 continuation of Andy Burnoff's work. So tonight there'll be an Estimathon here and again it's going to be like on Tuesday there'll be pizza at 5.30 in the dining tent followed by the Estimathon here around 6.15 or so. So please come to that and I'm very pleased to introduce Eli Lubrov who's the CEO and founder of the Desmos Corporation and we're really honored to have him here and you take it away. Amazing thank you. First off our folks able to hear me okay front and back. Oh boy awesome shout out to the sound crew there might be a few other times that I put you to the test. So we've got an hour together and I want to warn you in advance that I've packed maybe three hours of material into this hour so buckle up get excited and I'd love to start if you could see if you can find someone with a computer and let's try to get like two people at each computer just for internet reasons and go to student.desmos.com and type in this code and we're going to be going back and forth to this activity some it's also going to be open for the next two weeks for the second two hours of material that we're not able to cover so that you can go back and look at it but the opening question have folks gotten this code down? Oh no is it internet or is it Desmos? Phew okay that I'm fine with. Alright well we are going to try to do this internet free then but as you get ready just memorize this or write it down on a piece of paper try to get in when you can maybe form larger groups. Oh that's a great idea. Alright F-E-U-J-A-S forgive my yeah you don't need to use an account feel free to join anonymously and yeah there's going to be plenty of opportunities to come back to this you don't need to be at a computer this is just a way for me to get some information back from you it looks like oh yeah we've got 44 people successfully in and the opening screen was just a little bit I wanted to know about you so how you're doing today how well you know Desmos so that I can know how much to tailor the content about Desmos and how much you like Desmos so I can know if I've got a friendly crowd. I'm also hoping to see some correlations between familiarity and enjoyment of it and sure enough look at that looks like the more you know it the more you like it which means we've got either some people who are saying what they think I want to hear or a good product but we're not actually going to be focused on Desmos for a big chunk of this. I wanted to start by focusing on a different technology company a little one that some folks might have heard of called Google and we're going to start with a thing that I noticed when we were trying to build the percent feature in the calculator and noticed that if you type in 30 plus 20 percent into Google into the calculator the result it gives you is 36. So to start I'm curious if you agree with this result so just to show of hands who agrees that 30 plus 20 percent is 36 and who thinks that it's something different and someone who thinks it's something different say what you think it is we've got 30.2 over here also 30.2 does anyone think it's something different than 30.2 any guesses why Google doesn't return 30.2 it does 30 and then 20 percent of 30 and then adds them together and my speculation for that actually I want to hear your speculation for why it does that yeah that's what most people are looking for when they type something like plus 20 percent probably because they're trying to leave a tip and probably they got some complaints from small businesses that people are leaving 20 percent tips or sorry 20 cent tips oh ruin that joke what a disaster but here's the problem when you do 30 plus 20 percent is that this does not scale very well and so I want you in this activity or if you're not in the activity just with your partner I want you to guess what Google returns if you type 30 plus 20 percent plus 20 percent and no cheating no looking at the site but let's collect some answers talk with your partner noisy room I want to know what you think Google thinks that 30 plus 20 percent plus 20 percent is oh alright so I'm seeing a bunch of results up here a lot of folks who think it's 43.2 a lot of folks who think it's 42 some folks who are maybe in the process of typing or else think that it's four I'm so sorry is that my audio can you still hear me okay alright technology that thinks with us and not for us is the topic today um so here is my hint to the room is that nobody has yet gotten the right answer there's not a single person who has done what Google does when you do 30 plus 20 percent plus 20 percent I still don't see it oh no we've got two people who have what Google does alright but before I before I reveal someone want to make the case for 43.2 a 20 percent increase on top of a 20 percent increase someone want to make a case for 42 adding it adding it again both of these seem like very reasonable algorithmic choices um anyone want to make a case for 37.2 I actually love this response and I've never seen this one before who did this that is awesome it's a 24 percent increase so maybe it put the parentheses around the 20 percent of plus 20 percent and then it applied that all um so the thing that Google actually does here is 36.2 correct response um and what is wild about this is I'm going to let you on to the next screen and show you some other results that it does if you put parentheses around the 20 percent plus 20 percent you don't get that brilliant 37.2 instead you get 30.4 but what if you put parentheses around the first two terms then you get our 43.2 and so in trying to implement a tip calculator Google has managed to break associativity commutativity um basically all of all of math um and so my big takeaway from this is that I'm not sure I yet trust Google to drive my car for me that's that's my that's my main takeaway um and so a challenge for folks as I do a little bit of a rant which is the next part of this presentation if you would rather is you can spend some time on Google and see if you can actually figure out what the spec was like some product manager somewhere said here's how we need percent to behave and try to describe it try to write an algorithm that gives these absolutely bananas results so that is your project if you don't want to listen to me um rant about technology in classrooms and outside of classrooms um all right so this is a video I'm not actually sure this is going to work and I'm going to skip over it anyway but I recommend that you look this up this is um Eric Schmidt I think who's the CEO of Google talking about his vision for Google um in 2010 and he's describing oh please that's a great question see if you can figure it out and then write the algorithm to solve it um yeah Google does very different things when you multiply by percents than when you add percents which is also pretty hilarious um it's a it's a wild wild algorithm um but here's Eric Schmidt CEO of Google 10 years ago talking about his vision for the future of Google which presumably is their vision for technology where he says let's say you're near a museum and you search for a hot dog we can return the result for a hot dog that's right near where you are and he says we've got all this other information we know where you've been before we know where you're going we know who your friends are maybe soon enough you don't even need to ask us for a hot dog we can just tell you hey I bet you're hungry and want a hot dog there's a stand 15 feet away and he says this as if it's the panacea we should all be so excited and I hear that and I am terrified um this is technology thinking for us instead of with us and working itself into corners as a result and this is not just Google um this is kind of the Bible of designers in Silicon Valley a book called don't make me think um a common sense approach to web usability and I can trust this with some of the progenitors of computational technology who had just the exact opposite idea we want to be thinking as much as we possibly can and we want computers to help us with that so I think for example of Douglas Engelbart has anyone heard of of this gentleman um he is the inventor at park which was the Stanford lab of the mouse of the first constructive geometry technology of networked software and the vision that he laid out um is this idea of a research center for augmenting human intellect computers are there to help us think new thoughts in new interesting ways not to preempt not only the getting of the answer but even the asking of the question which is where it seems like so much of technology is headed today on Facebook on Instagram recently just switched so that instead of you choosing what it shows you it shows you what it thinks you're going to want to see because that's more effective for tiktok this is the direction so much technology is going and I hate it um so my goal is the exact opposite and so I want to talk about what it feels like when technology thinks with you instead of for you and the first premise that I have for this is that the thing that is at the center is then the person instead of the technology itself and how can you tell when you're looking at a product if it is there to think with you versus for you and I think one easy way is to look at its advertising how does it market itself and is it like showing off how cool the tech is or is it showing off how cool the stuff is that you can do with the tech and I want to give you an example of each so here's a product that maybe some of you have heard of and maybe my internet will be good enough to play called photomath has anyone heard of photomath well the audio is working the video is not um that's a bummer all right it's an ad and you can imagine what this ad is showing it's someone with their camera over a textbook and it takes a picture and magically it shows you the answer to that problem and a lot of really interesting thought went into making that video but all of it was by the programmers who built this software none of it was by the people who use it and I contrast this with something like Geometers Sketchpad has anyone here used Geometers Sketchpad absolutely brilliant software and in a lot of ways feels like kind of the shoulders that Desmos has gotten to stand on and if you look at any of their marketing materials over 40 years it is always showing interesting examples of what someone did with the software it's never like celebrating how brilliant the insight was of designing the software it's celebrating the interesting ideas that come out of it and this to me is incredibly central to where I want technology to fit in society and also inside of classrooms and it comes back to something that one of the strands here has gotten to experience and I'm nervous to even show this slide but I put it in every presentation which is the dimensions of equity by Rochelle Gutierrez who I have no idea if you're in the audience but if you are I'm such a fan of your work and it has guided so much of what we do and one of the premises here is that when folks talk about equity and apologies so much if I'm getting this wrong when folks talk about equity they often talk about things like access and achievement which is just saying inside of the system that currently exists how are you fitting into that and they don't talk about this other axis the critical axis of identity and power do you have the tools to actually change the system and so to me technology that is doing your thinking for you is absolutely not equipping anyone with tools to change that technology itself and this shows up inside and outside of classrooms so this is the first premise for me and I promise the rants over soon and we're going to get to play with some math soon you can get on the internet is that technology that thinks with you instead of for you is about the person and not about the product it lets you do interesting things so let's look at some of these interesting things technology that thinks with you instead of for you is transparent and I've got the funniest video to play that you're not going to get to see but I'd encourage everyone in this room to do a Google search for the for number Wang has anyone ever seen this sketch so premise is it's a game show where it's just people shouting out random numbers and either they win or they don't win and there's just no insight into how it's happening but it's hilarious and for so many people this is what math is is it's a game of number wing like did I say the thing that you were looking for I have no idea how and for so much technology it's the same did you type in stuff in exactly the right format and I'm going to tell you I got it right or wrong but technology that thinks with you is transparent to what it's doing and we'll see some examples of this soon here's that just a teaser look at how funny this looks it's going to be so funny oh it will not the other key part about technology that thinks with you not for you is that it lets you build incrementally it doesn't wait for you to get to the final product before you can see how well it's working and one example of this that I love is actually something like Excel or a spreadsheet where you can imagine a spreadsheet that it starts with just typing in the numbers and then you're like I wonder what would happen if this one depended on this one and you incrementally can improve and you can incrementally see what's happening and also anytime you open up someone else's spreadsheet you know exactly what they did it's all there you can see every bit of it and so what I want to do is switch over to the calculator and I want to show you a few examples of this in in action and let you play around with it as well all of this applies to other tech I just happen to be more familiar with Desmos than I am with most tools and also there aren't as many tools that fit this design philosophy as I would like so I'm going to pace us to play around with a couple different examples and I'm going to show you up here on my screen we're just going to get a blank calculator can folks see this okay I'll make it a little bit bigger and we're going to just play around and try to ask progressively more and more interesting questions so here's one I'm going to graph y equals x squared I get a parabola everyone has seen this a thousand times yeah all right I'm going to put it in standard form x squared plus bx plus c everyone has seen this a thousand times I bet if I ask this room what happens if I changed a folks would have a pretty good sense did this match your expectations and if I adjust c folks have a pretty good sense did that match your expectations and if I adjust b it's going to do something kind of weird did that match your expectations for some people not for everyone does anyone want to try to formulate a hypothesis for what is going on when I change b so you can either try to describe where the parabola is going or you can try to come up with an explanation for why it is going in that way and I want you to just type in the same thing into your calculator play around with it for a few minutes with your partner and see what you're able to discover just by incremental building on top of this graph I'm going to walk around while you do it all right I want to show you two fun things I saw while I was walking around that you can build on top of we're going to spend a little while on this one because there is so much depth the parabolas there's so much depth so one fun thing I saw was someone completing the square I think it was up there on the left completing the square and discovered that what you're going to look for here is let's make a one just so that we can make this as simple as possible to complete the square and we're going to find the x-coordinate of the vertex is going to be at negative b over 2 did I get that right so I'm going to try graphing that and just check it so I'm going to say that the x-coordinate of the vertex is at negative b over 2 and let's confirm that it follows it sure enough you could try to expand this to make it so that it works even when we change a so right now these are we kind of going to become disconnected as I move this I think it's maybe b over 2 a does that sound right nice and so now as I change a this is also going to follow so that's one description of why it's moving right and left and I think what I saw going on up there is trying to figure out where it's moving up and down so that was a blast here's another thing I saw one group do which is that you can instead of having b be a single value you can have it be a list of values and we're going to play with this in a second to try to predict what's going to happen so here instead of b being negative 1.2 I'm going to use list notation so I'm going to do brackets and I'm going to have it be every value from negative 5 to 5 and I get this kind of picture so you can also play around with that and now you can explore many hypotheses at once all right so keep going I just wanted to show those two techniques and see if there was any insights that came out of those all right a few things I saw walking around this time so one I heard a question about how does it decide what values to use in the list and a hypothesis that it's just doing the integers that is its default but you can actually make it more dense it will follow whatever linear sequence you start with and so here if I start at negative 5 and then do something like negative 4.5 it will now count by halves if I do negative 4.9 it'll count by tenths so we can do anything that we want with those lists I saw a few folks who seem to hypothesize that the path that the vertex follows is itself a parabola do folks agree with that anyone want to share what parabola they think it is hit it negative a x squared plus c nice so we know it's going to be symmetric so we probably think there won't be any b term and this looks quite plausible I'm curious if someone is able to prove that this is the parabola it follows but there's oh hit it a little louder please perfect so we found the the coordinate the x coordinate was that negative b over 2a you plug it in and you end up with c minus b squared over 4a squared and so that tells you that it's going to follow this parabola is that right did I hear that right perfect um so that is correct there's a thing that I've always wondered though like that's descriptive of what's happening but I don't have an intuitive sense for why algebra sometimes gives an intuitive sense it doesn't always I tried following doing the following thing when I first graphed this and it opened up my eyes and I didn't see anyone else do this so maybe this is going to be a surprise for folks what happens if I graph y equals just bx plus c and now as I change b what's staying the same what changes and what does that tell you about where the parabola should place itself and what a fun connection between algebra and calculus right this is looking at the slope the linearization at x equals zero all right so this was just one example where maybe folks who thought you knew a ton about parabolas maybe learned something I don't know I did and maybe the software helped you um helped you think about parabolas more deeply did I see a hand up okay great um let's try one more example of this before I set you free with some uh tools and techniques that that might be uh that might be kind of fun to use um so this was when we first introduced integrals and we do integrals numerically um in part because writing a symbolic integrator is as far as I know impossible uh to get perfect and really hard to get close um and we found uh a technique it's actually not that old it's maybe 10 or 12 years we're doing numeric integration really quickly um and one of my uh beliefs is that technology that thinks with you it has to be extremely responsive it has to be so fast to try out a hypothesis that you want to try out many of them so what I'm going to do here is try doing the integral from like let's define a function first sorry f of x I'm going to do my favorite one to show in a demo and I'm going to do the integral from zero to x of f of t dt and it's going to now plot this antiterivative function but what I wanted to do here was say what would happen if instead of being the integral from zero to x it was the integral from one to x hypotheses what's going to change if I change the zero to a one it's going to shift it's going to shift vertically yeah um let's try it sure enough shifts down a little bit did people expect that okay awesome what happens if I change it to two it's going to shift again but it's always keeping the same shape this is a little bit why uh we talk about the plus c when we're doing any derivatives but I wanted to go a little bit deeper and try hooking this up to a slider so we do a slider integral from a to x and we're going to see that it's just kind of bopping around popping up and down so we've got an intuition um built up over a lot of calculus for why it is that the shape stays the same but it shifts up and down does anyone have an intuition for how it picks where it's going to be vertically what's that you can integrate from zero to a to find out where it's going to be for what a is that's one technique I love it any other thoughts in the back yo it will be zero when x is a let's try that and suddenly we get a picture I'm going to put a comma zero and watch this it's just shifting the parabola to be at that point and that's so natural when you think about it this way but when I first saw that I was like of course that's where it has to go it has to position itself so that it's zero here because we know the integral from a to a is zero um all right so on the next screen play around with integrals a little bit if you would like see if there's anything that you can learn from that and then I'm going to go deep into lists and polygons and colors and we're going to just mess around with math for a little bit if folks are okay with that oh yeah so put a comma after it yep exactly oh heck yeah pushing it to the limit that's actually your choice so if you click and hold on the icon here um yeah click and hold on it yep then you can choose which directions it drags all right I'm going to pull us together are folks having fun playing with the calculator is that a good way to do this all right then I'm going to teach you some techniques that I imagine very few people in this room have uh played with I hope um all right so first we're going to do this thing where everyone raise your hand if you have ever heard of Desmos and then keep it raised if you have taught using Desmos and keep it raised if you have uh used sliders in Desmos before today um and I guess yeah re-raise it if you've never taught but did you sliders um and have used lists before today and have used dynamic colors before today and have used list comprehensions before today all right there's going to be something for everyone here we go um so I'm going to open up folks so that you should feel free to just ignore me and play around instead of uh following and I'm going to show you what I've done for the next five screens actually yeah I'm just you're unrestricted freedom go crazy um so here I've got a couple screens that are going to demo and I'm going to also be doing this live a couple of the features that we're going to look at um so this one is about using colors dynamic colors and lists um and then I've got one using uh polygons and list comprehensions um and then a playground screen that you should feel free to play with or just go to Desmos and then a few examples of graphs from actual students that combine all these features together and so part of the transparency part of technology that I think helps folks think is that you can actually see everything that went into it it looks like magic I couldn't have made that graph from scratch but everything that they did is there for the viewing um so you can uh learn from it and build on top of it and add your own thoughts um but let's just uh do a little bit of some features that maybe folks haven't seen as much we're going to start with colors um but first a little backstory so uh we get a lot of emails every single day of people who are mad about various things um the number one thing that people are mad about is the fact that math is really hard on computers and floating point math sometimes gives you the wrong answer point one plus point two is not exactly equal to point three in base 10 if you've got a finite number of bits the second most common annoyance is the answer we give for zero to the zero does anyone have strong feelings about zero to the zero nice this is my kind of room someone want to share your very strong feelings about zero to the zero undefined we've got a strong feeling for your undefined anyone else have a strong countervailing theory or feeling what is it indeterminate form also similar I heard an argument for one would you like to make an impassioned case for one uh okay so one is the limit of x to the x as you're approaching let's hear an impassioned case oh that's fun in polynomials the ones digit they're always counting as zero to the zero there isn't a hole at zero for every time that we graph a polynomial I like that one I like the binomial theorem claim I like the thought about it's like you're multiplying it by itself this number of times but you never even started so like what else could it possibly be um but the third most common complaint that we used to get is that we didn't graph the color yellow um people would say I really need yellow in my graph and we would say build your own calculator that was our that was our stock response um if you really want yellow we didn't do it in part for accessibility reasons um but also we didn't want uh kids spending all of their time thinking about the colors we wanted them thinking about math and then we realized there was a way to have it so the kids would think about both which is that they need to use math to make their colors so here's how we did that I'm gonna graph an equation I can open this I've got these six default colors every single one of them is um compatible with the WCAG standards except for orange which is why we skip over it and that was a compromised position with our unruly users um but if you wanted to find a color we're going to need to assign it to a variable and there's two different color encoding schemes that you can use one of them is RBG I always forget if it's RBG or RGB which one was the supreme court justice uh and hero and which one is a color scheme um it is RBG nope GB oh man after all that um and then each of these is a number between zero and 255 because that is two to the eighth and that's a nice compact way to encode something for a computer um and so zero zero zero says there's no red no green no blue it comes out black and 255 255 255 says that there's um full saturation of all three of them and it comes out white um and you can also make them anything in the middle so I'm actually going to do RGB like this um and I can adjust them and watch this change color I'm going to hook it up to this just so that it's a little bit more visible so here's our color I'm also going to make it really thick so this is going to be 20 pixels wide so you can see it and now we can watch what happens when I change the amount of red and the amount of green and the amount of blue and you can make some interesting commerce uh combinations the other color scheme that could be interesting we're going to make a different color this is HSV anyone know what the HSV stands for I always forget hue saturation value what do those three things mean no idea hands thrown up correct um so I think that the hue is uh the color and it's uh between zero and 360 which has nothing to do with bits or bytes but has to do with our arbitrary how many degrees there are in a circle and then s and v are both between zero and one and one of them is something like how much black there is in the color and the other one is something like how much white there is in the color um but I it's it's a little bit confusing but you can play with it and watch what happens so I'm going to change this to that new color and I'm going to go full saturation and I'm also going to go full value and then as I drag this you'll see it goes around the color wheel and for any one of these colors I can make it fully white by getting rid of all of the saturation and I can make it fully black by getting rid of all of the value and it's a better color scheme for various things but you can play around with these but what I wanted to show you is that part of the dream of this tool is that all of the features stack on top of each other so that you can do that progressive building as you go and so I'm going to combine this with lists so what I'm going to do here is make this a little bit thinner let's make this like five pixels wide and I'm going to have this not just be sine of x but sine of x plus um let's go like 0.51 all the way up to 10 something like that um actually I did this way wrong I'm going to make a list that is from 0 to 10 and I'm going to have it be the sign of that list divided by two so anytime you have a list you can also divide it it divides all the elements you can exponentiate it exponentially exponentiates every element here I've added um like this but I'm also going to want to make a array of colors and the way I'm going to do that let's make this a little bit thicker just so that it shows up a little bit better um and instead of hsv all being 0 to 1 here I'm going to have our saturation be the list as well and it's going to be l divided by 10 and now suddenly each one of them also has its own color and so we can see what happens if I change v against all of these and maybe we'll get a little bit of a better sense of how those two things relate or I can try going around the color wheel and seeing what stays the same and what's different so spend a few minutes if you would like just playing around with colors combined with lists try to make something fun and feel free to take inspiration oh please is there an easy way to uh easy no but possible yes um so there's a set of list elements that we can do um let me try to do this on the fly they say you should never do that forgive me if this goes badly um is there a way to manipulate the list so that the last element becomes the first element and so what I'm going to do is make a list here l is the values 1 through 10 um and just so that we can see what's happening I'm going to plot l against l like this and I'm going to make a new list that uh shifts that list by 1 so I'm do you want the element from the end to come to the beginning or the element from the beginning to go to the end either way all right so we can use a syntax here that says take the second element to the last element and then I'm going to join the first element so we can do this and now if I plot l l2 we're going to see that it's a shifted version of the list so you can get subsections of a list and then you can also join two lists together um so that combo should give you what you want but yeah play around with lists play around with colors for just a couple minutes and feel free to take inspiration from any of the graphs that are on future screens where students are using this combination of dynamic colors and lists to make things that look like they're rotating in three dimensions or solve that problem that we spent a while working on yesterday of the apollonian circle packing and I'm going to pull us back together in seven minutes so you gotta click and hold on that icon and then it's now a new color that's available to you it's that bottom one so if you click on that now it's going to be that one I'm going to demo that actually good call um for folks who haven't found how to change color that is not your fault that is my fault and the answer is that the icon next to every expression if you click and hold on it it gives you options and one of them is that list of colors so you need to click and long hold on the icon next to an expression and that'll let you change the color absolutely so same idea where for r you would make r a list and so try something like zero yeah you can now if you just turn it off then that'll be fine um and try doing that times 36 so times is going to be shift eight perfect and now you'll get a big range yeah yeah you can do it to r and g and b um and actually I chose the wrong it should have been times 250 sorry times 25 because the number should go from zero to 250 you can't do that yet because that would require us to be uh programming on the gpu which is coming but not yet um but you can sort of by making a list of values and then using that exactly exactly so I'll actually demo how to do that in one second good call yep yeah that's going to take some work well I would just click like another slider yeah so I think um you'd want to figure out how to write all of those in terms of a different parameter and then you can drag it it's not even irritating at all so my r changed to the radius oh yeah I can't I like I don't like that if you click on the display then it just like won't show it and that'll be fine okay so it can still use it that's like there's a few special variables oh it is changing the color yeah okay yeah I knew that I knew that r was a special variable so but it also can be referenced yeah it means both okay so it's like both plotting and referencing it's a very weird one that is cool thank you cool or annoying perk of mathematic notation one or the other nice yeah so for I would do I did yeah and so you could re-reference it yeah just change s equals um from that to that like 1 dot dot dot 10 divided by 10 because we want it to be a value between one and 10 and then it will map to each one all right I'm going to pull us together with apologies because I want to show one more feature which was a question from over here and then I'd love to go on to the next section and I think I've got 12 minutes left is that right all right we are I think gonna successfully fit three hours into one hour um partly by going extremely shallow so question was how do you have the color depend on the position of a point or a value um and it turns out that that is not possible inside of our system yet but that's never stopped the students who use it from figuring out a way to do it anyway and we added a feature that I think you're going to find uh useful I hope which is that lists are very one-dimensional and you can still do very interesting things with them right like I could make a list here of the values 1 through 10 or negative 10 to 10 and I could graph something like l and l squared um and this is great I managed to plot a bunch of points but it's still fairly one-dimensional and the thing we might often want to do is for example make a grid of points and play around with that grid of points so we added a feature we imported it from um python and some other computer languages um which is called list comprehension where you can make a list out of other lists so the other thing you might have noticed is that when you like add a list with a list you end up with a list that's the same length it doesn't give you the cross product that's the size of the whole matrix because that would have been absolutely bananas for a lot of things we tried but you might want to do that anyway and so here's how we can do that we do our brackets and I'm going to do something like x y 4 x is 1 to 10 and y is 1 to 10 and then uh when I zoom back to a reasonable viewport I'm going to get a whole grid of points um and so now what you could do is define colors based on for example the x and y values should I try this on the fly this might be a disaster all right I'm going to try it I'm going to make this zero point one up to one in that direction and I'm going to make the zero point one up to one in this direction I'm going to zoom way in I'm like this let's make these points all a lot bigger just so that they show up nice on our screen so instead of being nine I'm going to make them uh 20 30 something like this oh man I've never tried this all right I think I think we're going to be able to do it um I'm going to make this a thing that we can reference and I'm going to make a set of colors which I'm going to say that my colors are let's do HSV I'm going to want this one to be a slider that can go from zero to um 360 so I've got my same classic colors let's hook it up to here so now as I change h this is going to change and I'm going to make it so that the x coordinate gives us the saturation and the y coordinate gives us the value and so I'm going to do I don't know if this is going to work l dot x l dot y and we managed to make a grid so this is the way that you can have the color depend on both of the coordinates if you want to so the idea all of this is that you could open up this graph you could see it you could build on it and you could maybe make something uh like some of what we've seen in the rest of this activity um so I'm going to actually leave this behind here and feel free to play with it all you want later feel free to accost me I think I've got q and a right after this anything that you run into that you're curious how it works or especially any complaints what I wanted to do is go on to the last piece for me of technology that thinks with you and this one is a little bit of a stretch to fit into this talk but I wanted to do it anyway which is the importance of technology being accessible um and for me this means many many different things um you should be able to show up and use it the first time even if you're not a confident math learner that's really important to me but the specific I wanted to talk about here is accessible from the perspective of students with learning differences and students with various disabilities physical or mental disabilities and the one that we think about a lot is students who are blind and students who are visually impaired um and part of the reason for this comes back to a story I heard from our lead accessibility engineer who is himself blind um and he was describing what it used to be like if you were a blind student trying to learn algebra and the answer was that you just basically ran into a wall like the the pinnacle of technology for a blind student would be a piece of braille graph paper so picture this you've got graph paper and it would have um like raised axes and raised viewports and then a piece of clay with a little wire in it that you would use to form the graph that you wanted to show so someone would say graph y equals x squared and you would try to approximate it using clay but there wasn't any dynamic technology to do it so there was no way to check it there was no way to get feedback there was no way to try that thing where you move a slider and you notice how it changes um and so he said his mission was to try to make it so that uh vision impaired and blind students could also have dynamic interactions with the math um and to me this was such an eye-opener of technology can actually help you think deeper thoughts it can help you explore broader domains it often doesn't it often does the opposite but it can and for some populations the technology just wasn't there to help explore those domains um so i just want to show you a little bit about what steve built because i'm just insanely proud of it um there are a few different features uh around for example talking to you as you're typing so you type out an equation or an expression and it will evaluate it but my favorite one is um audio trace where you can graph a line like this go into here click the play button and it will play it as sound and this is uh we'll see if this works let me know if you can hear it did that come across and i can change this i can make it something like 10 sign of x and we can see how that's going to change the shape or sound of this curve the audio is way better on your own computer i promise um there are a few just like very fun details that steve built into this like it's um stereoscopic and so if you're wearing headphones it starts sounding like it's coming out of your left ear and ends sounding like it's coming out of your right ear um and when there's intersections between graphs he recorded himself making a little like pop sound with his cheek you know that thing and so it pops when it hits some of these um but one of the reasons that i'm so obsessed with accessibility is that it turns out that when you build an accommodations it ends up helping everybody um and my favorite metaphor for this not metaphor example um was all of the laws around what's called curb cutting which is the idea that um if a sidewalk is this elevated a wheelchair isn't going to be able to get up over it and so anytime that you have public access you need to make sure that there's a cut in the curb um and this was i think pushed really hard by advocacy groups for folks with limited mobility but it turns out that anytime you're dragging a roller bag you're probably grateful for this and anytime that you're looking down at your phone and walking and don't trip and fall on your face you're probably grateful for this um and this audio trace is an example of that where we've seen classrooms do just such interesting things with audio trace and it opens up some new understandings and i wanted to show you one very straightforward example of this and then some not so straightforward examples um which is it's kind of counterintuitive and definitely a convention that we plot things from left to right and that slope being positive looks like this um and i bet that when you were learning this and i bet that when folks who are teaching we're teaching this and when folks who are teaching teachers who are going to teach this you're going to run into this um and it's not that obvious that the slope of this is one that it's increasing that it's going up that it should go from left to right um but as soon as you play this as sound you notice something about it right and if i were to switch that if i make it negative it's going to sound really different and it's going to go down and so one of the things that i've heard is that when folks are teaching slope turning on audio trace is this light bulb moment for some of the students in their class um the other fun thing about it is that students now get really into i'm trying to make it do their bidding and so instead of just i want to know the shape of this curve they say for example i want to make a graph that plays the moonlight sonata and then they do or they say that they want to do oh man let me find it this one um anyone want to guess what song this is oh it's going to come out so bad on the audio oh man this is going to be a bummer but let's try it anyway someone managed to write a full rick roll as one equation look at that that's a singular equation and one of the best uses of what's that function i think it's a logistic curve does that look right i don't know one of the best uses of that i've ever seen um so encourage folks to feel free to play around with this feature also um and see what it reveals for you but i think that is oh i time this perfectly wow um i think that's uh that's everything that i wanted to share today um my takeaways that i'm hoping for you and then there's a closing screen where feel free to add reflections um feel free to enjoy the calvin and hobbs comic on that last screen but this is my first draft of what i think it requires for technology to think with you instead of for you and the things you can look for are that it centers the human instead of the product you can look at what is the marketing is it all about how cool the tech is or is it all about how cool the stuff you can do with the tech is um it's really transparent that's the way that you can think thoughts that you can see and for me the like pinnacle of transparency is pencil and paper which i maintain is one of the best pieces of technology humanity has ever come up with um it lets you build incrementally it lets you try a thing and then build on top of it and then build on top of that and build on top of that that's the only way to have a really really profound deep thoughts or maybe the best way um and it's something that we do in math all the time it's something that we've been doing for three weeks in our morning math um and finally is accessible because this is our way of making sure that technology thinks with all of us not just with some of us and every time that we build new accessibility tools it ends up benefiting every single person um so that's me thank you for having me here um i've heard such wonderful things about this conference did not disappoint and i think over to whoever is in charge thanks very much Eli so that was wonderful