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## COMPREHENSIVE REVIEW FOR MIDDLE SCHOOL MATHEMATICS

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**COMPREHENSIVE REVIEW FOR**MIDDLE SCHOOL MATHEMATICS 2013**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: www.studyisland.com www.ixl.com(links provided throughout) www.mathmaster.org(links provided throughout) and PSSA Coach workbook Created by: Jessie Minor**EXPERIMENTAL PROBABILITY!**IN ORDER TO CALCULATE EXPERIMENTAL PROBABILITY OF AN EVENT USE THE FOLLOWING DEFINITION: P(Event)= Number of times the event occurred Number of total trials Coach Lesson 30**EXPERIMENTAL PROBABILITY!**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. 4 Experimental Probability IXL**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. 80 96 144 192 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? 10% 25% 30% 50%**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. PRACTICE THEORETICAL PROBABILITY! • 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 = ______% 60 Coach Lesson 29**RATE/ UNIT PRICE/ SALES TAX!**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 = 72 1200 + 72 $1272 Unit Prices IXL COACH LESSON 4**PRACTICE 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? $7.99 x 3 = $23.97 $23.97 x 0.06 = $1.4382 Sales Tax = $1.44**DISTANCE FORMULA!**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**PRACTICE THE DISTANCE FORMULA!**• 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**PRACTICE USING THE DISTANCE FORMULA!**Gilda’s family goes on a vacation. They travel 125 miles in the first 2.5 hours. If Gilda’s family continues to travel at this rate, how may miles will they travel in 6 hours? Distance = rate x time 300 miles Michael enters a 120-mile bicycle race. He bikes 24 miles an hour. What is Michael'sfinishing time, in hours, for the race? d = rt A 2 B 5 C 0.2 D 0.5**RATIOS & PROPORTIONS!**• Ratio: comparison of two numbers. • Example: Johnny scored 8 baskets in 4 games. The ratio is 8 = 2 • 4 1 • 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 Ratios Word Problems IXL COACH LESSON 7**FRACTIONS!**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: 5 + 8 12 9 12 9 2 6 3 3 12: 2 2 3 2 3 9: 3 3 3 x 3 x 2 x 2 = 36 Common denominator = 36 3 x 5 = 4 x 8 = 15 + 32 = 47 36 36 36 36 36 Least Common Denominator IXL COACH LESSON 1**MULTIPLYING & DIVIDING FRACTIONS!**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 2nd fraction 3 ÷ 5 4 8 = 3 X 8 = 24REDUCE!!! 4 5 20 1 1/5 Multiplying Fractions IXL Dividing Mixed Numbers IXL COACH LESSON 2**PRACTICE MULTIPLYING FRACTIONS!**1 X 7 3 X 5 6 5 X 4 9 5 49 13 5 8 4 9 1 91**Multiplying & Dividing Mixed Numbers!**When multiplying or dividing mixed numbers, always change them to improper fractions, then multiply. Example 1: 1 ¾ x 1 ½ = 7 x 3 = 21 4 2 8 Example 2: 12 x 2 ½ = 12 x 5 = 60 = 1 2 2 2 5 8 30 Dividing Mixed Numbers IXL**Dividing Mixed Numbers!**When dividing any form of a fraction, change the division to multiplication, then reciprocate the 2nd fraction. Example: 1 ¾ ÷ 1 ½ = 7 ÷ 3 4 2 7 x 2 = 14 = 4 3 12 11/6 Dividing Fractions IXL**LEAST COMMON MULTIPLE!**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: 24 32 2 12 2 16 2 2 6 2 2 8 2 2 2 3 2 2 2 4 24: 2 2 2 2 2 32: 22 22 22 32 2 2x2x2x3x2x2 = 96**GREATEST COMMON FACTOR!**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 32. 22 22 22 32 2**PRACTICE LCM AND GCF!**What is the least common multiple of 3, 6, and 27? A 3 B 27 C 54 D 81 What is the greatest common factor of 12, 16, and 20? A 2 B 4 C 6 D 12**PRACTICE LCM AND GCF!**What is the greatest common factor (GCF) of 108 and 420 ?A 6B 9C 12D 18 What is the least common multiple (LCM) of 8, 12, and 18 ?A 24B 36C 48D 72**ABSOLUTE VALUE!**ABSOLUTE VALUE: the number itself without the sign; a number’s 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**PRACTICE ABSOLUTE VALUE!**If x=-24 and y=6, what is the value of the expression |x + y|? A 18 B 30 C -18 D -30**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 8 4x = 16 • divide by 4 x = 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 don’t have variables. Distributive Property IXL COACH LESSON 20**Associative**Commutative Associative & Commutative Property! • 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 Properties for Multiplication IXL Commutative Property for Addition IXL**Stem and Leaf Plots,Box – and – Whisker Plots**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 2 3 4 5 Leaf 1 3 5 6 6 8 0 0 2 2 4 5 6 6 8 9 0 0 1 1 1 2 3 3 5 7 7 8 0 1 1 2 3 5 8 Stem-and-Leaf-Plots IXL COACH LESSON 24**Stem**2 3 4 5 Leaf 1 3 5 6 6 8 0 0 2 2 4 5 6 6 8 9 0 0 1 1 1 2 3 3 5 7 7 8 0 1 1 2 3 5 8 MODE—The number that occurs the most often—The mode of these scores– is 41. RANGE—The difference between the least and greatest number—is 37. MEDIAN—The middle number of the set when the numbers are arranged in order—it is 40. MEAN– Another name for average is mean. FIRST QUARTILE OR LOWER QUARTILE —The middle number of the lower half of scores—is 32. THIRD QUARTILE OR UPPER QUARTILE—The middle number of the upper half of scores—is 47. Upper quartile- 47 Lower quartile- 32 COACH LESSON 27, 25**Box-and-Whisker Plot!**First quartile or lower quartile Upper extreme Second quartile or median Third quartile or upper quartile Lower extreme Inter quartile Range**PRACTICE STEM & LEAF/ BOX & WHISKERS!**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, 39 Stem 2 3 4 Leaf 3 5 6 7 7 7 9 0 1 5 9 0 5**PRACTICE STEM & LEAF/ BOX & WHISKERS!**Below are the number of points John has scored while playing the last 14 basketball games. Finish arranging John’s 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 26 14 17.5 Range: Mode: Median: 5 9 4 4 4 6 6 9 1 2 2 0 1 1**Order of Operations!**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**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 + 4⁴) B (9² + 16²)² C (7²)² D (9 + 16)² More Practice! 1.) 3 + 2(4 x 3) 2.) 12 - 15 - 3 3.) (22 + 14) – 6 4.) 64 – 8 + 8 3 + 2(12) 3+ 24 27 -3 -3 -6 56 + 8 64 36 – 6 30**PRACTICE ORDER OF OPERATIONS!**Simplify the expression below. (6² - 2⁴) · √16 A 16 B 64 C 80 D 108 Order of Operations Math Masters 12 4.) √144 = 1.) 2³ = 2 x 2 x 2 = 8 8 5.) √64 = 2.) 3⁴ = 3 x 3 x 3 x 3 = 81 3.) 4² = 4 x 4 = 16 Order of Operations IXL**FINDING THE MISSING ANGLE OF A TRIANGLE!**Finding b: Since the sum of the degrees of a triangle is 180 degrees, we subtract the sum of 65 + 50 = 115 from 180 180 - 115 = 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° …so Angle c = 115° Finding a: To find a we do the same thing. 180 – 50 = 130 …so Angle a = 130° a 50° 65° b c Measuring Angles IXL**Practice finding the measure of <A in the triangle ABC**below! A B C 30° m<A + 90 + 30 = 180 m<A = 60 °**A square has 4 angles which each measure 90 degrees.**D A 45 45 What is the total measure of the interior angles of a square? 45 45 C B 360 °**Pythagorean Theorem!**To find the missing hypotenuse of a right triangle, we use the formula… A² + B² = C² C² = A² + B² C² = (6)² in + (8)² in C² = 36 in² + 64 in² C² = 100 in² √C²= √100 in² C = 10 in² Hypotenuse Height = 6 in Base = 8 inches Pythagorean Theorem MathMasters**AREA OF A TRIANGLE!**A = base x height 2 Area = base x height 2 A = 10in x 8 in 2 A = 80 in² 2 A = 40 in² Height= 8 in Base= 10 in Definition of height is a line from the opposite vertex perpendicular to the base. Area of Triangles & Trapezoids IXL COACH LESSON 12**PRACTICE FINDING THE AREA OF A TRIANGLE!**AREA = ½ (BASE X HEIGHT) A = ½ bh Area = ½ bh A = ½ (2ft)(4ft) A = ½ 8ft A =4 ft² Height= 4 ft Base= 2 ft**FINDING THE AREA OF A PARALLELOGRAM!**h b Area = b x h**AREA OF A RECTANGLE & A SQUARE!**Area of a RECTANGLE = Length x Width Area of a SQUARE = Side x Side Example: 2ft 2ft 4ft 2ft A = l x w A = s x s A = 4ft x 2ft A = 8ft² A = 2ft x 2ft A = 4ft² 4ft² 8ft² Area of Rectangles Parallelograms IXL**CALCULATING PERIMETER!**PERIMETER IS THE OUTER DISTANCE AROUND A FIGURE. 9 FT 3FT P = a + b + c + … P = 9FT + 9FT + 3FT + 3FT P = ____ FT 27**CALCULATING PERIMETER AND AREA OF COMPOUND FIGURES!**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 2FT 7FT 3FT TOTAL AREA = 12FT² + 15FT² = 27FT²**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**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**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**RECOGNIZING ADJACENT ANGLES!**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? 2 3 1 4 Complementary Supplementary Vertical Adjacent Angles IXL**REVIEW: CLASSIFYING LINES!**• Supplementary angles:sum is 180 degrees • Complementary angles:sum is 90 degrees • Straight angle: equal to 180 degrees Complementary Supplementary Vertical & Adjacent Angles IXL**PRACTICE GEOMETRY!**What is the total number of lines of symmetry that can be drawnon 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