What's the chance you'll get a four when you roll the dice?
Let's review what we want to do to make math fun
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
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:
Think of teaching as an art. Develop your own style just as
an actor, dancer, painter, or writer does.
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.]
"You're playing Monopoly with your buddies, and you own Park Place and want to buy Boardwalk. When you last rolled the dice, you found yourself just four squares away from Boardwalk.
"What are your chances of rolling a four the next time it's your turn?"
[Let the kids guess, and write down their answers on the blackboard.]
"Tomorrow we'll start the class by figuring just how lucky you have to be to buy your heart's desire."
[Start as follows at the beginning of the next class.]
"Yesterday we were playing Monopoly, and were four squares from Boardwalk. We wanted to know what our chances were of rolling a four with the dice when it was our turn again.
"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. Here the problem is, we have two dice and want to know our chances of getting a four. How can we get a four with two dice? ... Roll two two's? Very good. [Write that on the board]. Any other way to get a four? ... Roll a one and a three? Very good. [Write on board]. Any other way? ... When we said 'Roll a one and a three', we meant 'Get a one on die 1 and a three on die 2'. How else can we get a four? ... How about if we get a three on die 1 and a one on die 2? [Write on board.]
"Now we think we understand the problem. There are three ways to get a four when we roll two dice, and we have to figure the chances of one of those three possibilities occurring. Now we must plan how to get an answer. How do we figure our chances of rolling one of those three combinations? ...
"Have we seen a problem like this one before? ... No? One scheme for solving a problem we haven't seen before is to try to think of a similar problem that's easier. Anyone have an idea of a similar easier problem? ... How many sides does a die have? ... Six, that's right. What could we use that has less than six sides? ... How about a coin? How many sides does a coin have? ... Two.
"Before we tackle the dice problem, let's look a similar problem with coins. If we flip one coin, what's the chance we'll get a head? ... That's right, one chance in two, or 1/2, or .5. We can use a simple diagram to show what's happening. [Draws it on the board]. We start here on the left, and flip one coin. We either get a Head, which takes us to the top branch, or a Tail, which takes us to the bottom. The two 1/2's show the probability or likelihood of each of these choices.
"Now let's make the problem a little more complicated. Let's flip a second coin, and ask: what's the chance of our getting a Head on one coin and a Tail on the other? Can anyone guess the answer? ...
[Looking at the drawing on the board] "Let's use the same scheme we used when we flipped one coin. When we flip the second coin, we 'start' after we've flipped the first -- either here where we had got a Head, or below where we got a Tail. Suppose we got a Head. What are the possibilities of the second coin? ... Right. Either a Head or a Tail. [Draws the upper two branches]. And what are our chances of getting a Head with this second toss? ... One-half. That's right. Every time we flip a coin, no matter what the last flip was, our chance on the next flip is 1/2 for a Head, and 1/2 for a Tail. The coin has no idea what happened last time it was flipped. So even if the last 100 flips were Heads, the chance of a Head or Tail on the next flip is 1/2. [Put 1/2 on upper branch.]
"What's the chance of getting a Tail on this second toss? ... Right. One-half again. [Put 1/2 on lower branch.]
"Now suppose our first flip got us a Tail. What's the chance that the next flip will give us a Head? ... Right. 1/2. And a Tail? ... Right. 1/2 again. [Draw the two 1/2's on the lower part of the diagram.] So with two coins, there are four possible results. Two Heads, a Head then a Tail, a Tail then a Head, and two Tails.
"What's the chance of getting two Heads? [Point to the top branch. ... Right! One in four, or 1/4th. How did you get that? ... Well, our chance was 1/2 of getting the first Head, and 1/2 of getting the second. The chance of getting the first, then the second is 1/2 times 1/2, or 1/4. What's the chance of getting each of the other three results? ... Right! 1/4. For each possibility, we multiply the possibilities along the path.
"Another way of looking at the problem is to realize that we're equally likely to get any of the four results. So the four chances must all be equal. But what must the add up to? What must be the total of the four chances? ... The chance we'll get two Heads, or a Head and a Tail, or a Tail and a Head, or two Tails, is 'for sure'. What probability number goes with 'for sure'? ... That's right, one. So the four chances must be equal and must add to one. So each of them must be 1/4.
"Now we have a plan. We'll treat the dice just like we did the coins. Let's carry out the plan. How many branches should I have from the starting point? ... How many results can I get from the first roll of the die? ... That's right, six. [Draws the six branches and their results.] What's the chance of my rolling a one? ... That's right. One time in 6 I'll get a one, so the chance is 1/6. What's the chance of rolling a two? ... Right 1/6. What's the chance of getting a three or a four or a five or a six? ... Right. 1/6 for each. [Writes the 6 1/6ths].
"Now for the second roll of the dice. How many branches will we have from each of these six branches of the first roll? ... If we roll a one on the first roll, how many possible results will there be from the second roll? ... Right, six. So if we have six results from each of the six first-roll results, how many results will there bealtogether? ... Right. 36. Six time six.
"Do we have to look at all 36 results? ... What are we interested in? ... What total do we want, for the two rolls? ... That's right, 4. We're trying to roll a 4. So: do we have to look at all 36 results? ... If the first roll gives us a 6, can we have a total of 4 for two rolls? ... Of course not. So we don't have to look at what happens after we roll a six. What first-rolls DO we have to look at? ... Right. Just a 1 or a 2 or a 3. If we roll a 4, 5, or 6 on the first roll, we can't get a total of 4.
"Suppose we rolled a one on the first roll. What must we roll the second time, to get a total of 4? ... That's right, 3. And what's the chance of getting a 3 on that second roll? ... One in six, or 1/6. (Remember, the second die doesn't 'know' what the first one did. No matter how the first tie turned, the chance of getting a 3 with the second is 1/6.) What's the chance, then of our getting first a 1 and then a 3? ... How did we calculate the chance of getting first a head, then another head? ... We multiplied the chances, didn't we? ... So what's the chance of getting first a 1 and then a 3? ... That's right, one in 36 or 1/36 -- 1/6 times 1/6. [Draw the branch from the first one, and mark one-three with 1/36.]
"What's the chance of rolling first a two, and then another two? ... Right. 1 in 36, or 1/36. How about first a three, and then a one? ... Right. 1/36 again. [Draw these last results]. In fact, how many total results can we have from the two rolls? ... Right, 36. And are all of the 36 results equally likely? ... Yes, they are. And what must all the chances add up to? ... Right, 1. So if they're all equal, and they add to 1, what must each of them be? ... Right. 1/36.
"Now we're almost finished. What's the chance of rolling a 4? ... What's the chance of rolling either a 1 and 3, or 2 and 2, or 3 and 1? ... How did we find the chance of getting a Head-Tail or a Tail- Head? ... Right we added the chances of each of them together. So what's the chance of rolling a 4? ... Right. We add 1/36 + 1/36 + 1/36 and get 3/36, or 1/12. One time in twelve we'll get a four when we roll two dice.
"How can we check our result? Any ideas? ... Here's one sort-of-check. Suppose we're 10 squares from Boardwalk and want to know our chances of getting there in one roll of two dice. What rolls will give us a 10? ... A five and a five. Right. Any others? ... A six then a four. Any others? ... A four, then a six. Any others? ... No. No more. So there are three ways of getting a 10, just like there are 3 ways of getting a 4. How do we compute the chance of getting a 10? ... What's the chance of getting two fives? ... Right. 1 in 36, or 1/36. So what's the chance of getting a 10? ... Right. 1/36 + 1/36 + 1/36 = 3/36 = 1/12. The same as for getting a 4. That's a sort of check, anyway."
So there's a first example. How can you find more examples -- 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 to find them. A particularly good link is Cut-the Knot.com --an award-winning site full of good ideas. And a particularly useful book is Teaching Mathematics--a Sourcebook by M.A. Sobel and E.A. Kaletsky. Finally, 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.
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