WEBVTT Kind: captions Language: en 00:00:01.490 --> 00:00:06.720 Well hello there. Good morning or good afternoon or good evening whichever of 00:00:06.720 --> 00:00:14.610 those applies to your current context and welcome to OAME 2020 virtual. My name 00:00:14.610 --> 00:00:19.439 is Nat Banting and for the next 30 to 40 minutes ish we are going to be 00:00:19.439 --> 00:00:25.410 talking about teaching algebraic flexibility. And so before we kind of 00:00:25.410 --> 00:00:30.179 dive into it on a very interesting platform and a time constraint, I just 00:00:30.179 --> 00:00:34.860 wanted to highlight a couple things. The first is a word of thanks to both the 00:00:34.860 --> 00:00:40.079 organizing committee for OAME 2020 of which I was hoping and planning on 00:00:40.079 --> 00:00:45.210 attending and then also the emergency planning committee as it were for the 00:00:45.210 --> 00:00:48.930 virtual conference. I think this is just such a fantastic opportunity that's been 00:00:48.930 --> 00:00:52.949 put together for us to to learn together even though we have to be a part. And so 00:00:52.949 --> 00:00:57.329 while we think about that, I think if you are interacting on a digital space I 00:00:57.329 --> 00:01:02.190 will encourage you to use the hashtag OAME2020 on Twitter even if you're 00:01:02.190 --> 00:01:08.670 watching this much later than the original conference date. I think we can 00:01:08.670 --> 00:01:14.610 still gain some value through interacting in that digital space. And so 00:01:14.610 --> 00:01:19.130 as I said my name is Nat Banting I am a mathematics teacher in Saskatoon, 00:01:19.130 --> 00:01:24.450 Saskatchewan. I've been out to OAME a couple times now. I've taught in 00:01:24.450 --> 00:01:30.750 Saskatoon high schools for 10 years. I've been a district math coach and this year 00:01:30.750 --> 00:01:34.710 I'm on faculty at our local university, the University of Saskatchewan and I'm 00:01:34.710 --> 00:01:38.009 teaching pre-service teachers, future mathematics teachers and that's been a 00:01:38.009 --> 00:01:42.090 very interesting transition and I am transitioning back to going into 00:01:42.090 --> 00:01:46.140 the classroom in the fall. So it's all kind of a amidst these crazy times 00:01:46.140 --> 00:01:50.759 there's a bunch of change in my world all over the place. And I just want to be 00:01:50.759 --> 00:01:54.630 absolutely explicit from the beginning that as a very young teacher I grew up 00:01:54.630 --> 00:01:59.640 with a online support network of sorts filled with ambitious mathematics 00:01:59.640 --> 00:02:03.899 teachers. And that to a large extent has shaped who I am as a teacher today and 00:02:03.899 --> 00:02:08.280 so I will encourage you to kind of take these three avenues to get in contact 00:02:08.280 --> 00:02:12.060 with me about anything that you hear or any other information that you would 00:02:12.060 --> 00:02:14.850 like. There's a Twitter. Please tag me if you have any questions. 00:02:14.850 --> 00:02:18.270 And this is a website where we can find all the materials we'll be talking about 00:02:18.270 --> 00:02:21.690 and then an email address if you have anything. Please don't think that this is 00:02:21.690 --> 00:02:25.860 a hollow invitation. I am sincerely hoping that you will take me up on this 00:02:25.860 --> 00:02:30.330 if anything's kind of burning in your brain because I wouldn't be where I am 00:02:30.330 --> 00:02:36.540 had it not been for the gracious collaboration of others. And so with that 00:02:36.540 --> 00:02:40.980 I think we're gonna start in a nice place and that is just with well-wishing. 00:02:40.980 --> 00:02:45.060 I hope you're doing well and I think one thing I've learned in 00:02:45.060 --> 00:02:50.040 the last little while here, while we've lost our regular context of teaching is 00:02:50.040 --> 00:02:54.870 that teaching very clearly for me at its core is about nurturing student 00:02:54.870 --> 00:02:59.370 well-being and that's mathematical well-being or otherwise. And that has 00:02:59.370 --> 00:03:03.570 never been clearer and so I want to premise everything I say today with that 00:03:03.570 --> 00:03:07.980 preload that that I want us to care for students above all. And maybe you're 00:03:07.980 --> 00:03:12.030 watching this after the threat of coronavirus has been minimized and if 00:03:12.030 --> 00:03:16.890 that's the case I hope that we are still thinking about student well-being above 00:03:16.890 --> 00:03:23.040 everything else. And so that's just maybe a gentle reminder as we move forward. 00:03:23.040 --> 00:03:28.459 I'm gonna base what we talk about today on a simple premise and that is my recent 00:03:28.459 --> 00:03:33.080 interactions in a classroom in an era of social distancing have made something 00:03:33.080 --> 00:03:39.410 abundantly clear. And that is I notice and maybe I'm more sensitive than ever 00:03:39.410 --> 00:03:45.090 when a student is being asked simply to digest content and not make sense of it. 00:03:45.090 --> 00:03:50.459 And I think before these times when I've been I had so much taken away from my 00:03:50.459 --> 00:03:54.810 teaching arsenal so to speak, I was able to kind of bury that fact. But now it's 00:03:54.810 --> 00:03:58.049 kind of standing out and it's right in my face and it's been bugging me 00:03:58.049 --> 00:04:03.600 recently. And so I think these ideas are even more important as we explore ways 00:04:03.600 --> 00:04:09.269 to have students act flexibly and make sense of their mathematics and not just 00:04:09.269 --> 00:04:14.310 kind of go to this very simplistic version because of our context. And I 00:04:14.310 --> 00:04:16.739 think this applies back to when we're face-to-face with learners as well 00:04:16.739 --> 00:04:22.380 eventually. So our purpose here is to explore two key characteristics for 00:04:22.380 --> 00:04:26.580 building algebraic flexibility. So by that 00:04:26.580 --> 00:04:32.340 what I want to do is kind of show you through a couple activities what I see 00:04:32.340 --> 00:04:38.939 as really really high yield practices or tenets that we can go through when we 00:04:38.939 --> 00:04:42.930 are thinking about having students act in flexible ways especially in those 00:04:42.930 --> 00:04:46.860 upper grades when we deal a lot with equations and functions and things of 00:04:46.860 --> 00:04:52.889 that nature where a lot of the work seems to distill into symbolic 00:04:52.889 --> 00:04:57.629 manipulation very very quickly. And so to do so we're gonna be using a digital 00:04:57.629 --> 00:05:01.800 platform as you know because you're here. So I can't interact with you like I 00:05:01.800 --> 00:05:05.940 would like to as a teacher but I can sort of interact with you and in in 00:05:05.940 --> 00:05:09.539 certain ways by by pausing the video and asking you to interact with certain 00:05:09.539 --> 00:05:14.219 tasks. So at any moment you can pause this video. That's the nice thing about a 00:05:14.219 --> 00:05:18.419 pre-recorded session. You can grab supplies if I ask you to grab them or 00:05:18.419 --> 00:05:20.969 you can let the dog out if it's scratching at the door. You can do any of 00:05:20.969 --> 00:05:25.080 those things. I won't be offended because I can't be offended. There's no way that 00:05:25.080 --> 00:05:29.189 I will know. I think the the conference might have a way that we can kind of 00:05:29.189 --> 00:05:33.300 chat on a document as we go through the first time but also those digital 00:05:33.300 --> 00:05:37.650 platforms a little bit are available to you or you can get in contact with me 00:05:37.650 --> 00:05:42.779 about anything that you that you see or any questions that you might have. So the 00:05:42.779 --> 00:05:47.039 majority of what we'll do, we'll do one Desmos activity kind of closer to the 00:05:47.039 --> 00:05:50.669 end, but the most of this will kind of be pause and work and come back and we'll 00:05:50.669 --> 00:05:55.560 sort of I'll try to anticipate what you're what you're going to do and we'll 00:05:55.560 --> 00:06:02.490 move on from there. So as a teacher, that all being laid, I think we should start 00:06:02.490 --> 00:06:06.509 in the best place that I know where to start and that's with a problem or a 00:06:06.509 --> 00:06:14.279 prompt. So this one came across my Twitter feed I think last week and I 00:06:14.279 --> 00:06:17.940 immediately fell in love with it. So what's happening here is you have 00:06:17.940 --> 00:06:22.770 some matches they're arranged like the digits on a clock face. And the question 00:06:22.770 --> 00:06:31.830 is, what's the largest number you can get if you move exactly to matches? So I do 00:06:31.830 --> 00:06:35.459 have some instructions for you before we get started. Number one you want to grab 00:06:35.459 --> 00:06:39.060 something where you can have like some work in front of you. So you're not doing 00:06:39.060 --> 00:06:42.000 this in your mind. I want some kind of written work. And as you do 00:06:42.000 --> 00:06:46.319 so I'm gonna be specific and ask you not to erase anything that you write. So just 00:06:46.319 --> 00:06:50.219 please keep everything in front of you. I don't know if that takes multiple sheets 00:06:50.219 --> 00:06:53.400 of paper or a very large surface, a dry erase surface, do something. I would like 00:06:53.400 --> 00:06:57.330 you to keep kind of a remnant of your thinking on this on this task. And also 00:06:57.330 --> 00:07:01.169 if you could I come up with a solution and then another one and then another 00:07:01.169 --> 00:07:03.960 one. If you could think through this in multiple ways and and just number them 00:07:03.960 --> 00:07:08.759 as you come up with them. So this was my first one, this is my second one, etc. 00:07:08.759 --> 00:07:13.050 I think that that process might be nice for you to go back. So at this time I'd 00:07:13.050 --> 00:07:16.469 like you to pause this video. Work through this for two to three minutes 00:07:16.469 --> 00:07:20.699 and kind of generate a few solutions. You can only move exactly to matches and 00:07:20.699 --> 00:07:26.360 make the largest number possible. Go ahead pause the video and give it a shot. 00:07:32.599 --> 00:07:36.569 Fantastic and so now hopefully what you've done is you've kind of come up 00:07:36.569 --> 00:07:42.569 with several ways maybe you had a goal in mind. I know when I started this task 00:07:42.569 --> 00:07:46.410 I thought about, is there a way that I can make nines? Nine seemed to me like a 00:07:46.410 --> 00:07:52.919 very powerful number. The zero feels like it's a waste of matches. So we have six 00:07:52.919 --> 00:07:56.130 matches being used on the zero and I would like to rearrange them in some 00:07:56.130 --> 00:08:00.810 sort of way. And so I've just come up with a few possibilities that I think 00:08:00.810 --> 00:08:06.630 people might try. And the first is this. And we end up with nine eight nine three 00:08:06.630 --> 00:08:10.860 which which is a significant improvement. Right so we're you know about, we're 00:08:10.860 --> 00:08:17.520 almost two thousand higher than we were at the previous arrangement. And I when I 00:08:17.520 --> 00:08:21.270 see this there was nine eight nine three, the problem is there's a very simple 00:08:21.270 --> 00:08:25.500 move to get nine eight nine five and maybe some of you are on there. There's 00:08:25.500 --> 00:08:29.069 also I believe a way to get nine nine nine nine. And I think some people are 00:08:29.069 --> 00:08:32.430 kind of operating under that and maybe you're screaming at the screen right now 00:08:32.430 --> 00:08:36.149 being like yeah like the nine is sitting right there. We can get nine nine nine 00:08:36.149 --> 00:08:41.219 nine. It's very straightforward. I'm gonna propose that all those people that have 00:08:41.219 --> 00:08:44.970 this sort of arrangement are operating under the assumption that we have to 00:08:44.970 --> 00:08:50.610 keep four digits. But it doesn't actually take a lot for you to realize that might 00:08:50.610 --> 00:08:57.750 not be the case. So I can move two toothpicks and I can get nineteen 00:08:57.750 --> 00:09:04.290 thousand thirty five. And I didn't break a rule. Like you might have broken the 00:09:04.290 --> 00:09:08.190 self-imposed rule that there has to be four digits but clearly the instructions 00:09:08.190 --> 00:09:13.170 were move two matches which I did and I get nineteen thousand thirty five. And 00:09:13.170 --> 00:09:18.149 this opens up a brand new set of possibilities which of course some of 00:09:18.149 --> 00:09:21.570 you may have figured out. It is possible to get a six digit number and I welcome 00:09:21.570 --> 00:09:25.800 those responses on Twitter when you figure out which one that is but I will 00:09:25.800 --> 00:09:29.370 move forward and I'll provide you another avenue of thought which I 00:09:29.370 --> 00:09:33.690 thought was particularly ingenious. And that was someone who suggested that you 00:09:33.690 --> 00:09:38.910 could move to matches and get 8 E 9 9. And so for those of you old 00:09:38.910 --> 00:09:43.779 school calculator enthusiasts will recognize the E as to the power of. 00:09:43.779 --> 00:09:52.209 So that is eight to the power of 99 which is significantly bigger than any five digit 00:09:52.209 --> 00:09:56.110 number. And so we have this kind of new set of rules. And I also will tell you 00:09:56.110 --> 00:09:59.769 there are more creative ways to get even larger exponents but you're guided by 00:09:59.769 --> 00:10:04.569 this new idea that exponents are allowed. And when they're allowed it opens up 00:10:04.569 --> 00:10:10.689 these brand-new sets of possibilities. From this I want to make two points. And 00:10:10.689 --> 00:10:14.410 neither of them have to do with algebra at this point but we will get there. 00:10:14.410 --> 00:10:20.709 I wanted to ask you this question to begin with to create two feelings. The first is 00:10:20.709 --> 00:10:24.939 that you had to make the decisions. All right you were the one that we're 00:10:24.939 --> 00:10:29.379 creating the object. Sure I gave you parameters, like the number of matches 00:10:29.379 --> 00:10:33.699 that existed on the slide, the number of moves you were allowed to make, and you 00:10:33.699 --> 00:10:38.110 had the goal to make the largest number. But it was your decision that activated 00:10:38.110 --> 00:10:40.990 your sense-making. So you were kind of in control about which matches would move 00:10:40.990 --> 00:10:45.100 where. You might be making underlying decisions based on I'm gonna keep four 00:10:45.100 --> 00:10:48.639 digits or I'm gonna try to make as many digits as possible or I'm gonna try to 00:10:48.639 --> 00:10:51.759 use an exponent. These sorts of things but you were making those decisions. 00:10:51.759 --> 00:10:55.980 That's the most important thing I want at the forefront today. The second thing is 00:10:55.980 --> 00:11:01.709 the difference between the solutions that I showed you was more powerful if 00:11:01.709 --> 00:11:07.180 it wasn't what you originally did. And why I asked you to sort of sequence 00:11:07.180 --> 00:11:10.689 through your solutions is I want you to go back and see if there was a moment 00:11:10.689 --> 00:11:15.399 where maybe you were operating under the what's the maximum four-digit number I 00:11:15.399 --> 00:11:18.819 could make and then you switched I wonder if I could make a five digit or a 00:11:18.819 --> 00:11:22.720 six digit or I wonder if I could use an exponent. There's a moment there where 00:11:22.720 --> 00:11:27.069 the difference is powerful. And we're gonna play on these ideas of making 00:11:27.069 --> 00:11:30.040 decisions and noticing difference because something about a difference 00:11:30.040 --> 00:11:34.059 sort of shoves it right into your awareness. And that helps as we've tried 00:11:34.059 --> 00:11:40.420 to build flexibility to understand that there are alternative options. And so now 00:11:40.420 --> 00:11:45.939 that those two things are at the forefront of your mind I'm hoping to 00:11:45.939 --> 00:11:50.259 transition us into an algebraic state of mind. And so I'm going to describe one 00:11:50.259 --> 00:11:53.620 structure that I built for my high school classroom when we started to get 00:11:53.620 --> 00:11:57.500 into those really symbolically driven, heavy 00:11:57.500 --> 00:12:01.640 algebraic units of study. Really once you get into that grade 11 level in 00:12:01.640 --> 00:12:07.970 Saskatchewan you see the focus switch into that precalculus route. And so then 00:12:07.970 --> 00:12:12.770 we get all these functions and equations that we need to deal with and I've 00:12:12.770 --> 00:12:18.530 noticed that there's a lot of operating with those algebraic objects that is 00:12:18.530 --> 00:12:23.570 very very rigid. And so can we take these two feelings, the one where you are 00:12:23.570 --> 00:12:28.040 designing decisions and the one where we can emphasize difference and kind of 00:12:28.040 --> 00:12:32.570 throw it right in front of our learners as a way to sort of abruptly jolt them 00:12:32.570 --> 00:12:38.420 into making sense with algebraic objects. Today we're going to use functions just 00:12:38.420 --> 00:12:41.360 because of time. I was going to show a little bit with equations and a little 00:12:41.360 --> 00:12:45.260 bit with functions and maybe some other ideas that the menumath community has 00:12:45.260 --> 00:12:48.740 come up with, but I think for today if we stick with functions, and if there's any 00:12:48.740 --> 00:12:52.700 other interest we can kind of handle that in an online space. So the first 00:12:52.700 --> 00:12:59.150 thing is the name menumath, the hash tag menumath, was just kind of sort of 00:12:59.150 --> 00:13:02.750 started on Twitter by a couple people. I had this idea. I blogged about it 00:13:02.750 --> 00:13:06.380 and then a couple people started making these. And I thought oh I should you know 00:13:06.380 --> 00:13:09.220 I favourited them. I should keep track of these. And then more and more people 00:13:09.220 --> 00:13:14.180 found this useful for their for their classroom and so eventually this hash 00:13:14.180 --> 00:13:18.230 tag menumath started to live at this website. So if you go there, you can even 00:13:18.230 --> 00:13:22.670 pause the video now and go and take a look to kind of see and prepare yourself 00:13:22.670 --> 00:13:26.750 for what I'm about to talk about. You can see a collection from all over the world 00:13:26.750 --> 00:13:30.500 of people who have taken this structure and built it for a different sort of 00:13:30.500 --> 00:13:33.410 object. Some of them are algebraic. There's a lot. There are numbers and 00:13:33.410 --> 00:13:36.290 shapes and all sorts of things. But they're built out of these two 00:13:36.290 --> 00:13:40.640 principles when it applies to algebra and so these two things I'm going to 00:13:40.640 --> 00:13:46.460 claim today are a key for me when we're trying to encourage flexibility. Ask your 00:13:46.460 --> 00:13:51.260 students to be designers and decision makers and number two provide them a 00:13:51.260 --> 00:13:58.550 context that emphasizes the difference between those designs. So if you go to 00:13:58.550 --> 00:14:03.020 the website you see this sort of image, welcome to menu math, etc, etc. It gives you 00:14:03.020 --> 00:14:09.380 a brief history and if you scroll down you get to a master list, master table of 00:14:09.380 --> 00:14:12.800 all the different menumath activities that have been designed. And so I kind of 00:14:12.800 --> 00:14:16.430 screenshotted this just with ones for functions because that's gonna be our focus 00:14:16.430 --> 00:14:20.060 today. But if you scroll down you'll find others from all sorts of grade levels 00:14:20.060 --> 00:14:25.400 and of course you're always welcome to submit one via the website as well. 00:14:25.400 --> 00:14:28.339 I love working through these and kind of posting and seeing the collection grow. 00:14:28.339 --> 00:14:32.150 Most of them also have been translated into French and so that's nice for those 00:14:32.150 --> 00:14:36.200 of you who teach in a bilingual classroom. They're there for you. They're 00:14:36.200 --> 00:14:40.400 already already reliably translated so you don't have to do that work. We are 00:14:40.400 --> 00:14:45.140 going to focus, to kind of describe how these work, on one of them, the systems of 00:14:45.140 --> 00:14:50.510 linear inequalities. So this was one of the first ones that I worked through 00:14:50.510 --> 00:14:54.560 when I was trying to find a way to get students to act flexibly while building 00:14:54.560 --> 00:14:58.880 systems of linear inequalities. If you were to click the handout link there it 00:14:58.880 --> 00:15:02.420 would bring you to a document that looks something like this. And all menu tasks 00:15:02.420 --> 00:15:07.160 look something like this. So the most important part of the menumath tasks 00:15:07.160 --> 00:15:11.330 are the eight constraints that are detailed in the table here. Constraints A 00:15:11.330 --> 00:15:18.410 all the way through H. These are eight characteristics that I want the system 00:15:18.410 --> 00:15:23.720 of linear inequalities to have but I don't need a single system of linear 00:15:23.720 --> 00:15:30.290 inequalities to have all eight. Instead the students are free to build as many 00:15:30.290 --> 00:15:36.770 systems of linear inequalities as they want as long as they satisfy each of 00:15:36.770 --> 00:15:43.220 these eight at least once. However the menumath task asks them to 00:15:43.220 --> 00:15:49.459 do this in as few systems as possible. So they are actually asked to possibly 00:15:49.459 --> 00:15:54.620 satisfy all eight constraints with one system of linear inequalities. If that's 00:15:54.620 --> 00:15:59.690 not possible, the next best case would be to satisfy it with two separate systems 00:15:59.690 --> 00:16:04.900 of linear inequalities. And both of those systems don't need to have every 00:16:04.900 --> 00:16:10.310 characteristic. They just need to cover all the bases, all eight bases, between 00:16:10.310 --> 00:16:16.730 the two of them. So for example, before I set you to work on this, and hopefully 00:16:16.730 --> 00:16:21.860 you haven't paused the video and gotten way ahead of me, this is a possibility 00:16:21.860 --> 00:16:25.030 for a system of linear inequalities that a student 00:16:25.030 --> 00:16:30.880 might use to begin a solution of this menu. So what I've done here is I have 00:16:30.880 --> 00:16:37.210 listed all of the eight constraints, sort of summarized them here. And then I have 00:16:37.210 --> 00:16:41.950 played around in Desmos just really quickly with two functions or sorry its 00:16:41.950 --> 00:16:46.150 not functions, inequalities. So this is y<3x+4 and you 00:16:46.150 --> 00:16:50.470 can see it's graphed in red and then below that you have y>1/2x+1 00:16:50.470 --> 00:16:56.290 And you can see the system of linear inequalities on the 00:16:56.290 --> 00:17:00.400 right-hand side there. It has certain characteristics. And I've taken the 00:17:00.400 --> 00:17:05.740 liberty of bolding them and turning them green. So all of these characteristics 00:17:05.740 --> 00:17:14.410 are satisfied by this system. Okay so can you see where one of the functions has a 00:17:14.410 --> 00:17:19.540 y-intercept of 4? Right we can see that the red, not function sorry I keep saying 00:17:19.540 --> 00:17:24.070 that, the red inequality has a y-intercept of 4 which is sitting over 00:17:24.070 --> 00:17:28.060 here. Okay. This one, doesn't, the blue one doesn't also need to have the intercept. 00:17:28.060 --> 00:17:33.580 It was one inequality had to have the y intercept of 4 so that's satisfied. 00:17:33.580 --> 00:17:38.260 Can you see that all of these inequalities have positive slopes? That they result in 00:17:38.260 --> 00:17:42.460 an unbounded solution region and that there are more dashed lines than solid 00:17:42.460 --> 00:17:47.800 lines. Those are four of the eight are satisfied with this system. However this 00:17:47.800 --> 00:17:52.780 system has left four of the eight unsatisfied. So the student would then be 00:17:52.780 --> 00:17:56.650 forced to do one of two things. Number one you could kind of tinker or play 00:17:56.650 --> 00:18:01.240 with this system that I've given you to hopefully satisfy some of the other 00:18:01.240 --> 00:18:05.350 requirements. Maybe you want to try and move it so the entire solution is in 00:18:05.350 --> 00:18:09.430 quadrant one or maybe you want to move it so it contains the origin. Whatever 00:18:09.430 --> 00:18:14.530 you want to do. You have to change it so all eight are satisfied. Or you could 00:18:14.530 --> 00:18:19.480 leave this system of inequalities as good, as satisfying four of the eight, 00:18:19.480 --> 00:18:26.280 start fresh and build a new system of inequalities that is targeting these 00:18:26.280 --> 00:18:31.570 unsatisfied constraints. But now you're on to your second keeping in mind that 00:18:31.570 --> 00:18:37.270 your goal is to do this in as few systems of linear inequalities as 00:18:37.270 --> 00:18:40.659 possible. So that's hopefully a start. The key piece 00:18:40.659 --> 00:18:44.169 for these is you don't have to, and sometimes you have to stress this to 00:18:44.169 --> 00:18:49.269 students, you don't have to do it in one object. You don't have to satisfy all 00:18:49.269 --> 00:18:55.480 these eight constraints with one system. You can always satisfy a good portion of 00:18:55.480 --> 00:18:59.409 them until you're satisfied with that. Set that one aside and then try to build 00:18:59.409 --> 00:19:03.669 from scratch on the constraints that haven't been satisfied. But in the end 00:19:03.669 --> 00:19:07.600 your job is to do this and convince me that you've done it in as few systems as 00:19:07.600 --> 00:19:13.240 possible. And so with that being said and those instructions laid, I would like you 00:19:13.240 --> 00:19:17.710 to pause the video and maybe sketch out in front of you, don't worry too much 00:19:17.710 --> 00:19:23.559 about the rigor of the actual inequalities. Maybe you just want to do 00:19:23.559 --> 00:19:26.320 this with sketches on the Cartesian plane. Maybe you want to throw it into 00:19:26.320 --> 00:19:29.889 Desmos. Kind of like I showed you there and make sure that that you have all 00:19:29.889 --> 00:19:33.789 your bases covered with it with a high degree of precision. That's excellent as 00:19:33.789 --> 00:19:37.990 well. Again it's kind of at your own pace here because of the online format. So 00:19:37.990 --> 00:19:43.720 that's nice. Go ahead and pause now and please solve my systems of linear 00:19:43.720 --> 00:19:49.679 inequalities menu in as few systems as possible. 00:19:51.480 --> 00:19:57.519 Great! So hopefully you've spent a good portion of time. I know it is not 00:19:57.519 --> 00:20:02.950 necessarily a quick exercise and this is nice, depending on the menu that you're 00:20:02.950 --> 00:20:06.460 working with, some students can see a way to a solution right away. 00:20:06.460 --> 00:20:11.259 Some students are okay with adding one more system just to satisfy that last 00:20:11.259 --> 00:20:15.399 sort of nagging constraint that they didn't get in their first ones. And some 00:20:15.399 --> 00:20:19.090 students will sit there and stare at it, and tweak it, and move it, and model it 00:20:19.090 --> 00:20:23.710 over and over and over again until they don't have to add that extra one. 00:20:23.710 --> 00:20:28.990 They want to get it in as few as possible. Here in my experience with these things 00:20:28.990 --> 00:20:32.169 over the last couple years are some common students strategies. And it would 00:20:32.169 --> 00:20:37.929 be amazing if you could kind of see yourself in this activity. The first is 00:20:37.929 --> 00:20:41.919 one of the most popular things that students do, especially if this is the 00:20:41.919 --> 00:20:46.059 second or the third menu task that they've done, is they tread very lightly. 00:20:46.059 --> 00:20:50.500 And the first thing they try to do is establish a minimum number of 00:20:50.500 --> 00:20:55.900 objects that they need to build. So is it possible? Is it worth chasing the perfect 00:20:55.900 --> 00:21:02.590 solution? Can I get all eight constraints satisfied with only one system? Right and 00:21:02.590 --> 00:21:07.060 so how they do this is they choose characteristics and they ask themselves 00:21:07.060 --> 00:21:11.800 if it's even possible for both of those things to exist together. And so if we go 00:21:11.800 --> 00:21:17.650 back to our systems task, there's not anything, and by design, there's not 00:21:17.650 --> 00:21:22.210 anything that sort of jumps out of the screen at you. There's not something like 00:21:22.210 --> 00:21:27.070 that says the entire solution region is in quadrant one and another constraint 00:21:27.070 --> 00:21:31.780 isn't something like the entire solution region is in quadrant three. If both of 00:21:31.780 --> 00:21:35.920 those needed to be satisfied, we know for sure they can't happen together. 00:21:35.920 --> 00:21:39.490 Therefore we're already looking at designing two objects to satisfy the 00:21:39.490 --> 00:21:43.180 menu. But we don't see that here. I highlighted these three because I 00:21:43.180 --> 00:21:47.050 think these have a very nice interplay. And and this is often where the students 00:21:47.050 --> 00:21:51.820 end up spending their time. Is it possible for these three things to 00:21:51.820 --> 00:21:56.830 happen together? And of course here my job isn't to tell you yes or no. My job 00:21:56.830 --> 00:22:00.180 is to design the the environment, which I did, and hopefully you've already 00:22:00.180 --> 00:22:04.309 struggled with this if you spent some time with the video paused. 00:22:04.309 --> 00:22:09.010 How could you have an inequality that has a y-intercept of 4?Only one in 00:22:09.010 --> 00:22:12.910 the system has to have that. All inequalities have positive slopes. 00:22:12.910 --> 00:22:17.470 We actually satisfied both of those in my sample. If you remember the sample from 00:22:17.470 --> 00:22:23.670 here, this satisfies both of those. So you see there's a y-intercept of 4 and 00:22:23.670 --> 00:22:29.110 everything is positive there but the origin was not in the solution region. 00:22:29.110 --> 00:22:35.040 So is it possible to add F? Could I tweak that system that I built there to add F 00:22:35.040 --> 00:22:40.360 into the constraints that have been satisfied? And in doing so what you're 00:22:40.360 --> 00:22:43.600 doing is you're sort of establishing, what's my goal? What's a realistic goal 00:22:43.600 --> 00:22:47.290 here? What things play well together as far as the constraints and what things 00:22:47.290 --> 00:22:53.020 don't? And notice these decisions are all based on, you are required to make them, 00:22:53.020 --> 00:22:57.520 and there's sort of this incessant tinkering that's going on. Maybe you give 00:22:57.520 --> 00:23:04.039 up on B, C & F combining. B C and D. We know those can combine nicely. 00:23:04.039 --> 00:23:11.989 Can add one to it? Can I go B,C,D, and maybe add G? Is that possible? And of course as I'm 00:23:11.989 --> 00:23:17.179 sure many of you are already aware, we can definitely alter that system to also 00:23:17.179 --> 00:23:20.929 contain G. Actually we don't have to alter it at all. The original example 00:23:20.929 --> 00:23:25.940 that I gave in this presentation includes G because both of the 00:23:25.940 --> 00:23:29.989 inequalities were oblique lines. And I do that intentionally at around the 00:23:29.989 --> 00:23:33.649 seventeen and a half minute mark here. You can go back and catch me in a 00:23:33.649 --> 00:23:37.220 bold-faced lie. That wasn't an accident. That was on purpose. I just wanted to 00:23:37.220 --> 00:23:41.059 make sure that you were in fact paying attention at this point of the 00:23:41.059 --> 00:23:44.279 presentation. The other one is very interesting 00:23:44.279 --> 00:23:47.940 because I like the reference but none of the students get it anymore and I'm 00:23:47.940 --> 00:23:52.739 really hoping that this audience will understand the game of snake. And if you 00:23:52.739 --> 00:23:58.679 were anything like me as a burgeoning math nerd, you spent some time playing on 00:23:58.679 --> 00:24:04.769 coolmathgames.com. This was sort of our Mecca in our computer lab at school. 00:24:04.769 --> 00:24:08.309 And this was the game of snake. You can still go play this game. You know what, if you 00:24:08.309 --> 00:24:12.809 wanted to take a break from this webinar, and go onto coolmathgames.com and have 00:24:12.809 --> 00:24:16.649 a couple rounds of snake, I would not blame you in the least. The game is very 00:24:16.649 --> 00:24:22.169 straightforward. There's a red dot. Well a red pixel that appears on the screen and 00:24:22.169 --> 00:24:26.429 you are controlling the green snake. You start quite small. And you go and eat the 00:24:26.429 --> 00:24:32.039 red brick. I think it's supposed to be an apple. When you eat the brick your snakes 00:24:32.039 --> 00:24:36.960 length grows by one. And then you must continue to meander around the screen to 00:24:36.960 --> 00:24:42.090 kind of get the next apple. The trick is you cannot stop moving. And so your 00:24:42.090 --> 00:24:46.499 biggest obstacle becomes yourself. All right. And every time you eat an apple 00:24:46.499 --> 00:24:52.440 you slowly grow. And that's what I mean with how these students attack the menu 00:24:52.440 --> 00:24:56.909 math problem. They start with a constraint that's familiar to them. 00:24:56.909 --> 00:25:02.249 So maybe that constraint is has a y-intercept of 4 and then they see if 00:25:02.249 --> 00:25:07.350 they can tweak their system to add in another one. Swallow another one up. 00:25:07.350 --> 00:25:12.809 And so in our example, we already swallowed four of the eight constraints. Can we 00:25:12.809 --> 00:25:17.220 tweak it, not throw it out, can we just swallow another red pixel just to kind 00:25:17.220 --> 00:25:20.999 of grow in the number of constraints? And of course the ultimate goal here is to 00:25:20.999 --> 00:25:27.960 slowly bloat your way into the perfect solution of the menu where one system of 00:25:27.960 --> 00:25:32.220 linear inequalities satisfies all eight of those. But you don't try to tackle it 00:25:32.220 --> 00:25:36.929 all at once. You slowly are playing what I call the game of snake, you swallow one 00:25:36.929 --> 00:25:41.549 constraint. Okay fantastic. Is there a way to gain the next without losing? And you 00:25:41.549 --> 00:25:46.590 watch sort of your solution grow into a super solution so to speak. And that 00:25:46.590 --> 00:25:51.659 brings us to our last one. And this is what one student once called the hero 00:25:51.659 --> 00:25:56.970 solve. So he actually did this to me once on a menu that I was doing 00:25:56.970 --> 00:26:00.990 with fractions. And so he came up to my desk and he told me Banting, I found the 00:26:00.990 --> 00:26:05.100 hero solve. And I didn't really know what that meant. I could kind of assume. 00:26:05.100 --> 00:26:09.510 But I've come to call this the super example. So this is very similar to the snake 00:26:09.510 --> 00:26:13.679 strategy except where there is no incremental like hit the red pixel. Hit 00:26:13.679 --> 00:26:17.789 the red pixel. Slowly grow. Slowly gain a constraint. Slowly tinker and gain a 00:26:17.789 --> 00:26:26.100 constraint. This is a reasoned and logical attack on all eight parameters 00:26:26.100 --> 00:26:31.169 at once. So once they once they know that there's no obvious 00:26:31.169 --> 00:26:36.720 contradictions, they sit and they try to imagine a single solution without that 00:26:36.720 --> 00:26:41.610 kind of incremental approach to that one single solution. And so when you watch 00:26:41.610 --> 00:26:45.450 students do this, a lot of times there's not a lot on their workspace. And so what 00:26:45.450 --> 00:26:48.750 I would encourage you to do is don't jump to conclusions that they're stuck 00:26:48.750 --> 00:26:52.470 but ask them what they're trying to do. Ask them if there's a constraint that 00:26:52.470 --> 00:26:56.640 maybe is the easiest that they're trying to tackle. They often will tell ya those 00:26:56.640 --> 00:27:00.600 ones are easy. We're stuck here. And oftentimes if you build these things 00:27:00.600 --> 00:27:05.190 well the super example gets tripped up. And something that isn't immediately 00:27:05.190 --> 00:27:09.840 obvious but becomes very very difficult as you start to kind of design these 00:27:09.840 --> 00:27:12.840 things. And that's what I'm hoping happened to you when you did the system 00:27:12.840 --> 00:27:18.120 of linear inequalities. Okay and like anything, these are really nice because 00:27:18.120 --> 00:27:23.039 they take a fair chunk of time but that time is very dense with reasoning. And 00:27:23.039 --> 00:27:26.850 the reason for that I believe goes back to the two things we're talking about to 00:27:26.850 --> 00:27:32.220 build flexibility. The students are constantly making decisions and there's 00:27:32.220 --> 00:27:36.120 so much difference at play here. There's difference between the constraints. 00:27:36.120 --> 00:27:40.770 There's difference between what can go in to the object like the system in the 00:27:40.770 --> 00:27:43.980 example that we did. There's difference of opinions if you have these people 00:27:43.980 --> 00:27:48.330 doing it in groups. And so that difference really causes them to have 00:27:48.330 --> 00:27:53.610 this very rich and flexible interchange between between the models. And it's not 00:27:53.610 --> 00:27:58.049 a simple, here's a model can you please describe it? It's here's something I 00:27:58.049 --> 00:28:01.890 would like a model to contain can you please design it? And there's a there's a 00:28:01.890 --> 00:28:09.210 big difference there. What we're gonna do kind of as our last activity, as we wind 00:28:09.210 --> 00:28:12.990 down in time, is one of the actually, I think the 00:28:12.990 --> 00:28:17.279 first menu task was a was a parabola menu task. This is where this idea sort 00:28:17.279 --> 00:28:20.250 of came from. So you'll notice that this menu task is a little bit different than 00:28:20.250 --> 00:28:24.779 the last one. You're not building systems of linear inequalities. You're building 00:28:24.779 --> 00:28:31.679 quadratic functions. And there's no longer eight requirements there are 00:28:31.679 --> 00:28:37.130 actually ten requirements which maybe makes the hero solve very very difficult. 00:28:37.130 --> 00:28:43.679 And instead of just sort of pausing the video on this one and having you work on 00:28:43.679 --> 00:28:49.309 it on paper, what happened was through another online collaboration, I 00:28:49.309 --> 00:28:56.610 interacted with the genius of Desmos computationaler Jay Chow and he built 00:28:56.610 --> 00:29:01.799 something for us to work with in this remote setting. And so what he did, and 00:29:01.799 --> 00:29:06.330 you can go here, if you're not familiar with Desmos, I would encourage you to 00:29:06.330 --> 00:29:10.470 become familiar with Desmos and maybe this is a really nice first step to 00:29:10.470 --> 00:29:15.629 becoming a participant in your first student Desmos activity. You go to this 00:29:15.629 --> 00:29:20.730 URL student.desmos.com. If you just can navigate here in your web browser and 00:29:20.730 --> 00:29:28.230 you enter in this code VN4 8FN. What will actually happen when you do that is 00:29:28.230 --> 00:29:33.629 you'll enter into a screen that looks like this. This is the menu. But the nice 00:29:33.629 --> 00:29:39.000 part about Desmos is when you were doing those inequalities, it's very possible 00:29:39.000 --> 00:29:42.750 that you kind of just made sketches of graphs and then you kind of would worry 00:29:42.750 --> 00:29:46.980 about writing the equations for those inequalities later. In desmos what Jay 00:29:46.980 --> 00:29:52.350 has done is as you enter in the functions not only are they graphed 00:29:52.350 --> 00:29:57.240 dynamically right in front of you but he has turned all the of all the 00:29:57.240 --> 00:30:01.679 constraints that you have satisfied green. So what you can see by doing this 00:30:01.679 --> 00:30:06.330 is that by entering the function y equals x squared, not only do I see it 00:30:06.330 --> 00:30:10.889 here, but I know that that never enters quadrant 3, has a positive intercept, 00:30:10.889 --> 00:30:17.340 vertex on the y-axis, etc, etc, etc. Now based on the coding there are also some 00:30:17.340 --> 00:30:20.220 very interesting avenues for conversation. Like maybe this is 00:30:20.220 --> 00:30:23.340 something that your students want to talk about. Does this function really 00:30:23.340 --> 00:30:25.890 have positive y-intercept? I'm less interested 00:30:25.890 --> 00:30:32.429 in the technical correct answer in this case and more in the little caveat, in 00:30:32.429 --> 00:30:38.160 the corner case for conversation. So what I'm gonna ask you to do is instead of 00:30:38.160 --> 00:30:44.730 doing this on pencil and paper is to navigate to Desmos. Log into the 00:30:44.730 --> 00:30:48.240 student account and I will leave this active for as long as possible. So if you 00:30:48.240 --> 00:30:54.120 are watching this a long time after OAME virtual 2020 that's okay. You should 00:30:54.120 --> 00:30:59.610 be able to log into this class code and play around with this parabola menu. 00:30:59.610 --> 00:31:05.340 Until you can satisfy it in as few quadratic functions as possible. Now keep 00:31:05.340 --> 00:31:09.600 in mind, now I can see your work because I can go to the teacher dashboard and 00:31:09.600 --> 00:31:13.200 see how you did it. So if you claim you do it in two, I'll be able to verify that 00:31:13.200 --> 00:31:17.190 claim. Go ahead and pause the video and navigate to a browser and kind of just 00:31:17.190 --> 00:31:21.000 play around with this one until you can satisfy all ten. And see maybe you go for 00:31:21.000 --> 00:31:24.919 the hero solve. Maybe you notice yourself playing the game of snake as you slowly 00:31:24.919 --> 00:31:30.510 develop and adapt to fit more and more constraints. Maybe you notice some right 00:31:30.510 --> 00:31:34.320 away that cannot exist together. Right you find that oh we're gonna have to at 00:31:34.320 --> 00:31:38.730 least two because such and such and such and such can't both happen. That's great. 00:31:38.730 --> 00:31:41.790 maybe you can see that reasoning. And then come on back and we'll just talk 00:31:41.790 --> 00:31:52.620 really quick and wrap up the webinar for today. Great yeah you can save that 00:31:52.620 --> 00:31:54.570 activity. I believe you can add it to a collection 00:31:54.570 --> 00:31:58.200 of your own and all those things. If you wanted to use it with your class. It's a 00:31:58.200 --> 00:32:03.419 very nice thing for a distance setting but I also obviously, these were born and 00:32:03.419 --> 00:32:06.179 face to face settings and I think they're excellent wherever you want to 00:32:06.179 --> 00:32:11.010 build that flexibility. One quick summary before you can move on to all the other 00:32:11.010 --> 00:32:15.780 great opportunities that are provided for you in this virtual conference and 00:32:15.780 --> 00:32:20.220 that is this. Just to just to loop back on it, I think there's so much power that 00:32:20.220 --> 00:32:25.320 goes into flexible thought here and we've just chosen algebra as our 00:32:25.320 --> 00:32:30.330 platform. And the reason for that is as a secondary teacher I've often felt the 00:32:30.330 --> 00:32:34.830 pressure of yeah that's really great, that's really creative math and I see 00:32:34.830 --> 00:32:37.889 that, and and admit that and that that's 00:32:37.889 --> 00:32:42.059 awesome, but I often found those higher-level teachers, the ones that are 00:32:42.059 --> 00:32:45.809 teaching the grade 11s and the grade 12s and the honors say, that's really nice 00:32:45.809 --> 00:32:49.830 but where I am my hands are kind of tied. And I think what they were doing is 00:32:49.830 --> 00:32:53.940 operating in that one direction where here's your model can we describe the 00:32:53.940 --> 00:32:58.289 heck out of it and then we can learn it. I think if we flip that on its head and 00:32:58.289 --> 00:33:02.340 we focus on the students making the design decisions we really can get them 00:33:02.340 --> 00:33:07.259 to operate flexibly by tinkering with those the design decisions. It's kind of 00:33:07.259 --> 00:33:12.360 a very simple thing that can help us become more flexible in our algebraic 00:33:12.360 --> 00:33:16.259 courses. And again the second is that power of difference. And you can use this as 00:33:16.259 --> 00:33:21.269 your teaching move where you introduce a new solution or a new way of thinking 00:33:21.269 --> 00:33:24.690 about it and then you can kind of leave the students kind of shocked and awed in 00:33:24.690 --> 00:33:28.350 that difference. And so that plays a role here but also the menu is really nice 00:33:28.350 --> 00:33:32.279 because there's so many options provided to them. The difference between the 00:33:32.279 --> 00:33:37.919 constraints is thrown into their awareness. They cannot escape it. And so 00:33:37.919 --> 00:33:43.769 sometimes that's a that is the best way to build that sense making is when you 00:33:43.769 --> 00:33:47.580 have to make that conscious choice. This change will sacrifice this constraint 00:33:47.580 --> 00:33:52.740 but will satisfy this constraint and so it's not sort of a random tinkering. 00:33:52.740 --> 00:33:57.090 It's a very purposeful one and that's where I think we see a lot of the flexibility. 00:33:57.090 --> 00:34:02.549 And so when we're doing menumath is just one one way, and there are other 00:34:02.549 --> 00:34:05.970 ways that you can highlight these two key things with designing decisions and 00:34:05.970 --> 00:34:11.010 emphasizing difference. But here what I think menumath does really nice is it 00:34:11.010 --> 00:34:14.250 highlights the compatibility and impossibility. So right away you can look 00:34:14.250 --> 00:34:18.210 at those constraints and think those ones work really well together and those 00:34:18.210 --> 00:34:22.770 ones maybe are impossible. They cannot both be satisfied and that doesn't wreck 00:34:22.770 --> 00:34:27.629 the task. It just changes the way that students are going to operate. And I have 00:34:27.629 --> 00:34:31.589 had students eliminate one and say you know what I have a hero solve with 00:34:31.589 --> 00:34:35.639 seven and I'll kind of throw in the last one with the 00:34:35.639 --> 00:34:39.960 with the last constraint because there still is this sort of desire to do this 00:34:39.960 --> 00:34:44.609 with as few as possible. It really shows a lot of flexibility. In the end they're 00:34:44.609 --> 00:34:48.810 practicing a lot all their parameters. So when you were building those parabolas 00:34:48.810 --> 00:34:52.290 you're thinking to yourself how do I shift that to the left? Or how do I shift 00:34:52.290 --> 00:34:56.550 that to the right? Or how do I, what parameter controls the direction of 00:34:56.550 --> 00:35:01.110 opening or the axis of symmetry? These sorts of things. So they are getting that 00:35:01.110 --> 00:35:06.270 repeated practice that helps build that fluency. And that's what I mean about the 00:35:06.270 --> 00:35:10.370 proficiency with critical features of algebraic models. Whether that be 00:35:10.370 --> 00:35:14.670 vocabulary as we see there with the systems. Like what is a solution region? 00:35:14.670 --> 00:35:19.830 What makes the dashed line versus the solid line? Reintroduction to positive 00:35:19.830 --> 00:35:26.100 slope. Can two positive slopes be in the first, third, second, fourth quadrant? These 00:35:26.100 --> 00:35:30.450 things like this. They're exploring those possibilities. And the last one there is 00:35:30.450 --> 00:35:35.940 interesting because for those students that want to have something for sure 00:35:35.940 --> 00:35:39.930 before they write anything down. They don't tinker as much on paper but they 00:35:39.930 --> 00:35:44.400 do tinker in their minds. This is really nice for that visualization and that 00:35:44.400 --> 00:35:49.350 connection between the algebra and the and the geometry which is what Desmos 00:35:49.350 --> 00:35:52.500 does so well. So a lot of these students when you build something like a system, 00:35:52.500 --> 00:35:55.950 they're imagining what that might look like to have a y-intercept at 4 and 00:35:55.950 --> 00:36:00.000 include the origin and be unbounded. So that is that is another sort of 00:36:00.000 --> 00:36:04.830 flexibility that is built into this. And of course I'm not saying menumath is 00:36:04.830 --> 00:36:08.610 the only way to do this but I think it's kind of one nice tight succinct package 00:36:08.610 --> 00:36:13.080 that has all these benefits that's been sort of intelligently designed around 00:36:13.080 --> 00:36:17.580 these two important features of algebraic flexibility. And it's always 00:36:17.580 --> 00:36:23.640 nice as a teacher of those algebra dense courses to have these structures that 00:36:23.640 --> 00:36:28.890 students are familiar with and maybe even create themselves that can have 00:36:28.890 --> 00:36:35.010 them operate as creative designers of mathematics and not just digesters of 00:36:35.010 --> 00:36:39.000 mathematics. That's sort of the the ultimate goal. And so again if you need 00:36:39.000 --> 00:36:42.690 to get in contact with me, this is how you can do that. And again if you go to 00:36:42.690 --> 00:36:48.150 natbanting.com/menu-math, you can get right to that page with 00:36:48.150 --> 00:36:51.870 all the menus that are there. And of course I would welcome any creations 00:36:51.870 --> 00:36:56.540 from you or your students with menus and I'll sort of just slowly curate them and 00:36:56.540 --> 00:37:01.090 polish them up and I will have I'm translated into French and post it 00:37:01.090 --> 00:37:05.350 onto the page so we can share with others. So thank you so much for 00:37:05.350 --> 00:37:12.840 listening and I hope you enjoy the rest of this 2020 OAME virtual experience. 00:37:12.840 --> 00:37:18.250 One last thank you to the to the planners of this for the opportunity and 00:37:18.250 --> 00:37:23.370 enjoy all the mathematical thinking that awaits. Thank you very much.