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COMPREHENSIVE REVIEW FOR MIDDLE SCHOOL MATHEMATICS 2013

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COMPREHENSIVE REVIEW FOR MIDDLE SCHOOL MATHEMATICS Purpose: Mathematics Review for 7th Grade (Can be used as enrichment or remediation for most middle school levels) Contents: Concept explanations & practice problems. Sources: PA Standards-PDE website. Additional Reinforcement: (links provided throughout) (links provided throughout) and PSSA Coach workbook Created by: Jessie Minor

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IN ORDER TO CALCULATE EXPERIMENTAL PROBABILITY OF AN EVENT USE THE FOLLOWING DEFINITION: P(Event)= 3 Coach Lesson 30 Number of times the event occurred Number of total trials EXPERIMENTAL PROBABILITY!

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Example: A student flipped a coin 50 times. The coin landed on heads 28 times. Find the experimental probability of having the coin land on heads. P(heads) = 28 =.56 = 56% 50 It is experimental because the outcome will change every time we flip the coin. EXPERIMENTAL PROBABILITY! Experimental Probability IXL 4

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5 PRACTICE EXPERIMENTAL PROBABILITY! A spinner is divided into five equal sections numbered 1 through 5. Predict how many times out of 240 spins the spinner is most likely to stop on an odd number. F.80 G.96 H.144 I.192 Marilyn has a bag of coins. The bag contains 25 wheat pennies, 15 Canadian pennies, 5 steel pennies, and 5 Lincoln pennies. She picks a coin at random from the bag. What is the probability that she picked a wheat penny? F.10% G.25% H.30% I.50%

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Coach Lesson 29 6 THEORETICAL PROBABILITY! The outcome is exact! When we roll a die, the total possible outcomes are 1, 2, 3, 4, 5, and 6. The set of possible outcomes is known as the sample space. Find the prime numbers of the sample space above– since 2, 3, and 5 are the only prime numbers in the same space… P(prime numbers)= 3/5 = ______% PRACTICE THEORETICAL PROBABILITY! 60

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RATE: comparison of two numbers Example: 40 feet per second or 40 ft/ 1 sec UNIT PRICE: price divided by the units Example: 10 apples for $4.50 Unit price: $4.50 ÷ 10 = $0.45 per apple SALES TAX: change sales tax from a percent to a decimal, then multiply it by the dollar amount; add that amount to the total to find the total price Example 1: $1,200 at 6% sales tax = 6 ÷ 100 = 0.06 x 1,200 = $1272 COACH LESSON 4 Unit Prices IXL 7 RATE/ UNIT PRICE/ SALES TAX!

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$7.99 x 3 = $23.97 $23.97 x 0.06 = $ Sales Tax = $ Example 2: Rachel bought 3 DVDs. Using the 6% sales tax rate, calculate the amount of tax she paid if each DVD costs $7.99? PRACTICE SALES TAX!

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Distance formula: distance = rate x time OR D = rt Example 1: A car travels at 40 miles per hour for 4 hours. How far did it travel? d=rt d=40 miles /hr x 4 hrs d = 160 miles. We can also use this formula to find time and rate. We just have to manipulate the equation. Example 2: A car travels 160 miles for 4 hours. How fast was it going? d = rt 160 miles = r (4 hours) 160 miles ÷ 4 hrs = r 40 miles/hr = r COACH LESSON 23 9 DISTANCE FORMULA!

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DISTANCE = RATE X TIME WITH THIS FORMULA WE CAN FIND ANY OF THE THREE QUANTITIES, RATE, TIME, OR DISTANCE, IF AT LEAST TWO OF THE QUANTITIES ARE GIVEN. If the time and rate are given, we can find the distance: EXAMPLE: How far did Ed travel in 7 hours if he was going 60 miles per/hour? d = rt d = 60miles/hr x 7 hrs d = 420 miles Or if the distance and rate are given, we can find the time: d = rt 420miles = 60 miles/hr x t (420 miles ÷ 60 miles/hr) = 7 hours 10 PRACTICE THE DISTANCE FORMULA!

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Michael enters a 120-mile bicycle race. He bikes 24 miles an hour. What is Michael's finishing time, in hours, for the race? d = rt A2 B5 C0.2 D PRACTICE USING THE DISTANCE FORMULA! Gildas family goes on a vacation. They travel 125 miles in the first 2.5 hours. If Gildas family continues to travel at this rate, how may miles will they travel in 6 hours? Distance = rate x time 300 miles

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Ratio: comparison of two numbers. Example: Johnny scored 8 baskets in 4 games. The ratio is 8 = Proportion: 2 ratios separated by an equal sign. If Johnny score 8 baskets in 4 games how many baskets will he score in 12 games? 1. Set up the proportion 8 baskets = x baskets 4 games 12 games 2. Cross multiply & Divide 4x = 8 ( 12 ) 4x = 96 x = 96 4 x= 24 baskets COACH LESSON 7 Ratios Word Problems IXL 12 RATIOS & PROPORTIONS!

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ADDING AND SUBTRACTION – FIND COMMON DENOMINATORS! Use factor trees, find prime factors, circle ones that are the same, circle the ones by themselves. Multiply the circled numbers. EXAMPLE: : : x 3 x 2 x 2 = 36 Common denominator = 36 3 x 5 = 4 x 8 = = COACH LESSON 1 Least Common Denominator IXL 13 FRACTIONS!

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14 PRACTICE FRACTIONS!

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Multiplying fractions : cross cancel and multiply straight across ¹ 4 X ¹ 5 = 1 ¹ 5 ² 8 2 Dividing fractions : change the sign to multiply, then reciprocate the 2 nd fraction 3 ÷ = 3 X 8 = 24 REDUCE!!! COACH LESSON 2 Multiplying Fractions IXL Dividing Mixed Numbers IXL 15 MULTIPLYING & DIVIDING FRACTIONS! 1 1/5

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3 X X X PRACTICE MULTIPLYING FRACTIONS!

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When multiplying or dividing mixed numbers, always change them to improper fractions, then multiply. Example 1: 1 ¾ x 1 ½ = 7 x 3 = Example 2: 12 x 2 ½ = 12 x 5 = 60 = Dividing Mixed Numbers IXL Multiplying & Dividing Mixed Numbers!

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When dividing any form of a fraction, change the division to multiplication, then reciprocate the 2 nd fraction. Example: 1 ¾ ÷ 1 ½ = 7 ÷ x 2= 14 = Dividing Fractions IXL 18 Dividing Mixed Numbers! 1 1/6

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LCM : Least Common Multiple : the smallest number that 2 or more numbers will divide into Example: Find the LCM of 24 and 32 You can multiply each number by 1,2,3,4… until you find a common multiple which is 96. Or you can use a factor tree: : : x2x2x3x2x2 = LEAST COMMON MULTIPLE!

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GCF~ GREATEST COMMON FACTOR : The Largest factor that will divide two or more numbers. In this case we would multiply the factors that are the same. 24: 32: Example: 2x2x2 = 8, so 8 is the GCF of 24 and GREATEST COMMON FACTOR!

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21 PRACTICE LCM AND GCF! What is the least common multiple of 3, 6, and 27? A3 B27 C54 D81 What is the greatest common factor of 12, 16, and 20? A2 B4 C6 D12

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What is the greatest common factor (GCF) of 108 and 420 ? A 6 B 9 C 12 D 18 What is the least common multiple (LCM) of 8, 12, and 18 ? A 24 B 36 C 48 D PRACTICE LCM AND GCF!

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ABSOLUTE VALUE: the number itself without the sign; a numbers distance from zero The symbol for this is | | Example: The absolute value of |-5| is 5 The absolute value of |5| is 5 Absolute Value IXL 23 ABSOLUTE VALUE!

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24 PRACTICE ABSOLUTE VALUE! If x=-24 and y=6, what is the value of the expression |x + y|? A18 B30 C-18 D-30

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DISTRIBUTIVE PROPERTY! A(B + C) = AB + AC (We distributed A to B and then A to C) Solving 2 step equations: 4(x + 2) = 24 4x + 8 = 24 subtract 84x = 16 divide by 4x = 4 Remember when solving 2 step equations do addition and subtraction first then do multiplication and division. This is opposite of (please excuse my dear aunt sally,) which we use on math expressions that dont have variables. COACH LESSON 20 Distributive Property IXL 25

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Always has parentheses A ( B X C) = B (C X A) FOR MULTIPLICATION A + (B + C) = B + (C + A) FOR ADDITION A X B = B X A FOR MULTIPLICATION A + B = B + A FOR ADDITION 26 Associative Commutative Properties for Multiplication IXL Commutative Property for Addition IXL Associative & Commutative Property!

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We use stem and leaf plots to organize scores or large groups of numbers. To arrange the numbers into a stem and leaf plot, the tens place goes in the stem column and the ones place goes in the leaf column. Example: We will arrange the following numbers in a stem & leaf plot: 40, 30, 43, 48, 26, 50, 55, 40, 34, 42, 47, 47, 52, 25, 32, 38, 41, 36, 32, 21, 35, 43, 51, 58, 26, 30, 41, 45, 23, 36, 41, 51, 53, 39, 28 Stem Leaf Stem and Leaf Plots, Box – and – Whisker Plots Stem-and-Leaf-Plots IXL COACH LESSON 24

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MODE The number that occurs the most oftenThe mode of these scores– is 41. RANGE The difference between the least and greatest numberis 37. MEDIAN The middle number of the set when the numbers are arranged in orderit is 40. MEAN – Another name for average is mean. FIRST QUARTILE OR LOWER QUARTILE The middle number of the lower half of scoresis 32. THIRD QUARTILE OR UPPER QUARTILE The middle number of the upper half of scoresis 47. COACH LESSON 27, Leaf Lower quartile- 32 Upper quartile- 47 Stem

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Box-and-Whisker Plot! Lower extreme First quartile or lower quartile Second quartile or median Third quartile or upper quartile Upper extreme Inter quartile Range 29

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Make a stem and leaf plot from the following numbers. Then make a box and whiskers diagram. 25, 27, 27, 40, 45, 27, 29, 30, 26, 23, 31, 35, PRACTICE STEM & LEAF/ BOX & WHISKERS! Stem Leaf

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Below are the number of points John has scored while playing the last 14 basketball games. Finish arranging Johns points in the stem and leaf plot and then find the range, mode, and median. Points: 5, 14, 21, 16, 19, 14, 9, 16, 14, 22, 22, 31, 30, 31 StemLeaf Range: Mode: Median: 31 PRACTICE STEM & LEAF/ BOX & WHISKERS!

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Note that there are not any variables in the statement. This is why we use order of operation instead of the Distributive Property. COACH LESSON 5 32 Order of Operations!

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More Practice! 1.) 3 + 2(4 x 3)2.) ) ( ) – 64.) 64 – PRACTICE ORDER OF OPERATIONS! Karen is solving this problem:(3² + 4²)² = ? Which step is correct in the process of solving the problem? A (3² + 4) B (9² + 16²)² C (7²)² D (9 + 16)² 3 + 2(12) –

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Order of Operations Math Masters Order of Operations IXL 34 PRACTICE ORDER OF OPERATIONS! Simplify the expression below. (6² - 2) · 16 A 16 B 64 C 80 D ) 2³ = 2 x 2 x 2 = 2.) 3 = 3 x 3 x 3 x 3 = 3.) 4² = 4 x 4 = 5.) 64 = 4.) 144 =

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FINDING THE MISSING ANGLE OF A TRIANGLE! 65° 50° a bc Finding b: Since the sum of the degrees of a triangle is 180 degrees, we subtract the sum of = 115 from = 65 …so Angle b = 65° Finding c: If b = 65 to find c we know that a straight line is 180 degrees so if we subtract 180 – 65 = 115° …soAngle c = 115° Finding a: To find a we do the same thing. 180 – 50 = 130 …so Angle a = 130° Measuring Angles IXL 35

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Practice finding the measure of

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A square has 4 angles which each measure 90 degrees. 45 DA C B 37 What is the total measure of the interior angles of a square? 360 °

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Hypotenuse Height = 6 in Base = 8 inches C² = A² + B² C² = (6)² in + (8)² in C² = 36 in² + 64 in² C² = 100 in² C = 10 in² Pythagorean Theorem MathMasters 38 Pythagorean Theorem! To find the missing hypotenuse of a right triangle, we use the formula…

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Height= 8 in Base= 10 in Area = base x height 2 A = 10in x 8 in 2 A = 80 in² 2 A = 40 in² Area of Triangles & Trapezoids IXL COACH LESSON AREA OF A TRIANGLE! Definition of height is a line from the opposite vertex perpendicular to the base.

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Height= 4 ft Base= 2 ft Area = ½ bh A = ½ (2ft)(4ft) A = ½ 8ft A =4 ft² 40 PRACTICE FINDING THE AREA OF A TRIANGLE!

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h b 41 FINDING THE AREA OF A PARALLELOGRAM!

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Area of a RECTANGLE = Length x Width Area of a SQUARE = Side x Side A = l x w 4ft 2ft A = 4ft x 2ft A = 8ft² 2ft Area of Rectangles Parallelograms IXL 42 AREA OF A RECTANGLE & A SQUARE! Example: A = s x s A = 2ft x 2ft A = 4ft² 4ft² 8ft²

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PERIMETER IS THE OUTER DISTANCE AROUND A FIGURE. 9 FT 3FT P = a + b + c + … P = 9FT + 9FT + 3FT + 3FT P = ____ FT 43 CALCULATING PERIMETER! 27

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To find the area of a compound figure, we simply have to find the area of both figures, then add them together. 6FT AREA = LENGTH X WIDTH A = 2FT X 6FT A = 12FT² AREA = LENGTH X WIDTH A = 3FT X 5FT A = 15 FT² 44 CALCULATING PERIMETER AND AREA OF COMPOUND FIGURES! 7FT 3FT 2FT TOTAL AREA = 12FT² + 15FT² = 27FT²

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CONGRUENT ANGLES & CONGRUENT SIDES! Congruent angles and sides mean that they have the same measure. Use symbols to show this! Complementary Supplementary Vertical & Adjacent Angles IXL 45

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Complementary angles : angles whose sum equals 90 degrees Supplementary angles: angles whose sum equals 180 degrees Right angle: angle measures 90 degrees ---symbol Acute angle: angle less than 90 Obtuse angle: angle greater than 90 degrees Congruent: when two figures are exactly the same Similar: when two figures are the same shape but not the same size Regular: when a figure has all equal sides Line of symmetry: when a line can cut a figure in two symmetrical sides COACH LESSON 17 46

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Parallel lines: lines that never touch--- symbol Perpendicular lines: lines that intersect---symbol Skew lines: lines in different planes that never intersect Plane: a flat, 2-Dimensional surface, formed by many points A point (0-Dimension); A line (1-D); A plane (2-D); A solid (3-D) Vertical angles: angles that share a point and are equal Adjacent angles: are angles that are 180 degrees and share a side COACH LESSON 18 47

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Adjacent Angles: Angles that share a common side In the figure below: ANGLES 3 AND 4 ARE ADJACENT ANGLES. ANGLES 2 AND 3 ARE ALSO ADJACENT ANGLES. What are some other adjacent angles? Complementary Supplementary Vertical Adjacent Angles IXL 48 RECOGNIZING ADJACENT ANGLES!

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REVIEW: CLASSIFYING LINES! Supplementary angles: sum is 180 degrees Complementary angles: sum is 90 degrees Straight angle: equal to 180 degrees 49 Complementary Supplementary Vertical & Adjacent Angles IXL

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What is the total number of lines of symmetry that can be drawn on the trapezoid below? Circle One: A.) 4 B.) 3 C.) 2 D.) 1 Which figure below correctly shows all the possible lines of symmetry for a square? Circle One: A.) Figure 1 B.) Figure 2 C.) Figure 3 D.) Figure 4 Symmetry IXL 50 PRACTICE GEOMETRY!

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Calculating Volume of a Quadrilateral! 4 ft 5 ft 3 ft Volume IXL 51 V = 5ft x 3ft x 4ft = 60ft³ [Volume= units³ or cubed units]

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Two figures are similar if they have exactly the same shape, but may or may not have the same size. The symbol is 52 Identifying Similar Figures! A BC X Y Z For example: ABC XYZ Which angle is similar to angle B? Angle: _______ Y

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Diameter: distance across the center of the circle (double radius) Radius: the distance half way across the circle ( ½ diameter) Chord: line that cuts the circle and does not go through the center of the circle Sector: a pie-shaped part of a circle made by two radii Segment: the area of a circle in which a chord creates Circumference: distance around the outside of the circle COACH LESSON Arc: a connected section of the circumference of a circle

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Inscribed angles: angles on the inside of the circle formed by two chords Central angles: angles in the center of the circle formed by two radii COACH LESSON 15 54

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55 PRACTICE FINDING THE CIRCUMFERENCE OF A CIRCLE! If the circumference of a circle s 16Π, what is the radius? Hint: C= 2Πr A4 B8 C16 D32

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56 PRACTICE FINDING THE AREA OF A CIRCLE! If the diameter of a car tire is 30 cm, what is the area of that circle? Round your answer. Hint: Area = Π x r² *USE = 3.14 A cm² B 314 cm² C 7,070 cm² D 707 cm²

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A duck swims from the edge of a circular pond to a fountain in the center of the pond. Its path is represented by the dotted line in the diagram below. What term describes the duck's path? A chord B radius C diameter D central angle 57 MORE PRACTICE!

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Rules: Negative + Negative = Negative = -7 Positive + Positive = Positive = 7 Negative + Positive = ? (Keep the sign of the larger integer & subtract) = -1 Add & Subtract Integers IXL 58 Adding Negative Numbers!

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Rules: Negative x Negative = PositiveNegative ÷ Negative = Positive -4 x -2 = 8-4 ÷ -2 = 2 Positive + Positive = PositivePositive ÷ Positive = Positive 4 x 2 = 84 ÷ 2 = 2 Negative x Positive = NegativeNegative ÷ Positive = Negative -4 x 2 = ÷ 2 = Multiplying & Dividing Negative Numbers! Multiplying & Dividing Integers IXL

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Negative integers further to the left of zero have less value. Positive integers further to the right of zero have greater value. Example: -3 IS GREATER THAN -6 COACH LESSON 3 60 Comparing & Ordering Integers! NEGATIVEPOSITIVE

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Use the following symbols for inequality number sentences: < less than -4 < 2 less than or equal to 3 4 > greater than 6 > 3 greater than or equal to One-step Linear Inequalities IXL 61 Inequalities!

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To solve for a variable in an equation, the variable must be alone on one side of the equals sign. Use a model or an inverse operation to solve a one step equation. Example: 3x = 24 Step 1: Divide by 33x = 24 on both sides3 3 of the equation x = 8 COACH LESSON 21 Two-step Linear Equations IXL 62 Solving One-Step Equations!

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We can translate math sentences to numbers and symbols only Examples: Translate: five more than(5 + n) Translate: three times a number(3 x n, or 3n) When you combine both: five more than three times a number 5 + 3n or 3n +5 COACH LESSON Modeling Mathematical Situations!

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Functions: inserting a value in for x to find y or f(x) Example: f(x) = 2x + 4If x = 2 Then f(x) = 2 (2) + 4 f( x) = f(x) = 8 So y = 8 A function is when we put a value in and get an answer out. COACH LESSON 20 Evaluating Functions IXL 64 Functions!

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Scientific notation x 10 (This means to move the decimal six places to the right.) x 10 becomes 4,057,000 Expanded notation --- numbers written using powers of 10 Example: 4,234 = (4 x 10³) + (2 x 10²) + (3 x 10¹) + (4 x 10 ) = 4,234 Any number raised to the zero power equals = 1 Any number raised to the 1 st power equals that number. 8 ¹ = 8 65 Scientific Notation!

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METRIC SYSTEM & CONVERSTION! START at the unit you currently have, then move the decimal to the unit youre looking for. Example 1:4 kilometers = 4000 meters Example 2:36 millimeters = 3.6 centimeters COACH LESSON Kilo Hecto Deka Meter Liter Gram Deci Centi Milli

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67 PRACTICE UNIT CONVERSIONS! The students in a math class measured and recorded their heights on a chart in the classroom. Keiths height was 1.62 meters. Which is another way to show Keiths height? A cm B cm C 162 cm D 1,620 cm A drawing of the Greensburg Airport uses a scale of 1 centimeter = 300 meters. Runway A is drawn 12 centimeters long. How many meters is the actual length of the runway? F300 G360 H3,000 J3,600

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Weight Unit Conversions! Use the chart and move the decimal point. Gram = weight Meter = distance Liter = volume For U.S. Customary measurement, conversions are on PSSA charts provided during testing time. 68

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The flower box in front of the city library weighs 124 ounces. What does the flower box weigh in pounds? * Hint: 1 pound = 16 ounces A7 ½ B7 ¾ C868 D PRACTICE WEIGHT UNIT CONVERSIONS! Which of the following is a metric unit for measuring mass? A meter B liter C pound D gram

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70 PRACTICE MORE UNIT CONVERSIONS! A scientist measures the mass of a rock and finds that it is 0.16 kilogram. What is the mass of the rock in grams? A 1.6 grams B 16 grams C 160 grams D 1,600 grams

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1.Always list the conversion. 2.Identify the correct multiplier. 3.Set up the multiplication problem with units being opposite (top & bottom) 4.Multiply & Simplify For example: Change 240 feet to yards a)First list the conversions:3 feet OR 1 yard 1 yard 3 feet b)Since we want yards multiply by 1 yard 3 feet c)So 240 feet x 1 yard 13 feet d)Then 240 feet = 80 yards COACH LESSON 9 71 Unit Multipliers!

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A ratio is a comparison between two numbers. Two ratios separated by an equals sign is called a proportion. COACH LESSON 7 Ratios IXL 72 Ratios & Proportions: To solve a proportion, we cross multiply and divide. Example:4 = 2 5 = x 4x = 10x = x = 2 ½

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73 Rational & Irrational Numbers An Irrational Number is a real number that cannot be written as a simple fraction. A Rational Number can be written as a simple fraction. Irrational means not Rational. Example: 7 is rational, because it can be written as the ratio 7/1 Example (3 repeating) is also rational, because it can be written as the ratio 1/3

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Practice Irrational Numbers! 74 Which of these is an irrational number? A -2 B 56 C 64 D 3.14 Which of these is an irrational number? A 3 B C 7 11 D 1 9

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FractionDecimalPercent Place number over its place value and reduce Divide by 100Multiply by = x 100 = 75% 125 = x 100 = 12.5% 150 = 3 = 1 ½ x 100 = 150% Converting Rational Numbers! COACH LESSON 4 75

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Points on a Coordinate Grid! Quadrant I Quadrant II Quadrant III Quadrant IV COACH LESSON 16 Ordered pair: [3, 2] 3 is x value and 2 is y value Point of Origin [0, 0] 76

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A scale is the ratio of the measurements of a drawing, a model, a map or a floor plan, to the actual size of the objects or distances. Example: An architects floor plan for a museum exhibit uses a scale of 0.5 inch = 2 feet. On this drawing, a passageway between exhibits is represented by a rectangle 3.75 inches long. What is the actual length of the passageway? To find an actual length from a scale drawing, identify and solve a proportion. Drawing = Drawing Actual Let p = the actual length in feet of the passageway Use cross products to solve the proportion 0.5 = p 0.5 x p = 2 x p = 7.5 p = 15 COACH LESSON 14 Scale & Indirect Measurement MathMaster 77 Scaling!

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SOLVING PROBLEMS USING PATTERNS! Example: Erin is collecting plastic bottles. On Monday she has 7 bottles, on Tuesday she has 14 bottles, on Wednesday she has 21 bottles, and on Thursday she has 28 bottles. If the pattern continues, how many bottles will she have on Friday? 1)Notice the pattern: 7, 14, 21, 28 2)Write the different operations that you can perform on 7 to get 14. a)7 + 7 = 14 b)7 x 2 = 14 3)Check these operations with the next term in the pattern. c) = 21 d) 14 x 2 = 28 4)Find the next term in the pattern to determine how many bottles Erin will have on Friday. 5) = 35 COACH LESSON 19 78

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Estimation! Estimating involves finding compatible numbers that will make the numbers easier to operate. Leos yearly salary is $51,950. Estimate how much money Leo makes in one week. $51,950 is about $52,000. Divide the compatible numbers. $52,000 divided by 52 = $1,000 COACH LESSON 10 79

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Histogram is a bar graph without the spaces between the bars. Bar graphs have spaces to show differences in data. COACH LESSON 26 Interpret Histograms IXL 80

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Double and Triple Bar & Line Graphs are used to show two sets of related data. COACH LESSON 25 81

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We can use trends or patterns seen in graphs to make predictions. COACH LESSON Making Predictions!

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