How can our students take a full year of High School Geometry and not know the circumference of the world. Have we failed them in some way? Have we failed the education of our youth in some way? Shouldn't this be required knowledge like knowing who won World War II (WWII....on Jay Leno the other night...someone on the streets said that Hulk Hogan won WWII...lol).
How can you live on this earth and not be curious about how many miles there are around the world. I must admit that some of the better students do know it is about 3,000 miles between LA and NY (as the "crow flies"....i know all the babyboomers did that math in their head back the 1960's and 70's....3,000 miles and if i drive at 60 mph for 10 hrs a day...it will take me ?? how many days ?)
The guessing starts at 1,000,000 miles !! Some even say more than that....then i put up 100,000 miles on the board. Half the class says more, the other half less. And these are juniors in one of the best prep schools in Florida, over 600 on the Math SAT's and taking Honors Precalculus (forget the seniors in Honors Calculus they know even less than the juniors, and they will know even less after college).
So we establish that from the east coast to the west coast of the USA it is about 3,000 miles. (and many of these kids have flown to Colorado to go skiing...so they have at least have looked at a map of the USA even if it is just the one on the back of the airplane seat in front of you).
We go across the Pacific Ocean (a big ocean) and across China and into Russia. Across Europe and finally to the Atlantic Ocean and back to NY or Florida. (And these kids all have laptops and google earth !!)
Now most students in the class say it is LESS the 100,000 miles. We are making progress. And of course some student asks "is this on the next test ?" and "how is this important ?" and "I thought we were in Math class ?" and "This has nothing to do with PreCal ?" and "can i go to the bathroom ?" and "my mom is calling me on the cell, can i answer it ?"...the usual.
The Greeks in the BC (before Castro...no, before calculators...lol...no, no Before Christ) knew the world was round. It was obvious, just look at the sun and the moon. Just watch the sail of ships as they go over the horizon. In fact, the Greeks had determined that the circumference of the earth was actually 25,000 miles !!
They did some measuring using Euclidean Geometry of parallel lines and determined the ratio of 7 degrees/ 360 degrees was equal to 500 miles/ circumference of the World.
7/360 = 500/ Circum.
like Sherlock Homes would say..."elementary my boy, elementary"
So why do our students not know this as they prepare to graduate. And it gets worse for the general public. I taught adult math at the University for 30 years and i can tell you that adults do NOT even know that A = pi radius square. Even worse, they do not know how to find the area of a triangle. Mainly because they memorized some formula rather than understanding the the triangle occupies half the rectangle and every one knows the rectangle is width times height. But i have digressed.
So indeed the World's circumference is just 25,000 miles and Disney was correct when he sang "It's a small world after all".
Now living in Florida one should know that the Space Shuttle travels at about 17,000 miles per hour. So it takes the Space Shuttle only about 90 mins to make an orbit. When it says we can not land at the Cape, we will try on the next go around, i always thought that meant 1 or 2 days. But what it really means is we will be back in LESS then 2 hours !!
Saturday, January 15, 2011
Friday, January 14, 2011
7 Bridges Corollary
Another problem very close to the 7 Bridges is the one where one must cross every wall once and only once. Again very hard to imagine without a picture.
There is a famous spot in Florida in the middle of NOWHERE called Yahoo Junctions. It is where the Turnpike and Rt 60 meet (just south of Orlando). The only thing that is there is a restaurant where everyone stops to use the bathroom.
On the counter they have this game where you have to cross every wall once and only once. But it is impossible. It can be proven impossible with Euler 7 Bridges odd and even vertices solution.
The cook and waitress had been trying to beat that little game for 20 yrs. They were so relieved when i told them it was impossible, that they give me FREE coffee whenever i stop there on the way to Miami from Tampa (of course now adays we take I-75 instead...so that stop is even deader now then it was back then).
Let see draw a Rectangle then a line through the middle parallel to the top and bottom. So now you should have two long rectangles.
Next we add 3 walls. 2 on the bottom perpendicular to the base, making three rooms on the bottom.
And one more little wall on the top rectangle perpendicular to the top base making 2 bedrooms. For a total of 5 bedrooms. (2 bedrooms on top of 3 bedrooms)
Now try to cross every wall once and only once...GOOD LUCK !! IF you get it right i will buy you a coke.
(check out the picture at the bottom of this page)
By the way, if you move the wall on the top over to the right, it can be done.
There is a famous spot in Florida in the middle of NOWHERE called Yahoo Junctions. It is where the Turnpike and Rt 60 meet (just south of Orlando). The only thing that is there is a restaurant where everyone stops to use the bathroom.
On the counter they have this game where you have to cross every wall once and only once. But it is impossible. It can be proven impossible with Euler 7 Bridges odd and even vertices solution.
The cook and waitress had been trying to beat that little game for 20 yrs. They were so relieved when i told them it was impossible, that they give me FREE coffee whenever i stop there on the way to Miami from Tampa (of course now adays we take I-75 instead...so that stop is even deader now then it was back then).
Let see draw a Rectangle then a line through the middle parallel to the top and bottom. So now you should have two long rectangles.
Next we add 3 walls. 2 on the bottom perpendicular to the base, making three rooms on the bottom.
And one more little wall on the top rectangle perpendicular to the top base making 2 bedrooms. For a total of 5 bedrooms. (2 bedrooms on top of 3 bedrooms)
Now try to cross every wall once and only once...GOOD LUCK !! IF you get it right i will buy you a coke.
(check out the picture at the bottom of this page)
By the way, if you move the wall on the top over to the right, it can be done.
Thursday, January 13, 2011
Fun Math - The 7 Bridges
I have been teaching high school math for over 30 years and also at the University. Many of my students have returned and told me that i was the best math class they ever took. Mainly because i forced them to think and because i made math fun.
One of my tricks is to introduce math concepts by using history. So here is a collection of my favorite Math stories or discovery.
Today in class we were discussing mathematical modeling. And the first person that came into my mind was Euler. The greatest mathematician of all time. Now you can image if most students do not even know who was president during the Vietnam War, how are they going to know Euler !!
We just celebrated his 300 year anniversary. He is pictured on the 10 Franc in Switzerland where they put people who have contributed to civilization, not Presidents like in the US. But then of course we are lucky we have George Washington on the one dollar bill or no one would know who our first president was.
So one of the most interesting Geometry questions in history is the famous "Seven Bridges of Koningsberg". What ? Say what ? Never heard of it ? It is amazing that high school students who have had a FULL year of Geometry and never heard of Euler nor the famous problem of the 7 bridges. I am sure the Europeans students know of Euler and they probably also know who was the American president during the Vietnam War. But then US students do not know the Circumference of the Earth either !! What do they teach them in Geometry then. Don't they know that Geo-metry means the "measure of the earth" and that the Greeks knew the Circumference of the earth before Christ was born. (let me see was that 2,000 years ago or 3,000 years ago...where is my calculator)
Worse yet when we try to discuss the circumference of the world, the first thing they ask is "will this be on the test". Never mind that we live in Florida and that the space shuttle circles the earth every 90 minutes (from Red October a classic movie). I show it in class every year...at least the good parts.
Getting back to the 7 Bridges. It is hard to describe it here with no drawings and if you are really interested, i am sure you know how to google search it. I will put a link at the bottom for the over 50 crowd.
Euler told his students that they need to make a mathematical model of it. To get away from the real world rivers and bridges that get in the way of the thinking.
Then we have a discussion of the well known barn picture that must be drawn with straight lines and in one smooth motion. Again hard to demonstrate here (maybe my son will show me how to draw on this blog....i will try to add the picture to the margin).
One can quickly see that there are "odd" vertices and "even" vertices. And in order for the game to work you must start on the bottom (an odd vertice) and end at the other bottom also an "odd" vertices. And it you have one odd, you must have another odd. And only 2 odd vertices will work.
Well the 7 Bridges has 4 odd vertices so it is impossible.
Then it gets real exciting when the town decided to add a Bypass Bridge (number 8 bridge) outside of town. Now there are only 2 odd vertices and it can be done.
And it is DONE !! every hour on the hour by tour buses in Koningsberg.
One of my tricks is to introduce math concepts by using history. So here is a collection of my favorite Math stories or discovery.
Today in class we were discussing mathematical modeling. And the first person that came into my mind was Euler. The greatest mathematician of all time. Now you can image if most students do not even know who was president during the Vietnam War, how are they going to know Euler !!
We just celebrated his 300 year anniversary. He is pictured on the 10 Franc in Switzerland where they put people who have contributed to civilization, not Presidents like in the US. But then of course we are lucky we have George Washington on the one dollar bill or no one would know who our first president was.
So one of the most interesting Geometry questions in history is the famous "Seven Bridges of Koningsberg". What ? Say what ? Never heard of it ? It is amazing that high school students who have had a FULL year of Geometry and never heard of Euler nor the famous problem of the 7 bridges. I am sure the Europeans students know of Euler and they probably also know who was the American president during the Vietnam War. But then US students do not know the Circumference of the Earth either !! What do they teach them in Geometry then. Don't they know that Geo-metry means the "measure of the earth" and that the Greeks knew the Circumference of the earth before Christ was born. (let me see was that 2,000 years ago or 3,000 years ago...where is my calculator)
Worse yet when we try to discuss the circumference of the world, the first thing they ask is "will this be on the test". Never mind that we live in Florida and that the space shuttle circles the earth every 90 minutes (from Red October a classic movie). I show it in class every year...at least the good parts.
Getting back to the 7 Bridges. It is hard to describe it here with no drawings and if you are really interested, i am sure you know how to google search it. I will put a link at the bottom for the over 50 crowd.
Euler told his students that they need to make a mathematical model of it. To get away from the real world rivers and bridges that get in the way of the thinking.
Then we have a discussion of the well known barn picture that must be drawn with straight lines and in one smooth motion. Again hard to demonstrate here (maybe my son will show me how to draw on this blog....i will try to add the picture to the margin).
One can quickly see that there are "odd" vertices and "even" vertices. And in order for the game to work you must start on the bottom (an odd vertice) and end at the other bottom also an "odd" vertices. And it you have one odd, you must have another odd. And only 2 odd vertices will work.
Well the 7 Bridges has 4 odd vertices so it is impossible.
Then it gets real exciting when the town decided to add a Bypass Bridge (number 8 bridge) outside of town. Now there are only 2 odd vertices and it can be done.
And it is DONE !! every hour on the hour by tour buses in Koningsberg.
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