 Okay, well welcome everybody. Thank you all very much for joining us this afternoon. Today we're going to talk about probability topics in the middle grades. And my name is Maria Hernandez. I teach at the North Carolina School of Science and Mathematics, and I am very lucky to have Alan Maloney here with us from the Friday Institute for Educational Innovation over at NC State University. Well, we'll just see how lucky she is. This is our seventh session, our webinar this year, and the final one, unless I hear otherwise. So this has been kind of a mixture of different types of mathematical topics and discussions about pedagogy in terms of the common core state standards. And we've actually talked about topics that range from middle school to high school, and this was a big request when I kind of put it out there to the curriculum specialists in the area. I asked, I kind of put it together a list of things that I thought might be useful for folks and then they gave me input too. And there was a resounding yes, we would love to know more about probability topics in middle grades. And since I am not an expert in that area, I went and found an expert for you all. So Alan is going to take it over from here and let you know what you're in store for for today, and then I'll jump in later on when I'm requested. If you could please make sure that your system is muted if you're just joining us so that you don't get the feedback. And also there's a chat window available for you to participate. So feel free to chat in that chat window. I have Carol Stern here from the distance education department over here at NCSSM. She's monitoring that chat window so that we can see when you have questions. Also when we ask you to play along, if you would please give us some response in the chat window, that would be great. Or if there's any difficulty, if you're having some difficulty, let us know. And I will turn it over to Alan. Thank you. Hi everybody and welcome. You will see that Maria is the pro with this webinar business. I've only done a couple of them so far. So if I start looking blankly at the screen because I don't know where I'm going, you'll understand why. We're going to talk from the standpoint of student, really student learning and how to encourage student learning and conceptual understanding in the topics of chance and probability. I will be working primarily from the turn-on CCMath site that we've been developing here at the Friday Institute. But here's our goals for the session. We may get through all of them or most of them, but in any case we will have the entire PowerPoint, including any material we do not, that we either go over quickly or do not get to available for you to use afterwards. I just thought it would be useful to recognize that this is pretty shaky ground for a lot of people. We will do a sampling of observations as we go, but you can't just go through this stuff randomly. Maria's smiling. I'm glad you appreciate it. With a little luck, we'll get some good outcomes on this. And we think, frankly, that the odds pretty good that you'll develop a complimentary knowledge to your previously radical knowledge of probability, but all in all, you just have to investigate this stuff. And with those tongue-in-cheek observations, we'll get into the real meat of the matter. One thing it's really important to notice is that if you've read the common core standards and you've dug into them enough to find where probability and chance are sort of tucked in there, you'll notice that probability and chance topics are really under sold. But they really are critical for development of student reasoning, and one of the reasons for this is that in sixth, seventh, and eighth grade, not very heavily prior to that, but in sixth, seventh, and eighth grade, there's very heavy treatment, much heavier than it used to be in middle grades of topics about data and distribution and statistics. You combine these with the probability and chance understandings. And by the time you get to high school, these topics get merged into a much deeper treatment of statistics and probabilistic reasoning than had been done previously in conventional high school standards and curricula. It gets into the topics of statistical inference, probability density functions, sampling distributions, all of which require both an understanding of data distribution and statistical reasoning and probabilistic reasoning. So we're going to go through some of the sort of the critical principles from the early foundational understandings of, from the standpoint of student learning research. I mentioned issues of pitfalls and thinking about probability and worked through some examples and some problem setups that seem to be give you high leverage in generating discussion and growth of student understanding in classrooms. There's really only four standards in the Common Core that treat probability. They're all in seventh grade. They've been confined to that and the Common Core standards really do not address issues of how students probabilistic understanding, understanding of chance and randomness of those topics developed in earlier grades. Instead it jumps immediately to calculating and analyzing probability from a quantitative standpoint and defining it numerically. And finally it sort of defines it, if you read this text, it's sort of in the subtext, defines it more or less in terms of fractions, although it does mention ratios. The point of view we take from the learning sciences, the research end of things here, is that students really need to deal with their informal understandings of chance and probability. And a big part of understanding, of developing their understanding of these topics is to build it up to a more formal, more systematic treatment and quantitative treatment of probability reasoning. But it doesn't start there, it starts with their informal understandings. And as I'll mention in a couple of places later on, we've added bridging standards to support instruction in these more informal earlier topics. So I've mentioned this a little bit already. We come in at, in my group, and a lot of this work has been started and generated by Jer Confrey, who is now with Amplify Learning. But we've been working for a number of years now on issues of learning trajectories. The simple definition of this is research-based descriptions of student learning. And in particular we pay attention to their conceptual development across time, and by time we mean in this case years. And that fits in well with the way the Common Core Standards are built, because they are built to support conceptual understanding over years as the mathematical understanding accumulates and deepens. So the three main underlying features for learning trajectories are that we emphasize big ideas that develop gradually over time, describes the transition that students bring in from their prior knowledge, what they bring into the classroom with them, to more sophisticated what we call target understandings or domain goal understandings, the top-level mathematical principles that are the takeaway messages for so much of mathematical learning in schools. What we're trying to do is identify the intermediate understandings, how they can contribute to conceptual growth, and how to recognize and build on these. Let me skip ahead just one bit here. Let me show you this map. This is the map of the K-8 standards that we've developed and is on the site of turnonccmap.net, what you'll see is 18 different sets of standards that are identified as the learning trajectories. I encourage you to go to the site and investigate it. If you click on any one of the hexagons you'll get a set of descriptors which are detailed descriptions of how the standards work embedded in these learning trajectories. And there are, I'm going back now in this in the PowerPoint here, we unpack the common core standards with a set of five general types of what we call descriptors. Cognitive principles, strategies, inscriptions, and misconceptions that students exhibit as they build their understandings of topics. We try to identify mathematical distinctions that help students understand differences between different types of different approaches to mathematics and multiple models of understanding mathematics. Wherever we can identify coherent pieces of structure that help students support their reasoning, we've done that. And then finally I mentioned earlier the bridging standards which are essentially additional standards that we've added into this. They're not part of the common core but we've added them as part of the learning trajectories to help understand how instruction will help places where there need to be additional pieces of instructional treatment embedded in these learning trajectories, something more than just the common core standards have. So this is an overview of our chance and probability learning trajectory. There are really two main sections of it. One is probability of simple events and probability of compound events. What we'll be doing today is only treating probability of simple events. The compound events add on levels of complexity but there's just too much to cover in a short period of time. These are the standards that are found in the common core standards document itself and we've broken them out by each of their individual substandards. So there's really number 7.sp.5 through 7.sp.8 and this is how they appear in the hexagon map that I showed you a moment ago. So several points as students move from informal to more formal understanding and I'm going to have several slides through here that are sort of takeaway messages. One is that again the learning trajectory is a way of understanding how students can progress from informal reasoning to more formal complex reasoning and in this case about chance events and how these chance events can be modeled. Emphasize that there has been some tension in over the last several decades about whether probability and statistics topics are actually mathematics or they're just sort of applied numeracy. Point out and emphasize and say over and over again probability is in fact part of mathematics. It's simply that it's not a deterministic part of mathematics. We understand from the research that early reasoning and probability really involves making distinctions about among other things among terms and concepts and helps students to identify where their understanding is informal or vernacular and where it needs to be more systematic and quantified. So we're going to here build on our framework to eventually end up comparing theoretical and empirical probabilities of what we call events. And eventually this builds up into as students progress through this set of understandings a way to define and model increasingly complex events, conduct investigations, determine and infer probabilities and eventually make evidence-based decisions in situations where there's a great deal of uncertainty and part of the part of the art of probability theory is to understand where you can identify places of lower uncertainty versus have to deal with issues of higher uncertainty. Oops sorry about that. Slipped. Here we go. I want to come back to this slide a little bit later so we don't have to pay so much attention but this is sort of a summary of examples of the descriptor elements that I described a little bit earlier that are in the probability warning trajectory. So let's start a conversation. I'm going to let's see. I want to just check. Maria are you still able to hear me and tell me something? Yeah I'm unmuted and I can hear you. Okay great. I just wanted to I assume that you'll be while I'm looking at the the power point here you'll be able to flag me if people are asking questions. Yes I can do that. Carol's walking. Great so so everybody in the in the audience out there in our discussion here please don't hesitate to interrupt with questions and so forth and we'll we'll try to give you responses as we go. So for the moment let's let's think about how you start a conversation. What is the language of probability the kind of informal language that we use in situations involving chance or uncertainty? What kind of things do you hear? It's just luck you know. It just lets you all read these along. It's pulled up a bunch of things the kind of thing that you might hear. Could hear it in a classroom, could hear here on tv for instance that one. And all of these exhibits various ways of expressing probabilistic reasoning some of it formal most of it informal but most of it based on some perception of what it is to have a chance event in in the mix. One of my favorites I'd rather be lucky than smart. So kids live in this world of probability probabilistic reasoning uncertainty it's usually informal a lot of it they get their experience from games. So the reasoning about these things all the time we believe from the standpoint of learning sciences and this is backed up by a considerable amount of very very good research over the past 15-20 years. The kids are very capable of learning to reason more formally about probability as well as about statistics well before seventh grade. The common core standards don't call for that but it does not mean that instruction should not include this as one goes forward. And even now some of these things may seem like they're old-fashioned or old hat but situations that are used as context for modeling probability are these I've listed here the simple things that we're familiar with every day and we've all had in a class one way or another coin flipping using spinners we'll talk about spinners a little bit later drawing items from a collection without being able to see the items so you really don't know what's in there and you're trying to figure out what is in there. More advanced formalized probability modeling typical kinds of examples that people start with are drawing cards from a deck of cards these are just basic things that have been done for literally hundreds of years. The weather if you think about how the weather is talked about it's above average a higher probability higher chance lower chance well are these theoretical probabilities is there some probability one can determine or is it something that's based on an empirical model all of us know it's based on an empirical model and I invite all of you to just think about other situations that you're familiar with that you may have used in your classes or just used in talking with your own kids and other things you're aware of. If you've read USA Today you see lots of very simple probabilistic statements chance statements and so forth so I'm going to if anybody wants to add in something else we've missed here that you think are really important please please chime in and pause for a moment here. These are terms that are used over and over again probability and reasoning about probability but for many of us whether we've got a lot of experience with this or not instructionally or professionally they're pretty squishy there are things that we think we understand we have pretty good feel for when you get down to actually quantifying them or formalizing them or systematizing them they may it may turn out that you are not as secure in your understanding I know that's been the case for me and it's really it all having a little bit of squishing this gives you the opportunity to I would say sympathize empathize with your students and also engage them in discussion of what these terms really really mean and how do we move to a more formal almost if you would controllable way of dealing with them but a more systematized way of dealing with these terms and I've listed here a number of very common pitfalls or misconceptions that students will exhibit will express about probability for instance that the probability of an event is simply one over the number of events and we'll get to talking about what an event actually is in a few moments another one is that the number of objects in a population determines the likelihood of an event when in fact it's the ratio of a certain one or more set of outcomes that comprise an event relative to the total number of possible outcomes in a population it doesn't have necessarily to do with the size of the or the number of objects in a population it may be the number of objects in the sample may determine a more robust or secure definition of probability in a given sample people typically confuse the odds of an event with the probability of an event whereas the odds are really a ratio of two different probabilities rather than a probability itself here are two good ones that are going opposite directions for independent trials on an experiment an outcome in one trial increases the likelihood of the same outcome in the next trial and vice versa and in the other direction you have an outcome that happens now you're not going to get that outcome again that's the reason when in fact if they're independent events actually it doesn't matter what's happened before the probability of the next event the next outcome is exactly the same and here's one that gets to the heart of people's understandings of randomness and predictability the sense that if something is random it's completely unpredictable and it's random if you can't predict the result with 100 certainty when in fact the study of probability is all about understanding the structure of events that have an element of chance associated with them and you probably can think of many others that you've seen in your own classes with your own students and we won't tell anybody if there are misconceptions that you've stumbled upon that you find that you've had as well because i think all of us have this have various kinds of misconceptions where we crossing the boundary between what chance and probability feel like in your daily experience versus the system that ties to formal mathematical treatment of chance events so out to all of you who are participating this list of five items here are these chance events and if so why are they chance events if your role is excited die is that a chance event if you close the door is that a chance event assuming you close the door with enough force to close it i think we'd all recognize that if you deal a card face up from a deck of 52 different cards the whatever card comes up is itself a chance event because you don't know which card is going to come up there 52 different possible outcomes i like the next two in particular the event of snow occurring in Fayetteville in February that feels like a chance event because you never really know whether it's going to snow or not however the event of snow occurring in Fayetteville in June that doesn't feel like such as chance event chances are pretty darn low that that's going to happen the chance events are really there are events for which there are one or more possible there are multiple possible outcomes that you can't know for certain are going to happen the theme of chance and randomness versus predictability is something that keeps recurring in probability probability theory probability studies is a really good idea in classes to revisit this as you go through as you and as your students learn more and more about probability as well as statistics because it one's understanding of these topics changes the more you learn about probability it's always good to get this sort of metal level discussion going repeatedly in a class and here's where also the kind of the situation where we begin to distinguish events from outcomes event is something that is happening but outcomes are one or more possible ways that event could happen to get the same event and that these are critical terms and probability reasoning so here's a problem for your to think about so i hand you a bag of marbles and i tell you there are two colors of marbles inside there's black marbles and there's white marbles so here's an event you draw a black marble now that's not saying you are drawing a black marble that's the event of what's the probability of drawing a black marble out of that bag here's another event the event of drawing a white marble out of the bag are these two events equally likely ponder that for a moment and i'm also giving you these all of these scenarios are things that you could do in your class and we should generate quite a bit of discussion among the students and maybe even debate and that is that's where you want to be because that helps the students figure it out consider it from different standpoints and wrestle with their informal understandings so here's another situation so this time i'm going to give you a bag of marbles i'm going to tell you there are two colors inside there's black and white but now i'm going to tell you the bag has 30 marbles 20 of them are white and 10 of them are black so what's the probability of drawing a marble from that bag that's the definition of event number one and what's the probability of drawing a white marble from that bag okay now it's participant participation time somebody tell us how you think about each of those events so here's where we encourage you to type in the chat window there are lots of folks out there so if you guys could help us play along with us here i'm going to pop out and take a look at the probability of drawing a black a black marble is that what we're looking at now you're going back to this uh go back to the previous one yeah let's go back to the well when i pop out it gives me the whole screen okay we're going back in here go one more alan so we can see the two questions or the two please there we go sorry thanks the probability of drawing a marble and then the probability of drawing a white marble nobody's going to participate we're getting a quiet crowd here i'll start calling on people that's the way i am that's right that's right we're not going any further until somebody talks that's right somebody's gotta help us Marie alan can you hear me this is Teresa yeah Teresa my chat window has disappeared when you started it disappeared that's why i can't enter anything oh can you look at the top there's a green line that says viewing alan meloni's desktop and you can pull it down and see the click on the chat thank you very much you're welcome okay from uh you see yep yeah we've got some responses from Barry Barry said 100 for the first question and 33.3 percent for the second one and then we also have a response from Teresa she says one you will always draw marble so probability is one alan did you hear those responses yep i did i did and then also from Barry it says should be 66.7 for the second yep and this is one of those situations where unlike dice when you have marbles in a bag the probability of a simple event of drawing simply one marble out of the whole bag depends on the ratio of the marbles in the bag but the probability is the number of ways you could draw the black marble compared to the number of ways you draw any marble and the answer to this question of course is no they're not equally likely and this also gives you a sense of in standard seven sp5 they talk about the probability of an event being between zero and one and the one or the 100 percent is the baseline for something that happens all the time that is completely certain as in drawing a marble from a bag of marbles and okay scenario number three bag of marbles black and white 30 marbles 10 or 10 black 20 white now the probability of black versus white are the two events equally likely you all out there know that they're not equally likely and you undoubtedly are going okay we're going to go through this again huh and the students have to wrestle with the question of is the probability of drawing one the same as the probability of drawing the other and why getting to the issue of the difference in the number of each so the chance of drawing one at random not knowing what's there and just grabbing one that you can't see depends on from the theoretical standpoint at least depends on the relative numbers of one color versus the other color in the bag i would say that let's go to the last issue so here's another thing that students can sometimes answer and that the probability of an event or the probability of two different events is going to be the same it's going to be 50 50 or it's going to be equally likely but that's not without knowing i mean that's without knowing what the event is but sometimes students will simply say if you're going to um draw one marble from a bag regardless of the color they will simply say because there are two kinds of marbles the probability is one half that you're going to get one or they're equally likely to draw one as another and that's an intermediate understanding it can generate give you the opportunity to generate a lot of discussion the kids in the class will probably debate that among themselves given an opportunity so this this set of questions really gets at the idea that well we have you have to you have to model these situations probability event depends on other issues that you know and how would you know what the probability is you help you have the opportunity to find what the things you need to know are in order to define the probability of any particular event and again a lot of these situations are just really our ways of signaling to you places where students this is before you even get to the to any kind of rigorous or systematic treatment of defining the numerical values of probabilities but varieties of contexts and situations that are slightly or they're at least slightly different that give students a chance to reason about what you need to know in order to determine what the likelihood or the probability of something is and one of the things you need to know is for any given event how many outcomes comprise that event for instance if you're going to draw two marbles out of one of those bags of marbles then how many ways could you get you can get a white marble and a white marble you get a white marble and a black marble you get two black marbles different outcomes will have different likelihoods and each of those outcomes feeds into the event if you if you are drawing if you want to know the probability of drawing two marbles both of which are black that's a different event than drawing two marbles one of which is black and one of which is white how many possible outcomes are there overall versus how many outcomes are there that could be that could qualify as satisfying the particular event one thing the common core standard sort of takes for granted is the issue of sample space the sample space has to be determined very carefully by students who are determined probability because it is the number of outcomes in the sample space that is in fact the denominator or the base ratio in or the base value in a ratio that determines the probability so let's talk about let's switch context now and talk about a roll of a die so we've got a six-sided and in every case we'll say we'll assume that it's a fair die unless we say otherwise so the question is the situation is that you're making a single roll of a six-sided die the event is rolling a two what's the probability of rolling a two what are the possible outcomes well you could roll a one or it could roll a two all the way up to six so there's six possible outcomes the outcomes for the event that we stated there's only one outcome you can get a two if you don't get a two you don't get a two or you get a two so the probability of rolling a two is one outcome that is rolling a two out of possible of six outcomes for the actual um the roll so the probability of getting a two is one over six now it's expressed as a fraction but it's really a ratio of one possible outcome relative to all of the possible outcomes so here's an interesting situation we're going to add a die so now we have two six-sided dice um we want to ask the question what's what is it we're going to roll a three that's the event what's the probability of rolling that three and how do you think about this critical thing about this now we're starting to think about um quantifying probabilities we've made this move to understanding what outcomes are what outcome spaces are sample spaces what's the probability of the event of rolling a three well we have to figure out what the what the outcome space is you could roll a two you can only roll that by getting a one on one die and a one on the other die you roll a three by getting a one on one die and a two on another die or the two on one die and a one on the other die and so on and so forth so i've listed the number of possible outcomes for rolling each particular value of the total of two dice there's total of 36 possible outcomes as long as you're distinguishing between one die and the other die if you roll a double five you don't know which one is one and which one is the other so there's really only one way to roll a double five now there are two ways to roll three the 36 ways to roll two dice so the probability of rolling three is two out of 36 or one to 18 so we're going to go back to the same question or same situation but now we're going to pose a different different question to the to the students two ways of counting the outcomes you can distinguish between the different valued single dice or not and this question comes up over and over again came up for us when we were writing the learning trajectory itself do you count when you're if you're rolling eight do you count one and seven and seven and one is two different outcomes or only one after all you can't really distinguish between the dice if you're rolling at the same time right come up with a one and a seven what difference would it make here's a question for you all the difference would it make to the theoretical probabilities of rolling any particular value and here's where i'm going to call on Maria because i'll leave this i'll leave the desktop up so i won't be able to see your see the interface for the for the session so what do you think your students would say to this question so alan do we really want a one and a seven is that what you're talking about when you're talking about the diet that we just have a maximum of oh no no i'm sorry i i mean just in general what difference this is sort of a thought question at this point but it can be calculated what difference would would any particular what would it make what difference would it make to any particular event if you counted the dice uh the situation of of two different dice as two different possible outcomes or one does that make sense yeah okay so another way of asking this question is so we've posed this question how would you go about determining whether there is a difference to count those two outcomes as two different outcomes or only one and similarly for any other situation in which you could have two different two different combinations if you guys have got guys have comments or responses to that let us know just chat in the chat window not a comment uh from um in farmer it would make a different number of outcomes if the dice were different colors ah good point so that would be a way of actually defining the dice as the two dice as different dice so would that be the situation in this case would that be the situation where you would count two different outcomes for one and seven or two different outcomes for two and three or any two different numerical values on the dice or is that only one it may seem a sort of trivial question but it's important to make that this it's these are one of the kinds of distinctions that's really important for learners to grapple with and recognize why they're saying why they're making these justifications so um when you're chatting in the window if you select send to everyone and everyone will see your chat items and um and farmer added uh you would then have 36 outcomes instead of 18 and I haven't counted the other way in in a long time is it actually 18 somebody do a quick count so if you think about it in terms of different dice you have your six doubles right right double five yeah that's what somebody's just counted Teresa counted that for an outcome of seven we distinguish between three and four being rolled versus a two and a five oh that's a good question so a so for seven a total of seven Teresa asked if we're distinguishing between three four two five yes yes we are distinguishing between that two and a three not distinguishing between three four versus four three right that's right so 18 is right Alan because you have the six double ones and then you have six um times two which is 12 is that right seems like there might be fewer we can't count we're having trouble counting this one that's right it's it's one of the challenges of probability right is actually counting the current outcomes right of all that everyone agrees it's it's two different values we would settle on one value we think it's 18 so you can actually determine what difference it makes in the probability so and we have another question here it says but what if two dice are thrown not at the same time ah good point so what if class discussion that was actually the way i thought about it at first i thought well if we throw the two dice at the same time and we're always distinguishing between the two because we can see the two dice as opposed to throwing one then throwing the other one and not and not caring about whether it's three four four three and then we have another comment couldn't we think of the two dice just as we would a die and a spinner that's interesting so you have a spinner with six numbers on it is that what you're saying yeah so would that be would that be an equivalent model because really you have you're basically asking for two independent events that comprise a single event that one of which is one you get one value out of six possible values and the other one you get one value out of six possible values right so then cherry says wouldn't that give us back 36 yes yeah that would that's right i like that example because it does kind of help distinguish between the two it's not like i got to figure out if i'm counting you know if i have to remember what happened on one die next next time i throw the second die okay go back to the counting of the possibilities if you don't distinguish between the dice i'm coming up with 21 because there's one way to get two there's only one way to get three two ways to get four which would be one and three and two and two two ways to get five three ways to get six three ways to get seven three ways to get eight two for nine two for ten and one each for eleven and twelve what about four ways to get five well let's see one and four and two and three but four and one and three and two are the same right because we're not distinguishing between the dice in that case oh she changed it to three yeah she changed it to three right she's got it so i'll leave this for a question for your classes but you can design a way to answer what is the correct way to determine the probability of rolling any value two through twelve with two six sided dice that are identical and what the students will discover is that the probabilities that empirical probabilities if they do enough repetitions enough trials of this will end up showing will end up converging toward the probabilities that you get if you count two different values if you distinguish between the dice excuse me i only saw a part of what sherry just put up there what was the whole remark questions will definitely come up in the classroom it's good to have an idea of where you're heading with your class right right and and one of the values of learning trajectories the way that we have these outlined in the in the turn on cc math site and what learning trajectories try to do in particular is as you understand what the research says about how students and this is research in classrooms as well as outside of classrooms if enough work gets done you will be able to if enough work on the learning trajectories if they're detailed enough teachers will get a much better idea of what to anticipate is likely to come up in the classrooms ahead of time now students will come up with something new all the time but the range of possibilities this is itself a sort of a probabilistic question sorry the range of probabilities a range of possible responses is not infinite it's relatively predictable and the more you know about how the students tend to build their understandings the more you can anticipate and be prepared for what they're likely to say and that's not to simply necessarily give them a an answer to correct them that's not what i'm talking about but you can anticipate how to build the discussion and then what we call the classroom discourse among the kids in ways that they can build their own understanding of it and eventually come to you know wherever possible come to consensus that is actually matching a next level up of conceptual understanding that they can also build by making arguments and bringing evidence to the table about why these answers are correct or why they're valid why they work in a given situation that's one of the really powerful things about learning trajectories so um let's move to the a little more discussion of the issue of modeling because this is this follows from this question let's see if we go back here this question of designing this a way to answer the fault this question it really involves making a probability investigation and for most intense and purposes these are the steps that most people will use in doing basic probability investigations and this is one of these coherent structural components coherent structure that i referred to earlier that surfaces in in the probability learn for a chance in probability learning trajectory uh how do students engage in probability investigations and there's a routine the details may be different in every different situation but there's a way to design and follow these and design these investigations and carry them out you pose a question refine the question so it seems like it's going to be able to be answered define what events you're actually asking about then define the outcomes and the outcome space for that event and in doing so you establish the theoretical probabilities so if you have a spinner for example i'll go back to the the dice example you know the outcome space is there's six of six outcomes for rolling a single die you can ask the question well what's the probability of rolling an odd number so there's three possible outcomes for rolling an odd number you can roll a one or a three or a five versus all of the outcomes you could imagine which is six so the probability of rolling an odd number is one six right oh no sorry three six it's a 50 chance then and here's where the stuff really gets interesting is you generate data and then you answer the you try to answer the question using the experimental probabilities you've made some certain assumptions in step four about about the the nature of the event itself and the nature of the tool you plan to use to generate the empirical probabilities and a lot of the issues in probability reasoning are comparing what you get from an experiment relative to what you believe you would get if everything was perfect then you were able to run well give away the secret here infinite numbers of trials and the final art of this and where you can generate all kinds of discussion in the classrooms is interpreting these results you get the arguments get going fast and furious but it allows kids to surface what they understand and what assumptions are making that are so inherently important to the whole process of modeling a situation a situation Maria your turn calculator simulation you ready oh can't hear you if you can pass me the ball then we'll all right sorry about that calculator no problem let's see now let's see i have to figure out how to get back here now let's see i have to get my interface back pull that tab down maybe and then you can see participants you see the green tab at the top of your screen there we go got it and pass it me wrong i'll end up passing the ball to somebody else i'm sure let me open my window a little bit also while you're working on that too we have a couple of comments and questions it says could you identify the difference and or similarities in the terms outcome sample space and in the end they come up can you read that they come up in the study of probability i mean he's selecting a letter from the word mathematics okay say the question again now that i understand the context yep could you identify the differences and or similarities in the terms outcomes sample space and events that come up in the study of probability right right for example selecting a letter from the word mathematics okay so if you have it partly is going to depend on how you imagine it but if you imagine all those letters as individual objects and you've got them randomly assorted and you just grab one the event would be what your result would be you define a particular result you want to get and then look at the probability of that but that that event that results may maybe um there may be multiple ways of getting that result and those each of those different ways of getting that result is an outcome for that event the entire sample space or outcome space is all the possible ways of doing the action you're going to do whether or not you get the particular result you're looking for does that make sense and rei i have the ball back to you looks like you're ready to share your desktop right okay now i'm unmuted and i think i'm ready to share my desktop um so what i'd like to do is just do an example um because i thought it might be kind of nice to show you some different kinds of technologies um but i think like alan has talked about i think it's important for kids to have a chance to conduct some of these simulations with some type of hands-on activity at first so of course we can't do that with a webinar but anything that you can do to bring into the classroom to simulate some of these particular ideas is i think is very valuable for the kids to kind of get their hands on it also it's interesting when you think about designing some type of activity and ask the kids to even help you design an activity that would help you simulate for example if you don't want to bring coins in the class and have people flipping coins all over the place you can get kind of crazy but if you ask kids to think about could there be something easier that we could do here in the classroom to simulate it they might come up with an idea like we could have paper bags with different colors of pieces of paper in there and if we thought about having the same number of green pieces of paper as white pieces of paper then that might be an easy way to simulate something like that it was just an idea i had because i know it can be kind of crazy when you think about hands-on activity and so what i'd like to do then is think about after you've given the kids an opportunity to touch something with their hands to move to the technology as a second step and i wanted to show you a couple of different ways to do a sample problem and then we're going to pool our data and collect the data as a whole so what i'm going to do is if you have a ti-84 um Anya or an 83 we can run a simulation where what we're going to do is we're going to toss a coin we're going to pretend like we're tossing a coin 50 times and we're trying to think about how many heads we might expect if we toss a coin 50 times so i'm going to escape out of here so i can show you i have this in the powerpoint so that when you get back to your classroom if you want to um you know if you don't remember all the all the um button pushes the keystrokes you'll have that but i'm going to escape out of here the simulator uh or the emulator and i'm going to show you how we can do this let me get out of this actually was playing with something earlier and i'm going to show you this other app too but if you have um an 83 or an 84 sorry i'm just going to get out of there then what we can do and this is actually kind of i was talking to a person who's teaching statistics here and that's one of the things that they actually ask kids to do in ap statistics is to model some kind of simulation with either some technology or some other way so if you think about um tossing a coin one of the ways we can simulate that is to use a random number generator let me turn off the set plot let me turn off the set plot sorry i was just playing with the calculator and what we can do is use a random number generator to generate an integer between quote unquote between zero and one inclusive so that means that we would be generating numbers sorry um we would be generating either zeros and one so to do that i've got the calculator stuff in there for you but what we're going to do is we're going to hit the math button so if you're playing along with me if you go to math and you arrow over to probability then there are all these different things under probability here ran generates a random number and you can give it a range but what i want is number five which is ran it and the way this works is is expecting oops sorry i picked the wrong one math probability you can either hit the number five or arrow down to five this emulator is pretty slow so i'm just going to hit the number five and it's expecting uh it could be expecting just two parameters normally what you could do is say the number from zero and then you put a comma and one so that means it'll generate a random integer between zero and one inclusive and then if you wanted to do it several times you could put another parameter in there but i'm not going to do that i'm just going to show you how it works so i just hit enter and enter and enter and if i think about it if i say well i'm going to let one represent tossing coin and getting a heads and i can count up to do this 50 time and count up how many heads i get so that's certainly one way to do it that could be arduous and time consuming but the good news is you could put a third parameter in this command that says i want you to do this 50 times and then we can actually collect the data and put it in one of our list so that's what i'd like to do next what i'd like to do is go to math arrow over to probability and then choose random and we're going to say from zero comma one comma and we're going to generate 50 of these numbers but if i just hit if i hit enter now then i'll get these numbers kind of spewed across the screen which i can do and what i'd like to do is store them into a lift so if you hit your store button which is right on the keypad here in the bottom left hand corner can y'all see my mount yet so if i hit store and then i hit second l1 and hit enter then it will take all of the data and put it in my my list so if i hit stat edit then in that first list these are just ones and zeros and and because i've decided to model this in this way where one represents the head and the zero represents the tail now it's really easy for me to count up the number of heads because i can just sum the elements in the in that list and again i have those keystrokes for you and you'll have the archive webinar if i'm going to fast i'm going to get them out of here and i'm going to go to math no sorry i'm going to go to uh list so let's go to the list operation so hit second stat which is list and go over to math different math and there's a sum command which is number five so we'll do some the elements in l1 and then hit enter so you can see that for these 50 trials i i tossed the coin uh 50 times i got 21 hits so this is kind of an easy way to get your kids if they have 83s or 84s in their hand to to generate these numbers i'd like to just take a quick poll can you just do a right raise your hand if you have access or your kids have access to this type of technology in the classroom because i know for middle grades not everybody has this maybe if they're algebra or yeah algebra one teachers or actually see how they raise your hand um oh but you could raise your hand okay um Dean says yes Teresa says yes that's right so we have a couple yeses in there because i don't want to be like talking about something that no you get back to your classroom you don't have that technology but you can do this you know have kids do this several times even one kid could do 50 of these or maybe five trials and then what's nice about this is you can pull the data all together and um going back to the ideas of some of the topics in sixth grade when you think about creating i think of it as a histogram but you could just think of it as a dot dot spot if i go back to the power point then we could collect the data together and what we're doing is along the horizontal axis i would put the number of heads that had occurred so i could actually sample the room i could say well who got you know 19 heads or 10 heads and they did this when they did this 50 times and so we can kind of get a range across the horizontal axis maybe somebody got as few as 10 heads and somebody said they had the uh 73 so i don't know if you could do this on the 73 if you want to do it it was lots of fun oh great that's great i'm not familiar with the 73 so then you could put up here you could actually just do a dot plot where everybody would come up and on the board they could put a little dot on the board um to kind of represent the histogram so this what this i think this activity does is gives the kids an opportunity to kind of see what some of the things alan was talking about was that you know this idea of experimental probability is not going to be exactly the theoretical probability in fact when i did this and i collected the data i got very few 25 and that would be something that we would kind of work towards in terms of an expected value and a comment from jenny you can download the probability simulator oh great right so i have that here too and i know we're kind of running that time and we wanted to show you some other technology so let me just show you real quick that if you have there's an app on the 84 that's the probability simulator and if you hit alpha p you'll get you down to the p and i um if you go to this probability simulator it's pretty cool like someone just mentioned you can toss a coin and you can actually roll a die and make the um you can actually make it like a a weighted die an unfair die and if i go over here to set you could say well i want to set the trials to 50 and you could go down here and change this to clear table that way we won't be accumulating these we're just going to clear them out every time and again i have this in the powerpoint that's the keystroke but if you do this and you can say okay these top keys are what what you're looking at now if i say toss then it has this nice little coin toss there and it's showing you each toss and you can see it's building a histogram for the number of heads and tails and then if you use the arrow key not the trace key but the arrow key you can count how many heads you got there and then you've got 28 heads that's kind of a nice quick way to do the simulation free on the um 84 if you have an A4 and it is so critical that the students work with different kinds of representative representations just as it is with almost any other kind of mathematics um any so the faith because they reason you'll you'll reason differently depending on the representation it'll again each representation is a model in itself and it shows certain things and it doesn't show other things working across multiple representations helps students understand that aspect of modeling it also helps students reason about different aspects of the probability at the probability investigation so i know we're running out of time and some people might have to leave us but i just want to make what we can go ahead and keep talking a little bit but what i want to do is let everybody know that we will have all these materials available on the website so in case you do have to sign off but there are a couple of other things i just wanted to show you actually just one other um technology that i wanted to show you and then alan i don't think we're going to get to the spinner activities i think that will probably have to wrap up but one of the other things that i just wanted to show folks just for you to kind of explore on your own of these nctm core math tools and you can download these and pull the whole suite down onto your machine i've talked about them before at other webinars really nice tools one is a simulation tool that has a one where you can toss coins and then like like the activity that i just ran it showed you there how many heads and tails you got for a particular experiment so you know play with those and then these were just some of the um this was what i showed you just now on the probability simulator activity so um in terms should we just cut to the very end there alan since we're ready yep okay so alan's got all these great simulator i mean these spinners um investigations that are available on the shoder website if y'all haven't seen those and i think we just want to talk about this last kind this last slide so i'll let you talk about that sure and this is what i said we're going to come back to a little bit later we flashed this slide up before and just i just put it here to remind you that what we've tried to do is give you a little walk through how to think about the learning trajectory as a way of progressing from informal reasoning generating lots of discussion in the classroom by lots of different activities in order to generate and consensus as well as instructional settings where you are helping them become more systematic and to build ways of conducting more complex reasoning about chance events and about increasingly complex events as well um the reason is the reasoning becomes inherently more formal and mathematical the early reasoning is really important to establish the ability and the willingness of students to make distinctions and to question these distinctions if they can't make sense of them so that they make more sense out of these increasingly complex situations and we of course we haven't even touched compound events and they become they can be very very tricky so the the challenge is to systematize and quantify our reasoning about chance events while always keeping in mind that we are modeling situations that are in either in the world or invented to help us make sense of these events that are to one degree or another random but certainly have multiple outcomes eventually we can make uh the students can understand how increasingly they can use this type of reasoning to determine probabilities infer them from events that they see in the world and make evidence-based decisions based on situations that have uncertainty as an inherent part of them and let's see i was going to mention oh yeah oh thanks marim um this is a recap of a couple of the five different descriptor elements that you'll see uh that we use to organize the the descriptors in the turn on ccmath.net site um the an example of student misconceptions is that random means unpredictable or without any structure that's one thing that comes out and repeatedly shows up in student reasoning um among the underlying cognitive principles that help us understand that are continuing underpinning cognitive themes of probability is that probabilities are ratios of outcomes versus outcomes in the entire outcome space and that probability is a modeling and a comparison between theoretical situation and empirical situations to help us understand how far we have to go to be able to be as i say in some of the slides that we haven't talked about satisfied that what we're finding in the experiment in fact does or does not in fact match the theoretical probability and that helps you understand uh answer questions like is a die fair or is a spinner fair and how would you know um distinguishing events from outcomes is one of those one of the sort of basic distinct mathematical distinctions we make in the probability field uh the structure uh example of the coherent structure that runs as a theme through this is the structure of investigations which i showed in the slide earlier and then you'll see as if you get into turn on cc math the bridging standards it's set up the first two standards are actually bridging standards that are set up to help guide you through the more informal parts uh and the vernacular parts of students understandings of probability before you even get to the first official standard in the common core um seven sp five that talks about quantifying probabilities so there we are and i think now we've got yeah we've got a bunch of a bunch of resources in there we've talked about and uh all of these are available for free without any costs that uh and uh are accessible to you for use in classes and allen's folks have put together a a great list um that's in a also in a pdf document that will make available on our website these are a couple of things that we have used today and then this live binder site it kind of changes depending on what you go look for but i found a couple of of nice um activities for these two um links and then the illustrative math project also has some good stuff for k through a in this area also in terms of professional development in the summer these are just some things that i know are coming up the nctm summer institute there's a summer institute in new warlands i believe for middle grade there's one at the end of the summer in washington dc for high school meredith college has some math and science institutes that are running this summer so i just thought i'd put these up there and you can do a google search on them the melt workshops which melts an acronym for something at Appalachian state that has to do with some i think they did some really successful workshops last summer for common core state standards and then the philips exeter academy high school workshop happens at the end of june and i'll be presenter at that and i'll also be a presenter of the nctm summer institute for high school folks so it thank you all very much for coming thank you alan so much for being here with us and i just want to let you guys know that carol and i are very interested in your feedback so she will send you a link to a survey where you'll have an opportunity to talk to us about this particular webinar but all of the webinars if you've been to other webinars and you want to just give us some feedback we're trying to think about future webinars for next year you know what people need will we be able to have the resources to provide someone to do these throughout the school year and if there are particular topics that you're interested in as well feel free to give us that input alan did you want to say anything else no that's got it covered we'll probably just tweak the powerpoint a little bit to make any little corrections that need to be made and then we'll post it and y'all can have at it okay thank you all very much thanks for those of you who participated and asked questions and gave us answers have a wonderful afternoon thank you everybody