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Intro to Mathematics Nathan Frey Arithmetic Concepts Number systems, Base ten system, Order of operations, Fractions, Decimals, Percents, Ratios, Rates and Proportions Number Systems Base 10 (decimal) system Order of Operations • PEMDAS • Parentheses – any grouping symbol including brackets, absolute value, fraction bar, etc. • Exponents – also includes radicals like square roots, cube roots • Multiplication/Division – left to right • Addition/Subtraction – left to right Decimals • Adding and Subtracting Decimals • Multiplying and Dividing Least Common Multiple and Greatest Common Factor • Find the GCF and LCM of 60 and 42 Least Common Multiple/Denominator (LCM/LCD) Least Common Multiple/Denominator (LCM/LCD) Greatest Common Factor (GCF) Greatest Common Factor (GCF) Fractions • Adding/Subtracting – need a common denominator Multiply Fractions • Change all mixed numbers to improper fractions, multiply numerators, multiply denominators, and reduce the fraction Divide Fractions • Rewrite division as multiplication by the reciprocal (flip the second fraction) Converting Fractions, Decimals and Percents • To convert a fraction into a decimal do the division. 4/5 means 4 divided by 5 • To convert a decimal to fraction read the decimal to the correct place value .25 means 25 hundredths • To convert a decimal to percent multiply by 100 (move decimal 2 places to the right and add percent sign) • Percent % - means out of 100. Change a percent to a fraction over 100. • To convert a percent to a decimal divide by 100 (move decimal 2 places to the left and drop the percent sign) Common conversions =‘[-p\ Ratio (p. 174) • A ratio is a comparison of two quantities using the same units • Example: In a sample group of patients there are 15 men and 20 women. What is the ratio of men to women? 15/20, 15:20, 15 to 20 or simplified ¾, 3:4, 3 to 4 • What is the ratio of women to men? • Practice on p. 176 Rates (p. 178) • Rate is a comparison between two quantities using different units written as a fraction. • Example: You travel 250 miles in 4 hours. The rate is 250 miles/4 hours or simplified 125 miles/2 hours • Unit rate is when the denominator is 1 • 250 miles/4 hours = 62.5 miles/hour • Practice p. 180 Proportion (p. 182) • A proportion is an expression of equality of two ratios or rates • A proportion is true if both sides are equal when written in simplified form • 3/6=1/2 is true 6/10=10/15 is not true • Solve a proportion by cross-multiplying • Ex 1: 9/6 = 3/n Ex: 5/7 = x/20 • Practice p. 186 Metric System (p. 372) Converting metric units by using unit multipliers Setting up percent problems • • • • • Translate keywords into math language “of” means multiply “is” means = “what” tells us to use a variable If you see “what percent” use p and remember to convert answer to percent • If you see “what number” use n and remember your answer should just be a number • Change percents into decimals Percent Problems • Translate as an equation. Use decimals. Percent of Change • Difference divided by the original amount Simple Interest (p. 248) • Amount of Interest=Principal x Annual Interest Rate x Time (in years) • A=Prt • Maturity Value = Principal + Interest • Monthly Payment = Maturity Value/Length of loan (in months) • Ex. 1: Find the maturity value of 5 year $1000 CD with 5% simple interest. • Ex. 2: Find the monthly payment for a 2 year loan of $1000 with 7% interest. Proportion problems • When setting up a proportion, make sure that you set up both sides comparing the same way • Apples to oranges = apples to oranges • Not apples to oranges and oranges to apples p Geometry Types of angles, properties of triangles, types of polygons, solids, coordinate geometry, congruence and similarity, perimeter, area, and volume Intro to Geometry • • • • • • • • Point - 0 dimensions Line – 1 dimension Line Segment – part of a line with 2 endpoints Ray – part of a line with 1 endpoint Plane and plane figures – 2d Space and solids – 3d Parallel lines Intersecting lines Types of Angles • Angle – formed by two rays with common endpoint, measured in degrees • Right angle – 90 degrees • Perpendicular lines – two lines that form a rt < • Acute angle – less than 90 degrees • Obtuse angle – more than 90 degrees • Straight angle – 180 degrees • Complementary angles – add up to 90 • Supplementary angles – add up to 180 • Vertical angles – formed by intersecting lines, across from each other • Adjacent angles – have a common endpoint and common side Angles formed by parallel lines • How many angles are formed from two parallel lines and a transversal? • • • • • • Alternate Interior Alternate Exterior Same-Side Interior Corresponding Vertical Adjacent • • • • • • • • • • • • Alternate Interior 3 & 6, 4 & 5 Alternate Exterior 1 & 8, 2 & 7 Same-Side Interior 4 & 6, 3 & 5 Corresponding 1 & 5, 3 & 7, 2 & 6, 4 & 8 Vertical 1 & 4, 2 & 3, 5 & 8, 6 & 7 Adjacent 1 & 2, 2 & 4, 1 & 3, 3 & 4, 5 & 6, 6 & 8, 8 & 7, 7 &5 • • • • • • • Triangles Base – one side of a triangle Height – perpendicular to the base Angles add up to 180 Right Triangle – has a right angle (90 degrees) Hypotenuse – longest side of a right triangle Acute Triangle – all angles less than 90 degrees Obtuse Triangle – has one angle more than 90 degrees • Equilateral – all three sides congruent (same length) • Isosceles – two sides congruent • Scalene – no sides congruent Quadrilaterals • • • • • • Parallelogram – two pairs of parallel lines Rectangle – a parallelogram with 4 right angles Rhombus – a parallelogram with 4 equal sides Square – both a rectangle and rhombus Trapezoid – only one pair of parallel sides Isosceles Trapezoid – a trapezoid with congruent sides • Kite – has two pair of consecutive sides that are congruent Solids • • • • • Prism Pyramids Cylinder Cones Sphere TEST 1 MATERIAL ENDS Midpoint Formula • Given two points the midpoint is halfway between them and is found by taking the average of the x’s and the average of the y’s S,m,x,mcll x xz Pythagorean Theorem • For right triangles Polygons • • • • • • • • • Triangle – 3 sides Quadrilateral - 4 Pentagon - 5 Hexagon - 6 Heptagon -7 Octagon - 8 Nonagon - 9 Decagon -10 Dodecagon -12 Perimeter • • • • • Perimeter is distance around a polygon Shortcut formulas: Rectangle P = 2L + 2W Square P = 4s Circumference is the name given to the length around a circle • C = 2πr Area • Area is how much space a shape contains measured in square units • Rectangle A = L x W • Square A = s2 • Parallelogram A = b x h • Triangle A = ½ b x h • Circle A = πr2 Perimeter and Area of composite figures Congruent Triangles • Congruent triangles have all the same angles, and all the same side lengths. Congruent means they are exactly the same. • SSS • SAS • ASA • AAS Volume • Volume is the measure of space in an enclosed surface. • Rectangular Prism V = L x W x H • Cube V = s3 • Sphere V = (4/3)πr3 • Cylinder V = πr2h • Cone V = (1/3) πr2h Intro to Algebra • Variable – letter standing for a number • Constant- a known number • Coefficient – number being multiplied by a variable ex. 2x, the 2 is the coefficient • Term – part of algebraic expression separated by plus sign • Like terms – have the same variables with the same exponents Properties • Commutative property – a property of addition or multiplication that allows one to change the order a+b=b+a ab = ba Ex: 1+2=2+1=3 2x4=4x2=8 Properties • Associative property - a property of addition or multiplication that allows one to change the grouping (a + b) + c = a + (b + c) (ab)c = a(bc) Ex: (1 + 2) + 3 = 1 + (2 + 3) = 6 (2 x 4) x 3 = 2 x (4 x 3) = 24 Properties • Distributive property - a property of addition and multiplication a(b + c) = ab + ac Ex: 2(3 + 5) = 6 + 10 = 16 2(3x – 4) = 6x - 8 Translating Verbal Expressions into Algebraic Expressions • • • • • Addition Subtraction Multiplication Division Raise to a power TEST 2 MATERIAL ENDS One step equations Slope/Rate of Change • Slope measures the steepness of a line • Slope = rise/run • Horizontal lines have 0 slope • Vertical lines have undefined slope Linear Equations • y= mx + b • m is the slope • b is the y-intercept (where the graph crosses yaxis) • Ex. 1: y = 2x - 3 • Ex. 2: y = -x + 2 • Ex. 3: y = ½ x Solving Systems • A system is two or more equations to solve simultaneously • Solve each equation for y • Graph each equation • Find the points of intersection Descriptive Statistics • Statistics concerns data and methods of representing and analyzing data Pictograph Circle graph or Pie Chart 9% 27% 15% 18% 20% 11% Scatterplot 70 60 50 40 Series1 30 20 10 0 0 1 2 3 4 5 6 7 Bar Graph 70 60 50 40 Series1 30 20 10 0 1 2 3 4 5 6 Line Graph 70 60 50 40 Series1 30 20 10 0 1 2 3 4 5 6 Histograms Statistical Measures • Arithmetic mean (average) • Median – the middle number – If the data set contains an even number of values, take the mean of the two middle numbers • Mode – the most frequent number – There may be one mode, no mode, or more than one mode Box and Whisker Plot • Find the median of the data • Find the first quartile Q1 by taking the median of all the values below the median • Find the third quartile Q3 by taking the median of all the values above the median • The range of the data is the difference between the highest number and the lowest number • The interquartile range is the difference between Q3 and Q1 Box and Whisker Construction 21, 21, 23, 24, 35, 35, 36, 38, 40, 41, 46, 48 Counting techniques • Multiplication principle for independent events: If one step can be done in m different ways and the second step can be done in n different ways, then together they can be done in m x n ways. • Example: If you have 6 shirts, 4 pants, and 2 pairs of shoes, how many different outfits can you make? • At a restaurant, you can choose one of 3 salads, one of 5 entrees, and one of 2 desserts. How many different meals? • A student ID is the students first initial, last initial, then 2 digits. Permutations • Permutations are ordered arrangements • Factorial – multiply all the numbers up to and including the number – n! = n x (n-1) x …x 3 x 2x1 • nPr – the number of ordered arrangements from a group of n with r objects selected Permutation Examples • 3 people are running a race, how many different ways can they place? • 4 people are running a race, how many different ways can they place? • 5 people are running for president, vice president, and secretary. How many different possibilities? • 6 people are running for president, vice president, and secretary. How many different possibilities? Combinations • Combinations are unordered arrangements • nCr – the number of unordered arrangements from a group of n with r objects selected Combination Examples • Five people are running for city council with three to be elected. How many different possibilities? • Six people are running for council with two to be elected. How many different ways can this be done? • You decide to make a fruit salad using three of the following: blueberries, bananas, strawberries, peaches, grapes. How many different ways? Probability • Probability is the mathematics of chance and measuring uncertainty • Experimental vs. Theoretical probability • Sample Space is the set of all possible outcomes • Event is one or more outcomes • Probability Probability • Probability is a ratio between 0 and 1 • A probability of 0 means the event is impossible • A probability of 1 means the event is certain to happen • Example 1: What is the probability of rolling a number greater than 7 with one die? • Example 2: What is the probability of rolling a number less than 7 with one die? • Example 3: What is the probability of rolling a prime with one die? Rolling two dice • List all possible outcomes of rolling two dice • How many ways can we get each number? 1 1 2 3 4 5 6 2 3 4 5 6 Probability with 2 Dice • • • • • • P(even) P(roll a 4) P(roll a 7) P(roll a number less than 5) P(roll a number less than 4 or greater than 9) P(roll a prime) Flipping a coin • A coin is flipped 2 times. How many different ways can this be done? • A coin is flipped 2 times what is the probability of getting at least one head? • A coin is flipped 4 times. How many different ways can this be done? • What is the probability of getting exactly two heads?