After the Pythagorean crisis with the square root of two, Greek mathematicians tried to avoid working with numbers. Instead, the Greeks used geometry to demonstrate mathematical concepts. A line can be drawn any length, so straight lines became a sort of non-algebraic variable.
You can see an example of this in The Pythagorean Proof, where Alexandria Jones represented the sides of her triangle by the letters a and b. These sides may be any length. The sizes of the squares will change with the triangle sides, but the relationship is always true for every right triangle.
The story of mathematics is the story of interesting people. What a shame it is that our children see only the dry remains of these people’s passion. By learning math history, our students will see how men and women wrestled with concepts, made mistakes, argued with each other, and gradually developed the knowledge we today take for granted.
In a previous article, I recommended books that you may find at your local library or be able to order through inter-library loan. Now, let me introduce you to the wealth of math history resources on the Internet.
Math concepts: subtraction within 100, number patterns, mental math Number of players: 2 or 3 Equipment: printed hundred chart (also called a hundred board), and highlighter or translucent disks to mark numbers — or use this online hundred chart
Set up
Place the hundred chart and highlighter where all players can reach them.
How to play
Allow the youngest player choice of moving first or second; in future games, allow the loser of the last game to choose.
The first player chooses a number from 1 to 100 and marks that square on the hundred chart.
The second player chooses and marks any other number.
On each succeeding turn, the player subtracts any two marked numbers to find and mark a difference that has not yet been taken.
Play alternates until no more numbers can be marked.
Almost all math problems call for the student to assume one thing or another. Without assumptions — definitions, postulates, axioms, common notions, or whatever you want to call them — mathematics of any kind is impossible. Tony at Pencils Down (who plans to be a math teacher when he grows up) reminds us that, necessary though it may be, we are stepping on dangerous ground when we assume:
The title which I most covet is that of teacher. The writing of a research paper and the teaching of freshman calculus, and everything in between, falls under this rubric. Happy is the person who comes to understand something and then gets to explain it.
No peeking! This post is for those of you who have given the trisection proof a good workout on your own. If you have a question about the proof or a solution you would like to share, please post a comment here.
But if you haven’t yet worked at the puzzle, go back and give it a try. When someone just tells you the answer, you miss out on the fun. Figure it out for yourself — and then check the answer just to prove that you got it right.
One of the great unsolved problems of antiquity was to trisect any angle using only the basic tools of Euclidean geometry: an unmarked straight-edge and a compass. Like the alchemist’s dream of turning lead into gold, this proved to be an impossible task. If you want to trisect an angle, you have to “cheat.” A straight-edge and compass can’t do it. You have to use some sort of crutch, just as an alchemist would have to use a particle accelerator or something.
One “cheat” that works is to fold your paper. I will show you how it works, and your job is to show why.
Here are a few more tidbits from math history, along with links to relevant Internet sites or books, and three more math puzzles for you to try. I hope you find them interesting.
All of the books I link to should be available through your library loan system. But if you do feel inclined to buy any of them (or any other books I mention in my blog articles), I would appreciate your using these links, so I get a small referral fee. That would help convince my husband that the time I put into transcribing these old math newsletter stories is worth something.
[In the last episode, Alexandria Jones discovered a mysterious treasure: three wooden sticks, like tent pegs, and a long loop of rope with 12 evenly spaced knots. Her father explained that it was an ancient Egyptian surveyor's tool, used to mark right angles.]
Back at the camp, Fibonacci Jones stacked multi-layer sandwiches while Alexandria poured milk and set the table for supper.
“Geometry,” Fibonacci said.
“What?”
“Geo means earth, and metry means to measure. So geometry means to measure the earth. That is what the Egyptian rope stretches did.”
Alex thought for a moment. “So in the beginning, math was just surveying?”
[In the last episode, Alexandria Jones, daughter of the world-famous archaeologist, caught her father's arch-enemy trying to uncover the Pharaoh's Treasure.]
…”I can’t believe it!” Simon Skulk threw down the last stone in disgust and walked away. At the mouth of the cave, he turned back and shook his fist. “You haven’t seen the last of me, Alexandria Jones.”
Her muscles aching, Alex sank to the ground and hugged her dog. The she gave him a little push toward the front of the cave. “Rammy, go get Dad.”
Ramus barked once and took off running.
Alex turned back to look at the Pharaoh’s Treasure. Where the last stone had stood was a hole. In the hole lay three wooden sticks, like tent pegs, and a long loop of rope with 12 evenly-spaced knots.
Once again, Rudbeckia Hirta brings us some funny-but-sad mathematics. The test question was:
Without factoring it, explain how the number
N = (1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11) + 1
can be used to argue that there is a prime number larger than 11.
Have you and your students enjoyed a project, game, or handout from my blog? If you would like to say "Thank you," you can buy me a cup of coffee (send a small donation) via Paypal or make a purchase at one of my Amazon.com bookstores:
Picture from MacTutor Archives.
