 I think Sean and I have tried to organize this, so we're probably going back and forth. Are you getting up? Yes, there it goes. Let's get our presentation up to the beginning. Anchor settings. We love anchor settings. We spent a lot of time debating what anchor settings or anchor tasks means, but we're not going to call it anchor settings today. So when we're looking at an anchor setting, we're looking at trying to find something that we can use to refer back to either as the whole year goes by or over the course of several years so that kids are familiar with the task and can build upon it. And with that, Sean, go ahead. Okay, so we're going to try to make this as interactive as possible, and we'll use the chat and also gestures to have you all interact with us, and also you can talk whenever you like. So have you ever implemented or wished to implement a rich mathematical task? I'm sure the answer is yes. We've all been to PCMI. So after a month of doing the rich task, how much did the students remember the math learned from the task? So what I want you to do now is I want, let's just go kind of in the chat box. You guys are going to type down when did the math content stick? When did it not stick? And I'm going to fill it in here on this presentation. So go and get your typing fingers ready and tell us when did the content really stick like a month later you checked and when did it not stick so well? All right, excellent. Thank you. All right, so nice lecture based, not so good. So we're saying teamwork's stuck, I think, no struggle time, like it. Okay, so we have a good list here of when things stick when they don't. What I'm seeing is a lot of when students have a chance to discover and own the material and then when the teacher's kind of doing all the work things don't stick. Here's another kind of viewpoint of how to make things stick. We've all heard of the elementary school series called Everyday Mathematics and the concept was oh if kids see it every day in their life then it will be it will stick and it will be important. Well, our presentation is kind of on not to say oh it has to be in your everyday life but what if the teacher can create a landscape over the years of learning mathematics that can create meaning throughout a child's life through their mathematical courses. So here we are. I was at the airport and I was perusing all the vending machines and there was a coffee machine, there was a candy bar machine, I came across this machine right here, this weird machine it said feed me two cards. Does everyone see the machine? Can I get a thumbs up if you're with me? Alright, so this weird machine it said feed me two cards and there were five cards in front of it. Numbers one, two, three, four, five. So I was like okay, I'm just going to feed it two cards and let's hear some audience feedback. What cards should I feed it first? Any ideas? Would anyone like to suggest some cards that I feed it? I'm not sure I understand the context. So there are five numbered cards and those are what we choose from? Yep. And so the machine, there's five cards and the machine says feed me two cards at a time. I need to eat two cards. Okay, I will arbitrarily pick one and five. Can everyone see my screen share of me drawing right now? Is that working? Yes. Yeah. Okay, so we picked one and five. Alright, and the machine eats them and on its little LCD display the machine says one plus five plus one times five and it spits out a new card. It spits out a new card and guess what the card says on it? It says eleven. Eleven? Yeah, it says eleven. So now, yeah, so it says eleven. So now we have four cards left. Okay. So guess what the machine says next? Feed me two cards? Yes, I'm hungry. Feed me two cards. So what could we do now? Could we use eleven, the eleven card? Sure. Somebody want to suggest, somebody else want to suggest two cards that it eats now? So you've got four cards, two, three, eleven and four, correct? Yes, I have four cards left. I have two, three, eleven and four. I'm going to say eleven and four since it's hungry. Give it the most time for its buck. I think it's the exact same thing. I like it. Alright, so we eat eleven, let me pick a different color for the second round, how about that? So we eat eleven and four and what's the machine's LCD display going to say? Four plus eleven plus four times eleven. And what card is it going to spit out? Fifty-nine. Oh man, you're quick. Okay, I'm assuming that's, I'm sure that's, I'll have to say correct, okay. Fifty-nine. Okay, so now we have three cards left. We have two, three and fifty-nine. So what we're going to do now is, let me get to this, here we go. So what we're going to do now, present, there we go, is what's the first mathematical question that comes into your mind? So in the chat, send us some of your questions you have about this machine or about what's going on, your mathematical questions. Let me delete these before you read them. There we go. Yeah, I go for it. In the chat. Awesome. That's the highest possible card. So what happens when you run out of cards? What's the lowest card we can get? By the way, has anyone done this problem before? So what cards can we make along the way? Great questions. Okay, we have a good amount of questions to address right now that you all just came up with. Find the smallest sum of cards, ooh I like it. So what I'd like to do now is, how many combinations are possible? Nice. What I'd like to do now is for you all to investigate these together or on your own. And let's mainly stick with what's the highest possible card we can make right now. So can we all investigate that maybe five, ten minutes? We can talk, whatever, share screens if you want, I'll stop sharing. Okay, go. And please speak up if you have any questions. And you forgot to tell us to get calculators too. Oh. Oh yeah. Yeah, don't worry, use calculators. We'll answer this early in the morning, or we're trying to find our highest sum, right Sean? Yeah, and let's pretend that the highest sum you get is how much money the machine spits out for you. After all five cards are gone, so at the end. Yes. The machine can only eat two cards at a time. But you want to get down to just the end and feed the last thing and know what my last card is that it spit back out at me. Yep. Okay. So I came up with an answer. Can I ask a clarifying question, are we trying to get the highest sum or the lowest? Well in this context, let's pretend that whatever number you get at the end is how much money the machine gives you. So let's try to get the highest. Yeah, then type whatever you get in the chat and maybe we can see who wins. Imagine in a classroom you get many more different answers without such mathematical brains here working. Now, if you're curious about what's going on in the chat, don't spoil it by just looking if you're not done yet. If you're curious, can you explain what's going on? Maybe the power of more than arithmetic? I don't want to spoil anything, but I'm thinking about the purpose of the talk and anchor settings. And I'm wondering whether an activity like this, there's a lot of richness to it, but I'm wondering whether teachers have had experiences where they do this and all the students really remember is like trying out different combinations of cards so they don't actually remember the deeper mathematics underneath. And so it was like a fun and successful lesson, but didn't actually help them. Yeah. So where's we going to talk about where this goes and we're going to take it. And what we're doing right now is kind of a sixth grade thing. But actually it's interesting to me because, Chuck, I think a lot of times teachers are just so worried about getting to the end that they don't give their kids time to do. Like, I need this time right now to just plug in numbers and play before I start working on any kind of algebra or anything else. Like, I really need just some play time and some time that is OK that I'm not coming up with an arithmetic conjecture yet. It's really morning math. Yeah. Yeah. Yeah. So if you're ready to take it to the next level, what if we put in we could put in like five variable cards, but that sounds hard. What if we start with just three variable cards and what variables should we use? Any ideas? ABC. ABC. All right. The machine eats an A, a B, and a C card. And I'm not telling you what order yet. Can you figure out what it spits out and write it in the chat? And maybe you should specify which you pick first. So like, are you doing A machine B first and then doing machine C? Or are you doing some other order? Does everyone understand the question? Any questions about that? Yep. So F of AB equals A times B plus A plus B. Yep. So what's FAB, or what's G of F of, I don't know, FAB of F, A, whatever. You know what I mean. Okay. So something that's not typically talked about also in algebra classes is, I'll keep working if you're working. We are on a schedule. I don't want to, I want to give Cindy plenty of time is, okay. So what concept are we working with right now? What like math concept are we working with? Choosing which order to associate these things with. Part of it seems like commutativity. Okay. Like does the order in which we pick these things and do these things make any kind of difference? Okay. So if I picked the four and 11 card from before, does it matter if I picked the 11 card in the four card? No, that didn't seem to matter. So that's kind of the commutativity, right? Yeah. Four and 11 is the same as 11, four. Yeah. But really what are we dealing with with, I see someone wrote F of F of A of AB of C. What is that compared with F of F of AC of B? Yeah. That's sort of what we do. We associate two things first and then another. Is it an associated or did you just say associate to throw this off? I don't know. Yeah. Yeah. You're right. It's a associative property. Okay. Wait, we showed okay, so we kind of showed in the chat, we showed that this binary operation is associative for three elements. or 100 or n. No. Great. We did some algebra. We kind of showed it works for three, but when in algebra class do you actually show associativity for like 100 objects or n. So now we're gonna shift the talk to a more algebra of one or algebra two focus and we're gonna look at re-representing this binary operation. So I'm just gonna, normally if we had a lot of time I'd have you get here organically, but let's just go through it together. Let's look for some patterns. So we, there we go. So we have one, two, five. Okay, so we take two and three and what is the machine spit out? Eleven. Eleven. Okay, we take four and five. What is machine spit out? 29. 29. Hey, eleven looks a lot like something neat. Eleven looks a lot like, oops, three times four minus one. Weird. Four and five is 29. That looks a lot like five times six minus one. Whoa, what's going on here? How can I figure out, how could I instantly figure out if the machine eats five and seven? Anyone see a pattern yet? How could I instantly figure it out much easier? Seven times eight minus one. Yeah, I think six times eight minus one. Oh, sorry, six times eight. I know you knew it. Thank you. Yeah, six times eight minus one and so that's 47. So if my machine eats x, y, what's another way I can think of writing that? Quantity x plus one times quantity y plus one minus one. Okay, now can you all compose this with z and tell me what you get and please leave it in this kind of form. Let's not leave it in our old form of x plus y plus xy. Let's leave it in our new form of, it looks like, something like this. So see if you can figure this out. Let's compose this with z and see what happens. And let me know if you have any questions. I already have a question. Okay. Are we doing it for consecutive cards? Oh, consecutive cards doesn't matter. Sorry, I just don't hit those on accident. If you take any two cards, I think what we're talking about, like, let's pick. I picked two and four. Great. So two and four. And we're claiming that two machine four is the same thing as, two machine four is going to be the same thing as three times five minus one. So that pattern is something that we're assuming that we've derived already. What pattern? The new equivalent. Yeah. Yeah. Okay, so that's the part you said we'd have a lot more time and we figure out. Yeah, sorry. The pattern didn't hold. Right, so we're running a little low on time. I just wanted to make sure I wasn't missing something else because that's where I got. Initially, we we defined our machine function to be X machine Y to equal X plus Y times X times Y. But now we're defining it to be the same thing. What we're saying it's XY equals X plus one, quantity X was one times Y plus one minus one. And I'm curious what happens when we compose this with Z now? Take this whole chunk and compose with Z. Let's take a look at your chat. Is anybody finding something cool? Yeah, okay, awesome. So I see in the chat, you guys have, we just kind of appended a Z plus one on the end. What happens if I take that and compose it with W? Any guesses? We had a W plus one. We had a W plus one. Does it matter what I compose? What order I compose things in? No, because multiplication is commutative, right? We can move all those packets around. Those plus ones around. Okay, so what am I getting at with this? We can take this concept and simplify it for elementary schoolers. It doesn't have to be a binary operation. It can be we just add three to your number each time. Tell me what comes out of the machine. We can take this concept and take it in middle school and we can put fractions and decimals in it with a binary operation and like Chuck was saying, it's kind of fun, but we're not getting at the algebra yet. We can take this later to do the algebra we just did or even matrix algebra on it. So what we're going to do is reflect right now on this kind of anchor setting and how we, one second. One second. Yeah, so what mathematical concepts can we anchor to this topic? So you can tell me or write them in the chat. Yeah, so let me know. How would we structure this progression and if we got some kind of leadership position in our district or K-12 school, how would we maybe tell the teachers, hey, try this out. Proving disc, proving conjectures. All right, so why don't you guys just share. Okay. And what about just arithmetic for little kids? Which I can never spell. Equivalent expressions. Hey, Sean. Yeah. My trick for spelling arithmetic is a rat in the house might eat the ice cream. That's funny. I learned that when I was a kid and I always use it because I can never spell arithmetic. I always have to remember a rat in the house might eat the ice cream. I always think of it as arithmetic. Okay, so we came up a great list. We're gonna kind of define what is an anchor. It's a setting that learners can laminate a mathematical concept to by systematically revisiting these settings throughout the year or years. We connect math through a meaningful or a memorable theme. So like we said, K to 5, we can have in-out machines take one card at a time. 6 to 8 in-out machine. Now we're looking at rationals and then 9 to 12. We can explore other exciting fun things. So I'm gonna give it up to Cindy now who is gonna take over. Okay, it's good if I turn my mic on. So anyway, I just want to give one more quick example of what we mean by an anchor setting. And there was actually a really good one from Math 2 because Tracy was asking me about, oh, how do you like the way CPM set up tangent ratios? And I said, I loved it. And I loved it because, and I'm gonna actually, I don't think I'm gonna try and get, can everybody see that? Can you see my line? Okay, all right. So what, for years you spend, you spend time with students trying to explain what slope is. And a lot of times you use slope, you use your slope triangle. I'm gonna try and draw it there. There is a slope triangle. So you're explaining them that you have, down, there you go, that you have, this is working, it's slow text, but it works, that you're trying to explain to them that basically you're gonna go rise over run, that you look at a series of triangles and try to explain to them that, you know, different slopes will have different slice triangles. And they eventually learn that, you know, delta y is delta x. Okay, you can expand upon this and you start looking at congruent triangles, congruent triangles, and then similar triangles along these lines. And then the key thing is they've been spent so much time learning about these slope triangles that you get to trick. And this is where it gets cool. They're very familiar with slope triangles. And then you say, okay, let's look at a line and let's look at delta y. You're looking at delta y, you're looking at a series of triangles along the series of slope triangles along the same line. And you start noticing that all those triangles are not only similar, but they have proportional sides. And lo and behold, they all have the same slope and they all have the same because you've learned about similar triangles. They're basically their angle is the same. So they're similar. So their angle is the same. So if their angle is the same, that means their delta y or delta x is the same. And that's how you introduce the tangent ratio. And the orientation of the triangle was amazing because all the kids understood slope triangles because they had been drilled into them. They had practiced with it. They had used it. They had played with it not only as geometry, but they completely understood it. They got it. If they orientate their triangle in this fashion, they got that the tangent ratio is going to be the opposite over the adjacent here. And I thought that was a really good example of a kind of an anchor setting is because they started playing out with triangles earlier in their years, and then they learned about slope. And then they started playing with right triangles and how to orientate it. And they immediately understood, oh, I'm going to basically I'm looking at the slope without hypotenuse and what the ratio is. And they merely got the tangent ratio. In fact, I was incredibly impressed at how quickly they got it. And then from there, you can just move on to cosine and sine. And it's probably so far as an example of an anchor task. I hope that was a really good one. I also, does anybody here have used MVP? Okay, math vision project. There is a Ferris wheel explanation for sine and cosine. That could also be used for both younger grades and older grades. You can use it for different concepts and build it up into trig and how the trig goes, how sine and cosine gets graphed. They'll have a beautiful thing for that. So I'm actually looking for our so I think we were going to move on to that was a couple of our examples of what an anchor task is. And what we would like you guys to do right now is spend a few minutes and try and come up with what you think you might use as an anchor tax in your classroom. Like come up with an example, something that has either worked. And you can think about an anchor task as being something that might go from kindergarten all the way to calculus here, or something that you might use within the year, like, like the our slope triangles here would be over the course of when they start learning slope up to trig. You can look at it that way. So we'd like you to spend a few minutes and come up with a couple ideas and put them type them into the good chat here. Cindy, can you just repeat like what is it the the ideas that you want us to come up with? Yes, try and think of a task or a an activity that you might use to for them to work on a certain concept in your grade right now, but may be able to be simplified or made more complicated for later so they could see it again in the following years or use it earlier. And if anybody wants like a seed to start with, challenge yourself to think about how bikes could develop from kindergarten to calculus. Using bicycles in the classroom. It's hard to type this in the chat. But in calculus, one of the cool. One of the cool proofs for polynomial derivatives is like, for instance, you want to know the derivative of X squared. You make an X by X square and increase the length and width by by DX and then analyze it. And so I was I was thinking those area models are a nice anchor representation that you can use, you know, in elementary school and learning area and you can use them in middle school and talking about distributive property and multiplication and all that. And then in high school with factoring polynomials and multiplying polynomials and then in calculus with actually, you know, driving polynomial derivatives and stuff. It's a pretty powerful. If you take an X by X square and you add DX to both sides. No, let me let me draw. Thank you. Is this similar to what the area models that CPM uses because they're doing they're doing tons of very models right now, they're going to use it to introduce factoring. I think he's drawing and that was a really. Yeah, I use areas of models a lot with my sixth graders when we're working with equivalent expressions and adding and multiplying variables and oh my gosh, it's just so helpful. Yeah. So they start those area models when they're in elementary school with multiplication and then it just leads all the way up through the distributive property and multiplying by no meals all the way up through through calculus. Yeah, I had a kid the other day we were doing a multiplication problem like four and a half times three and a fourth. I don't remember what it was. And he actually drew an area model to do the problem and it was so much easier than you know, finding the equivalent fractions and multiplying and simplifying and it was cool. CPM also has a really cool way of showing just multiply and I'm sorry my brain went froggy. This may have been exactly what you said, Jen, but about multiplying with fractions of cutting, cutting a unit rectangle and it shows a really cool way that a lot of light bulbs turn on for students of why they're actually doing it. Awesome. And maybe using that language of area model sticks with kids. Whatever you can do to continue the experience. So light bulb goes on again. Well, do you guys want to hear about some bikes? All right, let's do this. So here we go. I'm actually going to plug in my computer. All right. So yeah, we came up with some great ideas. So if you've done exploding dots before, you can imagine that can be a nice anchor task going from arithmetic operations before they even know the algorithms in kindergarten, all the way up to polynomial algebra. I just watched the James Tanton thing. I won't go into that. But yeah, basically, yes. So I talked to James about this and he said that he does not recommend exploding dots for elementary kids. That it's more something you come back to after you have some understanding to see it in a new way. That's something that he's told me. I know. Nice. I've had no idea. That makes sense. All right. Yeah. So maybe you don't just have this be the first exposure to arithmetic. Let's talk about bikes. So I use bikes in my classroom a lot. I have a bike trainer and I have kids bring their bikes in. And we will put a kid on, I won't say in this video, not playing. We'll put a kid on a bike trainer and basically having pedaled at a constant rate. Not one second. Let me pull up this video. This one. Yeah. And then we'll ask the class, the kid pedals at a constant cadence. We'll ask the class, how fast would the kid be riding right now if they're on the street? There we go. Wait for it to load. Come on. Need activity. I'm not sure if this is going to load, but luckily I'm out work and we just access all these things right here. There we go. Can you guys? Hold on a second. Let me just share this one. There we go. Can you all see this? Yeah. So the kid's pedaling at a constant cadence and we're asking the class, how fast would he be moving? Here she'd be moving if they were just on the street right now. And they come up with all their different ways of doing it. And you saw them counting how many times the tire went around. That's pretty tough. They're measuring the tire. So we're talking about circumference and everything. Then they have to go home and basically measure every part of their bike and draw their bike to scale and then count all the teeth on their chain rings and find every gear ratio that they have on their bike. So now we're getting at equivalent fractions and equivalent ratios and actually way too many decimal points. So how precise is enough? And by the way, the gear ratio of a bike is simply the number of teeth in the front chain ring divided by the number of teeth in the rear sprocket. And this is a really tough concept for people to get at first. But once they get it, it is sweet. Let me just draw a little diagram. So say something if you know how gear ratios work already. Does anyone know how gear ratios work on a bike? All right, so here we go. Let's clear this. We have our first of all really cool thing about a bike. And this extends all the way down to elementary school is a bike. Since the late 1800s has, oops, bikes have just been two triangles. Well, this is really actually a quadrilateral, but it's kind of triangle like. And then your fork comes out like this. You got a tire and a tire. And this has been the design since after the penny farthing. And this is actually called the safety bike, because if you fall off it, you don't really hurt yourself on those really tall bikes that look like this. If you you sat right up here and if you fell off, you got really hurt. So anyways, here's how gear ratio works. You have a chain ring right here, and let's say it has 40 teeth. And by the way, every chain ring sold in the States, the distance from tooth to tooth is one half inch. Every bike doesn't matter. And on every chain, the distance from link to link is one half inch. Okay, so we have 40 teeth in this front chain. And let's pretend that we have 20 teeth in this back chain. Well, if I turn my crank one full time around, what I've done is I've pulled 40 teeth worth of chain. I've pulled 40 teeth worth of chain. And so that means my back tire has gone around two times. And so we call this gear ratio 40 to 20, a.k.a. It's a gear ratio of two. So you'd want a high gear ratio so that one crank will give you many, many spins. Yeah, potentially. Can you see what I'm googling right now? We're still on the drawing. Okay, so I need to change it. So Sean, is this with this gear ratio, because now I feel either enlightened or foolish or both. Is the gear ratio of two, is that correlated directly with the actual gears we use on the bike that make it, you know, so we go up the hills or... Yes. Okay, it makes sense. Yeah, let me just show you this. This is really cool. Everyone see this bike right here? Yeah. It has 104 teeth. This bike went 100 miles per hour. It has 104 teeth in the front chain ring and about 11 in the back. So each turn of the crank is like nine rotations of your wheel, which would be like the hardest thing ever to pedal from standing still. And so mountain bikes have much lower gear ratios so that they can go up mountains easier, whereas this touring bike for speed has a really high gear ratio, aggressive gear ratio. So okay, let's talk about how this applies in the classroom and mathematically. So, one second, you present. Okay, so they find, these are all the gear ratios. Do you see, here's my rear gears on my bike. The smallest one has 11 teeth. The largest has 28. Here's the rear gears, the front gears or chain rings. And we can divide each of the fronts by each of the backs and we get all these gear ratios. Notice we have some equivalent gear ratios. So I show how that works. You can put post it notes on your pedal and then on your back wheel and show that correspondence. So then you can have kids make some direct variation functions relating cadence. That's how many times your right knee comes up per minute. So cadence is RPM to speed, given a fixed gear ratio and a given tire diameter. 27 inch tires are kind of standard. And then what you were just talking about, Aisha, was when do I shift? Here's a step function. When you shift, if you want a cadence of 60 to 80 rotations per minute of your crank or your right knee. This is like a healthy cruising cadence. Professional cyclist ride around 100 RPM. And you can see we start off real slow, only going five miles per hour right here. And in that we can be in our lowest gear ratio. It's around 1.6. But once we start going faster, and the cool thing is you can skip gears. You can go up here, and then jump up to the gear. And the different colors are different chain rings, which is kind of takes longer to shift on a bike. Your left gears take longer. It's like... And then kids design some zany bikes. And they even designed this bike, which I built in my classroom. It's 10 feet long. But where does the bike project go? Well, in K to 2, we can talk about shapes. Kids can design and build a bike out of twigs and CDs or whatever they find. Grades 3 to 5. How many spokes are on your wheel? And to draw a wheel with all the spokes. How many spokes would be in half of your wheel? And a quarter of your wheel? How many speeds on your bike? There's multiplication. 6 to 8. It's a great thing for them to look at scaling and proportionality and unit conversions. Because we use French measurements for tires, like 700cc. I don't know what that means, but it's French. Cubic centimeters? Yeah, but it's no. It's something different. It's weird. It's a distance. Oh. I don't know what it is. Maybe I wrote it wrong. I don't know. But yeah. And so then this is really cool. 9 to 12. So fixed gear bikes. Can you guys see my mouse? Fixed gear bikes don't have brakes usually. And to brake, they just skid stop. But they always skid when their feet are horizontal. And there's no freewheel on these bikes. If you pedal backwards, there's no click, click, click, click. The wheel just goes backwards. And so they always brake at the exact same place on their tire depending if this rotates 180 degrees. So what happens over here is if you have a bad gear ratio, like 2 to 1, you will tear through your tire instantly because you only have one skid patch. So now we can get into modular arithmetic and ask, what's the best gear ratio such that you maximize the number and distribution of skid patches? Right? Because we have a 2 to 1 gear ratio. You're skidding on the same spot every time because your feet are always horizontal. And then we can talk about, you guys probably heard of the bike thing where if you have tire tracks in the snow, we can use calculus and tangents to figure out what direction they're traveling because of the way a bike works. That's also a Tantan video. It's quite cool. Yeah. So this is another example of an anchor progression, K to 12 that I have spread around my building. Let's try it. Kids engaged. Yep. And I have one more example of our lower schoolers build cardboard mini golf courses. So then I took that to algebra and used systems of equations graphing lines and I created this formula right here to calculate the slope of a line that rebounds off another line. But and it can extend all the way up to calculus with finding how lines rebound off parabolas and you'd have to find the slope of tangent lines, etc. So there's more ideas for your anchor tasks. Lin ball bike project. Yep. Sorry, there's a lot of me talking. Sorry about that. Let me know your thoughts. This is kind of an idea Cindy and me and Brooke were playing around with like, whoa, what if we anchored stuff throughout the years. I'd love to hear what you guys think about it. One of my goals for this year was to basically make an anchor task. It wasn't in so many words. This has really helped me because I want to have one focus task per unit at least as kind of the jumping golf point. And so something like this would be really good to culminate everything at the end as well. So thank you. Yeah, I agree with Aisha. I feel like I tried to do some sort of a task at the beginning to be to unit that I can keep coming back to throughout the unit. Oh, do you remember when we, oh, but I've never really thought about it in terms of an anchor task. Year to year, or even an anchor task from the beginning of the year to the middle of the year to the end of the year. I feel like I'm in kind of a unique position to implement this because if you're going across multiple years you have multiple teachers and a single teacher who's really into bikes is not you're not really going to be able to do that unless you can convince your colleagues to really buy into it. But writing a curriculum right now, we're writing high school curriculum, like that could actually be a possibility to actually have and then you see that with CPM to the, the more it's more representations than anchor tasks but they, they plant seeds early and those seeds grow throughout the years rather than just throughout a unit or one year. So that's kind of interesting something to something I feel is very relevant for me right now. MVP also that's the math vision project has something similar like when I talked about the Ferris wheel, they use that to introduce triangles and angular speed and then you go into trig with it. And it's the same thing that you see it repeated a little bit, not quite as extensive as some of Sean's examples but Yeah, sometimes when I feel like when I do problems I have to take them from beginning to end right away. And it's a kind of good reminder that there's nothing wrong with starting the year with something and then middle way through the year let's build on it a little more and little later let's build on it a little more I don't have to finish it right away I can wait. I love the bike thing but that's probably because I ride bikes so much. Oh yeah and the individual teacher has to bring themselves into the classrooms you want to develop trust and and all that so that's cool. I think the bike the bike discussion like I walked away with a conceptual understanding of how how the bikes work and the gears work I thought that was pretty cool and I think the kids and follow that conceptual understanding, you know at different grade levels. Yeah, it makes me want to investigate it a little bit more play with it a little bit more and yeah my kids leave class with their hands just covered in dirt every day like grease. They really need you can go to any like local co-op or bike place and just buy like a $10 bike and take it apart. So that they can they really need to touch like the gears and measure them and they need to like measure everything and figure out how the shifters work. Yeah. Awesome this has been fantastic Sean Cindy thank you so much. Yeah I would say I don't know if you are wrapping up soon but before we go. I want to get a screenshot of all of us waving if you want to be part of it if not before we go if we do this you can just turn your video off. And I actually already took a picture of that with my phone. Well I wasn't even better when we're all looking. Yeah we weren't waving like I thought it was something so what would I do with this I would put it on the PCMI alumni Facebook page just to get people excited about the next one. The next one is thumbs up. Okay we'll do thumbs up I can only give one thumb if I'm doing the screenshot. All right ready. All right we're good thank you. Well thank you all for taking the time to do math with us on a Saturday morning. Thank you for taking the time to lead us through your ideas. Awesome job. Sean are these the same slides you have that are posted from the summer. Not quite. Sean altered some of his when he gave in Chicago. But they're close it's close. Okay would you mind sending me a copy and then I can have that linked up to if we post this recording and everything so everyone can see all these awesome things. Yeah sure thank you. Yeah I like to go back and look at some of the ones that you skipped through that I missed or whatever I think to spend a little more time on it. And if anyone wants to lead the number card machine thing I'm going to put in the chat where it originally came from. It was Josh Zucker from math circles. Yeah. He spends like three hours on it. So I'll just post that right here. For this. Yes. Oh nice polynomial derivatives. So you could do it for a square. You can do it for a cube. And then kind of go for how it generalizes. So that's the idea. Thanks. So the video I just posted is where I got the idea of the machine card problem from. Josh Zucker always steals from the best. So I steal from him a lot. Yeah. And if you're not on Twitter get on Twitter right now there's so many good math ideas out there. I'm still not fond of Twitter. I look at your Twitter. I look at a few people's Twitter but it's just too short it's only what 144 characters. You link it to your blog and it's fine. Then it's fine. You just link it to your blog. I don't have a blog either. You got to create a blogger. It's free. Create a blogger. Well this needs self-creating a Twitter handle. Yeah I don't have a creative name that's really why I haven't done it yet. I'm a grass mash. Good God Chuck's here. All right I'll get on that today. All right well awesome. Thank you so much everybody. This was fun. I'm so excited that we got to do this. I feel I'm so excited to go in and especially use the card machine trick with my elementary math kids. Which aren't kids but you know still so excited. Yeah. Keep up the energy. Thank you Sean. Bye everybody. Thank you Sean. And go fuck guys. Thank you. And go fuck guys. Bye everybody. Bye everybody.