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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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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

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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

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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

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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

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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

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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

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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

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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

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

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

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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,

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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,

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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,

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Thanksgiving Chairs Copyright Steve Yurek March 17,

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Thanksgiving Chairs Copyright Steve Yurek March 17,

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Thanksgiving Chairs Copyright Steve Yurek March 17,

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Thanksgiving Chairs Copyright Steve Yurek March 17,

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Thanksgiving Chairs Copyright Steve Yurek March 17,

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

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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,

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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,

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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,

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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,

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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,

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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,

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But we know better. Copyright Steve Yurek March 17,

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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 = mpg Copyright Steve Yurek March 17,

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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 = mpg Copyright Steve Yurek March 17,

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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 = mpg Copyright Steve Yurek March 17,

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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 = mpg Copyright Steve Yurek March 17,

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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 = mpg Copyright Steve Yurek March 17,

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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 = mpg Copyright Steve Yurek March 17,

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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 = mpg Copyright Steve Yurek March 17,

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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,

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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,

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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,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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Copyright Steve Yurek March 17,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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But we know better. Copyright Steve Yurek March 17,

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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 = mpg Copyright Steve Yurek March 17,

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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 = mpg Copyright Steve Yurek March 17,

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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 = mpg Copyright Steve Yurek March 17,

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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 = mpg Copyright Steve Yurek March 17,

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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 = mpg Copyright Steve Yurek March 17,

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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 = mpg Copyright Steve Yurek March 17,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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, Harmonic Mean in Sketechpad

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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

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

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

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

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

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

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

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

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

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

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

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Copyright Steve Yurek March 17,

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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,

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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,

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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,

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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,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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,

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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,

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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,

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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,

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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, Sketchpad – One Last Time

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E Copyright Steve Yurek March 17,

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E ? Copyright Steve Yurek March 17,

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E Copyright Steve Yurek March 17,

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E As it turns out, this is very cool --- watch this Copyright Steve Yurek March 17,

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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,

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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,

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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,

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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,

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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,

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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,

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

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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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,

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