FOURTH: Think of teaching as an art. Develop your own style just as an actor, dancer, painter, or writer does.

     "I suppose most of you know how tall you are. Sammy, how tall are you?... Fifty-one inches? Good. But why did you answer in inches? Why didn't you give your answer in miles? If I ask how much you weigh, probably you'll give your answer in pounds or kilograms. Why don't you give your answer in tons, or milligrams?

     "I suppose most of you know how old you are. Anybody doesn't? How old do you think I am?... [Let them guess.] I notice you're all guessing in years. Why is that? You know what a decade is? Ten years, right? Why aren't you guessing my age in decades? Or why aren't you guessing my age in days, or hours, or minutes, or seconds?

     "Inches and pounds and years are what are called 'measuring units', and when we measure things we usually use units that give numerical answers not too big and not too small.

     "But we can always give answers in other units, if we like. Suppose we want to know our age in seconds. Your answer doesn't have to be exact, of course. If we insisted on exact answers, the answer would change every second! So let's guess our ages to within, say, a million seconds of our exact age. Who will guess first? ... James? [Write down several guesses on the blackboard.] At the beginning of the next class, we'll see how close these guesses were."

     "Okay, let's find out how old we are. Did anyone compute his age in seconds?... [If so, write down the answers, with names, on the blackboard]

     "Now let's talk about how 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. What's our age, in seconds? Sounds complicated. Can anyone think of a simpler problem like it? ... The problem is 'what's our age, in seconds?' A simpler problem might be 'what's our age, in .....' in what? In years? Ok. That's easy. Mary, what's your age in years? ...Twelve? You mean exactly twelve? Your birthday's today?... No? Then twelve and how many months, and how many days? ...So your age is 12 years, three months, and seventeen days? Good! There's a start.

     "But we want the age in seconds, right? Any ideas how we get from years, months, and days to seconds? ...No? Suppose we're talking about a fruit fly who has lived for exactly four minutes. What's his age in seconds?... 240? Right. How did you calculate it? ...You multiplied the number of minutes by 60, because there are 60 seconds in every minute. So [writing on blackboard] 4 minutes = 4 minutes x 60 seconds/minute = 240 seconds.

     "Now suppose we have a kitten born last night, who is 7 hours and 12 minutes old. How do we calculate her age in seconds? ...How do we calculate 7 hours in seconds?... How many minutes in 7 hours?... That's right 420 minutes. [writing on blackboard] 7 hours = 7 hours x 60 minutes/ hour = 420 minutes. So if the kitten has lived 7 hours and 12 minutes, how many minutes has she lived?... 432? Right 420 + 12 = 432 minutes.

     "Our kitten has lived 432 minutes. How many seconds has she lived?... How do we calculate it?... Right. 432 minutes = 432 minutes x 60 seconds/minute = 25920 seconds.

     "But Mary's age is 12 years, three months, and seventeen days. Do you see how we can apply the same methods to find her age in seconds?... If we knew her age in days, could we find a way to find it in seconds?... Can we find out how many seconds there are in a day?... Yes. So here's our Plan for solving the problem: we find Mary's age in days, and then multiply it by the number of seconds in a day.

     "We've devised a plan. The next step is to carry it out. First, let's find out how many seconds there are in a day. How many hours in a day?... 24. So 24 hours is 24 hours x 60 minutes/hour x 60 seconds/minute = 86,400 seconds in a day.

     "Now let's find out how many days in Mary's age. How many days in 12 years?... In one year?... 365. So 12 years = 12 years x 365 days/year = 4380 days (approximately--we'll forget about leap years). She's lived 3 months added to the 12 years. How many days in 3 months?... If we take 30 days per month, then the 3 months is 90 days. So her 12 years, three months, and seventeen days is 4380 + 90 + 17 = 4487 days.

     "We already figured that there are 86,400 seconds in a day. We found Mary has lived for 4487 days. So how do we find her age in seconds? ... Right. Mary's age in seconds is 4487 days x 86400 seconds/day = 387,676,800 seconds!

     "So we've carried out the plan by finding Mary's age in days, and the number of seconds in a day, and then multiplying those numbers together.

     "Can we check our answer? Any ideas?... If we knew how many seconds there were in a year could we figure her age in seconds?... Yes, if we knew her age in years. What is 12 years 3 months and 17 days, in years?... How many years is 3 months? ...3 months x 1 year/12 months = 3/12 years = .25 years. How many years is 17 days?... 17 days x 1 year/365 days = 17/365 years = .046 years. So in years, she's lived 12 + .25 + .046 = 12.296 years.

     "How many seconds in year?... We know there are 86,400 seconds in a day. Can we use that to figure the number of seconds in a year? ...Right 1 year = 365 days x 86400 seconds/day = 31,536,000 seconds. And Mary has lived for 12.296 years? So she's lived 12.296 years x 31,536,000 seconds/year = 387,766,656 seconds.

     "Our answers don't exactly agree? One is 387,676,800 seconds and the other 387,766,656 seconds? Well, they don't agree because we used slightly different methods -- approximations. In particular, in the first scheme we assumed there are 30 days per month. But the average number of days per month is 365/12, which is 30.41 days. Had we used that number, our first answer would have been the same as the second.

     "But we can round off both answers to 388,000,000 seconds. And we only wanted an answer within 1 million seconds of the right one, so that's ok.

     "We could have done some simplified calculations yesterday to guess our age. We could have made a rough calculation of the number of seconds in a year, and multiplied by our age. How to calculate seconds in a year? It's actually 365 x 24 x 60 x 60. Suppose we round off the numbers, and say it's about 400 x 20 x 100 x 50. 20 x 50 is 1000. 100 x 1000 is 100,000. And 400 x 100,000 is 40,000,000 second in a year. (Not bad. The actual number is 31,536,000). So in 12 years we've lived 12 x 40 million = 480 million seconds. That's not too far from 388 million, is it?"

     [Finish by looking at the answers some of the students reported at the beginning of the class. If the kids are all between 12 and 13 years old, all the answers should be around 400 million seconds.]

     We've supplied a few ideas here, but how will you find five ideas a week for a whole school year? It'll take a while, of course. But there are books full of ideas, and Web sites that can supply even more. Look below under Links and Books. Among other things, you'll find The Math Forum, which I recommend highly as a source of ideas. And if you go to our Book site, you'll find Martin Gardner's Mathematical Magic. Finally, if you subscribe to our hilarious free math newsletter, the Gnarly Gnews, you'll find new teaching ideas in every issue. Just click the Newsletter button down below to subscribe.

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Sincerely,