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Bell Ringer

Bell Ringer. Solve for x. 2x + 10 = 60 – 10 – 10 2x = 50 2 2 x = 25 . 30°. 2x + 10. Final Exam Review. Exam. The exam will be 40 multiple choice questions with 2 extra credit questions. You will have 1 hour to complete the exam. No extra time will be given.

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Bell Ringer

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  1. Bell Ringer Solve for x. 2x + 10 = 60 – 10 – 10 2x = 50 2 2 x = 25 30° 2x + 10

  2. Final Exam Review

  3. Exam • The exam will be 40 multiple choice questions with 2 extra credit questions. • You will have 1 hour to complete the exam. • No extra time will be given. • You may bring ONE sheet of notes to use on the exam.

  4. Topics The final exam will cover: 1. Inequalities 2. Probability 3. Area & Perimeter of Polygons & Circles 4. Angles & Lines 5. Exponents 6. Radicals 7. Polynomials

  5. Inequalities

  6. 1. Graphing Linear Inequalities . . . • < or > = Open Circle • < or > = Closed Circle • < or < = Shade to the left. • > or > = Shade to the right.  Less Than Greater Than

  7. 1. Graphing Linear Inequalities x < 3 Open or Closed? Right or Left? x > -4 Open or Closed? Right or Left? .

  8. 2. One-Step Linear Inequality -5 + x < -1 + 5 +5 x < 4 Now graph it!

  9. 3. Two-Step Linear Inequality 4x + 1 > -11 – 1 – 1 4x> -12 4 4 x > -3

  10. 4. Reversing the Sign! • Whenever we multiply or divide by a negative number, we must REVERSE the inequality sign. -2x < 6 -2 -2 x > -3 We have to divide by -2. So we have to reverse the sign.

  11. 4. Reverse the Sign -3x + 1 < 10 – 1 – 1 -3x < 9 -3 -3 x > -3 Reverse the Sign!

  12. Key Words “At least” means greater than or equal to (>) “No more than” means less than or equal to (<) “More than” means greater than (>) “Less than” means less than (<)

  13. 5. Word Problems Chris has $200 in his bank account. He makes $10 an hour at his job. He wants to save at least $400 to buy some chickens. What is the minimum number of hours Chris will have to work? 200 + 10h > 400 – 200 – 200 10h >200 10 10 h > 20 hours Wants more than this amount! Has already! Mo’ Money, mo’ money, mo’ money!

  14. 5. Word Problems Tom calls a cab which charges $2.50 plus $0.50 a mile. If Tom has no more than $20.00 in his pocket, how far can he go? $2.50 + $0.50m < $20 -2.50 -2.50 0.50m <17.50 0.50 0.50 m < 35 miles That’s a lie. I got big bank!

  15. Probability

  16. Probability Event – This is the selected outcome. Ex. If event A is the probability of rolling a 5 or higher, the probability is 2/7, so P(A) = 2/7. Complement – This is the probability of everything other than the event. Ex. In the example above, the complement is rolling 4 or lower, so the complement of event A is 5/7, or P(A) = 5/7. Probability of “A Bar”

  17. Coin Toss • If you toss a coin twice, what are the possible outcomes? HH, TT, HT, TH • What is the probability of two heads? HH, TT, HT, TH = • What is the probability of at least one head? HH, TT, HT, TH = It’s complement would be 3/4! 1/4 3/4 It’s complement would be 1/4!

  18. Two Independent Events • To find the probability of two independent events occurring together, multiply their probabilities! • Ex. Find the probability of tossing a coin twice and having heads occur twice. Probability of two heads! . 1 2 1 2 1 4 = Probability of Toss #1 coming up heads. Probability of Toss #2 coming up heads.

  19. Independent Events Ex. A coin is tossed and a card is drawn from a standard deck. a. What is the probability of tossing heads and drawing an ace? b. What is the probability of tossing tails and drawing a face card? . 1 13 1 2 1 26 = . 3 13 1 2 3 26 =

  20. Working with Factorials (4•3•2•1) + (3•2•1) = 30 (3•2•1) – (2•1) = 4 (4•3•2•1) (2•1) = 48 • 4! + 3! = • 3! – 2! = • 4! 2! = • = 6! 4! (6•5•4•3•2•1) (4•3•2•1) = 30

  21. Combinations (Formula) (Order doesn’t matter! AB is the same as BA) nCr = Where: n = number of things you can choose from r = number you are choosing n! r! (n – r)!

  22. Combinations (Formula) • There are 6 pairs of shoes in the store. Your mother says you can buy any 2 pairs. How many combination of shoes can you choose? So n = 6 and r = 2 6C2 = = 6! 2! (6 – 2)! 6•5•4•3•2•1 2•1(4•3•2•1) 30 2 = = 15 combinations!

  23. Permutations (Formula) (Order does matter! AB is different from BA) nPr = Where: n = number of things you can choose from r = number you are choosing n! (n – r)!

  24. Permutations (Formula) • In a 7 horse race, how many different ways can 1st, 2nd, and 3rd place be awarded? So n = 7 and r = 3 7P3 = = 7! (7 – 3)! 7•6•5•4•3•2•1 (4•3•2•1) = 210 permutations!

  25. Fundamental Counting Principle • You have a choice of 3 meats, 4 cheeses, and 2 breads. How many different types of sandwiches could you make? Multiply the choices! 3 • 4 • 2 = 24 different sandwiches

  26. Area & Perimeter: Polygons and Circles

  27. Key Vocabulary • Perimeter – The distance around an polygon. • Area – The amount of space inside a two dimensional shape.

  28. Perimeter • Estimate or calculate the length of a line segment based on other lengths given on a geometric figure. 8 + x = 17 - 8 - 8 x = 9 in 17 in 8 in Easy! x

  29. Perimeter • Compute the perimeter of polygons when all side lengths are given Add all the sides: 8 + 7 + 7 + 8 + 7 + 7 = 44 in 8 in 7 in 7 in 7 in Even I can do this! 7 in 8 in

  30. Area • Compute the area of rectangles when whole number dimensions are given. 25 in A = L • W A = 25 • 6 A = 150 in2 6 in Area of Rectangle = Length • Width

  31. Area • Compute the area and perimeter of triangles and rectangles in simple problems. Area of Triangles = 1/2 • Base • Height Height Height Base Base

  32. Area • Kyla mows lawns for $1.20 per square feet. How much did she charge to cut the lawn below? A = L • W A = 23 • 11 A = 253 ft 2 Price = 253•$1.20 Price = $303.60 11 ft 23 ft

  33. Distance of Sides • Find the missing value, x. 28 in x + 2x + x = 28 4x = 28 4 4 x = 7 inches x x 2x I get it. Add up the bottom sides to equal the top!

  34. Perimeter Find x: x + 5 = 17 – 5 – 5 x = 12 • Find the perimeter. x + 5 x – 1 If x = 12, then x – 1 is 11! 17 in Use x to find perimeter: 17 + 17 + 11 + 11 Perimeter = 56 inches

  35. Area Area of ‘A’ A = 10 • 7 A = 70 ft2 Area of ‘B’ A = 5 • 8 A = 40 ft2 A = 70 + 40 A = 110 ft2 • Find the area. 15 ft B 5 ft A 10 ft 8 ft 5 ft 7 ft I can find the area by cutting it!

  36. Area of Trapezoids • Definition – Quadrilaterals with at least one pair of parallel sides. Area of a Trapezoid = (Find the average of the bases and multiply by the height!) b1 & b2 are the top and bottom bases. h is the height. b1 h b2 (b1+ b2) 2 • h

  37. Area of Trapezoids • Find the area of the trapezoid below. 12 9 16 (b1+ b2) 2 A = • h (12 + 16) 2 = 126 ft2 = • 9

  38. Parts of a Circle Radius . Diameter Circumference Circumference – The distance around a circle. (Perimeter of a circle.) Radius – The distance from the center of a circle to any point on its circumference. Diameter – The distance from one side of a circle, passing through the center, to the other side of the circle.

  39. Circumference of a Circle The diameter of a circle is equal to twice the radius, or d = 2r Circumference of a circle is equal to the diameter multiplied by pi, or C = 2πr or C = πd

  40. Circumference of a Circle 5 C = 2πr C = 2π5 C = 10π or 31.42

  41. Circumference of a Circle Diameter is 25.5! 25.5 C = dπ C = 25.5π or 80.11

  42. Area of a Circle . A = π62 A = 36π A = 113.10 6 The area of a circle is equal to: A = πr2

  43. Word Problems 12 in diameter 12 in 12 in This is the better buy! A = 12 • 12 = 144 in2 A = π • 62 = 113.10 in2 Pizza World offers two types of pizzas: rectangles and circles. If each pizza cost $12.50, which is the better buy?

  44. Angles & Lines

  45. Parallel Lines • You can identify parallel lines by their equations! y = 3x + 7 y = 3x – 9 These two lines are parallel. Their slopes are the same! (Notice that they have different y-intercepts!)

  46. Perpendicular Lines • Lines that intersect at right angles (900) are perpendicular. • Perpendicular lines have slopes that are negative reciprocals. • The product of their slopes = -1. These two lines are perpendicular. They intersect at a right angle.

  47. Perpendicular Lines Negative reciprocals 1. What is the reciprocal of ? 2. What is the reciprocal of 3? 2 3 3 2 So the negative reciprocal is – ! 3 2 1 3 So the negative reciprocal is – ! 1 3

  48. Perpendicular Lines These equations are perpendicular: y = 2x + 8 y = - x – 5 y = - x – 7 y = x + 5 1 2 3 2 4 5 5 4

  49. Complementary Angles • Complementary Angles – Two angles that add up to 90° Angles X and Angle Y are complementary and add up to 90 °. X Y

  50. Try this… • Find the missing angle. x° 36° 90 – 36 = 54°

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