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OPERATIONS: Math information – notes from class ADD - SUM SUBTRACT - DIFFERENCE MULTIPLY – PRODUCT DIVISION – QUOTIENT ESTIMATION: Think rounding Rounding rules: 5 and up round up 4 and lower stays the same - estimate to make Math easier & quicker - use when we don’t need precise totals Estimating sum or difference – rule: round to the same place value Estimating product – rule: round to the leading digit (number) Estimating quotient – rule: round the divisor & then find compatible number INTERGERS negative and positive numbers that represent real life amounts positive – gain, deposit, above negative – loss, debit, withdrawal, below Absolute value – how far away from “0” the integer is Opposite – two numbers that are the same amount from “0” Adding integers: same sign – 1. add the absolute value of the integers 2. Use the common sign different signs – 1. subtract the smaller absolute value from the larger absolute value. 2. use the sign from the bigger absolute value Subtracting integers: 1. change the subtraction to addition 2. change sign of the integer that follows. 3. Follow rules for adding integers Multiplying / dividing integers: same sign will be positive different sign will be negative positive positive = positive negative positive = negative negative negative = positive ORDER of OPERATIONS – Parenthesis PATTERNS – Exponents Multiplication/Division left to right Addition/ Subtraction left to right Numbers going down - subtract & divide Numbers going up – add & multiply Example: 256, 128, 64, 32 decreasing so you would use subtraction or division 32, 64, 128, 256 increasing so you would use addition or multiplication NUMERICAL EXPRESSION – includes numbers and operations (numbers sentence) - do not have = signs 15-8 1679 x 2345 VARIABLE – a letter that represents one or more numbers - do not use “o” as a variable (can be confused with zero) VARIABLE EXPRESSION – includes variables (letters), operations, numbers 7m +13 134- 7t TERMS- parts of the expression LIKE TERMS – identical variable parts ex. – 6b 4b CONSTANT TERMS – your numbers that do not change Simplify expressions by combining the like terms Example – 2d + 4 + 6d + 12 8d + 16 EQUATION – two expressions separated by an equal sign - solution for an equation is the number substituted for the variables - What number minus 8 equals 4? - b–8=4 +8 +8 b= 12 EVALUATE – solve the problem Determine the operation Get the variable by itself Do the inverse (opposite) operation What you do to one side of the = sign, you MUST do to the other side of the = IS – equal, equal to, a total PROPERTIES DISTRIBUTIVE- use with multiplication distribute a(b + c) = ab+ ac COMMUNATIVE- use with use with multiplication and addition #’s have moved a+b+c=c+a+b ASSOSCIATE - use with use with multiplication and addition (a + b) +c = a + (b + c) associate FUNCTION – relationship between numbers input (x) output (y) function rule y = x + 5 DATA – information that we gather Ways to represent data: makes data easier to read & understand pie chart & circle graph pictograph frequency table line plot line graph bar graph histogram double bar graph FREQUENCY TABLE - how often something happens LINE PLOT- LINE GRAPH - shows change over time BROKEN SCALE – use when the data starts at a large number BAR GRAPH - comparisons of specific numbers INCREMENTS - the number you go up by HISTOGRAM – differs from bar graph because the bars touch DOUBLE BAR GRAPH – compares two pieces of data Must include a key CIRCLE GRAPH - percentage (%) of the whole (total number) AVERAGES – all of the following are averages mean – add up all the numbers and divided by the number of numbers added 7, 1, 2, 6, 1, 7 7+1+2+6+1+7=24 24/6 = 4 4 is the mean median – the middle number after you order the numbers from least to greatest. When it is an even number of numbers you must add the 2 middle number together and divide by 2. This will then be the median. 1, 1, 2, 6, 7, 7 2+6 =8 8/2= 4 4 is the median mode – the number that occurs most often; you may have more than one mode and may be no mode; the mode must occur at least twice. 1 and 7 are modes for the above set of numbers GEOMETRY – Chapter 9 in textbook line – extends without end in two opposite directions; 0 end points ray – has one endpoint and extends without end in one direction. segment – has 2 endpoints; parallel lines – lines that never meet intersecting lines – meet at a point perpendicular lines – lines that meet at right angles ANGLES made by combining 2 rays VERTEX- point where lines meet Measured in degrees VERTICAL ANGLE - angles that are opposite and equal COMMPLEMENTARY ANGLES – 2 angles that make 90 ̊ SUPPLEMENTARY ANGLES – 2 angles that make 180 ̊ 180 ̊ = a straight line CLASSIFYING TRIANGLES Size : Scalene - no equal sides Isosceles - at least 2 equal sides Equilateral – all sides are equal Angles: Acute – has three acute angles Right – has a right angle Obtuse – has one obtuse angle All angles of a triangle = 180 ̊ PARALLELOGRAM – 2 pairs of parallel sides POLYGONS – a shape with many sides TRIANGLE – 3 sides QUADRILATERAL – 4 sides Angles all add up to 360 ̊ REACTANGLE – 4 right angles, parallelogram TRAPEZOID – 1 pair of parallel sides RHOMBUS – 4 equal sides; parallelogram SQUARE – 4 equal sides; 4 right angles; parallelogram POLYGONS – REGULAR POLYGONS – all equal sides and all equal angles IRREGULAR POLYGONS – not the same TRIANGLE- three sides PENTAGON – 5 sides HEXAGON – 6 sides OCTOGAN – 8 sides DECAGON – 10 Sides DIAGONALS- segment that connects 2 vertice TO FIND THE TOTAL DEGREES OF A POLYGON: Take the number of sides and subtract 2 (n-2) example: hexagon 6 – 2 = 4 triangles 4 × 180 ̊̊ = 720 ̊ SIMILAR AND CONGRUENT FIGURES CONGRUENT – same shape and same size SIMILAR - same shape but different size CORRESPONDING PARTS - parts of polygons that match LINE OF SYMMETRY – divides a figure into 2 parts that match exactly AREA- The amount of surface covered by a figure. Area is measured in square units such as square feet (ft²) or square meters (m²) PERIMETER- The distance around a figure. FORMULAS TO FIND AREA & PERIMETER: RECTANGLE - A = l · w PARALLELOGRAM – P = 2(l +w) area = base · height A=b·h TRIANGLE – Area = ½ · base · height A = b · h ÷ 2 Height is formed at a right angle CIRCLES –Has no straight lines. RADIUS – the distance from the center to any point on the circle. DIAMETER – The distance across the circle through its center The diameter is twice the radius. CIRCUMFERENCE – The distance around the circle. We use pi (3.14) when we calculate the area of a circle. PI - How many times the diameter goes around the circumference of a circle; ratio of the diameter to the circumference of a circle VALUE of PI – 3.14 A = π r² symbol for PI – π C= πd or 2πr CLASSIFYING SOLIDS SOLID- closed figure that is 3 –dimensional SPHERE – all points on the sphere are the same distance form the cente CONE – one vertex and a circular base PRISM – a solid with 2 parallel bases that are congruent polygons Base names the solid PYRAMID – solid made up of polygons The base can be any polygon and names the pyramid. The other polygons are triangle and meet a common vertex. CYLINDER – 2 bases that are congruent and parallel circles. FACES – sides of a figure VERTICES – point where edges meet EDGES – segments where faces meet Dotted line in a drawing indicates the edge you can not really see SURFACE AREA - sum of all the areas of all the faces Find the area of each face & add together SA = 2(lw) + 2(lh) + 2(hw) VOLUME – the amount that would fit inside v=l ·w · h answers are always in cubic units (units ³) PRIME FACTORIZATION – a number written as the product of prime numbers Think of FACTOR TREES The prime factorization of 54 is 2 x 3 x 3 x 3 or 2 x 3 FACTORS – Two numbers multiplied together to make another number. Ex. – 6 x 4 = 24 6 and 4 are factors of 24 DIVISIBILITY RULES: A number is divisible by 2 if it is an even number. A number is divisible by 5 if it ends with a 5 or a 0. A number is divisible by 10 if it ends with a 0. PRIME NUMBER - has only 2 factors – one and itself ; 2 is the smallest prime number; one is NOT a prime number as it has only 1 factor. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 … COMPOSITE NUMBER – have three or more factors 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, … GREATEST COMMON FACTOR (GCF) – The highest number that divides exactly into two numbers. Will be smaller than your numbers ex. GCF of 18 & 12 is 6 6 is smaller than 12 or 18. LEAST COMMON MULTIPLE (LCM) – The smallest (non-zero) number that is a multiple of two or more numbers. Will be bigger than your numbers ex. LCM of 40 & 32 is 160 160 is bigger than 42 or 32 Finding the GCF using the list method: The factors of 12 are 1, 2, 3, 4, 6 and 8. The factors of 18 are 1, 2, 3, 6, 9 and 18. The common factors of 12 & 18 - 1, 2, 3, 6 The GCF of 12 and 18 is 6. Finding the LCM using the list method: The LCM of 2, 3, 4, & 6 is 12 Finding the GCF & LCM using the Venn diagram: 1. Make factor trees to find the prime factorization. 2. Complete the Venn diagram using the prime factorization. GCF – multiply the numbers in the middle 2 ·2 · 3 = 12 LCM – multiply across the Venn diagram 2 · 2 · 2 ·2 · 3 · 3 · 5 = 720 FRACTIONS – part of the whole Numerator - top number Denominator- number on the bottom of the fraction Equivalent fraction – fractions that are equal Mixed numbers – whole number and a fraction Improper fraction - Fraction in which the numerator is larger than the denominator Converting fractions – DECIMALS- a number that uses a decimal point followed by digits as a way of showing less than one. Repeating decimals –having a pattern of one or more digits repeated indefinitely. Bar on is only over the digits that repeat. Converting fractions to decimals: divide the numerator by the denominator Converting decimals to fractions: make sure fraction is in lowest terms Ones you should know: ¼ = .25 ½ = .50 ¾ = .75 think of money – quarters. RATIO: Compare the number of one thing to the number of another. Must be in lowest form RATE: a ratio of 2 measurements with different units Unit Rate- when the denominator is 1. The amount for one unit. Use this to compare PROPORTIONS – when an equation that shows 2 ratios are equivalent Use cross product to determine if they are equivalent. PERCENTAGES - per 100 100% means all. Example: 100% of 80 is 100/100 × 80 = 50% means half 50% of 80 is 50/100 × 80 = 80 Example: 40 5% means 5/100ths. Example: 5% of 80 is 5/100 × 80 = 4 How to find a percentage of a number: Converting percents to decimals and fractions: COORDINATE PLANES: Quadrants – the four regions of a coordinate plane. The vertical line is called the Y axis. The horizontal line is called the X axis. The axes intersect at the origin. ORDERED PAIRS – The first number tells you how many units to move to the left or right. The second number tells you how many units to move up or down. example: (4, -2) go to right 4 and then down 2. (see below) TRANSFORMATION: When a figure moves on a coordinate plane. We have an original figure, after the transformation we have an image. Image- The new figure after an transformation. labeled as : original image A A’ B B’ Translation (slide) All points of a figure move the same number of units and in the same direction. Reflection (flip) The figure is flipped over either the X-axis or the Y-axis. It must be same distance from axis. You must state line of reflection when describing. When reflecting over the X-axis, the Y value of the ordered pair will have an opposite sign. When reflecting over the Y-axis, the X value of the ordered pair will have an opposite sign. Example: A (-2, 1) A’ (-2, -1) B (2, 4) B’ (2, -4) C ((4, 2) C’ (4, -2) Rotation (turn)- where a figure is turned about a given point center of rotation – origin angle of rotation- 90 ̊, 180 ̊, 270 ̊, 360 ̊ direction of rotation- clockwise or counter clockwise Dilation (get bigger or smaller) PROBABILITY – outcomes – possible result of an experiment events – collection of outcomes favorable outcomes – the outcome you are looking to happen probability – chance or likelihood that an event may happen P (event) = number of favorable outcomes number of possible outcomes list as percent, fraction, or decimal When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6. The probability of any one of them is 1/6. Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble will be picked? Number of ways it can happen: 4 (there are 4 blues) Total number of outcomes: 5 (there are 5 marbles in total) 4 So the probability = = 0.8 or 80% 5 Two types of probability: Theoretical – the probability is based on what in theory should happen Experimental – probability is based on repeated trials of an experiment Independent event - if one event does not affect the likelihood the other event will occur. Dependent event – if the one event affects the other event Combinations – when order does not matter Permutations – when order matters TREE DIAGRAM - list of possible outcomes