|
Dear Visitor,
Let's review what we want to do to make math fun and
beguiling both for you and for your students.
FIRST: spend a short time each class period
(perhaps 5 minutes) on something mathematical which is puzzling or
fascinating.
SECOND: Let your students help you teach. Don't
force proofs or methods down their throats. Give them a lead, and then let
them make suggestions.
THIRD: Teach how to solve problems. Try Prof. George Polya's steps:
FOURTH:
Think of teaching as an art. Develop your own style just as
an actor, dancer, painter, or writer does.
[Start as follows at the end of class. When you see the ... that means you should wait for an answer.] "Let's try a bit of mathematical magic. Get out a piece of paper and a pencil. Now think of a number between 1 and 20, and add 5 to it. Multiply your result by 3. Now subtract 17 from that answer. Everybody done that? Now comes the magic. Tommy, what was your answer? ... 28? The number you thought of was 10. Sally, what was your answer? ... 4? The number you thought of was 2. James, what was your answer? ... 44? Sorry, you made a mistake somewhere in your calculation. Try again. Take your number, add 5 to it, multiply the answer by 3, then subtract 17. What answer did you get now? ... 43? The number you thought of was 15. Tomorrow we'll see how the magic was done. See whether you can figure it out! [Start as follows at the beginning of the next class.] "Yesterday you saw some mathematical magic. You remember I asked you to think of a number, add 5 to it, multiply your answer by 3, then subtract 17. When you told me what you got, I told you the number you had thought of. "For example, suppose the number you thought of was 6. [Write the calculations on the blackboard.] We add 5 and get 11. Now we multiply 11 by 3 and get 33. Finally we subtract 17 from 33 and get 16. You tell me '16', and I tell you the number you thought of was 6. Our job is to figure out how I do this magic. "How do we solve this problem? Our plan should be to [write on board] Understand the problem, Plan how to solve it, Carry out the plan, Check our answer, "So first -- Understand the problem. The problem is to explain how a magician can take your answer, and tell you what your number was. "Now we have to plan how to solve the problem. Anyone have any ideas? ... One thing we might do is make a table showing how the calculation goes for different 'numbers we thought of'. Here's a table. [Put it on the blackboard as you do the calculations.]
"If the number we chose was 1, we add 5 and get 6. 6 times 3 is 18. 18 minus 17 is 1. If the number we chose was 2, we add 5 and get 7. 7 times 3 is 21. 21 minus 17 is 4. If the number we chose was 3, we add 5 and get 8. 8 times 3 is 24. 24 minus 17 is 7. Charles, do the next calculation for us. ... Right. Choose 4. Plus 5 is 9. Times 3 is 27. Minus 17 is 10. [Add the line in the table for 5. The last line shown here is not yet on the blackboard.] [Stand back and look at the table.] "All right. The answers the magician gets are 1, 4, 7, 10, and 13. Can anyone see how he might, from those numbers, figure that the original numbers chosen were 1, 2, 3, 4, and 5? ... If the original number chosen was 6, what would the calculated answer be? Tell me by looking at the table, not by doing the calculations. ... The series of answers are 1, 4, 7, 10, and 13. What's the next number in that series? ... What's the difference between each pair of numbers? ... Right, it's 3. So what's the next number in the series after 13? ... Right, 16. Let's check. 6+5 = 11. 11 x 3 = 33. 33 - 17 = 16. "Each answer is 3 greater than the previous one. Any ideas on how to get the original number from the answer? ... "So far, we've just been trying different numbers, looking for a pattern. That's often a good way to start working on a problem. But we still don't quite have a plan for solving it. How can we change this pattern-looking into a plan? ... "We've been using actual numbers. Suppose our plan is to substitute a letter for the 'number chosen'. If we do the calculations using that letter, perhaps we will be able to see how to get from the calculated result back to the original number. "Let's say n is our original number. [Write it at the bottom of the table.] n can stand for 1 or 2 or 3 or 27 or 678 or any number at all. "We add 5 to n and get n+5. [Enter it into the table.] So if n was 4, n+5 is 9. Next we multiply that result by 3, and get 3(n+5). Then we subtract 17, and get 3(n+5)-17. "There's a way that can be simplified. Anyone know how? ... We can simplify 3(n+5) to 3n+15. For example, 3(4+5) = 3 x 9 = 27. But 3(4+5) also equals 3x4 + 3x5 = 12+15 = 27. Think of it this way. Suppose we have three kids, and each kid has 4 cats and 5 dogs, and we want to know how many pets there are altogether. One way of figuring is to find how many pets each kid has, and multiply by the number of kids. That's 3 kids times 9 pets per kid, or 27. The other way is to find out how many cats there are and how many dogs, and add those two numbers. There are 3x4=12 cats and 3x5=15 dogs, so there are 12+15=27 pets total. "So 3(n+5)-17 = 3n+15-17 = 3n-2. Right? ... If n=2, 3n-2 is 6-2 = 4. So it checks. "The magician knows, then, the number we give him is 3n-2. How can he calculate what n is, if he knows what 3n-2 is? ... Suppose he adds 2 to the number we give him. What will he get from that addition? ... What is 3n-2+2? ... Right. 3n-2+2=3n. So he adds 2 to the number we give him, and gets 3n. How can he calculate n if he knows 3n? ... Right. He divides by 3. "Here, then, is what the magician does. He takes the number we give him, adds 2 to it, and divides by 3. "Our plan was to use a letter for our number, and see where that led us. It helped us conclude we can find the original number by adding 2 to the answer, and then dividing by 3. How can we check that conclusion? One way is by checking is to see whether it works with the numbers on our table. Does it? ... Right. It does. For example, if the answer he gets is 10, he adds 2 to get 12, then divides by 3 to get 4, which was the number originally thought of." So there's another example. Would you like to find more examples to help make math fun for kids -- enough so you have a new one every day in the week for a whole year? There are a lot of books full of ideas, and many math links on the Web full of ideas. Click on the Book and Link buttons down below amid the Gofers to find them. Also, if you subscribe to our hilarious free math newsletter, the Gnarly Gnews, you'll find new ideas in every issue. Just click the Newsletter button down below to subscribe. Then there are our moneyback-guaranteed CD-ROM,
Solid Gold Gnarly Math, and our humorous Math Ebook The Historical, Humorous Math
Newsletter. Click one of the buttons down below to
learn all about how these products can make math fun for your kids, supply dozens of
teaching ideas, and cure any math phobias you might have. Click the Order
button below for the CD, or the 'Our Ebook' button for the The Historical..., and you can
have one of them working on your own computer in less than a week!
|
is full of teaching ideas. Or try our fabulous Ebook.
|
