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NHTM Spring Conference Nashua, NH March 17, 2014 Steve Yurek Lesley Cambridge, MA Copyright Steve Yurek March 17, 20141.

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Presentation on theme: "NHTM Spring Conference Nashua, NH March 17, 2014 Steve Yurek Lesley Cambridge, MA Copyright Steve Yurek March 17, 20141."— Presentation transcript:

1 NHTM Spring Conference Nashua, NH March 17, 2014 Steve Yurek Lesley Universitysyurek@lesley.edu Cambridge, MA Copyright Steve Yurek March 17, 20141

2 Napolis Yard Sale How much do you want for that set of 2004 Red Sox Baseball Cards? $5,000 Ill give you $1,000. $4,000 $2,000 $3,000 OK, its a deal Copyright Steve Yurek March 17, 20142

3 Napolis Yard Sale How much do you want for that set of 2004 Red Sox Baseball Cards? $5,000 Ill give you $1,000. $4,000 $2,000 $3,000 OK, its a deal Copyright Steve Yurek March 17, 20143

4 Napolis Yard Sale How much do you want for that set of 2004 Red Sox Baseball Cards? $5,000 Ill give you $1,000. $4,000 $2,000 $3,000 OK, its a deal Copyright Steve Yurek March 17, 20144

5 Napolis Yard Sale How much do you want for that set of 2004 Red Sox Baseball Cards? $5,000 Ill give you $1,000. $4,000 $2,000 $3,000 OK, its a deal Copyright Steve Yurek March 17, 20145

6 Napolis Yard Sale How much do you want for that set of 2004 Red Sox Baseball Cards? $5,000 Ill give you $1,000. $4,000 $2,000 $3,000 OK, its a deal Copyright Steve Yurek March 17, 20146

7 Napolis Yard Sale How much do you want for that set of 2004 Red Sox Baseball Cards? $5,000 Ill give you $1,000. $4,000 $2,000 $3,000 OK, its a deal Copyright Steve Yurek March 17, 20147

8 Napolis Yard Sale How much do you want for that set of 2004 Red Sox Baseball Cards? $5,000 Ill give you $1,000. $4,000 $2,000 $3,000 OK, its a deal Copyright Steve Yurek March 17, 20148

9 Napolis Yard Sale How much do you want for that set of 2004 Red Sox Baseball Cards? $5,000 Ill give you $1,000. $4,000 $2,000 $3,000 OK, its a deal Copyright Steve Yurek March 17, 20149

10 Napolis Yard Sale Which mean is at work here? The Arithmetic Mean (5,000 + 1,000)/2 3,000 Lets see what the arithmetic mean looks like geometrically. Copyright Steve Yurek March 17, 201410

11 Napolis Yard Sale Which mean is at work here? The Arithmetic Mean (5,000 + 1,000)/2 3,000 Lets see what the arithmetic mean looks like geometrically. Arithmetic Mean in Sketchpad Copyright Steve Yurek March 17, 201411

12 Thanksgiving Chairs The ratio of the height of the adult chairs in my Moms house to that of the kids chairs, is the same as that of the kids chairs compared to that of the dolls chairs. If the adult chairs are 49 tall and the dolls chairs are 16 tall, then what is the height of the kids chairs. Copyright Steve Yurek March 17, 201412

13 Thanksgiving Chairs The ratio of the height of the adult chairs in my Moms house to that of the kids chairs, is the same as that of the kids chairs compared to that of the dolls chairs. If the adult chairs are 49 tall and the dolls chairs are 16 tall, then what is the height of the kids chairs. Copyright Steve Yurek March 17, 201413

14 Thanksgiving Chairs The ratio of the height of the adult chairs in my Moms house to that of the kids chairs, is the same as that of the kids chairs compared to that of the dolls chairs. If the adult chairs are 49 tall and the dolls chairs are 16 tall, then what is the height of the kids chairs. Copyright Steve Yurek March 17, 201414

15 Thanksgiving Chairs Copyright Steve Yurek March 17, 201415

16 Thanksgiving Chairs Copyright Steve Yurek March 17, 201416

17 Thanksgiving Chairs Copyright Steve Yurek March 17, 201417

18 Thanksgiving Chairs Copyright Steve Yurek March 17, 201418

19 Thanksgiving Chairs Copyright Steve Yurek March 17, 201419

20 Thanksgiving Chairs Copyright Steve Yurek March 17, 201420 Geometric Mean in Sketchpad

21 Lets look at this often miscalculated problem For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201421

22 Lets look at this often miscalculated problem For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201422

23 Lets look at this often miscalculated problem For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201423

24 Lets look at this often miscalculated problem For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis average gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201424

25 Lets look at this often miscalculated problem For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis average gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201425

26 Lets look at this often miscalculated problem For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis average gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201426

27 But we know better. Copyright Steve Yurek March 17, 201427

28 For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. 1 gallon of fuel used For the next 10 mi. it registered 100 mpg. 1/10 gallon of fuel used Total distance traveled = 20 miles Total fuel used = 1.1 gallons 20/1.1 = 18.18 mpg Copyright Steve Yurek March 17, 201428

29 For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. 1 gallon of fuel used For the next 10 mi. it registered 100 mpg. 1/10 gallon of fuel used Total distance traveled = 20 miles Total fuel used = 1.1 gallons 20/1.1 = 18.18 mpg Copyright Steve Yurek March 17, 201429

30 For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. 1 gallon of fuel used For the next 10 mi. it registered 100 mpg. 1/10 gallon of fuel used Total distance traveled = 20 miles Total fuel used = 1.1 gallons 20/1.1 = 18.18 mpg Copyright Steve Yurek March 17, 201430

31 For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. 1 gallon of fuel used For the next 10 mi. it registered 100 mpg. 1/10 gallon of fuel used Total distance traveled = 20 miles Total fuel used = 1.1 gallons 20/1.1 = 18.18 mpg Copyright Steve Yurek March 17, 201431

32 For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. 1 gallon of fuel used For the next 10 mi. it registered 100 mpg. 1/10 gallon of fuel used Total distance traveled = 20 miles Total fuel used = 1.1 gallons 20/1.1 = 18.18 mpg Copyright Steve Yurek March 17, 201432

33 For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. 1 gallon of fuel used For the next 10 mi. it registered 100 mpg. 1/10 gallon of fuel used Total distance traveled = 20 miles Total fuel used = 1.1 gallons 20 / 1.1 = 18.18 mpg Copyright Steve Yurek March 17, 201433

34 For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. 1 gallon of fuel used For the next 10 mi. it registered 100 mpg. 1/10 gallon of fuel used Total distance traveled = 20 miles Total fuel used = 1.1 gallons 20 / 1.1 = 18.18 mpg Copyright Steve Yurek March 17, 201434

35 But lets look at this one a little differently Let R = Rate for the 1 st D miles Let r = Rate for the 2 nd D miles Then T for the 1 st D miles = D/R and t for the 2 nd D miles = D/r Total Rate = Total Distance / Total Time 2D 2D R = -------------- = ------------------ D D Dr + DR --- + --- --------------- R r Rr Copyright Steve Yurek March 17, 201435

36 But lets at this one a little differently Let R = Rate for the 1 st D miles Let r = Rate for the 2 nd D miles Then T for the 1 st D miles = D/R and t for the 2 nd D miles = D/r Total Rate = Total Distance / Total Time 2D 2D R = -------------- = ------------------ D D Dr + DR --- + --- --------------- R r Rr Copyright Steve Yurek March 17, 201436

37 But lets at this one a little differently Let R = Rate for the 1 st D miles Let r = Rate for the 2 nd D miles Then T for the 1 st D miles = D/R and t for the 2 nd D miles = D/r Total Rate = Total Distance / Total Time 2D 2D R = -------------- = ------------------ D D Dr + DR --- + --- --------------- R r Rr Copyright Steve Yurek March 17, 201437

38 Copyright Steve Yurek March 17, 201438

39 Copyright Steve Yurek March 17, 201439

40 Copyright Steve Yurek March 17, 201440

41 Copyright Steve Yurek March 17, 201441

42 Copyright Steve Yurek March 17, 201442

43 Copyright Steve Yurek March 17, 201443

44 Copyright Steve Yurek March 17, 201444

45 Copyright Steve Yurek March 17, 201445

46 Copyright Steve Yurek March 17, 201446

47 Copyright Steve Yurek March 17, 201447

48 Copyright Steve Yurek March 17, 201448

49 Copyright Steve Yurek March 17, 201449

50 Copyright Steve Yurek March 17, 201450

51 Copyright Steve Yurek March 17, 201451

52 Copyright Steve Yurek March 17, 201452

53 Copyright Steve Yurek March 17, 201453

54 Copyright Steve Yurek March 17, 201454

55 Copyright Steve Yurek March 17, 201455

56 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201456

57 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201457

58 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201458

59 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201459

60 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201460

61 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201461

62 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201462

63 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201463

64 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201464

65 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201465

66 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201466

67 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201467

68 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201468

69 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201469

70 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201470

71 So if D is irrelevant, then lets choose D = 1 That makes Copyright Steve Yurek March 17, 201471

72 1.What is the sum of the reciprocals of 10 & 100? 2.What is the mean of these reciprocals? 3.What is the reciprocal of the answer from part 2? Its a bit clumsy to do all this. Hooray for the formula For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201472

73 1.What is the sum of the reciprocals of 10 & 100? 2.What is the mean of these reciprocals? 3.What is the reciprocal of the answer from part 2? Its a bit clumsy to do all this. Hooray for the formula For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201473

74 1.What is the sum of the reciprocals of 10 & 100? 2.What is the mean of these reciprocals? 3.What is the reciprocal of the answer from part 2? Its a bit clumsy to do all this. Hooray for the formula For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201474

75 1.What is the sum of the reciprocals of 10 & 100? 2.What is the mean of these reciprocals? 3.What is the reciprocal of the answer from part 2? Its a bit clumsy to do all this. Hooray for the formula For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201475

76 1.What is the sum of the reciprocals of 10 & 100? 2.What is the mean of these reciprocals? 3.What is the reciprocal of the answer from part 2? Its a bit clumsy to do all this. Hooray for the formula For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201476

77 1.What is the sum of the reciprocals of 10 & 100? 2.What is the mean of these reciprocals? 3.What is the reciprocal of the answer from part 2? Its a bit clumsy to do all this. Hooray for the formula For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201477

78 1.What is the sum of the reciprocals of 10 & 100? 2.What is the mean of these reciprocals? 3.What is the reciprocal of the answer from part 2? Its a bit clumsy to do all this. Hooray for the formula For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201478

79 1.What is the sum of the reciprocals of 10 & 100? 2.What is the mean of these reciprocals? 3.What is the reciprocal of the answer from part 2? Its a bit clumsy to do all this. Hooray for the formula For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201479

80 1.What is the sum of the reciprocals of 10 & 100? 2.What is the mean of these reciprocals? 3.What is the reciprocal of the answer from part 2? Its a bit clumsy to do all this. Hooray for the formula For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201480

81 Lets look at a similar, yet different one For the first 10 miles of a trip, Petes gas mileage on his Harley Mini was 10 mpg. For the next 10 mi. it registered 100 mpg. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201481

82 Lets look at a similar, yet different one For the first 10 miles of a trip, Petes average speed was 20 mph What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201482

83 Lets look at a similar, yet different one For the first 10 miles of a trip, Petes average speed was 20 mph For the next 10 miles it was 100 mph. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201483

84 Lets look at a similar, yet different one For the first 10 miles of a trip, Petes average speed was 20 mph For the next 10 miles it was 100 mph. What was the Minis gas mileage for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201484

85 Lets look at a similar, yet different one For the first 10 miles of a trip, Petes average speed was 20 mph For the next 10 miles it was 100 mph. What was the Minis average speed for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201485

86 Lets look at a similar, yet different one For the first 10 miles of a trip, Petes average speed was 20 mph For the next 10 miles it was 100 mph. What was the Minis average speed for the entire trip? Your students would probably answer …… 55 mpg Copyright Steve Yurek March 17, 201486

87 Lets look at a similar, yet different one For the first 10 miles of a trip, Petes average speed was 20 mph For the next 10 miles it was 100 mph. What was the Minis average speed for the entire trip? Your students would probably answer …… 60 mph Copyright Steve Yurek March 17, 201487

88 But we know better. Copyright Steve Yurek March 17, 201488

89 For the first 10 miles of a trip, Petes average speed was 20 mph. 1 gallon of fuel used For the next 10 mi. it registered 100 mpg. 1/10 gallon of fuel used Total distance traveled = 20 miles Total fuel used = 1.1 gallons 20/1.1 = 18.18 mpg Copyright Steve Yurek March 17, 201489

90 For the first 10 miles of a trip, Petes average speed was 20 mph. It took 30 minutes For the next 10 mi. it registered 100 mpg. 1/10 gallon of fuel used Total distance traveled = 20 miles Total fuel used = 1.1 gallons 20/1.1 = 18.18 mpg Copyright Steve Yurek March 17, 201490

91 For the first 10 miles of a trip, Petes average speed was 20 mph. It took 30 minutes For the next 10 miles it was 100 mph. 1/10 gallon of fuel used Total distance traveled = 20 miles Total fuel used = 1.1 gallons 20/1.1 = 18.18 mpg Copyright Steve Yurek March 17, 201491

92 For the first 10 miles of a trip, Petes average speed was 20 mph. It took 30 minutes For the next 10 miles it was 100 mph. It took 6 minutes Total distance traveled = 20 miles Total fuel used = 1.1 gallons 20/1.1 = 18.18 mpg Copyright Steve Yurek March 17, 201492

93 For the first 10 miles of a trip, Petes average speed was 20 mph. It took 30 minutes For the next 10 miles it was 100 mph. It took 6 minutes Total time = 36 minutes (.6 hours) Total fuel used = 1.1 gallons 20/1.1 = 18.18 mpg Copyright Steve Yurek March 17, 201493

94 For the first 10 miles of a trip, Petes average speed was 20 mph. It took 30 minutes For the next 10 miles it was 100 mph. It took 6 minutes Total time = 36 minutes (.6 hours) Total Distance = 20 miles 20 / 1.1 = 18.18 mpg Copyright Steve Yurek March 17, 201494

95 For the first 10 miles of a trip, Petes average speed was 20 mph. It took 30 minutes For the next 10 miles it was 100 mph. It took 6 minutes Total time = 36 minutes (.6 hours) Total Distance = 20 miles Overall Speed =20 mi/.6 hr or Copyright Steve Yurek March 17, 201495

96 For the first 10 miles of a trip, Petes average speed was 20 mph. It took 30 minutes For the next 10 miles it was 100 mph. It took 6 minutes Total time = 36 minutes (.6 hours) Total Distance = 20 miles Overall Speed =20 mi/.6 hr or Overall Speed = 33 1/3 mph Copyright Steve Yurek March 17, 201496

97 Use the definition to verify your answer Sum of reciprocals (1/20 + 1/100) = 3/50 Average of previous answer 3/100 Reciprocal from previous answer 33 1/3 Now use the formula with a = 20 and b = 100 Copyright Steve Yurek March 17, 201497

98 Use the definition to verify your answer Sum of reciprocals (1/20 + 1/100) = 3/50 Average of previous answer 3/100 Reciprocal from previous answer 33 1/3 Now use the formula with a = 20 and b = 100 Copyright Steve Yurek March 17, 201498

99 Use the definition to verify your answer Sum of reciprocals (1/20 + 1/100) = 3/50 Average of previous answer 3/100 Reciprocal from previous answer 33 1/3 Now use the formula with a = 20 and b = 100 Copyright Steve Yurek March 17, 201499

100 Use the definition to verify your answer Sum of reciprocals (1/20 + 1/100) = 3/50 Average of previous answer 3/100 Reciprocal from previous answer 33 1/3 Now use the formula with a = 20 and b = 100 Copyright Steve Yurek March 17, 2014100

101 Use the definition to verify your answer Sum of reciprocals (1/20 + 1/100) = 3/50 Average of previous answer 3/100 Reciprocal from previous answer 33 1/3 Now use the formula with a = 20 and b = 100 Copyright Steve Yurek March 17, 2014101

102 Use the definition to verify your answer Sum of reciprocals (1/20 + 1/100) = 3/50 Average of previous answer 3/100 Reciprocal from previous answer 33 1/3 Now use the formula with a = 20 and b = 100 Copyright Steve Yurek March 17, 2014102

103 Use the definition to verify your answer Sum of reciprocals (1/20 + 1/100) = 3/50 Average of previous answer 3/100 Reciprocal from previous answer Copyright Steve Yurek March 17, 2014103

104 Use the definition to verify your answer Sum of reciprocals (1/20 + 1/100) = 3/50 Average of previous answer 3/100 Reciprocal from previous answer 33 1/3 Now use the formula with a = 20 and b = 100 Copyright Steve Yurek March 17, 2014104

105 Use the definition to verify your answer Sum of reciprocals (1/20 + 1/100) = 3/50 Average of previous answer 3/100 Reciprocal from previous answer 33 1/3 Now use the formula with a = 20 and b = 100 Copyright Steve Yurek March 17, 2014105

106 Use the definition to verify your answer Sum of reciprocals (1/20 + 1/100) = 3/50 Average of previous answer 3/100 Reciprocal from previous answer 33 1/3 Now use the formula with a = 20 and b = 100 Copyright Steve Yurek March 17, 2014106

107 Use the definition to verify your answer Sum of reciprocals (1/20 + 1/100) = 3/50 Average of previous answer 3/100 Reciprocal from previous answer 33 1/3 Now use the formula with a = 20 and b = 100 Copyright Steve Yurek March 17, 2014107

108 Use the definition to verify your answer Sum of reciprocals (1/20 + 1/100) = 3/50 Average of previous answer 3/100 Reciprocal from previous answer 33 1/3 Now use the formula with a = 20 and b = 100 Copyright Steve Yurek March 17, 2014108

109 Use the definition to verify your answer Sum of reciprocals (1/20 + 1/100) = 3/50 Average of previous answer 3/100 Reciprocal from previous answer 33 1/3 Now use the formula with a = 20 and b = 100 Copyright Steve Yurek March 17, 2014109 Harmonic Mean in Sketechpad

110 Whenever we are taking an average of an average (as long as the base remains constant), the harmonic mean will save lots of time. Proving that will take more time than we have here, but we can demonstrate this concept. Copyright Steve Yurek March 17, 2014110

111 Whenever we are taking an average of an average (as long as the base remains constant), the harmonic mean will save lots of time. Even if there are more than 2 components Copyright Steve Yurek March 17, 2014111

112 Sarah Jane bought cartridges for the various printers in her office and spent $200 for each type. She paid $10 for each of the cartridges for the black & white printers, $20 for each of the cartridges for the color printers and $25 for each of the cartridges for her supervisors printer. What was the average cost of a single cartridge? Copyright Steve Yurek March 17, 2014112

113 Sarah Jane bought cartridges for the various printers in her office and spent $200 for each type. She paid $10 for each of the cartridges for the black & white printers, $20 for each of the cartridges for the color printers and $25 for each of the cartridges for her supervisors printer. What was the average cost of a single cartridge? Copyright Steve Yurek March 17, 2014113

114 Sarah Jane bought cartridges for the various printers in her office and spent $200 for each type. She paid $10 for each of the cartridges for the black & white printers, $20 for each of the cartridges for the color printers and $25 for each of the cartridges for her supervisors printer. What was the average cost of a single cartridge? Copyright Steve Yurek March 17, 2014114

115 Sarah Jane bought cartridges for the various printers in her office and spent $200 for each type. She paid $10 for each of the cartridges for the black & white printers, $20 for each of the cartridges for the color printers and $25 for each of the cartridges for her supervisors printer. What was the average cost of a single cartridge? Copyright Steve Yurek March 17, 2014115

116 Sarah Jane bought cartridges for the various printers in her office and spent $200 for each type. She paid $10 for each of the cartridges for the black & white printers, $20 for each of the cartridges for the color printers and $25 for each of the cartridges for her supervisors printer. What was the average cost of a single cartridge? Copyright Steve Yurek March 17, 2014116

117 We can certainly determine how many of each type she bought and then divide that number into $600 ($15 15/19) Does this correspond with the harmonic mean of 10, 20 & 25? Copyright Steve Yurek March 17, 2014117

118 We can certainly determine how many of each type she bought and then divide that number into $600 ($15 15/19) Does this correspond with the harmonic mean of 10, 20 & 25? Using the Definition? Copyright Steve Yurek March 17, 2014118

119 We can certainly determine how many of each type she bought and then divide that number into $600 ($15 15/19) Does this correspond with the harmonic mean of 10, 20 & 25? Using the Definition? Copyright Steve Yurek March 17, 2014119

120 We can certainly determine how many of each type she bought and then divide that number into $600 ($15 15/19) Does this correspond with the harmonic mean of 10, 20 & 25? Copyright Steve Yurek March 17, 2014120

121 Can we generate a formula for the harmonic mean of 3 numbers a, b & c? Copyright Steve Yurek March 17, 2014121

122 Can we generate a formula for the harmonic mean of 3 numbers a, b & c? Copyright Steve Yurek March 17, 2014122

123 Can we generate a formula for the harmonic mean of 3 numbers a, b & c? Copyright Steve Yurek March 17, 2014123

124 Can we generate a formula for the harmonic mean of 3 numbers a, b & c? Copyright Steve Yurek March 17, 2014124

125 Can we generate a formula for the harmonic mean of 3 numbers a, b & c? Copyright Steve Yurek March 17, 2014125

126 Can we generate a formula for the harmonic mean of 3 numbers a, b & c? Copyright Steve Yurek March 17, 2014126

127 Can we generate a formula for the harmonic mean of 3 numbers a, b & c? Copyright Steve Yurek March 17, 2014127

128 Can we generate a formula for the harmonic mean of 3 numbers a, b & c? Copyright Steve Yurek March 17, 2014128

129 Can we generate a formula for the harmonic mean of more than 3 numbers? Copyright Steve Yurek March 17, 2014129

130 Can we generate a formula for the harmonic mean of more than 3 numbers? Copyright Steve Yurek March 17, 2014130

131 Can we generate a formula for the harmonic mean of more than 3 numbers? Copyright Steve Yurek March 17, 2014131

132 Copyright Steve Yurek March 17, 2014132

133 The trichotomy law states that when comparing any 2 numbers, a & b, then there are 3 and only 3 possible outcomes: a > b a < b a = b Copyright Steve Yurek March 17, 2014133

134 The trichotomy law states that when comparing any 2 numbers, a & b, then there are 3 and only 3 possible outcomes: a > b a < b a = b Copyright Steve Yurek March 17, 2014134

135 The trichotomy law states that when comparing any 2 numbers, a & b, then there are 3 and only 3 possible outcomes: a > b a < b a = b Copyright Steve Yurek March 17, 2014135

136 The trichotomy law states that when comparing any 2 numbers, a & b, then there are 3 and only 3 possible outcomes: a > b a < b a = b Copyright Steve Yurek March 17, 2014136

137 So will the 3 means ever all be equal? They will if the original numbers are equal. Copyright Steve Yurek March 17, 2014137

138 So will the 3 means ever all be equal? They will if the original numbers are equal. Copyright Steve Yurek March 17, 2014138

139 So will the 3 means ever all be equal? They will if the original numbers are equal. Copyright Steve Yurek March 17, 2014139

140 So will the 3 means ever all be equal? They will if the original numbers are equal. Copyright Steve Yurek March 17, 2014140

141 So will the 3 means ever all be equal? They will if the original numbers are equal. Copyright Steve Yurek March 17, 2014141

142 So will the 3 means ever all be equal? They will if the original numbers are equal. Copyright Steve Yurek March 17, 2014142

143 So will the 3 means ever all be equal? They will if the original numbers are equal. Copyright Steve Yurek March 17, 2014143

144 So will the 3 means ever all be equal? They will if the original numbers are equal. Copyright Steve Yurek March 17, 2014144

145 So will the 3 means ever all be equal? They will if the original numbers are equal. Copyright Steve Yurek March 17, 2014145

146 So will the 3 means ever all be equal? They will if the original numbers are equal. Copyright Steve Yurek March 17, 2014146

147 So will the 3 means ever all be equal? They will if the original numbers are equal. Copyright Steve Yurek March 17, 2014147

148 So will the 3 means ever all be equal? They will if the original numbers are equal. Copyright Steve Yurek March 17, 2014148

149 So will the 3 means ever all be equal? They will if the original numbers are equal. Copyright Steve Yurek March 17, 2014149

150 So will the 3 means ever all be equal? They will if the original numbers are equal. Copyright Steve Yurek March 17, 2014150 Back to Sketchpad

151 Copyright Steve Yurek March 17, 2014151 But what if a b? (lets keep them both >0) Lets assume that AM(a,b) < GM(a,b)

152 Copyright Steve Yurek March 17, 2014152 But what if a b? (lets keep them both >0) Lets assume that AM(a,b) < GM(a,b)

153 Copyright Steve Yurek March 17, 2014153 But what if a b? (lets keep them both >0) Lets assume that AM(a,b) < GM(a,b)

154 Copyright Steve Yurek March 17, 2014154 But what if a b? (lets keep them both >0) Lets assume that AM(a,b) < GM(a,b)

155 Copyright Steve Yurek March 17, 2014155 But what if a b? (lets keep them both >0) Lets assume that AM(a,b) < GM(a,b)

156 Copyright Steve Yurek March 17, 2014156 But what if a b? (lets keep them both >0) Lets assume that AM(a,b) < GM(a,b)

157 Copyright Steve Yurek March 17, 2014157 But what if a b? (lets keep them both >0) Lets assume that AM(a,b) < GM(a,b)

158 Copyright Steve Yurek March 17, 2014158 But what if a b? (lets keep them both >0) Lets assume that AM(a,b) < GM(a,b)

159 Copyright Steve Yurek March 17, 2014159 But what if a b? (lets keep them both >0) Lets assume that AM(a,b) < GM(a,b)

160 Copyright Steve Yurek March 17, 2014160 But what if a b? (lets keep them both >0) Lets assume that AM(a,b) < GM(a,b)

161 Copyright Steve Yurek March 17, 2014161 But what if a b? (lets keep them both >0) Lets also assume that GM(a,b) < HM(a,b)

162 Copyright Steve Yurek March 17, 2014162 But what if a b? (lets keep them both >0) Lets also assume that GM(a,b) < HM(a,b)

163 Copyright Steve Yurek March 17, 2014163 But what if a b? (lets keep them both >0) Lets also assume that GM(a,b) < HM(a,b)

164 Copyright Steve Yurek March 17, 2014164 But what if a b? (lets keep them both >0) Lets also assume that GM(a,b) < HM(a,b)

165 Copyright Steve Yurek March 17, 2014165 But what if a b? (lets keep them both >0) Lets also assume that GM(a,b) < HM(a,b)

166 Copyright Steve Yurek March 17, 2014166 But what if a b? (lets keep them both >0) Lets also assume that GM(a,b) < HM(a,b)

167 Copyright Steve Yurek March 17, 2014167 But what if a b? (lets keep them both >0) Lets also assume that GM(a,b) < HM(a,b)

168 Copyright Steve Yurek March 17, 2014168 But what if a b? (lets keep them both >0) Lets also assume that GM(a,b) < HM(a,b)

169 So, for any a,b where a and b both >0 AM(a,b) > GM(a,b) & GM(a,b) > HM(a,b) then AM(a,b) > GM(a,b) > HM(a,b) Copyright Steve Yurek March 17, 2014169

170 So, for any a,b where a and b both >0 AM(a,b) > GM(a,b) & GM(a,b) > HM(a,b) then AM(a,b) > GM(a,b) > HM(a,b) Copyright Steve Yurek March 17, 2014170

171 So, for any a,b where a and b both >0 AM(a,b) > GM(a,b) & GM(a,b) > HM(a,b) then AM(a,b) > GM(a,b) > HM(a,b) Copyright Steve Yurek March 17, 2014171

172 So, for any a,b where a and b both >0 AM(a,b) > GM(a,b) & GM(a,b) > HM(a,b) then AM(a,b) > GM(a,b) > HM(a,b) Copyright Steve Yurek March 17, 2014172

173 So, for any a,b where a and b both >0 AM(a,b) > GM(a,b) & GM(a,b) > HM(a,b) then AM(a,b) > GM(a,b) > HM(a,b) Copyright Steve Yurek March 17, 2014173 Sketchpad – One Last Time

174 Copyright Steve Yurek March 17, 2014174

175 Copyright Steve Yurek March 17, 2014175

176 Copyright Steve Yurek March 17, 2014176

177 80 50 30 Copyright Steve Yurek March 17, 2014177

178 80 50 30 E Copyright Steve Yurek March 17, 2014178

179 80 50 30 E ? Copyright Steve Yurek March 17, 2014179

180 80 50 30 E 18.75 Copyright Steve Yurek March 17, 2014180

181 80 50 30 E 18.75 As it turns out, this is very cool --- watch this Copyright Steve Yurek March 17, 2014181

182 The Harmonic Mean turns up in many fascinating places Can you reconcile its place in the telephone problem? What is the harmonic mean of 30 & 50? Its 37.5 What the…..??? How can the wires meet at a spot that is higher than the shorter pole? Is there a relation between the answer and the harmonic mean? Copyright Steve Yurek March 17, 2014182

183 The Harmonic Mean turns up in many fascinating places Can you reconcile its place in the telephone problem? What is the harmonic mean of 30 & 50? Its 37.5 What the…..??? How can the wires meet at a spot that is higher than the shorter pole? Is there a relation between the answer and the harmonic mean? Copyright Steve Yurek March 17, 2014183

184 The Harmonic Mean turns up in many fascinating places Can you reconcile its place in the telephone problem? What is the harmonic mean of 30 & 50? Its 37.5 What the…..??? How can the wires meet at a spot that is higher than the shorter pole? Is there a relation between the answer and the harmonic mean? Copyright Steve Yurek March 17, 2014184

185 The Harmonic Mean turns up in many fascinating places Can you reconcile its place in the telephone problem? What is the harmonic mean of 30 & 50? Its 37.5 What the…..??? How can the wires meet at a spot that is higher than the shorter pole? Is there a relation between the answer and the harmonic mean? Copyright Steve Yurek March 17, 2014185

186 The Harmonic Mean turns up in many fascinating places Can you reconcile its place in the telephone problem? What is the harmonic mean of 30 & 50? Its 37.5 What the…..??? How can the wires meet at a spot that is higher than the shorter pole? Is there a relation between the answer and the harmonic mean? Copyright Steve Yurek March 17, 2014186

187 The Harmonic Mean turns up in many fascinating places Can you reconcile its place in the telephone problem? What is the harmonic mean of 30 & 50? Its 37.5 What the…..??? How can the wires meet at a spot that is higher than the shorter pole? Is there a relation between the answer and the harmonic mean? Copyright Steve Yurek March 17, 2014187

188 Copyright Steve Yurek March 17, 2014188

189 Copyright Steve Yurek March 17, 2014189

190 Copyright Steve Yurek March 17, 2014190

191 Copyright Steve Yurek March 17, 2014191

192 Copyright Steve Yurek March 17, 2014192

193 Copyright Steve Yurek March 17, 2014193

194 Copyright Steve Yurek March 17, 2014194

195 Copyright Steve Yurek March 17, 2014195

196 Copyright Steve Yurek March 17, 2014196

197 Copyright Steve Yurek March 17, 2014197

198 Copyright Steve Yurek March 17, 2014198

199 Copyright Steve Yurek March 17, 2014199

200 Copyright Steve Yurek March 17, 2014200

201 Copyright Steve Yurek March 17, 2014201

202 Copyright Steve Yurek March 17, 2014202

203 Copyright Steve Yurek March 17, 2014203

204 Copyright Steve Yurek March 17, 2014204

205 Copyright Steve Yurek March 17, 2014205 j=1 n π Σ i=1 n ajaj

206 Copyright Steve Yurek March 17, 2014206

207 Copyright Steve Yurek March 17, 2014207

208 Copyright Steve Yurek March 17, 2014208

209 Last season Cody got 60 hits for a Fenway batting average of.400. For his AWAY games he also got 60 hits, but his batting average was only.300. What was Codys batting average for the entire season? Copyright Steve Yurek March 17, 2014209

210 Jacks company will award an end-of-the-year bonus to any employee whose total yearly sales represent at least 10% of the companys sales. During the last fiscal year, Jacks sales were a consistent $50,000 for each quarter. However his quarter 1 sales represented 10% of the companys sales. For quarter 2, his sales represented 6% of the companys sales. For quarters 3 & 4 they represented 8% and 30% respectively. Did Jack earn his bonus? Defend your answer. Copyright Steve Yurek March 17, 2014210

211 In a recent county election poll, voters in each of the 5 districts were asked whom they support. For each of the 5 counties, 473 voters expressed their support for Mr. James W. Beam: These results represented 19% of voters in District A, and 25%, 18%, 57% and 31% in the other 4 districts. Mr. Beam claims to have the support of 30% of the county. Is he correct? If no, then by how much is he off? Copyright Steve Yurek March 17, 2014211

212 Ollie Charles Dickens (known to his friends as OCD) will have a great day if he can average exactly 60 mph on his way to work. The 1 st 6.3 miles are along back roads, while the final 6.3 miles is traveled on straight as an arrow freeway. School buses, wet leaves and a touch of ice slowed the back road portion to only 30 mph. How fast must Mr. OCD travel on the freeway, so that he can have a great day? Copyright Steve Yurek March 17, 2014212

213 Each day Violet sells 72 each of 9 fruits: Apples – Oranges – Pears – Plums – Kiwi – Nectarine – Pomegranate – Peaches & Tangellos and respectively they represent 3/4, 2/3, 1/2, 3/5, 3/7, 8/9, 6/13, 9/11 and 1/2 of the amount of each fruit that she bought. At the end of the day she donates any unsold fruit to a food pantry. What percent of her daily fruit purchase goes to the food pantry? Copyright Steve Yurek March 17, 2014213

214 The EPA has mandated that, by 2015, the total % of all models of all vehicles produced by any manufacturer must get at least 35 mpg. Four of the 5 models of the Great Wall Auto Corp have tested to get 30 mpg, 38 mpg, 42 mpg and 25 mpg. If they all use the same test track, what must the fuel mileage of the 5 th model be in order for GWAC to be allowed to manufacture automobiles in the US? Copyright Steve Yurek March 17, 2014214

215 Periodically the water in Sparkletown is tested for impurities. Recently 7 samples were tested: Four 25 gallon samples from spots near each of the four corners and three 50 gallon samples from varying spots in the middle of the reservoir. The corner samples registered 84%, 87%, 89% and 85% pure, while the center samples had readings of 96%, 99% and 93% pure. On the state report, Sparkletown Public Health reported that the purity rate of the 250 gallons sampled was ________ percent. Copyright Steve Yurek March 17, 2014215

216 Periodically the water in Sparkletown is tested for impurities. Recently 7 samples were tested: Each of 4 samples from spots near each of the four corners revealed 25 gallons of pure water and each of 3 samples from varying spots in the middle of the reservoir yielded 50 gallons of pure water. The corner samples registered 84%, 87%, 89% and 85% pure, while the center samples had readings of 96%, 99% and 93% pure. On the state report, Sparkletown Public Health reported that the purity rate of the 250 gallons sampled was ________ percent. Copyright Steve Yurek March 17, 2014216

217 Two telephone poles are a feet and b feet tall, and they are positioned so that they are F feet apart. When guide wires from the tops of each extend to the bases of the others, they intersect at a specific point E. How high is E above the ground? Copyright Steve Yurek March 17, 2014217

218 If Thelma can paint a house, by herself, in 8 hours, and Louise can paint the same house in only 5 hours, then how long will it take for them to paint the house if they work together? Copyright Steve Yurek March 17, 2014218

219 Harry can mow a lawn all by himself in H minutes, while David can mow the same lawn in D minutes. If they both work together, then how long will it take for them to mow the same lawn? Copyright Steve Yurek March 17, 2014219

220 Thank You All For Being Here This presentation will be uploaded to the NHTM Conference Website Or you can write to me at Copyright Steve Yurek March 17, 2014220


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