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The majority of the following narrative is generated mainly from the 5th set, 8th grade mathematics class. I felt it necessary to capture other interesting narratives that happened outside of the 5th set class as well.
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PART I: THEORETICAL VS. EXPERIMENTAL PROBABILITIES AND THEIR USES
LESSON 1: Introduction To Probability & Backgammon
DAY 1 (2/10)
This begins the first day of the implementation of this unit. My cooperating teacher explained that his students will be interested and are ready for a nice change. Sure enough students were quick to become engaged in conversation about what comes to their minds when they hear words such as "chance", "luck", or "probability". The intention behind this introduction is to surface students' mathematical thoughts and abilities on probability. I felt it necessary to have students guide the discussion as often as possible.
Teacher: What comes to mind when I mention words such as "chance", "luck", or "probability"? Brian: What are the chances of winning something? Like when you play games and stuff like that. Christian: Everything is luck with that stuff. Teacher: How so? Christian: No one ever wins anything. Teacher: Well, why don't we talk about that. It seems like all of you have played carnival games before. Why don't people ever seem to win. Brian (and other voices): Because it's rigged. How the heck are you suppose to throw something this big into a container this small. I hate those stupid games. Teacher: Nice example. I agree with you guys, some of these games are unfair-or as you have stated rigged. What makes something unfair? Mike: When you have a slim chance of wining. The odds are against you. Tim: You have a greater chance of loosing than winning. If that didn't happen than the booth would lose allot of money. Teacher: Good point. You guys don't think skill is ever a factor? Dave: It depends though on the game. Like some games are based on skill, others just luck. Teacher: Ok. Then what is luck based on? Dave: Luck, it's hard to explain-luck you know, winning or loosing. Teacher: Anybody want to help Dave? Tim: Luck, is like something beyond your control. Teacher: I agree, but how do you know when something is based on luck versus skill? |
At this point, the class had a tough time trying to explain what "luck" really is and whether it coexists with probability. This is a rather big issue for young students to inquire about. However, I'm glad to have captured the opportunity for them to attempt answering such "hard to explain" questions. In their exploration of Backgammon, I will ask them to take a closure look at the game and try to compare where they see luck as a factor versus where they see probability as a factor. In other words, when can chance be recognized and used to win the game in making decisions.
Dave: Whenever someone mentions that word "probability", I think of those colored balls in a bag and trying to find the probability of choosing a specific color. Teacher: Funny you should say that, I sort of have that vision permanently ingrained in my brain. Don't you worry though, I promise you'll have other things to think about "probability" by the time I'm through with you. Ame: My parents play the lottery all the time. I think it's such waste of time. Teacher: Why? Ame: Because your chances of winning are one in a trillion! Teacher: Wow, that some chance!! Kristen: I heard that you have a greater chance of getting hit by lightning than winning the lottery. Teacher: Where do you think that statement came from? How could someone arrive at that conclusion? Brian: Well, first of all I'm sure that some sort of scientist group have calculated someone's' chance of getting hit by lightning and than you just compare it to winning the lottery. Teacher: Well I guess so, there probably is some statistics on the number of people that get hit by lightning per year? I unfortunately do not know how many people win the lottery in a year. Does anyone know? Christian: Maybe like 5-7 people or so? Maybe less? Teacher: I'd probably guess about 3-4 win the whole pot. Teacher: Where else do you guys hear about things that are tied with "chance"? Jessica: I play basketball and my foul shot average is something like 1 out of 5. Teacher: What does that mean? Jessica: It means that I have a 20% chance of making a foul shot-I'm not very good. Teacher: Well, your young, I'm sure with more practice you'll get better. Speaking of making shots. I'm going to give Jessica this dime and ask her to try to get it in this cup. But first, Do you guys think she'll hit or miss? Some students yell hit while the majority yell miss. Teacher: Oh please, how do you guys know? Matt: Well, she's not very good at basketball, but at the same time that's not really hard to do. Teacher: Go ahead Jessica make the shot. Jessica misses the first time. The class responded "told you so". Teacher: Ok. Let's have her do it again, what do you think will happen. More students said she'll miss. However, she ended up making the shot. Teacher: Wow, Ok so what happened? Many voices: "luck"! Teacher: Luck huh, okay let's have her try again. What do you think will happen. Interesting thing happens here-more students said she'll make it! Fortunately she did. Teacher: What if we had Jessica do this another time. What would happen. Brian: She'll miss, the other tries were just luck. |
This small little activity really got the kids riveted. More importantly, it gave me insight to how the students tended to utilize the outcome of previous shots to make a prediction of what would occur next. Students again referred to making the shot as luck. I think this is because they haven't seen Jessica perform enough trials in order for them to resort to skill.
Brian: Ms. Maine, why is it that when you toss a coin it can only land hands or tails? How come we don't include landing on it's side. Teacher: Good question Brian-any reactions? Tim: That's a dumb question-the chance of it landing on it's side is nearly impossible-so that's why we can't count it. Teacher: Why can't it land on it's side? Tim: It won't balance, it will just fall. Teacher: You're right-the mechanics of coin make it very difficult for it to land on it's side. How many times do you think you would need to toss it until you get it to land on it's side? Brian: About a million. Which doesn't matter anyway, because by that point you'll have 50% heads and 50% tails. Teacher: Excellent point Brian!! Do you guys see that? How many tails to you think you'll get with a million tosses? Many voices: "half", "500,000" |
I was very impressed with this particular conversation. However, I realized that only a fraction of the class really were in sink with Brian. Yet, I could see the "wheels" start to spin in their heads as they continued to think about the probability of a coin landing on it's side and why we usually do not consider it an option. Moreover, as seen by the dialogue students seem very comfortable using percentages and ratios to describe the chance of winning or getting something.
Teacher: What if we tossed a marshmallow? What are the chances of it landing on it bigger side? Tim: What kind of question is that? Kristen: 50% Tim: Either it lands on it's side or it's top- 50%. Teacher: Do you guys agree with that. Mike: Yup. But it might not be that exact. Brian: Frankly, I don't think anything is ever exact in probability? Teacher: Why is this so? Brian: Well, if we are talking about chance, chance it's not definite-it could or could not happen. Teacher: Nice intuition Brian. What do you guys think. Kristen: Probability and chance are not definite. Like suppose in the Coca Cola game they say you are suppose to get a free coke in one out of every 8 cans. So like if I buy 8 cans-I should get a free coke-really, I mean how often it that true? Tim: Yeh, I mean if you bought a 24pack there's no way you'll get 3 free cans. Teacher: Nice example, your right. I think the packaging may not necessarily allow you to literally get a free coke with every 8 cans. I guess what we would need to explore is how random the process is. Christian: Yeh I mean how are they getting that answer 1 out of 8? How did they make sure that every 8th can will have a cap that says you win a free coke. Ame: And than what cans go together-like is it done fairly? |
Now this particular conversation was very impressive as well. They were really thinking and applying my line of questioning to real life examples. The students showed that they have some basic understanding of investigating really what games like the Coca-Cola Game really mean. One very important concept behind their thinking is that they recognized that probability is not a definite outcome. In other words, they know that it acts as way of determining how likely something will/will not happen.
However, one big misconception I know they have is assuming that everything has a 50% chance-either you win or you loose. In other words, students tend to ignore that a percent should be placed on the likelihood of the event literally occurring. This is only associated with the idea of winning and loosing.
Although it may seem like these conversations caused "chaos" in the classroom, it provided me with some great insight into what the students already know and what will need to be worked on throughout the investigation. More importantly, the students were able to display some analytical skills and probabilistic concepts that I can always refer to when we begin to formally learn about theoretical and experimental probabilities.
Since I felt pretty good with what has begun to surface, I decided to begin the introduction to the game of Backgammon. I had a goal of not spending more than 2 days in learning this game. I explained to the students what we will be doing over the next couple of weeks.
| Teacher: You guys, brought up some great points that I will refer to as we proceed in this unit on probability and statistics. We will be doing some interesting activities that investigate the purpose for certain concepts in these areas. One major theme over the next number of weeks will be thinking about topics in probability and statistics that help us predict future outcomes or events. I want you to begin viewing mathematics as a tool or key to exploring many topics in our lives. To start off we will be playing a game called Backgammon. Over the next couple of days I want you to think about any forms of strategy that is involved with this game. Before I begin to teach you the rules of this game, I want you to think about how many of you have said that luck is something you can't control. Well, suppose that there is strategy to winning this game. I want you to think about whether such strategies will change your chance of winning and to what extent? What role does chance play in this game? |
I distributed backgammon boards to every students. Some students brought in their own which was a big help. I had made out paper chips for students to use. I introduced the game using the overhead and the chalkboard. The class learned how to set up the game and I discussed the rules of the game. As students began playing, I went around the room individually teaching the game to groups who were confused . I had students who were familiar with the game go around helping others as well. The paper chips were difficult for students to use as they would easily fly away if someone sneezed let alone moved. Thus, I told students to bring in pennies and use the head/tail sides to differentiate amongst the two players.
DAY 2 (2/11)
This day was devoted to playing backgammon. I found it difficult to make sure that every student was on the right track. My cooperating teacher provided additional help as well. Most students were able to finish one game of backgammon in this period. However, I felt a little bothered by the fact that about 10 minutes before the period ended many students got up and walked around while others tried to finish their game. On this particular day, not too many students showed enthusiasm about the game. However, most of the students did get the general rules and objective of the game. At this point, I gave them their first Reflection Worksheet (Appendix D) for homework. I also decided that a great way to get these kids enthused would be to bring the point of why they are being asked to learn this game into light. I decided to jump into the scenario with only 1.5 days of learning the game. I thought that the scenario would really get these students to think a little deeper into the game and reach a stronger understanding of the rules. It is interesting to note here that while I was planning these sets of lessons, I originally thought that we would spend 1 week learning this game! I'm so glad that I was able to reflect on really what is important and what I trying to get at with this game.
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LESSON 2: Backgammon Scenario
DAY 3 (2/12)
This marks the middle of the week before break. The students were starting to get that "I can't wait till break" excitement- a little on the hyper active side. I collected their Reflection Sheet (App. D) that they had for homework and placed the Backgammon Scenario up on the overhead (Appendix C):
Teacher: Okay guys, I know some of you have a better handle on this game than others. Today, I'd like to present you with a specific scenario. Now, believe or not, while I watched most of you play, a large number of you had this kind of situation happen. Here's the board as it stands. You are the black chip. You are on your way to clearing the table. You have these white chips coming your way. You rolled a (4,3). What move are you going to make. What are your choices? Christian: You can move the black chip 7 Teacher: Okay that's one option. I'll put it up on the board, the rest of you tell me what other options there are. Ben: You can move it 4 and than move some other black chip 3 or the reverse. Teacher: I agree, so now we've got 3 choices. Anything else we can do? Brian: You know what, Ms. Maine, you don't have to move it at all. You can move other black chips as well. Erin (sitting next to Brian): I think we have a less chance of getting hit-I think Brian is right. Christian: No, what are you talking about, you are close to winning the game-go for the chance and move that one chip 7. If those white chips don't get you, you can get them. Kacey: I think 7 is too much to move that one chip. I think we should move it 3 or 4 to stay on the "safer" side. Teacher: (as the class begins to get very involved), okay guys, I love all the thinking that's going on here. I'm not going to make a comment on anything that was just said because, you all have excellent points. I think this is a good time to get in groups, take some poster paper and tell me what move you would do and why. When your done, you can present it to the class. Use pictures, tables or computation to help you explain your position. Go to it! |
This was exactly how I wanted the class to start off with. The students were all thinking from a probabilistic point of view. There were other students in addition to Brian and Erin who were able to deduce that the "safer" move would be not to move. If the chip is not moved, the white chips would need to get an 11 to hit the black chip (a probability of 2/36). While I was walking around the room to observe students. I noticed that very few were attempting to figure out what the probabilities really were. Yet, they would use reasoning such as "less chance of getting hit", "take the risk at get home", "even numbers are going to come up more often". I was very impressed by there intuitive thinking skills. I also realized that the students really enjoyed working in groups to come to a consensus of what the best move would really be. Moreover, much of the discourse that was occurring amongst students really forced a crucial goal in the learning of mathematics- "backing up your thinking". Some groups even began rolling two dice to see what kinds of numbers they would get! What is even more exciting is that the students weren't going to be happy until they could convince themselves that their arguments made sense. Here are a few examples of what students offered as explanations:
Kacey & Emily: Don't move, the white chips will come up on you-but they won't get you and than you can get them! Teacher: How do you know you won't get hit? Emily: Because you have less of a chance. These white chips have to get an 11 to hit you. That's a big number. Usually the smaller numbers are what I keep getting. Teacher: You tested it? Kacey: No, but when we play the game, I don't remember getting an 11 allot. Teacher: Do me a favor, think about a way of finding out the chances of getting an 11. Tell me when you think you've got something. Morgan &Matt: It's a 50% chance of getting a 4, so we said go for it! Move 7. Teacher: Where did 50% come from? Matt: We rolled 20 times, and according to the rules of backgammon we got a 4, 10 times. But who cares-take the risk-you've got a 50/50 chance! Teacher: Do you think that if another group rolled 20x they would get a 4 10 times as well? Morgan: (a little hesitant), well, I mean yeh-why not? They would at least get close to it. Teacher: What if you rolled it 30 times? Morgan: We'd still get 1/2. Teacher: Okay-you very well could. But how do we know for sure what your chances are? Matt: You can't know for sure, it's the luck of the dice. Teacher: You mean you can't control what the dice say-I agree, but what about after you've rolled-is there a way of looking at the numbers and getting an idea of the probability of getting hit? Morgan: Well, I guess we have to list all the pairs and see what seems to come up the most. Teacher: Nice thinking, Morgan, hold on to that thought-I think we very well may use it! Ame: We decided that if you move 4, than the white chips needs a 7 to hit you. Since seven is odd, it won't come up as often and you'll be a little closer to home and probably on the safer side. Teacher: So you think it's good to get your guy moving across the board. But, tell me more about your view on the odd number 7. Ame: Well, look how many ways can you get a 7, doubles won't get you it, only (2,5) and (4,3), (1,6). That's it- like 3/12. Teacher: Mmm. Interesting. Your right, those are the ways to get a 7. How did you get 3 out of 12? Ame: there are two dice with 6 choices on each so 6x2 = 12. Teacher: Anyone want to comment on that? Brian: I thought it would be 30. You gotta list all the pairs! Teacher: Any comment on what Brian is saying? |
Ame was one of the last presentations. At this point, many students begin playing around with trying to develop a list of possible outcomes. However, some were more focused on rolling the die an "x" number of times to see what happens. At any rate, this particular lesson was quite enjoyable for both I and the students. Every student had something valid and substantial to share with the class. I'm not so sure that every student really understood how impressive their arguments were. Since their work was so valuable, I decided to hang the students'' posters on the back wall. Some of the rationale the students had could easily be referred to when we begin to move towards closer exploration of theoretical and experimental probabilities. Thus, the posters would act as a frame of reference for the students to remind them that they indeed, had the right idea! Since class was ending soon, I decided to have the students think about ways we could find out if our predictions were true. In other words I asked them to think about how one could find the exact chance of getting hit on the move they chose. This would begin our next class.
DAY 4 (2/13)
I regret, not assigning the homework as written work! When I asked students with what they came up with, they had nothing to say! It was evident that they didn't take time to think about it. This was my fault in the sense that with this age group, unless an assignment isn't literally assigned on a piece of paper or book, the students' simply won't consider it as a "real" assignment. At any rate, the students' had surfaced more than enough to begin a discussion of examining how we could try to find out the chances of the moves in the scenario. My goal for today was to get through finding the theoretical probabilities of each of the moves and than comparing it to what students' predicted as the "best" move.
Teacher: Okay, yesterday we started examining the question: How could you find out the chances of getting hit. Well, today I want to proceed with this and see if we can develop a process that indeed will allow us to check our predictions. Now I know some of you began mentioning fractions and percents when doing your poster presentation. Others, used comparison, in what numbers would come up more often. For example an odd number may not have as many choices as an even number (at least according to the rules of backgammon). Why are probabilities talked about in terms of fractions or ratios? Tim: Because you are comparing the chance of getting something to not getting it. Dan: Yeh, it's like the percent something will happen out of 100. Teacher: I agree. But what about the fractions that some people came up with. Why are fractions used? Erin: Can't you think of it like say the probability of getting a six on a die is 1/6 because you only have one way of getting a six and there are 6 choices on a die so 1/6. Teacher: Erin, that's exactly what I'm looking for-let's use something that you guys learned in 6th grade, a simple die. What are the chance of rolling any number? Class: "1/6" Teacher: okay, can you use words to describe that fraction? What is 1/6-Erin said it-can anyone repeat it. Brian: It's how many times you can get any number. Teacher: okay, you guys agreed that the probability of getting a 6 in one roll of a die is 1/6. What does the 6 represent in the fraction? Brian: it's 1,2,3,4, 5, 6. Teacher: what are those numbers representative of? Brian: all the numbers on the die. Teacher: exactly-it's all the choices we have! Now, can someone explain to me what 1/6 means in terms of rolling the die. Matt: It means that if I roll it once, I'll have a 1/6 chance of getting the number, say like the number 2. Kacey: You can get a 2 at least once in 6 rolls. Teacher: Matt and Kacey have got it. Are you all with us. (most heads are nodding at this point). Ben, since you guys know that the chance of rolling a 2 on one dice is 1/6- and you didn't have to test it, is it possible to figure out the moves in our backgammon game the same way? Ben: yeh. Teacher: okay, I'm going to pass out this worksheet (Appendix E) that will help us find out what the safest move really is. Now follow along and let's see if we can get some actual answers-I don't know about you but the suspense is killing me! (class sighs....) Now, let's look at our scenario. Say we decide to move 7. The white has to get a 4 to hit us. What are the chances of getting a four? What do I need to do? Brian: well, since we are talking about 2 dice and not one we got to figure out all the pairs that come up like double ones, double twos, etc.. Teacher: Okay, so what am I trying to do? Christian: you need to find out all the ways of getting a four and compare it to all the possible pairs. Teacher: Perfect. Now, I want you guys to take 2 minutes and see if you can develop your own system for finding all the choices we have with 2 dice. Anybody, want to take a guess at how many it will be. Class: 12, 30, 24 Teacher: well, let's go ahead and see what we come up with. |
At this point, I walked around the room to find that most of the students were not using the counting tree. Rather they were listing any pairs that came to their heads. Some were doing all the ones first and pairing it up with all the numbers and than doing the same for the remaining numbers. I also know that I had a couple of problems that would need to be addressed. The first was to explain that something like (1,3) and (3,1) are not the same. The second is to convince them that 12 is not the total number of possible outcomes. These two situations are problems that many students have. They think: "since one die has 6 choices, two dice will have 12 choices". The (1,3), (3,1) scenario was going to be a tough pill for these students to swallow as well.
| Teacher: okay, 2 minutes is up-go ahead guys start shouting out pairs and I'll right them on the board. |
I intentionally decided not to immediately teach them the counting tree or the punnette square just yet. Rather I wanted students to realize that listing all the possible outcomes with out any framework or structure may result in an incomplete list. I want the students to fall into asking themselves: "how do we know if we covered all the possibilities" -especially when their is no organization. The students began listing numbers.
Teacher: hey hold up, I can't right this fast, I wrote down (1,2), but what about (2,1)? Gaelyn: It's the same. Teacher: Does everyone agree??? Class: yeh. Teacher: No one wants to challenge that? Christian: How can you-it's the same thing. Teacher: Okay, everyone stop for a minute-this is really important. What is 1 + 2 =? Class: 3 Teacher: What about 2 + 1 =? Class: 3 Teacher: So how many ways do we have of getting a 3 right now? Class: 2 Teacher: Now tell me if you are convinced that (1,2) and (2,1) count as 2 separate choices? Brian: Well, I guess that makes sense. Emily: But it's still the same-you get a sum of 3. Erin: Why do we have to count it as separate? |
If someone were to take a snap shot of the class at this point, one would see that probably half the class accepted this passively, while the other half really were not convinced. I decided to use a white die and a brown die to give them another way of looking at it.
Teacher: Okay these are two dice that are different colors. Suppose I get a 2 on the white dice and a 1 on the brown dice. That's the (2,1) outcome. Now can't I get a 1 on the white dice and a 2 on the brown one? Matt: yeh but our dice is not colored. Teacher: I know that but suppose I gave you colored dice, or suppose I took spray paint and painted one of your dice-would that change things? Matt: I guess not since it's the same dice. Teacher: Right, I'm only changing the color. Who's with me. |
The students needed time to think about this for a moment. A few more people began to understand. But, for some reason I wasn't convinced just yet! Off the top of my head I decided to give them another way of thinking about it.
Teacher: okay, guys, I'm not convinced that you guys are with me. Let's try this. Brian you are on the track team right? Brian: yeh Teacher: Suppose you are running against one person. So someone will get first place, and the other will get 2nd. Right. Class: yeh Teacher: Folks, how many ways can Brian place? Brian: I can get first place and he get second. Teacher: OR Brian: okay or I can get 2nd and he get 1st-but that will never happen. Teacher: Erin does that make sense. Erin: yeh, I see what your saying-so that means we have to do the same for all those pairs of numbers. Teacher: you got it kid! So let's continue. Give me more pairs. |
The class continues to increase the list until no one has anything left to offer.
Teacher: How do you know if we have everything-it's hard to look at this list and see what's missing. Did anyone organize things differently. If so come up to the board and show us.
Two students came to the board and actually had counting trees. When the students saw the trees, many of them quickly remembered with a "oh yeh". The counting tree was relatively easy to explain. In addition, I taught them the punnette square method as well. It reminded them of either their science class or the multiplication table. Students took to this method pretty well also. At this point, students saw that in fact they had 36 possible outcomes.
Teacher: Well look at the diagrams on the board. How many pairs do we have? Class: 36 Teacher: Now, once again what does 36 stand for Matt? Matt: That's all the choices with 2 dice. Teacher: You got it, it's the total possible outcomes. Now, guys look at this list. What are the chances of getting a 4-according to our outcomes? |
The students begin to mention the pairs as I would circle them. In backgammon, any pair of double is worth 4 times. This needed to be reminded to the students. They arrived at 18/36.
Morgan: Holy Cow-we were right Ms. Maine. Yesterday we told you it was 50%! Teacher: You better believe it! What's even more fascinating is you did it without doing what we just did. What does this fraction mean in your own words: Morgan: 18/36 is 1/2 so you have a 50% of getting a 4 in the game. Teacher: Matt, what is another way of thinking about this fraction? Matt: It's like out of 36 possible a 4 comes up 18 times-just like your grid says. Teacher: very good. Ben-what else is 18/36? Ben: I don't know. Teacher: Well, if someone tells you that you have a 1/2 chance of winning something-what does that mean in terms of your chances specifically. Ben: 50% Teacher: yeh-but use the fraction and put it in words-can someone help him. Christian: I think I got, it means that 1 out of every 2 rolls you'll get a 4. Teacher: Thank you-Perfect. Did everyone hear that! |
One of my main focuses, have been to get the students' to really understand and interpret mathematics in their own words. If this is not accomplished than a fraction just becomes a fraction with no meaning. In other words, what have they learned?
Teacher: Okay, now we have this fraction 18/36. Kacey what does the 18 represent? Kacey: The number of times you get a 4. Teacher: Good, now in general would you guys argue if I said that the top number is the number of times an event occurs-such as 4. Class: Nods in agreement Teacher: Okay, now what does the bottom number represent-the 36. Brian: all the possible choices. Teacher: You got-it's all the possible outcomes or choices. |
The interpretation of the fraction is written on the board. I explain to the students that what they have just found is called a theoretical probability.
Teacher: When someone says in theory you have a 50% or 1/2 chance of getting a 4 what does that mean. Morgan: It means that your exact chance is 50%-but that doesn't mean it's going to happen. Christian: It's like in theory, it should happen but may not with one roll. Teacher: I like it-you guys are good-now tell me anybody, how much do you guys trust a theoretical value of 1/2? Addie: Well, you just take a chance and see. Teacher: See what? Addie: If in fact it will happen-the 4 coming up. Teacher: Okay, but where did this theoretical come from? Ben: Didn't we get it from looking at all our choices and then counting all the times a 4 came up. Teacher: Exactly-a theoretical probability is based on calculating all your possible outcomes. You are actually using mathematical analysis to figure out what should happen theoretically. Did I ask you guys to roll dice at all today? Class: No. Teacher: Exactly for a theoretical probability, we didn't have to roll the dice. We figured it out mathematically. Now, you guys are on a roll today. Go ahead and finish the rest of the worksheet in class today and for homework. You just need to find the theoretical values for the rest of the scenarios. I didn't make you read the worksheet while we were exploring the probability of getting a 4. However, tonight I want you to make sure you can piece everything together and read the sheet in it's entirety. |
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LESSON 3: Theoretical Probability of Backgammon Scenario
DAY 5 (4/14)
This is the last day before winter break. The students were high strung today. However, most all of them did their homework. Because it was the last day before the break, I wanted students to reflect on the theoretical probabilities they have found and compare them to their own predictions. I thought it would be nice to give them a little "game playing time" as well. I wanted the students to have the opportunity to try playing the game once more with the idea of applying what they have gathered about theoretical probabilities of the dice.
Class started with comparing the answers the students had and discussing their thoughts:
Teacher: okay, so you all should have figured out a percent and a fraction for the 3 remaining scenarios. Any thoughts on the numbers you got? Actually, what is the "safest" move according to the theoretical probabilities you found? Emily: The safest move is the move where the chance of getting hit is 1/18. Teacher: Which move is that? Emily: Just stay where you are. Teacher: Does anyone disagree with that? What does the fraction or percent stand for? anyone? Ben? Ben: It means the chance of getting an 11 is 1 out of every 18 rolls or 2 out of 36 rolls. Its like a 6% chance of getting an 11. Teacher: So your chances are pretty slim. Nice interpretation Ben. How many of you said not moving is the better idea when we did our presentations? Less than one half of the class rose their hands. Teacher: Those of you who chose other moves? What are you thinking right now? Kacy: The probability of moving a 3 or a 4 really doesn't matter. Teacher: Explain to me what you mean Kacy. Kacy: I mean that if you move 3 you have a 16% chance of getting hit and if you move 4 you have a 19% chance-really there's not much difference. So it doesn't matter too much if you pick one or the other. Teacher: Your right those percent are pretty close. Christian: You know it doesn't even matter because those percents are really low to begin with so I would just go for the move-you only have a 16% chance of getting hit -BIG DEAL. Teacher: Hey, Christian is getting a little feisty! Let's think about what he said. Who disagrees with him? Tim? Brian? Erin? Brian: I think he's right-but we are still much safer if we stay. Let the white chip take the next roll and than nail him!! Teacher: That's one way of looking at it. But, let's think about this for a minute. How much do you guys trust these values? Like do you really think that the probability of getting an 11 means that it would take 18 rolls to get an 11? Christian: probably not-I mean it could happen more often. Teacher: Could it happen less often as well? Christian: I guess so. Teacher: So than what the heck is the purpose of this stuff? Ame? Ame: It sort of helps us with what may happen. Kristen: yeh-it's like we get an average of what may happen. Matt: It's not suppose to be definite. Teacher: What do mean by that. Matt: Well, we said that a probability tells you your chance of getting something-it does not necessarily guarantee it. Teacher: So how does what we did help you? Do you see any value in what we have done? Matt: yeh, because we got a chance to check if our guesses were correct. Teacher: Absolutely. What else? Anyone? Morgan? Morgan: When you play the game you can think about what the move is and see if the other guy will have a high chance of "hitting" you. Teacher: Excellent point. What do you guys think. Can you use this process to better your game strategy? Class shows general agreement. Brian: Ms. Maine, I don't think its necessary to do this with every move!!! I mean only in certain parts it may be good-like if its a close game. Teacher: I agree Brian, you don't need to use it with every move. However, I wanted you guys to see that analyzing the "luck of the dice" may work in your favor some times. So again tell someone what the benefits are of these theoretical values. Jessica: It helps you determine if something will happen or the chance of it happening. Teacher: Nice. Every body see that? |
At this point the class seems pretty much in sync with what theoretical probabilities were. However, I wanted them to begin talking a little bit more on how much does one trust these values. Students were pretty quick to conclude that the theoretical probabilities are what is expected to happen. However, most students I don't think are completely convinced that it will actually happen. I almost felt that the students believed that more often than not a theoretical probability may be off. I think this thought process will really fuel an interesting investigation of experimental probabilities. I thought this would be a good time to stop and allow for some play time.
| Teacher: Okay, you guys had some great insight into the rationale for theoretical probabilities and I think we all agree how such concepts can help us with our strategy when playing the game. So having said that, I'm going to allow you the rest of the period to play the game and really think about some of these things we talked about. Some of you might even play the game a little differently. Over the break try to play a couple games with your brothers, sisters, parents or friends. When you approach a move which seems difficult to make, try analyzing your choices and report to us next Monday on your experiences with the game. |
The class began playing the game. Interestingly enough, they were far more engaged in this game than when I first taught it! Students were definitely placing more thought into the game before they made there moves. Thus, they were in fact applying what they had learned-one of my more important goals in teaching mathematics.
2/17-2/21 FEBRUARY BREAK!!!!!!!!!!!!!!!!!!!!!!!!!!!!
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LESSON 4: Experimental Probability
At this time, I feel it necessary to discuss the event that took place in my first class of the day. It was their first day back from break. The students were pretty tired and quite. I on the other hand am rearing to go!!! I passed out the next worksheet on experimental probability (Appendix F). My goal was to begin with this worksheet WITHOUT THE INTENT OF ANY REVIEW!!!!!!!! What is even more distressing, is that the thought of "reviewing" what we did 2 weeks ago never crossed my mind! About 3/4 for the class did not remember what had happened before break. Yet, here I had this goal of having them get through the worksheet and realize the difference between a theoretical and an experimental probability! I tried to remind them of the probabilities that we discovered such as 18/36 for getting a 4. Students had no clue where the 36 came from! One girl though it represented the triangles on the backgammon board! Moreover, I had the students work in groups on the worksheet. Thus there really wasn't much classroom discourse going on. At the end we were able to discuss #1 and #2 on the worksheet. Yet, I found myself offering the answers. I felt completely miserable. The students had no idea what was going on! To make my start even worse Judy happened to be sitting in on that day.
At the end of class, I looked at Judy and asked for some serious help. My cooperating teacher explained to me that this was a typical day back from a break and that nothing really surprised him. Judy explained to me that there was nothing in the room that would help trigger the students' memories. I realized that I should of started class with perhaps a brainstorming session on poster paper and perhaps use the worksheet as a homework! Moreover, I should of at least used the back wall of the room to display the work they had done 2 weeks ago. At any rate, I had 3 more chances to rescue myself. The following is the continued documentation of the Set 5-a much better experience.
Teacher: Okay guys so how many of you played backgammon over the break? About 1/3 of the class rose their hands. Teacher: So did you guys find yourself playing differently or becoming better. Tim: I actually have been thinking about the moves and what someone would have to get to hit me. Teacher: Good. Why don't we start off refreshing our memories on what we did two weeks ago. I gave you a scenario. What did I ask you to do with it? Emily: Make a prediction or a guess at what the best move would be. Erin: Oh yea, we had to get into groups and present what our move would be. I remember. Teacher: Thank G-d-Who does not remember this? How did you guys decide what move would be the best? Ben? Ben: Well, it was like common sense, just move the chip so that you are less likely to get hit. Teacher: For some reason I don't remember it being that simple! But you are right, most of you tried to make a judgment call on trying to get home versus trying to avoid getting hit. Okay what next? Brian: We presented our posters? Teacher: Yes you did, and there up in the back of the room. (At this point I am in front of the posters). So we predicted and than what did I ask you to think about. Brian: We had to figure out our chances. Teacher: Chances of what? Ame: Chances of getting hit on the moves. Teacher: Yes, now what was the first thing we needed to do to figure this out. Does anyone remember? Try to reflect on simpler things that you've done already like the chances of getting 6 on a die. Christian: we had to do that list of pairs. Kristen: yeh and that table graph thing that you did. Teacher: You got it. What was the purpose of doing the lists, charts or trees? Kristen: to get everything. Teacher: Give me more. Kristen: It gave us all the ways of how numbers would come up. Teacher: Exactly it's all your possibilities or choices. So with the two dice, how many pairs did we come up with. Pull out your sheet from last week to help you out. Morgan: It was 36 pairs. Teacher: Yes, and what does 36 stand for? Class: all your possibilities. Brian: I remember the 36 was in the bottom part of a fraction and the top part was like a event that we were looking at. Emily: We used the probability of moving 7 which means getting hit by a 4-that's what is on my sheet. Teacher: Thanks Emily, Let's just use that example to take us through this. What is the probability of getting a 4 in backgammon? Class looked at their sheets and replied : "1/2" Teacher: Right, 18/36 which is 1/2. What does that fraction mean in your own words. Ame: It's a 50% chance of getting a 4. Teacher: okay Ame, that's right but let's stick with the fraction-spell it out to me using words. Ame: Well, out of 2 rolls you should get a 4 once which is half the time. |
It was very important to me to make sure that students understood the fractional interpretation. Student have known 1/2 is 50% since the 3rd grade. However, I wanted more from them as it relates to the situation at hand. This articulation of thought is crucial to making sure that students are truly understanding the purpose of fractions in the area of probability.
Teacher: Okay, is everyone with me thus far? (heads nod) Now, why do we call something like this a theoretical probability? Morgan: Because it what should happen but may not. Matt: It is there to give us an idea of what might happen, determine our chances. Teacher: very nice! So the probability of getting a 4 is 18/36. How much do you guys trust this? Is it true that for every 36 roles you'll get a 4, 18x? General Responses: "sometimes", "it depends", "in most cases". Teacher: How do we know this is true if I did not ask you to test it? How do we know if it will or will not happen? Brian: we have to test it. Teacher: Guys listen up I think Brian is on to something (at this point students begin to discuss amongst themselves whether they believe it is true or not). How would we test it Brian? Brian: By rolling it 18 times. Teacher: Why 18x? Brian: No wait I mean 36 times-since that's our possibilities. Teacher: Wait what if it doesn't happen in 36 times? More students begin to mention rolling it 100x Teacher: What is so special about 100? Morgan: You can get a definite proof that it is 1/2 b/c you do it allot of times. Teacher: So you guys are saying that a way to test a theoretical probability is by literally rolling the dice either 36 or 100x? Christian: Well, you could roll it any number of times and just keep watching what happens. I mean even in the first 36 rolls, I could close to 50%. The class confirm what Christian has just said. Teacher: Okay, but it seems like in general you guys are saying that all we need to do is roll the dice at least 36x-first to see what happens. What if we rolled it less? Brian: Well, we should get it 1 out of every 2 rolls since its 50% chance-but that's a theoretical and one roll is not going to prove a theoretical. Teacher: Let's talk about that for a moment. How do you guys think mathematicians came up with the chance of getting H or T when tossing a coin is 50%. Class Responds: "they tossed it like a billion times", "100x", "they did lots of tosses". Teacher: Wow, why do you think they did lots of tosses? Erin: To get some results. Like you have to keep testing to get at what you need. Teacher: Bravo Erin! I like that word "testing" or "experimenting". Okay, we are almost out of time. The sheet that is on your desk is for homework. You guys should not have a tough time with it. You did well today. |
What has just taken place is not only a review of what they have learned thus far, but the students' have managed to follow a discussion that not only treated an experimental probability as a way of checking the theoretical but also a way of developing theoretical. This was not stated in so many words. The students definitely had some great insight into the ability of proving events. I'm not even sure if the students are really aware of the transition they have made between experimental probability and theoretical probability and vice versa. However, this discourse was much more than what I had hoped for. Thus, I was quite pleased. What I also realized is that we sort of conducted a discourse that followed the worksheet (app. F). Yet, we really didn't refer to it. I assigned this worksheet for homework.
DAY 7 (2/25)
Today, we began class by discussing the homework. For #1 and #2 the students were mentioning allot of the points that have been coming up during our class discussion. Question #3 lead us right into conducting actually experiments in order to see how close we would come to our theoretical probabilities. At this point in the discussion of the homework, the students were convinced that a good way to start was just to begin rolling 36 times-since that was the way they interpreted the fraction "out of 36 rolls". I agreed with them that this would be a good way to start and asked them to turn to the chart on their worksheet where they can place all their data. I also decided to discuss the # of roles someone may choose to do later on in our investigation. Students got into pairs and began rolling the dice. I decided to focus on only testing the probability of getting a four according to the rules of backgammon. This would act as a model for the 3 other moves which they will be assigned to do on their own. As students arrived at their fractions they began giving me the results to place on the board:
| 18/36, 18/36, 16/36, 14/36, 19/36, 16/36, 17/36, 20/36, 18/36, 19/36, 17/36 |
As one can see, the class did pretty well. Moreover, I think this hands on experimentation really gave them a grasp on what the fractions mean. I began the discussion as follows:
Teacher: Okay, now just to make sure you are on the right track? Why am I making you do this? Various voices: "to test what should happen". Teacher: Right, we are testing our theoretical value of 18/36. Did we conduct an experiment to get this theoretical value? Class almost in unison: "NO" Teacher: So, now look up at the board, are you guys impressed with what your experiments resulted in. We got some people who actually got 18/36. Brian: Ms. Maine how come some of the values are not exact. Teacher: Yet, what is interesting Brian, is that they are not that far off! I'm glad you asked that question. If some people did not get 18/36 does that make the theoretical wrong? Brian: I guess not b/c it is close. Teacher: Suppose I were to make you do this 90 times. Then What? Christian: It will be closer to 1/2. Teacher: Why? Christian: Because it is more trials. Emily: Ms. Maine I just figured out the class average. It's 17.4. Teacher: Explain to me what you did. Emily: Well I just took the average of what everyone rolled-just the top numbers and divided by 11 and got 17.4 out of 36-not bad. Teacher: You are right. I never thought of looking it at that way. But indeed a class average is a good way of analyzing your results. Guys, I'm going to save you from rolling the dice anymore. Let's just pool all our results together. |
On the chalkboard, I begin to explain how many roles that actually occurred in the class as a whole. They followed along with me and punched in the amounts in their calculators to arrive at 48%.
Teacher: Mmm. You know for 396 rolls, why are we not at 50%? Morgan: First off, I think Tim and Dan who got 20/36 were rolling their dice across the floor. Brian: yeh so their dice rolled allot more each time-the rest of us just rolled it on the desk. Teacher: Wow, that is in fact a very good point. The way you roll the die can influence your results. Why else? Emily: I didn't count right-I forgot that doubles will also allow you to move four-so my fraction is incorrect-14/36 that's wrong. Teacher: Thanks Emily, you are helping us clarify what happened. So now what guys, Do you have faith still in theoretical probabilities. Jaessica: Yes, b/c it gives us a pretty good idea of what will happen in general. Teacher: Does the number of experiments you use matter when comparing it to your theoretical values? Kyle: Well, the more you do the closure you get to what is expected. Teacher: What if you guys only rolled 6 times. Do you think you will get a 4, 1/2 the time? Kristen: Well, yeh you might. Teacher: Suppose you do. Would you be absolutely convinced that if you roll another 6 times you get the same results? Various Voices were quick to disagree. They in fact began talking amongst themselves that you would have to get a large number of trials to get a consistent result. The more you do the closure you get to your theoretical. Before the class was going to end, I wanted to see if student could come up with an interpretation for the experimental probability fraction: Teacher: Remember when you guys were able to look at the theoretical value of 18/36 and tell me what the top and bottom number signify resulting in this kind of equation (begin to write it out on the board): Theoretical= # of times event occurs Total possible outcomes Well, let's look at what you guys got for the experimental probability. Let's use 17/36-what does the 17 stand for, where did it come from? Various Voices; the number of time we got a 4 Teacher: okay so it's the number of favorable or successful events-that's right (writes on the board). What about the 36? Various voices- "the number of rolls we did", "The number of times we rolled". Teacher: Exactly, it is the number of trial you performed. Thus, the experimental probability can be though of as: Experimental = # of favorable outcomes # of trials conducted |
The students were fairly quick at this point to contrive what the fraction really means. I think this is partly do the fact that they literally performed the trials, and counted the outcomes thus relating it directly to the fractions they came up with. I decided to stop, here to let the students soak in what they had done thus far. Interestingly enough, students began testing the other theoretical values without my direction. I allowed them the remaining of class time to compare the experimental values with the theoretical values. In general the class seem pretty convinced that these theoretical values do indeed make sense. My hope is that it will be brought out on the reflection sheet they will have for homework.
| Teacher: Tonight I want you to do this sheet for homework (Appendix G). It is another sheet that asks you to write out your thoughts so please use anything to help you articulate your thinking. |
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LESSON 5: Reflect and Summarize (DAY 8)
DAY 8 (2/26)
Thus far, I felt that although the students in general were following along with the activities and formulating some strong and important concepts there was still something missing. I felt that things needed to be more grounded-almost like a mini closure to experimental and theoretical probabilities. When I described this feeling to my cooperating teacher, he responded that the best way to "ground" the concepts in was to go straight to the notebook. He showed me his math notebook and said that he wants these students to have something that they can refer to whenever they need it. Moreover, he explained: "if you put it in the notebook, these students will consider it more official". Since I had not had much experience with giving notes, I decided to do exactly what my cooperating taecher suggested. He simply makes up the notes the night before and gives it to the students to write down in their notebooks. However, before I began with the notebook I decided to start summarizing some points that the students might have to offer on poster paper. The following boxes show the questions that were asked and the comments that the students made:
total possible outcomes
# of trials |
This was a nice start for the class. However, since I knew I had notes to give them I decided to stop and begin with the notes. There were a couple of items that I wanted students to discuss, however, I though I would just do it in the notebook. For example, I wanted students to reflect on what they know so far and decide whether or not everything has a theoretical value. I wanted them to also why some things have theoretical values while others cannot-this would bring us into equally likely and unequally likely events. I also decided to use the notebook to place other examples outside the dice. I felt that is important in order for students to get a true understanding of these concepts.
I A theoretical probability is what we can expect to happen if we perform an experiment many times. -Theoretical probabilities are found by using math to analyze all the possible outcomes of an event. Theoretical Probability: # of times an event can occur/ Total # of possible outcomes A first step in finding theoretical probabilities is to list all your possibilities in a given situation: Ex: Suppose you have two spinners A and B: What are the chances of spinning a black on one spinner and a white on another spinner? 1. Find all the possible outcomes of this event: Write down all your possibilities This can be found by making a list, a chart or a tree diagram: B W Total outcomes to choose from is 4 B | BB BW W |WB WW 2. Analyze the ratio. How many times can we get a Black and White scenario # of times event occurs/ total possible outcomes 2/4=1/2. Ex 2: List 4 tapes or CD's you have. If you had to pick one without looking what is probability you'll pick your favorite? Do all situations have theoretical probabilities? Tossing Marshmallow? Coke Can? ________________________________________________________________________ Experimental probabilities: A probability based on collecting data from conducting an experiment. If the experiment is conducted a large number of times you are likely to get closure to a theoretical value. To compute an experimental probability a ratio is formed: Experimental probability: # of the desired outcome occurs/total # of trials or turns Many uses of probability in daily life, such as weather forecasts and sports predictions, are based on experimental probabilities. Name examples of other things to experiment: tossing a can, your average. time for running a mile, foul shot average etc. |
The students had some nice examples of where they saw experimental probabilities. One student even proposed that the number of times a radio station plays a certain song in 2 hours is an experimental probability. However, when I asked students about whether or not everything has a theoretical probability they were a little thrown off:
Teacher: Do you guys think that everything has a theoretical probability? Various voice-mixed yes and no Teacher: Take the weather man-do we get mad at him/her when they are wrong? Various voices: "they are always wrong" "yes". Teacher: Why-why are they not exactly right all the time? Brian: Because it's like a 30% chance of rain-so they are really giving us a number from all their equipment that they use-but it doesn't tell us for sure what can happen. Teacher: You bet Brian. That is exactly right. So we can say that it's almost like they are using equipment to get at a experimental probability or chance of rain. So what other things do not have a theoretical probability? Class is sort of stuck at this point. Therefore I decide to proceed with another example to think about. Teacher: What about tossing a marshmallow? What are the chances it will land on its flat sides? Class was taken by surprise with this example. Christian: What kind of question is that. Teacher: Why can't you answer it? Christian: Well, I mean either it lands on its side or its flat side so 50/50. Teacher: Wait a minute-does everyone agree with that the chances for a marshmallow to land on its flat side is 50%-like a coin!!!!! A marshmallow has a 50% chance of landing on its flat side? Ame: No, it will probably land on its flat side more. Kristen: It depends if you are talking about big or small marshmallows. Teacher: I agree the mechanics of a marshmallow make it very difficult to determine the chances of it landing on its flat side theoretically. Christian: okay so we'd have to just keep tossing it and record the results. Teacher: okay guys suppose we do what Christian is saying. Let's say we get a result. Then what do we do with that? Do we have anything to compare it to? Like suppose we found out that the marshmallow landed on its side 30% of the time. Now what? Brian: well, than according to our test that's what the answer is. Teacher: Is that a theoretical answer-how do we know if it's a good value. Like we know that flipping a coin as a 50/50 chance of heads or tails. What about the marshmallow. Kristen: I see what your saying-it doesn't have a value to compare it to. Like we can only go by the number we get based on our own experiment. Teacher: Exactly, and why is this so class? Ben: Because nothing is proven with a marshmallow coin they have proved already. Teacher: Is everyone with me? See how a marshmallow will not have a theoretical value. The weight and shape of the marshmallow makes it too difficult to determine a theoretical value. What about a coke can. Many Voices: "it will land on it's side allot more times". Teacher: How did you guys know that? Matt: Because its hard to throw a can and have it land straight up. Teacher: I agree, it almost seems impossible-do you know the percent of that occurring? Class in general says "no". Teacher: So what does that tell you about the coke can and theoretical probability? Many Voices: "There isn't going to be one", "you would need to test it to see what the chances are". Teacher: If you test it, will that be your experimental or theoretical? Class: "experimental" Brian: Ms. Maine if we spent years testing it-we could perhaps get close to a theoretical value. Teacher: you are absolutely right-and this just confirms our notion that we can use experiments to get theoretical values. But we do have to do a great many experiments and analysis-just like the scientists do. |
I already had the foresight that students would view objects that have unequally likely events as 50/50-thus referring to either it can happen or not. In other words, they avoid having to offer what the theoretical value is. This is because they simple do not know it-yet they fail to realize that "not knowing" it, is actually more than legitimate with obscure objects such as soda cans, marshmallows and thumbtacks. I chose a homework sheet that I think will be very valuable to their understanding of when things exhibit a theoretical probability vs. when they exhibit a experimental probability (Appendix H).
Reflecting back on this particular class, the students were pretty active in the discussion. However, I realized that perhaps I really did not need to have students use "my" notes as a means of summarizing and compiling what we have learned thus far. This will be explained further in Part IV of this section. Moreover, there are some changes I would ad before going into Part II of this unit. These changes will be discussed in Part IV as well.
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