram ADB A is adjacent to BDC D C However, ADB is not adjacent to ADC because adjacent angles do not share any space inside the angle


Vertical Angles :Vertical Angles When two lines intersect, two pairs of vertical angles are formed Vertical angles are not adjacent Vertical angles have the same measurements Vertical angles are congruent A B AOC and BOD O are vertical angles C D


Complementary and Supplementary Angles :Complementary and Supplementary Angles Two angles are complementary if the sum of the measures of the angles is 90o Two angles are supplementary if the sum of the measures of the angles is 180o Complementary Angles Supplementary Angles 55o 35o 70o 110o


Corresponding, Alternate Interior, and Alternate Exterior Angles :Corresponding, Alternate Interior, and Alternate Exterior Angles 1 3 5 7 2 4 6 8


Perimeter :Perimeter The perimeter is the distance around a polygon To find the perimeter, add the lengths of all the sides of the polygon Ex. 15 in. 6 cm. 7 in. 4 cm. P = 7+15+7+15 5 cm. P = 44 in. P = 4+6+5 P = 15 cm.


Area of Squares and Rectangles :Area of Squares and Rectangles Area is always expressed in square units such as in2, cm2, ft2, and m2 The formula used to find the area of squares and rectangles is: A = l x w Ex. 8 cm. A = 8 x 3 3 cm. A = 24 cm2


Area of Triangles :Area of Triangles To find the area of a triangle you will used the formula: ½ x b x h b = base h = height Height Ex. 16 in. A = ½ x 26 x 10 10 in. A = ½ x 260 A = 130 in2 26 in. Base


Circumference :Circumference Circumference (C) is the distance around the outside of a circle Circumference is found by: C = 2 r or C = d Diameter (d) is the distance from one side of a circle to the other side, passing through the center Radius (r) is half of the diameter Pi ( ) is the ratio of the circumference of a circle to its diameter (3.14)


Area of a Circle :Area of a Circle The formula for area of a circle is: A = r2 Ex. diameter = 14 cm. A = 3.14 x 72 A = 153.86 cm2


Facts AboutTriangles :Facts AboutTriangles All the internal angles of a triangle add up to equal 180o If you know the measure of two of the angles the you subtract their total from 180 to find the third


Pythagorean Theorem :Pythagorean Theorem In a right triangle, the sum of the squares of the legs of the triangle are equal to the square of the hypotenuse of the triangle The formula for this is: a2 + b2 = c2 Ex. a = 3 c = 5 32 + 42 = 52 9 + 16 = 25 b = 4 25 = 25


Volume :Volume Measurement of volume is expressed in cubic units such as in3, ft3, m3, cm3, or mm3 The volume of a solid is the number of cubic units that can be contained in the solid The formula for the volume of a rectangular solid is: length x width x height (V = l*w*h)


Surface Area :Surface Area The surface area is the amount of material on the outside of a container The formula for the surface area of a cylinder is : SA = 2 r2 + 2 rh The formula for the surface area of a rectangular prism is: SA = 2h + 2b