Well, we're looking at the first sample problem set solution. And in this one, the student's answer is wrong because the statement is false. The problem is that this answer clearly assumes that zero is a natural number. And by definition the natural numbers are one, two, three on, that's historically correct as well so this is actually a false assumption. So I'm not going to give full max logical correctness. On the other hand, you will find some mathematicians who do include zero in the natural numbers so I'm not going to be too strong in the penalization here. But we're a community and mathematics takes place in the community when we're writing proofs. We do so, using the norms and practices of that community. We design our proofs to match the audience that they're intended for. So, writing a proof has to take account of which community you're in and who you're writing for. When I write for other mathematicians I do them very differently from the kind of proofs I write for students. When I write proofs for graduate students, I do it different. We have to adjust the kind of arguments we're giving depending on the community we're in and the people we're writing for. So this is a community aspect to write its true, it's not a matter of right and wrong always it's a matter of which one we communicate to. So what i'm going to do is I'm going to put two for logical correctness. I want to give some credit for saying well if you do assume 0's a natural number then this is the solution. But in fact we're in a community and we brought some accepted practices now. We've established a convention that 0 is not an natural number and in fact that is the standard convention within mathematics. So I think I've been pretty generous here in giving two, but I think that's fair. It's clear, it's absolutely clear. I mean, there really isn't an opening in this case, it's just, you're just doing it, it's so short and simple, so I'll have to give four for that. Conclusion is stated, reasons are given, in so far as there are reasons to give. Overall, I think I would have to give a zero. Because this is, from the point of view of what we're doing in this class, and the audience we're trying to communicate to when we write proofs, this is just not good. It really doesn't answer the question within the framework that we're working. Okay so that means I'm going to give a total of 18, I think that's a pretty good result. In fact I think that's generous given the students made a sort of fundamental mistake within the context of this course. Let's move on to question two. For number two, the student gives the right answer, but the proof is completely wrong, there is a fundamental misunderstanding here. The student has assumed that this is an existential quantifier, that there are five consecutive integers whose sum is divisible by 5. In fact this is a universal quantifier. What it means is that take any five consecutive integers then their sum is divisible by 5. So this is just not good, student doesn't really understand universal quantification, so we're going to have to give zero for everything here. This is just a basic misunderstanding, there's nothing we can give any credit for, we have to give a zero. When you really misunderstand something fundamental like what it means to have a universal quantifier as opposed to an existential quantifier, we can't possibly give partial credit or anything like that. This is just plain wrong, the student has no understanding of what we're trying to do. Pity, but there we are. Okay, number three now. Well first of all, the student doesn't actually say whether it's true or false, not strictly speaking. On the other hand, the student does start out and say I'm going to prove it, so it's obvious that this person believes it's true, which it is. So I'm not going to deduct any marks for not saying whether it's true or false, that's just a slip. It's got nothing to do with mathematical thinking I would say, they just missed out saying it. So there's something missing here but we're not going to worry about it, going to prove it by induction. Okay let's see, For n=1, true okay, suppose this is the induction hypothesis following this through this is all okay. Beginning to look a bit suspicious this one because this argument doesn't actually use the induction hypothesis. Which means it's not an induction, it's not an induction for everyone. Let's see, but one of n + 1, this is good, this is good, we'll come back to talk about that in a minute. Now that's good, so that's even. Well, certainly this is to prove the results, but it's not by induction because there was no use of the induction, that was never used. And you know there's another problem here, this actually says for any integer that includes the negative integers, induction only works with the positive integers. You can actually extend induction to work with the positive and negative integers, we haven't done it yet in this course but you can do that. But if you do that, then you're going to have to sort of take account for the fact that it's a little bit different for the positive and the negative integers. So as we've said, things are induction, and this is really standard. Induction is usually taken to deduce to mean the positive integers and the natural numbers. But the fact that you can use it for positive and negative is an extensions of it that's not what we're talking about. So there's a problem here, the induction as we've got it standard induction, standard mathematical induction doesn't prove things for all integers. So this is all [INAUDIBLE] it hasn't proved it, it's only proved it for the positive integers and more over it hasn't done it by using induction. So there's certainly problems with this. This is actually going to be tricky to grade this one because there is I'll come back to this one there's something good going on here okay. Logical correctness, I'm going to give one because there is a logical flow to this, it's not an induction proof except in some strictly formal vacuous sense but I'll give one. Okay, clarity, I'll give it two, in and of itself, it's clear but as a demonstration of of the truth of n, it's not clear. This is just out of clarity to knowing why this thing is true. Opening, again I'm going to give two because it sort of begins fine it's just that it's not relevant. It's not totally irrelevant, this is not completely off the wall but it's not really addressing the issue, so I'm going to give two for that one. State of conclusion, that was well done, I'll be 4 on that one, the conclusion was stated. Reasons, boy this is not an easy one. I'm going to give two for that, sort of, I'm on the fence here, reasons were given but they just weren't always the right reasons. On the other hand, this guy really, this is remarkable because if you up to here, the real key to proving this is to take that guy. And factor it as n times (n + 1) + 1. And then observe that one of these is even, and the other one is odd. Hence their product is even. So when you add 1, it's odd. So the proof of the result really just amounts to factoring this as n times (n+1) +1 and observing that that product is even, and then when you add 1, it's odd. That is the proof. Now the person has spotted that here. This is the key idea. And that was why I was generous here. On all of these I erred on the side of generosity, and I did it because the student had had the key idea. This was it, this was the key to the result. And I wanted to give credit for that. But everything else was just off the wall, almost off the wall, okay. Okay, overall, I'm going to have to give 0 because this simply doesn't get at the issue. The student has sort of seen it, but wasn't aware that he or she had seen it. So I can't give more than that. Okay, so I've got a total of what? 4, 6, all right, let's say, 11. Okay, just somewhat less than half marks. Okay, I'd alter it, but I think that's about right. I mean, I couldn't even get 50% for this one. I think that's as good as I could give, because I'm really just giving credit for this one particular observation. Incidentally, with these sample solutions, what I've done is I've compiled them from many solutions I've seen over the years giving this course. And I've sort of simplified them so they fit neatly onto a single slide. And that means we can use them as examples to look at. But if this had been an actual student paper, if these questions were all answered by the same student, then at this point I'd actually get rather suspicious because questions one and two were answered very badly, I mean really badly. And then suddenly a student comes up with a key idea. At this point I'd begin to suspect that the student had actually copied something from another student, either copied it wrongly or copied something that been done done wrongly by the other student. Because making this observation just seems so strange given what we've seen in questions one and two. And as I said, this is not an exam solution from a single student. It's a compilation to illustrate points of grading and evaluation, but in reality if I was grading a paper, I'd be suspicious. I would still give the grades the same way. You grade what's in front of you, but your suspicions are often aroused. And sometimes of course with a real course where there's actual credit being given for a certification and for a degree, say college credit, then you often end up calling the student into the office and going through it and double checking if the student really knows what they're doing. Okay, we've given 11 for that, I think the student shouldn't have any grounds to complain. They've actually got as good a grade as they're likely to get for this one. Okay number 4, we're proving by induction, what's going on? It looks as though the student is trying to prove some sort of a converse, namely that every number of one of these forms is odd, namely that 4n + 1 is odd, and 4n + 3 is odd. Well that's trivial. 4n is even, so you add 1, it's odd. You add 3 it's odd. And yet that's what the student seems to be proving, in which case they've missed the fact that it's obvious. So, what am I going to give for this? All I can give is 0 for this. It's just off the wall. There's nothing I can really give credit for. The student is trying to do something that's not asked for. What you're trying to show is that if you take a number which is odd, then it has to be of one of these two forms. You're not showing that any number of one of these forms is odd. So the student is not even answering the question. They're answering the complete converse of the question which is, in fact, trivial. So, no way I can give any credit for that one, okay. Let's just check and see if this is a valid argument. If it was induction, let's just see. That's okay if it's true for n. Well, okay, but since it's got nothing to do with what's been asked for, I'm not going to even look for giving partial credit for that. In a different context I may well have looked for partial credit if this had something to do with what was being asked for, but it's not. This is just completely irrelevant to what's been asked for. And as I've indicated when we set up the rubric, you can give partial credit so long as it's relevant to the question. But a valid argument that doesn't address the question at all just can't receive partial credit, and that's what's going on here. Where number 5, I have no idea what the student's trying to do here. It looks as though he or she has used that technique you often developed at high school of sort of trying to match to a template, looked at the question, looked through his or her notes to see something where this kind of thing is dealt with, and then throwing down expressions that seemed relevant at that time. In other words, using the template method, trying to match the question to a template, and then apply the template, which is as I've said many times in this course in the lectures already, templates tend not to work for this kind of material. You've got to think of the actual question. So forget trying to match a template and apply a method that you've seen and that you've got in your notes. Think about the actual question, and this has nothing to do with the actual question here. These might seem like relevant statements, but they're not. This simply doesn't really address it at all. There's nothing I can give here except 0s throughout, again, as in the previous question. You can't look for partial credit when what's written bears no real relationship to what's required. If there's some relevance, then you can give partial credit, but not in a situation like this. Clutching at straws, hoping that by scattering some sentences down that were copied from notes or from somebody else's notes you're going to get some grade, no, isn't going to work. Well, this is very much like the last one. It's a clutching at straws argument, putting things down that look as though they might be relevant on the hope that they'll get some partial credit. But it really just doesn't address the, wait a minute. There's something I kind of like about this. Suppose p, q is a, Let's look at this first bit. This is the opening part. There's a period missing. I'm not going to deduct any marks for punctuation, but let's put a period in. This actually shows some mathematical sophistication. There's something clever going on here. This makes sense as an attempt to start to prove something. Take a pair of twin primes and then show, with p greater than 5, and show that you can't extend them. So in terms of establishing a method whereby you're going to try and prove something, this actually is kind of impressive. Surprisingly so I've given the The other answers we've seen from this students. And remember this is actually a compilation, it's not a single paper. But if it was from a single student's paper then this would make me, I'd want to talk to the student because a student who starts off by saying this actually indicates some mathematical ability. So I'd want to determine if it was that student's own work, because if it was I would work with him or her in order to try and improve their overall performance. Because this actually shows some ingenuity. In fact, what I'm going to do is I'm going to give marks. I'm going to give full marks for the opening because this is impressive. I will give credit for a very good opening as a way of starting. I approve. Now, it turns out that the student's done nothing with this. It makes no sense. And so it's not logically correct. It's not clear at all what's going on. I have no idea what's going on here. Okay. I mean, I can't give credit for stating the conclusion because it's not addressing the question. I can't give credit for that either. And I'm certain not going to give credit for that. I'm just going to acknowledge the fact that this indicates some good mathematical sense, some ingenuity, and I want to give credit for that. But it's got nothing to do with what's being done. Okay so, you know, it sounds as though I'm contradicting what I;ve said earlier. I'm actually not. I'm trying to make a fine distinction because this is not totally irrelevant to proving this kind of thing. It's the kind of thing you might want to do. It makes to start that but then it's not delivered, it doesn't convert into anything sensible. So it's not an irrelevant thing to say. But it said it doesn't establish the results so I'm just giving the credit as an opening. Because it's possible that that would have been an opening of a correct answer. Well the students obviously using induction to prove these results, it doesn't tell us that fact. So we're going to be deducting marks for a variety of things here.. Let's just see if it is a correct induction. It begins with n equals one. That's okay. Assume it hold for n. Adding, now this is nice, explaining what's going on. So I'm certainly going to be giving partial credit for this one. [INAUDIBLE] 2n+1 brings them together, simplifies that way. Seventh. Okay, so there's some really nice stuff here. On the other hand there's some glaring holes in what's supposed to be a proof. Logically correct? It's absolutely logically correct, no problems there. It was clear. It states this part, it tells what's going on here, it observes that's the identity frame plus one. Completes the proof. So it was absolutely clear. No statement of the further, this was a method of induction and that's a big deal. So the opening simply isn't there. That's why part of the opening and last part of the opening is just a. We're going to prove this by mathematical induction and that's completely missing. Okay, this assumption here is part of the actual mechanics of the induction, okay. Stating conclusion, I'm going to go for 0 there and maybe I'd been a little bit harsh, but there's no statement of induction at the beginning and the conclusion in any case should have said by the principle of mathematical induction by induction the result follows. Simply going straight from the identity to n plus one which is proved on the assumption of the identity at n. That it completes the proof. But it doesn't. That part of itself doesn't complete the proof. That completes the proof that if it's true for n, then it's true for n+1. It doesn't complete the proof of this guy at all. That requires a principle of mathematical induction added to what you'd just done. So this is just not a good conclusion. There is no conclusion. I can't give credit for the conclusion to the overall proof. I mean, on the other hand I've been generous elsewhere. So I'm okay with that one. Reasons. I'm simply going to give two max, because locally reasons were given. But the big thing that was missing was induction, and that's a big reason that's involved in the argument. But within here, reasons were nicely given, so I'm going to give two for that. And that means overall I'm going to give two. So I'm going to get a total of twelve for this one and I think that's pretty good actually. Half marks for something like this. That's pretty good. The trouble is though the things that are missing are big things and so you're going to lose a lot of marks without induction proofs, draw upon a very important principle. And you really have to state that you're using induction. And you have to mention when you're using it. Okay. Incidentally there was no mention of where the induction proof used the induction hypothesis. But arguably that's okay because you're beginning with the identity and you're seeing what you're doing with it. So even though there was no statement in here that says by the induction of the hypothesis sort of doesn't make sense in this type of argument. All righty let's go on and look at number eight now. Well the answer to number 8 in many ways is quite similar to the answer to number 7. There's some good technical manipulations going on but it doesn't have the right structure of proof. First of all what's key here is the initial opening assumption that epsilon be greater than 0, be given, that's just not there. And I'm really going to ding that. In fact I'll put this down, now. The opening is missing. Okay. What about logical correctness. This is good. This is good. We can find an n so it's at that, by the assumption, by the given assumption. Then whenever that scales, that all follows. That works fine, which shows that, I'd be inclined to here want the student to put something like, by the definition of limits. At this level when you're talking about introductory material, introduction to real analysis, introduction to the concepts of limits and so forth, I think this really would need to be there. I mean for a professional mathematician you wouldn't because you can assume that they know what's going on. So in any case for logical correctness I'm going to give a four. Because logically it's correct. Clarity, I'm going to have to give 2, I think. Quite frankly, I don't think this is clear, it needs a little bit more explanation. At this level one would expect more explanation, a few reasons, a few hints. Okay, a professional mathematician can look at this and fill in the missing blanks and know what's going on. But my guess is, is most of you, the students in this class, would have to struggle to follow the student at this stage and to see what they're doing, especially given the fact that there's nothing by way of an opener to the thing and there's no reasons given. Stating the conclusion. I'm going to be gentle, the conclusion is stated. I'll worry about that when I talk about giving marks for reasons I think. But absolutely the conclusion is stated. Arguably this is part of the conclusion. And so when I look at reasons That I'm just going to give 2, because there's no reasons given anywhere in here, and in particular, it's not explained why it shows it. Which shows that? Well, at this level, a student is reasonably entitled to say, how come it shows up, why does it show up, what's going on? There's something missing. So some reasonable beginning. But at the very least, you need to alert people that this is simply the definition of a limit that's been applied here. So overall, [SOUND] I'm going to go with three. I could've gone two, but I'm going three, because this is actually correct. This is very slick, it's written very briefly and simply. There's explanations missing that could help a beginner, but this is actually pretty slick. And again, if this was an actual peer performer student, I'd really suspect that that student had had help or had copied the material. In doing so, they'd missed bits out, and they hadn't fully understood it. So, this is a situation where in a physical class, I'd want to sort of chat with the student and try and find out what he or she knew. Because if they were producing this themselves and making lots of silly mistakes, then I would want to work with them quite intensively, actually, to try and bring out their mathematical ability. So, what have I got now? I've got a total of 15 for this one, okay? Let's go and look at number nine. Well, the first thing I want to do here is get a picture of the student's answer. So let's see, A1 is going to be 1 over 2 to 1. A2 is going to be a third, to a half. A3 is going to be a quarter to a third, etc. So let me get a picture here. And let me draw the unit interval from 0 to 1. A1 is a half to 1, so this guy here is A1, okay. A2 is a third to a half, so let's put a third in. And there's A2. A3 is a quarter to a third, so let's put a quarter in. Well, the scale is off here, but nevermind. We're just going conceptually. A3! Already now, we see that the student's example doesn't work, because it doesn't satisfy the condition that An+1 is a subset of An. In fact, these are completely disjoined. So, the example is plain wrong. Dear, what a shame. Okay, so it's going to have to be 0 here. The example is wrong. It doesn't satisfy the requirements. Well, remember, we are trying to be helpful to a student. The point of evaluation is to provide feedback, to give credit for relevant work that is appropriate. So, having seen that there's some stuff here that looks good, I'll see if I can give credit. Because what's going on here is the student's carrying out the kind of argument that would be required to prove this result. Now, admittedly, they have made a mistake here, and we've dinged them for that quite significantly, they've got a 0 for that, the example's incorrect. But the rest of this is a kind of argument that you would need to give, or you could give, if the example was correct. If the students had given intervals that satisfy this requirement, then this is the kind of argument that you would need to do. So if these An's satisfied that requirement, if they'd been different definitions, this is the kind of argument you'd want to do.. So this is not irrelevant work here, there's a fundamental mistake, but in terms of what's done, this is actually clear. Since it's not relevant, I can't give full marks for clarity, but I can give 2 marks, I think, for that one. It's clear in of itself, it's just not clear within an example, it works. Opening, once you've got the example down, I mean, the opening first of all consists of defining the An and then of carrying out some little arguments. And that's done well. So, again, I'm going to give 2 for that. State the conclusion. The conclusion is absolutely stated. Again, this doesn't work for this example. So I'm just going to give 2. Ditto for reasons. Overall, all I can give is 0 because the example is plain wrong. The example does not satisfy the requirements. So I'm giving 8, which 33%. It's a shame, given that the student has done some decent stuff here. But, the fact of the matter is, if you're asked to prove an example, if you're asked to give an example to show something, and if your example doesn't satisfy the requirements, then it doesn't show it. So a third, I think, is probably as good as you could reasonably expect, given what we're asked to do. Where this answer to number 10 is perfectly correct. One could argue that this student should have said clearly, A to the n plus 1 is a subset of An for all n. But the argument itself is absolutely logical. It's rock solid. So, given the quality of the argument, I certainly wouldn't penalize for doing that. You may differ on that one, but as is pretty clear by now I'm sure, grading, even when you've got a rubric, grading is pretty subjective. Now it is the case that professionals, by their subjective grading, actually, achieve an objective result, because professionals end up remarkably similar in the grades they give, not exactly, but they're pretty close. So, there is this, it's a mixture of art and science going on here. But this is absolutely correct. I'm going to give, I'll put the marks down, first of all. I'm going to give 4 for everything. Again, if this was a single student's submission, I would be very suspicious and want to talk to that student. Because to have gone from some really bizarrely wrong answers, that suggests the student's got no understanding, to produce something of this level of sophistication, defies credibility as being a legitimate piece of work by one student. But let's just look at what this thing does. For any n, so we observe that 0 is in all of these sets. So 0 is in the intersection. Also, observe that anything other than 0 is eventually going to get missed out. And so, anything other than 0 won't be in the intersection, so the intersection consists purely of 0. This is slick, it's slick and it's as brief and succinct but correct as it is possible to be. So grading what's in front of me, for me, it's straight 4s all the way through. Because given the slick nature of this proof, I would be forgiving for having missed that. Yeah, that should be in. I mean, you should make that statement, especially given the audience that this is intended for, which is an introductory class, so you should really observe things like that. But this is straight across the board, correct. Okay, well, that's the first sample exam solution done. What have we given for this one? Let's see, we gave the totals, we've given for this one, for the whole sheet we've done, we gave 18, we gave a 0. Looking back on my list now, we've got an 11, we gave a 0, we gave a 0, we gave a 4, a 12, a 15, and 8, and a 24. That's what we gave, so altogether, we've given 92 out of a possible 240. So overall, for this paper, we have given 38%. Slightly more than one-third, okay? Which actually would be a passing grade for material like this, anything of the order of 35% or more would definitely be a passing grade for an introductory course. Remember that the grades are meant to indicate how close you are to being able to produce at the level of a professional. And no one expects someone in an introductory course to produce the kind of results that a professional does. So, for this kind of course, or at this kind of level, anything over the sort of 30, 35% mark is actually acceptable. And it would be unusual for a student to score sort of 70 or 80% if they were really genuinely a beginner. You do find students like that. You sometimes find students who score 90 and 100% at the beginning. They tend to end up becoming professors of mathematics. Okay, that's the first problem set solution done with.