All right, your brain may still be hurting from working on these discussion questions, but let's review them and talk about how you can help your students work through them. So, our first question was about a weather probe, and you'll notice the spelling out of things in English is over 100 degrees and we choose not to use the greater than signs, so that students would have to convert. Then also with the below 10 degrees, below zero. That's negative 10, right? So, what's the correct answer here? There's two, there is C and D. People say "But Dr. Simon: C or D wasn't an option." Yeah, this is a really great trick to use, because students like to get queasy. You'll find the first one is that correct thoughts good, but then you'll get really engaged discussion in the classroom and students start to be like, "What, they're talking about a different one, how is that?" It's just a fun thing to do to keep people on their toes and make life a little different. All right, let's talk about how you'll help students figure out that these are the correct answers. We did this previously with regular conditionals and that is you want to have students draw things out, even more important for compound conditionals. So, we've got a number line here that spans more or less the appropriate place, would be nice if it went over a 100, but we'll just draw on that. So, we're going to start off by drawing our original conditions and then we'll look at our possible answers. So, we said the temperature needs to be over a 100 degrees. So, that's the orange box on the right. So, that represents all of that part over 100 degrees, and then the box on the left, that's below 10 degrees below zero or below negative 10 degrees. So, we've got those there. So, now what we're basically looking at the difference between options A and B and option C and D, is that the first to use AND and the second two use OR. So, lets figure out AND versus OR. Is it AND? Well, is it possible to have a value that is both greater than a 100 and less than negative 10? No. We've got a box over here, we've got a box over here, you can't be a number that's in both of those places at the same time. We can tell that because we drew it out on the number line. OR says, "Okay, you can either be in the box on the right or you can be a number in the box on the left, you don't have to be in both places at the same time." So, hopefully this drawing is a little easier for students to visualize and see it's like, "Oh, you could be in one of the two places." Now, the last thing we want to get down to is why are both C and D are the correct answer? Well, really A and B are exactly the same thing and C and D are exactly the same thing, all I did was switch the order in which we asked, is temp greater than 100 or temp less than negative 10? It's important to know that, are these Boolean conditions both AND and OR have what we call the mathematical symmetric property and that the order in which you check the condition doesn't matter, right? It doesn't matter if you ask if temp is greater than 100 and temp is less than negative 10, or if you ask it in the reverse order, if temp is less than negative 10 or temp is greater than 100 and temp is greater than 100. Onto the next question, which of these expressions can possibly evaluate to true? Let's start off just by looking at that question, this is a spooky question, right? Can possibly evaluate to true, why would you say that? Well, this is really just a way to say, "Is there any numerical value at all for which these critical expressions could be true? " So, just note first off, let's look at the bottom one, it is correct and that is, if X is less than seven and X is greater than negative two. But let's go through, again, using our drawing approach and evaluate each one and draw it out. So, let's start with A, this one we're going to use two different colored boxes so that we can see the differences. So, X greater than five, that's going to be the orange box, and X less than five is going to be the blue box. Let's draw it on the number line, there we are. So, this isn't beautifully perfectly accurate because we're really only dealing with whole numbers here. But what you see is that the X greater than five, that's just a little bit to the right of the five on the number line, and the X is less than five is just a little bit to the left of the five. So, now we're asking about AND not OR, is it possible to have a value that is both in the orange box and in the blue box? No, we don't see any overlap. So this is the key thing to help students learn about AND. AND, if we express it on a number line, requires overlap. There needs to be some value for which it's true for both things. All right, let's look at the next one. Draw it on the number line. X greater than five, that orange box stays in the same place. X less than negative seven. Now, we've just moved that blue box down even further. So, these are, maybe it's even more obvious here, it's hard to say. Yeah, there's no overlap here. AND requires overlap. There's no way you can possibly have a value that is both greater than five and less than negative seven at the same time. All right, so finally we'll get down to how would you draw out our correct solution? What does overlap look like? Well, here's a way to make overlap look like on the number line. So, we have the orange is X less than seven, and then the blue is the X greater than negative two. I think it's pretty clear that you can say, "Hey look, there's some overlap here, there are some values on our number line for which both the orange condition and the blue condition can be true at the same time." So, this is a great way to help students visualize AND. Going along. Yeah, Boolean expression evaluation definitely exercising the brain here. This question was about the American grading system, about earning a C grade. It used a mathematical nomenclature which probably you will need to spell out for your students if you likely don't remember. But a C grade in US is a value in the 70's. So, we give it the close bracket 70 that means it can be equal to 70 percent but less than 80 percent. So, any number in the 70's but not 80. So, up to 79.999999 whatever. So, the grade would need to be greater than or equal to 70 and less than 80. The trick here is, we need to make our own greater than or equal to sign, because Snap doesn't have that neither does scratch either by the way. But that's kind of actually cool in this case because it has helped us having students do something that we need to learn to do in computer science a lot, and that is to break complex things down into smaller pieces. So, in this case, we're going to break the complex greater than AND or equal to, greater than, equal to, into a Boolean expression. So, on the right hand side of B here, we've got X greater than 70 or X equal to 70, and then that whole expression is, AND X is less than 80. All right, I think we can go on. This one really hurts. I almost actually coded it to make sure I was right but instead I just drew out a lot of things, and so there's going to be a lot of drawing here. So, let's make sure we understand or reading the question correctly. We're asking which expression would evaluate to true for x and y coordinates NOT in this box? So, if you squint at this grid here, I want everything that's not blue. Anytime, if I could click anywhere on here that's not in that blue box, I want that to be true. What's that going to be? So, only two options here, okay? So there's nothing difficult about the actual numbers in the x-coordinate and the y-coordinate, I'm giving those to you correctly, that's not the point of this particular thing. We want to focus students in on one thing, and that is, we calculate your x-coordinates, what they need to be and we calculate what the y-coordinate need to be. Do we want to OR those expressions or do we want to AND them? Before we start going into it, let's just remember what we brought up before, OR meant that things didn't have to overlap, AND meant things did have to overlap. But now we're not doing simple number lines really, we're doing two-dimensional. But in fact, if you break it down into the x and y coordinate, this way you can actually use that same number line drawing approach, it just looks a little different, so let's do that. Let's start by taking the expression, if your x-coordinate is less than negative eight or your x-coordinate is greater than negative four. So, if we're just considering x, x is just this part. So, I could have just drawn along that number line but it's for any value of y. So, I actually drew a box and I chose not to color them because then you couldn't see the numbers behind that number line anymore. So, there are two boxes here, the box on the left represents the expression x-coordinate less than negative eight and it's, again, for any value of y less than negative eight, or the x-coordinate greater than negative four. Got that right. All right, so can you squint at that and see how it looks like the same as our number line representation before? Good. All right. Now, it's going to get even more complicated. Let's, visually, let's switch and look at the y-coordinate, but I'm going to leave the x stuff up there. So, the y-coordinate we're going to show in purple, so try to ignore the red. If the y-coordinate is greater than negative two, so that's the purple box on the top, or the y-coordinate is less than negative four, so we have those two. So, now if we're just looking at those two different expressions about the x1 in the y-coordinate we've got that drawn out. Now the question is, when do I want my volume to be true? And as we said, if OR is the right answer, we know that part a little bit, but OR means that it doesn't need to be overlap between these two things, AND means there does need to be overlap. How do we convince ourselves there really is OR? I think the thing to do is to pick a point in some various places and some of them are in just the red box, some of them are in a red box and a purple box or maybe just in a purple box and decide if we need to have overlap, if you need to be inside both boxes, or if not being in that blue means you can be in any one or the other of the box but you don't have to be in both. So, here's some examples, OR no overlap required. So, let's put a point up here, this is definitely something we want to evaluate to true, it's outside the blue rectangle. It is in both the purple box and the red box, so that could work for AND or OR. But definitely that's good. We got that. At least in both of our boxes we got our expressions correct. Let's look at this one, this point, if you squint carefully, you will note it is not in any red box. It's x-coordinate is not outside, within that expression. It's not less than negative eight or greater than negative four. But is it a point we still want to evaluate to true?Yeah, because we just said, I wanted to evaluate to true anytime I click someplace that is not in that blue box. So, this right here is going to tell us that OR is what we need, because we're saying," You don't need to be in the overlapped box, you just need to be in one, either a purple box or a red box, could be both but doesn't have to be, it's OR not AND". We'll check another one here. This is the one where- this would be in the red box that is the x-coordinate less than negative eight, but it is not in any of the purple boxes because its y-coordinate is not greater than negative two and it's not less than negative four. So, but that is a point that we want to evaluate to true. So we got another confirming factor. Just to double-check the OR is right, that's what we want. Okay, challenge question, what would we be representing if instead our expression was AND? What parts of this screen would be places that would evaluate to true if we made it AND instead and we wanted overlap? Stop, think, think. I know you don't want to, you really don't want to, but just try it. Okay. Yes, that's what we would get. The places where the boxes overlap. So, top left corner, that's where purple and red overlapped, then we're missing that slice in the middle then when top right corner purple and red overlap similarly on the bottom, that's what it look like. I have no idea why you'd ever want to have them, I don't know, maybe something about streets or something. But anyway it was a nice logical exercise and yeah my brain hurts too.