My name is Melissa Silk and I'm from Sydney and I currently have a wear a couple of different hats in my employment I'd really like to thank Moodle for inviting me to talk to you today about STEAM and STEAM education I'm not quite sure how you found me, but it's really good that you did because What I would like to do as a teacher is to share some of the Curriculum development that I've been working on so that we can we can actually push some of that curriculum boundary-pushing together and I do have a relationship with Moodle which I Would just like to tell you a little bit of history about that. Okay, first of all, this is the school where I teach on the on the left The two images there's one of them looks like a teed box down the bottom It's a it's an inner-city campus at International Grammar School It's very close to UTS which is on the right and I also have a position at UTS in Studying first-year students in design I'm also a very very shaky doctoral student at UTS as well So because they're within five minutes walking distance of one another for me That's my that's my local area. They're my peeps and that's where I like to hang out a lot The other thing that I have is actually down there in the left-hand corner. That's a steam pop Which is a very small a little start-up that's got to do with Developing projects that are along the lines of embedding the arts into science technology engineering and math and by the arts I mean subjects such as music drama dance in in the Australian curriculum we we include Visual arts and media in the the area of the arts It's also broader than that So it's it's about bringing all of those really creative subjects in alignment or in conjunction or Intersection with science technology engineering and math It's what the steam acronym stands for and steam pop is Smashing together. It's a hybrid word that that I've developed for my company because pop was a project that I was very Very interested in pushing out as well, and I'll explain a little bit more about that later Right, I would just like to play you something now as a little bit of a starter Which is kind of strange. So just bear with me Oh, hang on a minute I'm not quite used to the system Hands up if you if you can think of it you've got a hand up over there Big loud voice, right, so it's about Yeah Right, so the impact of sound on on a surface. That's right. This is um, it was actually just hot off the press It's one of my disruptive Steam project. We've just finished that last week in year 8 at IDS So the entire cohort of year 8 spent some time learning about the mathematics and science of sound and then in partnership with little bits which are the Magnetic electronic components that snap together to create circuits They they have a partnership little bit for a startup from that based in New York and were developed at MIT and They were they also have a partnership with Korg synthesizers and so they have these hits called since hits which we Resourced the year 8 cohort with so they could work in groups and create their own synthesizers To create their own music with certain criteria to be addressed and those criteria was to do with the self Because we're talking about year 8 and all they're really interested in is themselves So most of the projects that I'll design for stage 4 and 5 Which is in New South Wales anyway from some sort of early high school before the senior years Most of the projects will be something to do with the the self identity Learning more about what you bring to a subject and what you what you take away And so if we can combine most more subjects together We've got the possibility of bringing more to a subject and taking way more away For our students the teachers were also involved in that project when I presented it to them Which was about nine weeks ago now and we had a pre-service teacher from the con the conservatorium of music She and her colleague were just in the last section of that video presenting that project back to her peers She's not yet graduated, but I have to say in The beginning of the project when she came on board She really didn't see the point of why we were looking at the mathematics of music and sound and why we would be bothering to Visualize that with scientific experiments And by the end of it. She had a very good understanding So the the gap that I can see in training teachers Which is where I would like to take these these ideas further is to be able to recognize or acknowledge That your subjects are now in mesh Okay, they they actually have an interconnectedness and it's up to the young teachers or the new teachers that we're training to be able to recognize the Possibilities and the potential of putting Embedded stem content into what is normally a disparate subject area somewhere within the arts or even English You know beyond that So I call that disrupting disrupting the content and I've been doing that for a while Just to take you back to the word pop in my little Company named steampop on the left here. We have an image which is an intaglio print So it's a fine artwork that was created by in the year nine student a little while ago in a project Within that was entitled the possibilities of the parabola. Now if I run a school Whoops, if I was to open a school and run a school I would choose one mathematical theory or function or Concept and I would teach my entire curriculum By that concept in each year group because I believe that you can so that was a parabola You can probably see why it was x squared in there if you're looking closely That was just treated as it in an artistic way more on that later next to it that flimsy looking object is actually a class set of numeric values associated with the the mathematics of mazes and labyrinths That was like the end product of another unit of work where we discovered what was going on in the development history and cultural aspect of mazes So these are disruptive Disruptive delivery this leads me back to moodle and I use moodle as an LMS at school where I teach I think I'm probably the biggest user because I quite like what it what the potential of moodle can do But quite a few years ago now maybe six years ago now I decided in my previous role which was head of design and technology at international grammar school. I decided that I wanted to get Respect the sustainability focus that the school had and still has and just get rid of paper exams We're required to examine our students, you know, which is probably something that's content. It's anyway So in years seven and eight in design and technology really an exam is is a quiz So I decided that we would just hold the exam online and the entire cohort would do their exam online The school didn't want to go as far as saying yes to the fact that I didn't want it scheduled I just wanted it open within a time frame and students could go away and do it at their ledger within that time frame But there was a bit of contention because oh, but what if they're going to talk to the person next to them? Well, that's okay in my opinion. It's still learning. There's some conference there that can go on You know, it could lead to the retention of knowledge. Who knows? It was disruptive but anyway, they said no to that So we have to set everybody up in the hall as you can see looked really regimented. I quite like it It looks they look like robots, you know, they had they had to be quiet because it was exam conditions They had to have working laptops what you can't see in that in that image is at the back end of that Just setting it up having all of the questions which were mainly closed answer questions There were probably 85% of it was closed. I think you can probably Guess why we didn't have to mark it Because it's a useless waste of time doing all that marking for questions that are multiple choice if you can do that online It's great And even that the questions that were extended responses were easier to mark because it was just all in the one spot There was no danger of losing bits of paper here and there and and whatever but what we can't see in that picture is the tech guys that were at the top of the of the hall Who were watching the spike in the in the network usage and and things cracking So and this was a while ago The the current head I'm not really sure if he will actually go go down this path with this exam But for at least five years We've been running with amendments the same type of exam and it's a beautiful way Well to test if you need to I'd still like to just take it completely outside of the exam schedule and the scheduling process, but I'm not in the position to do that anymore, but that was kind of fun So thanks Moodle for helping me with that one Most of the courses that I've delivered have been Paperless and they have been set up as just delivery on Moodle I was talking to somebody just earlier about the fact that I only know probably like you know 7% of what of the potential of Moodle in terms of teaching and stuff that you know It's been going on at this conference has opened my eyes as well But as a method of delivery, this is an older version This is this brings me back to the courses that I've put on Moodle The design and technology is 7 to 12 all of the coursework. I did there was on Moodle possibilities of the parabola was an entire course based in Moodle Thinking hyperbolically is where all of this began about or maybe eight years ago I started to look at putting curriculum together What about if we designed math and science units that would were actually sort of related to psychology or they were related to visual arts or they're related to music and I worked with the head of Mathematics at my school to develop this project. We were luckily in enough to get the permissions to run with this project Thinking hyperbolically and the exclamation mark is actually part of our our branding and we just ran with it for about three or four years in terms of Expanding the the possibilities of ways of teaching for year nine and ten So they're around 14 15 16 years old and there was no prerequisite that you had to be able in mathematics Or that you had to be considered creative in inverted commas what we found and I'll show you a little bit more about this No, I can't in fact what I really need to Dress here is that every course that I show you here is another entire presentation So I've tried to smash all this down for Francis I've tried to smash it all down into just enough time to give you a taste of what what we've done, but I've presented Thinking hyperbolically as a standalone because it's so dense with information as a steam course in other context You know contact me and I can come and do that for you happy to do that. What else have we got there? See music that music to two music programs that I'm working on the first one that you just saw Or the most previous one is that is our current one Some of you might have heard the cake tin lid that was a cake tin lid with some stand from the sand pit from preschool With it with an old speaker and a camera on a coffee stand. So it's like a science experiment Just really the simatics The science of sound of visualizing sound is where we got all these ideas from and you might have seen them Heard the vibrations of that cake tin lid. So it's not perfect But I can tell you now that the year 8 students that came and gathered around to watch those bits of sand jumping To the tune of the sounds that they made through their little bit synthesizers all about themselves There were some words in there that were describing themselves It was a really beautiful moment Actually, just last Thursday that we brought all of that together What's the last one on there? Galleries of work, of course, I put lots of galleries on Moodle so that we can share them. This is one gallery This is a in my opinion when you're when you're running a Curriculum that smashes ideas together you need to be able to exhibit that somewhere the best possible Example of that is high-tech high in San Diego where they they they're integrated learning is always exhibited We've got some fantastic sort of samples of Integrated on transdisciplinary work that's all around the school I have some of these samples from the possibilities of the parabola project up here Which I'd invite you to come and have a look at after after we finish or just you know before you go and have Your morning tea because it was a project that I designed that came out of the thinking Hyperbolicly course and I brought it into design and technology It was it did have outcomes those outcomes were mandated by the Board of Studies in New South Wales at that time So it did have to be within curriculum guidelines But just just stretching it enough to be something really conceptual Which doesn't happen much in design and technology at schools or in design courses at schools because They're mainly just about making stuff, you know breadboards boxes that sort of thing taking home It's really good because making is very important, but this one had criteria where the students had to make their Their objects that were vessels. I'll just hold on up here a vessel So it had to vaguely hold something And they had to make it with whatever we had lying around the workshop so we were not buying in materials It was a sustainable project. They had to make them entirely without glue they had to think about the ways of construction and Stability in the three s's of engineering, so we've got strength service ability and stability and They had to what produced them and then we exhibited them So we exhibited them luckily enough with the Australian design center that was called object at that point Gave us a little space in an exhibition space And so the students were able to exhibit their work in an external environment Which made them feel very very special and that's another thing that we need to do with steam Learning is to actually engage with museums and institutions that are outside of school To be able to exhibit what the students have produced inside of the school because it's important that that they get that Exposure, so I've got a couple of these here for you to have a look at Please feel free. There are some on the images up there We took that project into one when the desktop 3d printers were That came into schools. We bought one, you know, I learned how to use it and Wrote some instructions for the students based on the parabola Everyone knows the parabola. Yeah, the curve. Yeah, cool I spent at the back end. I've spent about a day writing really detailed instructions In the software that we that we had at the time which is vector work, which is predominantly Architectural software and we use that for architect projects But that was all we had and I thought well, you know, I'll give that to them It was before we'd really engage with anything that was free like you know tinker card and sketch up and this sort of thing So we had that in the school and I said, okay away you go. Here's the instructions They're online. They're on noodle and I had the students saying oh, but miss. What do I do now? Just read the instructions. That's all I did the entire lesson I've spent so much or invested so much time in creating instructions that I could understand as a newcomer to sort of parametric modeling So I thought well if I could do it, you can do it So I didn't actually help them. I said read the instructions and then they produced these forms and I think they're really beautiful This is a while ago. So These were presented back at the Australian Design Center in terms of a 3d printing PD professional development that I did with some other teachers and I realized at that time looking at the work that was being exhibited in that particular Exhibition which was called control P I Realized that my year 9 and 10 students had produced these beautiful forms Not by just pulling down some file from thingy verse and printing it but they produced them from scratch as the Americans like to say and They were really stunning and they were kind of way way beyond what what was being exhibited as Professional work by some of the other contributors to that Exhibition and I thought yeah, this is really disruptive. This is visible. This is science math visible Okay, but it really is about art and it's about conceptualizing Creative objects and creating aesthetic objects that are based in something really pure and original. I Hope that makes sense. The next thing that we did was we turned some of our parabolic curves into textile prints Don't know if you notice, but I was wearing this at dinner last night This is a textile print digital prints from the parabola. Okay, it's very nice Come and have a look at that. There's so much that you can do in all dimensions with with the with the one curve and That makes it visible and tangible. You can touch it. You can hold it. You can feel it You can wear it That led me to create a course in biomimicry These are just some of the objects. We're lucky enough to have a laser cutter in the school Which you know, I would like to use with students and teach students how to use it Basically, you can see some of the mathematics in there. So we're talking about iteration repetition Reflection all of those terms. I won't spend too much time on that It's just about exhibiting the work that students produce. I think that that was our pilot course for what eventually turned into paper-folding biomimetic course Now we're going to backtrack to we're actually going to lower lower the year groups now They've got year five and six now on the screen, but we're going to move forward This is the steam camp that we ran just recently just this year a holiday camp where kids that were interested in Visualizing their mathematics could come and you know have a play it was really hard work and Their brains were really really hurting and some of them were I think Resenting the fact that their parents had put them in this camp Which was meant to be fun and they had to think really really hard but it was so cool and some of them I would say probably 80% of them got some real value out of having an immersion couple of days in Really alternative types of mathematics Mathematical thinking you'll see there the the image of the well. We've got a hypercube. That's what the boys holding there first So we're talking about you know Dimensions we have some 10-segrity structures, which are just fabulous could talk to you for 40 minutes on them alone but next to that is the The representation in three dimensions of the maze artwork that I showed you at the beginning So we cut the mazes that they produce from the mathematics the numeric values that are associated with creating What is called a nine level labyrinth? And we put them together and it's about this big this little artwork and we use refracting or edge refracting Acrylic because it looks really cool and they're all fluro, which is also really cool and the little guy at the end There was I found this space to actually put the maze in which has some really really horrible ceramics in there So I worked out who belonged to them and said could I have the key and took them away and put the maze in there The little life on it and I found this guy just just staring at it after school one day It was in the winter so it's getting bit dark and I thought perfect got his permission and took a photo In fact, I know his mom so it's got her permission as well So I just thought that was a nice moment of this thing that glowed it came from the Perm of an idea that we had way back in thinking hyperbolically and it's still relevant We can still return to it and I will return to it again in other context So it has to be visible is what I'm stressing Whatever you produce has to you know in some way or form be aesthetic and visible tangible products now we did assess thinking hyperbolically and I'm just going to leave this view for a moment so that I can show you the assessment Outcomes that we had a look at for the teachers in the room just so that I'm actually creating a validity here Let's go in here So the assessment that we came up with for these kind of steam smashing together curriculum things were Outcomes such as this type of thing Some of them are quite dense some of them are easier to understand That one's very easy to understand that one's probably likely given This one not as easy to grasp if you're a parent reading a child's report and thinking hmm I wonder how they did in that outcome? But nevertheless it validated what we were trying to do and I think I have one more Okay, which is probably the most important in this You know steam zeitgeist that we've got at the moment stem to steam even Malcolm Turnbull left out the a At the beginning of the year which is interesting because the Australia Council for the arts then went oh what about the arts and Invited people to provide Provocations to their board so that they could respond to the innovation agenda that was released in March and I was lucky enough to be the education spokesperson Provided provocation at that time with two other people from the arts industry Here's the board. I don't know what they did with that because it was all very hot hot, but it was pretty exciting. I have to say So moving on from assessment Unfortunately at schools we always have to assess children students and And in fact at universities we all know we have to assess and sometimes it would be great to just throw that out the Window and just think well. What about the experience? What how what's the value in the experience? Couldn't we just assess that? What do you know? What can you tell me? What emotions did you bring to this? There's a whole bunch of stuff that you can research on the internet as you know To do with to do with steam learning or to do with the way that science And I'll include mathematics in the broad title of science and the way that science and the arts have been Intersecting for many many years, you know for thousands of years. This is just one example This is from the Museum of Modern Arts at MoMA in New York It's a fantastic website if you'd like to just note it down. It's totally interactive It's got some fantastic ideas and great examples So these are things that we would engage in during the research phase of any any unit of work related to steam Also other things that you find on the internet are actually really really interesting This is this one is just that I'm not going to play the whole thing, but it's a It's the explanation of turbulence as it relates to patterns and to then goes Dari-dari night, so I won't play the whole thing. Have a look at it So it's about turbulence flow and they relate it back to the artist and the artwork It's a really really lovely little video, but it's one of millions Um What I'm calling steam engines here are the collaborations that I think are important when you're designing any type of cross-curricular or transdisciplinary Experience for people to engage with and these are people of all ages So I've provided this slide to Fran maybe if you would want to if you want to get in touch And send it out. These are all links to people. They happen to be mostly women and that's not Conscious decision except that I do think that in in the tech industry that the balances is really is quite good But when I've been on my journeys in steam I've met a lot of women who are both educators or CEOs or directors of Certain steam incentives and they've really been standouts and they've been working with with guys as well But it's just been my experience to bring these people to your attention to Angelina Russo is the one on the top left. She's at the University of Canberra She's been working with a guy called Steven Barass to create visualized data for a long time now They're worth looking up. The links will be available to you. That's her data beanie. Okay, that was knitted on the 1970s hobbyist Knitting machine that was marketed towards girls because when everybody had a hobby so she's got one of them She knitted a data beanie. I think they've collected that data data data from the I think it was this the seal Migration somewhere in the Southern Ocean some some portion of land They've collected this data and they've created it into a beanie, which is kind of cool We've got Linda Keane who is from Chicago underneath that next dot CC is is her creative commons website that has got a myriad of steam related activities for Students who engage with she's fantastic Below that is Lucia Montage and she's standing next to John Mitre who was the kind of Father of design thinking at the Rhode Island School of Design very very influential people in terms of how science and the arts Meet and create, you know, beautiful forms Lucia works at a place called the nature lab Which is part of Rhode Island School of Design in North America? Which is just a wonderful place to go and visit have a look at their website Cindy Lawrence is the director the other CEO of Mo Math, which is the museum of mathematics in New York I'm really lucky to have been able to present some of the projects that we have developed at their museum If you're in New York, it's not it's five years old. He was open late 2012 It's coming up to being five years old and It's a really good museum that in brings students or brings anyone into contact with the way that mathematics can be visualised and played with In a great big complex. It's very very. It's a wonderful place. Please go there next time you're in New York and below that is actually a form one of the The polyhedron type of forms that was created by George Hart With a school group you can't quite see Anita you because she's in that group of people next to what's called a harm on a graph That she made with her little steam club and I met her at the Museum of Mathematics. That's why they're all grouped together and she introduced me to the work of George Hart who's a Mathematics Mathematics she's a lecturer somewhere in Stony Brook University on Long Island and he is a he's a wonderful Visualizer of mathematics as well for kids just above them is Cindy and one of my students just earlier this year when Cindy and her team visited Australia just by chance They're on their way to Singapore and it happened to be Chinese New Year So they couldn't do what they needed to do so they had four days in their calendar So they came to Sydney and they met some of our four unit mathematics students in that in that instance and then some of our year 8 students and had a bit of a We're lucky enough to get a bit of press around that visit, which was really was great fun. It was really good and then down below is Panna Panna of Vata Nana I think her name is but miss miss Panna says is her website And she deals with steam concepts for lower infant Age, sorry lower primary so infant schools. She's got a fantastic Fantastic body of work that is available and she's a real mover and shaker. I think she works in Taipei Those those are all available to you if you want later on So they're the engines that I've engaged with so the collaborators that I've had something to do with some of them And I just wanted to share that with you The reality is when you're teaching any steam Project is to actually base it into what work on this be applied So we've got the watercube as a really good example Chris bosses and Australian architect that they were the designers of the watercube all based on the Mathematics of soap bubbles That's a really good example. I'm not going to go too close into it We also have a look at nanotechnology and the development of Biomimetic forms in our research of a steam project. This is Leah Heiser's work and she's she currently works with biomedical groups in Melbourne as well as RMIT to develop information-giving forms that are on a nano scale that are based in some sort of natural type of Expansion so we we look at this in the context of origami the ones on the seed shapes on that on the left of the things That open and deliver information. She has other beautiful jewelry that is that is the delivery of insulin So that she's in development. So it's where artists meet Scientists where these beautiful examples of steam can occur and just being able to explore that with young people Really opens their minds in terms of what? What the possibilities are? So this is back to the music project.  That's fantastic. There's so much fun to play with So the little bits synth kits, you know, I knew nothing about Synthesizers before I investigated what these were so me and now a hundred and you know ten year eight students know a lot More about electronic sound, which is really really cool Whether or not they're all engaged, you know I can't I can't say but on the day that we were looking at the the production of their own voice in and And the correlations between their own sound made with a very short pop Which they recorded and then tried to meet in turn in an online mathematics software To be able to understand their the relationship between their sound and sine waves so where that occurs in maths So this is where we bring the maths teachers in and we say well This is what happening here. We've got frequency and pitch and this sort of thing and this is what it looks like mathematically So we created this kind of Group data based on those op sounds, which was kind of strange And that's but it worked. I think and they understood and that's when I did notice that there was like about 70% engagement probably like, you know Same as what's in the room right now But just playing with the kits please saw them in the video playing with these little kits and just making spooky sounds and Understanding what you're actually doing the science of those sounds is important because it's knowledge that you know Perhaps will be retained certainly will come up again at some point plus putting that into a context of what is the history of Synthesizers, you know, where did these sounds begin? You know, where did electronic music come from? I mean, they're all off listening to it now But where did it start? Why was it important? Where does it sit in culture and society and politics? Etc. Moving on there. They are having a bit of a go, you know, we sit on the floor muck around We required them to use their own devices in terms of their phones. They've all got a phone So we actually said well now that phone becomes a resource for this unit of work So you don't just a snapchat your friends at this stage you use it to record sounds We smashed around using a few iPads that we provided just to record sounds manipulate them Re-record that sort of thing. It was the only way we could troubleshoot around having only one kit between four kids You know, they're expensive as a resource and I have to say that not one piece of the kits was broken So they were very respectful, which was great of of This of the resources that we provided and let me tell you they're not an easy group They're not an easy bunch of year 8 students. They're probably the hardest that we've had at the school I'm telling you the truth Some more engines the thinking hyperbolically examples are on the left The engine that I'm talking about there is the fact that Myself and my colleague Jane Martin who is the head of mathematics were able to get permission from our principal at the time To run with this project that was completely, you know off the wall was very left field But she said yep go for it and it was very successful So at the end of our first year of teaching thinking hyperbolically We were able to actually have an exhibition of student work at the powerhouse museum, which is great And we could not have done that without the consent of leadership So where any of these incentives are going to work? We have to get all of our school leaders on board. This is the way that we go And they have to take risks and they have to actually be able to release teachers time to develop innovative Curriculum delivery and that's really important the research would show that as well on the other side is another Steam engine collaboration that happened only a couple of years ago And it's to do with a Japanese festival So I'm just going to play you a video of what that is and be quiet for a little while This is the shortened version of a longer documented Form from this day where other we interviewed other people the other stakeholders parents principal Etc. But this was a collaboration between Pre-service teachers, so we've got the University of Technology coming on board with another arts organization at the time They were called art waku waku because they were predominantly Japanese cultural arts organization and And our school and there was somebody else. I forget parents. I don't know part of the school community So we had a whole day of immersion steam projects going on based in a story That was the Japanese Tanabata story, which is about a constellation of stars at a particular time in the year It's a cultural festival in Japan. So we had this the science of astronomy We had the mathematics of a star. We had origami. We had making lots of making They we had from year one to year four so about 350 kids throughout the day in two sessions We started to make ninja stars because you know kids love them but we also made these things these are um Hyperbolic paraboloids as a few samples here So these are that fantastic piece of visible mathematics that you can play with and those when you put them together And understand what's going on in here create what's gotten the right the right form They will create a star of as many points as you want for instance. So that's a great project done that Beyond here as well.  Oh, they're great fun So we decided to have a little bit of a play with the idea of force and motion in kindy last year And we created Rube Goldberg machines, which are mechanical sort of intricate mechanical machines to actually sort of Have an outcome at the end of it All out of the recycled materials cardboard in the tubes this sort of thing But we thought well, what can we produce with that? It's okay to take a marble from here to here But what can we do with that? So we created a giant artwork and so the kindergarten students had to go through the design process and do all of their failing and then retrying To be able to take the marble through the Rube Goldberg machine and smash over a little cup of ink Or at a cold die Okay, the giant piece of white paper and create this artwork that you'll see is on the wall Then we thought we'll take this a little bit further We want to share this with the parents so we embedded the videos that we've taken by the year six students of the Kindergarten students we embedded them using augmented reality into the artwork and then invited the parents in and said okay Well, we're not going to explain it.  There you go And this is the outcome We had used augmented reality in a previous project, which I'll show you in a minute But that was a fantastic way of engaging parents in what their children are doing at school They're in kindergarten by listening to their stories without having the children come getting up and you know becoming Stage fright anyway, and so they were videoed They were much happier to video each other and then all we had to do was a bit of editing and then embed that into the Artwork to disseminate it you could also we said the digital file home So that many of the parents who might have grandparents or other family members or the wider community overseas All we had to do was just you know on share that that artwork image online and the AR would still work So we only made it live for the 30-day period that was part of you know the budget that we had But that's okay because I think things need to be a bit ephemeral Doesn't really matter if they don't last it was the experience is what we were after and it was a very positive one and Music if you just saw was also made by year eight So this year we actually incorporated little bits into the year one version of the force and motion unit of work Hello Oh, no, I just want to go back to this one So they're they're looking at force and motion again, but we use little bits technology to create these little machines That had to they had criteria. They needed to address again. It was the design process They were working with the incline plane, but they were also working with Sensors so their little machines had to be not motorized but had to work by gravity But enter our cave.  So we stepped it up We said okay now we're gonna learn about cogs We got all of the old Lego that wasn't being used from the Mindstorm kits and created these these these little cog Setups in groups and then we just put them all together and made sure they all kind of fitted vaguely and then we did some Mathematics and made some what's called dinosaur snowflakes. We're looking at fractions and scale and this sort of thing and within that that Collective artwork we've got a whole bunch of learning and then we invited, you know some of the members of the IGS community to come in so they would sit down and explain to those people what they were doing and again It was just a fabulous experience and one that meant that those kids suddenly started saying to their Teacher, oh, I love science. How can we do science today? We want to do more science. So I mean whether or not Projects will be that exciting that they do it. It needs to be recognized that they can be so the possibilities are there to actually step that up So you'll see all of the little bits components around the back. So all of our working, you know mechanical areas are around the back of the of the The smashed together units that we did But I think possibly and again, we've got year 8 students. They produce lots of music So I don't have to worry about copyright. We can just use their music, which is fantastic But the next project I think I'm proudest of Perhaps at a school level Because it is really beautiful and it was where we first engaged with augmented reality in the way that we would Produce a happening and have an experience for an audience again It's about visible learning. The learning is actually there on on show and the experience From the audience feedback is also part of the learning these masks were created by the year 9 and 10 visual arts class they were related to the asaro mudman tribe of New Guinea The music was created by the year 8 technology unit of work and music This is one of the tracks. They were they were inspired by the masks There were 15 masks so they had to be inspired by the masks and then we had a Happening we we just put this on again. It wasn't that there was going to be a beginning or an end to this happening You just came in got your device We had a few set up and had your own experience in this Location with the masks and the music and it's quite a delightful way We had a bit of in-pro there. You might have seen some students doing some in-pro. We did some musical Well sound it was about sound basically and a little bit of drama, but Everybody's in their own world there But they're all together in their own world So it's kind of a bit paradoxical It was a really beautiful and marvelous thing to experience and then we put the masks all around the school for a little while While again, the the AR was live so you could still access that music We had all of the other tracks because from the year 8 group also in a Soundcloud file so that they could they could also be accessed and we sent all of that home as well So parents could engage with with their students work I'm a bit of an overload. What are we going doing next ah? This is my current project that I'm working on with say it the kindergarten to year 4 because it is the International Year of the Pulse So we've been taking some microscopic images with our Bluetooth microscopes and the iPads of lentils and split peas and Smashing those through just you know a kaleidoscopic app Just and we're just at the beginning of this so I don't have the end result at the end result We're aiming to have a collaboration with another local primary school and create two textile projects probably a Quilt cover something like this and auction them and you know raise the money for charity But the main thing is that we're going to see what they do and we're going to see what we do We're going to Skype in and we're going to have a chat about that and again exhibit that work So the product is an aesthetic product that you're produced from this kind of scientific input I Think they're really beautiful when you look at these lentils and peas close up But you'll have to watch this space because that one's not completed yet but it's about the explorations and the possibilities and the Reaching out to other schools to come on board if you can or other universities to actually come on board with their pre-service teachers It's really important These are just a couple of images of what goes on in my steampunk's club And the sort of things that we access we don't actually have a location and we don't have a maker space But what we have is a sort of roving kind of Attitude towards resources that we can use at the school It's a really space-deprived school where I teach so there isn't any room to build me a maker space and There's no money either But what I have is a relationship with the people still in design and technology so we can use the laser cutter I have a relationship with the people in science so we can do some scientific, you know Work with light that we did for the shorts there That was actually scientific project the shorts and pencil cases were for National Science Week last year when it was International year of light or the theme for Science Week was was light waves So we produced photographic items that then we put into textile prints and then we made into items that again Could be sold at auction. We've got wearables. We've got electronics That the steampunks made by the design process So we were really after a quality outcome and not something that you would just think oh, yeah Meh, I'm just gonna throw that in the bin now. Okay, you want to go home and show your grandma What you what you've done or share it with everybody and then of course that leads me to the joy. All right joy Joy in making that's one of the things that I say There is so much joy in making all of these products up here that that I've exhibited to you were made by human hands Mostly in little hands at the moment. We've been working also on binary bangles Okay, so the whole bunch of mathematics in those black and white form. I'm gonna have to speed through we've been doodling 3d doodling the trees For a sustainable project So we're having to think about the plastic that we're using to doodle. So that's also for PLA Which is the the degradable one. I'm gonna quickly scoot on I need to get to the piece of paper Which is my project Okay, this is a big collaboration that I had when we presented our project called loomy fold at vivid and Those are all of the people that were involved So it was a great way to actually bring a lot of people together to provide an experience for people not unlike yourselves We did a teacher PD and then we did a general public Workshop that was just about making so everybody here would be making a little lamp which is somewhat like this guy here Okay, these biomimetic forms. This is actually a bit a bit degraded. It gets Handled a lot, but you're welcome to come up and have a look and we can light that up for you later That was a great project and the beauty of that project is right at the end when you light your lamp and you look up You've created a constellation of stars on the ceiling, which is really really wonderful That's a bit of a wow moment great. It's it comes from origami. So we always make sure that We relate these types of practices back to where they might be applied in the real world So on the one hand the same fold that we use in this this Lamp making is the same fold that is used in the heart stent Which is on the right hand side and similar types of folds and expansion have been developed for Things like the solar array built by NASA that goes into space So we have very very small and we have very very large so the scale of what you can do with Understanding the engineering what you make with your hands and you know the thinking hand what comes to your mind through your hands It's really important Not sure how I'm going for time, but I'm just going to scoot through so this is the project that we make I love my little animated gifts. That's kind of fun, but There's also something going on at the moment at the if you're from New South Wales the out of hand exhibition has got an Piece from this guy Matthew Gardner who uses the same concept that we use for the this lamp construction and But he's changed it he stepped it up a bit and he 3d prints some Articulated moving parts and he calls his project or robotics and it's actually on display right now in Sydney So if you are in Sydney the exhibition is called out of hand Another collaboration with another primary school again, we're thinking about patents We're thinking about binary thinking about mathematics, and we're thinking about making Okay, I'll just move on that's a school Just before I do this the IGS students where I teach down on the right-hand side Created their own lamps from their studies of eminence so they choose an eminent person and they do projects on This eminent person so this project-based learning then then we made there the image of those people on a lamp And we'd made a giant map of the world and place their lamp on the map of the world lit it And that was our constellation of knowledge. That's what we called it We skyped in with the girls that were at the other school and the questioning because they'd done a different type of project with A similar outcome and the questioning just culturally was really interesting for me to witness between the students So we didn't run that we let them talk to each other Which is a great way to actually really embed learning by asking peers, and they're all around ten years old I also took this project of Finland recently to the Bridges Art Maths Conference And these are a bunch of mathematicians doing the same thing and I can tell you right now That if you're ten years old you've bought about the same level of skill as someone who's 60 Who's been studying math for their entire life and is considered a world leader? That was really interesting We did two projects there this one, which is which we called a pattern squared Which is actually I'm moving that into being a binary bug Okay, that's a little artwork. You can have a look at the same type of fold which is called a glide reflection Which is actually patterned by using combinatorial probability and basically just flipping a coin So that's a project that I'd love to share with you at some time, but we don't have time And let me fold is the project that I'm most excited about because I'd like to actually take that further I'm currently looking for a way to be funded to take that outside of the privileged and take it out into the less resourced schools and be able to deliver these incredible kind of steam content in a way that is Meaningful for kids that don't have access to anything that would be that's my ultimate goal because I Want to spread the word I want to share that's why I'm really grateful for the opportunity of sharing with you today This is the last thing that I'm going to show you before we have any questions But before I do there's a lucky prize There's a little business card stuck to underneath one chair in here And I don't know which one it is because Petra put it on on the chair So have a feel under your chair, and I'm and you will receive a kit so you can go home and make a couple of lamps for yourself Who's oh there you go. You've got it. What's your name? What's your this is for you on On the on the steam pop website On the steam pop website you can go and find those kits for yourself But there are also two videos there that actually step you through the process of making the lamp because it's not easy It's about patience perseverance and the understanding of what you make it at the at the end of it It's just really quite remarkable in that these ones are really flexible What we call the loomie ball is really flexible, but the loomie tall which is a cylindrical shape. It's basically the same Pattern rotated 90 degrees, but it's incredibly strong So you'll need to test yourself on on questions. Why why is it why does that happen? Why is this? I'll just end today with a little bit of What happened at the vivid? That oh that much I'll end on this is some contacts Some if you like that's another slide that I've given to Fran that she can disseminate to to you I just put in a couple of links in there the young creators conference was something where we Took all of the work that we've developed in steam at IGS and put it all into a list into a website I like to go around and have a play with just about every web web site maker that I can just so I know which is the Easiest one for kids to use so they're all always you know different hosting But but that's got a lot more of the work that we've produced and down at the bottom I was lucky enough to receive the premier New South Wales Premiers teachers scholarship award Last year and I took a tour and met some of the people that I mentioned earlier And I've actually kept her there's another website Which is the log of that tour if you're interested you might be there's more information about Deam incentives going on basically in North America high-tech hires included in that that list of the tour anyway and I say the word freedom there because I think the stuff that I do you can free the kind of logical thinkers mind in terms of look I'm really really mathematical, but I can also create and it also can free the mind of the creative and think well I could actually start with the germ of an idea coming from science on that Okay, and I'm including engineering and technology in that Umbrella and so the freedom is to actually recognize the possibilities of what you can do Well, I keep doing that. I just want to play you one last thing here. We go Only about two So typically in a room full of people such as where we are now It's a little bit like a tourist video, but I apologize for that We just get a whole bunch of people making and sometimes I ask them to be quiet So I can record the sound of paper being folded by Which is also a really cool sound But the immersion that takes place when you're making something It's a really remarkable experience and that's what I'm most interested in and how long-lasting is that what do you take away from that? Would you remember in another year's time that you were actually working with them? Mathematic concept at that point or did you do remember the making? I know that students who studied with me Who have now just left school? But the possibilities of the parabola course Each one of them as I've met them in the corridor may or may not be teaching them again from from that time They they would if I said to them now, what do you remember most about that course? And it's a well the curve the parabola That's pretty good Nice moment when you do these workshops with a large group of people is when they light up It's a really beautiful thing human beings respond really well to light and If you can take them home what we did at vivid was we had a little bag that people went out into the festival with their bag of light that they made themselves and Because there was you know a limit on numbers obviously Everybody who was outside saying where'd you get them? So we'll have to run it again Pretty hard work, but it was really quite good. That's me I'd like to invite you to ask any questions, although I haven't we may have gone over time I'm not quite sure we have okay, but please Come and have a look at the at the at the things while they change the room around. Yeah, I'd like to say that was just inspirational I think that the combination of Taking that mathematical into everything else. It's just yeah It's a good thing. We can only do that in our middle course. It's just I mean I don't even know where to start but certainly something to think about so thank you so much And maybe if you want to try and grab during Melissa during coffee and you're out for the rest of the day I'm around until about two. Okay, so during coffee or lunch. I assume Melissa's gonna get Thank you very much so


Welcome to Toffy TV, it is the Everton News daily and the Danduma retains for Everton under 21s tonight. They are kicking off about right now, if you're watching this straight away, 7 o'clock kick off against Blackburn. He hasn't featured for Everton since limping off with an ankle injury at the end of January against Fulham and even though he's had a weird season with Everton. Everton could have done with him in the last couple of months, massively did really really could have done with him and it's just so slow really that he hasn't been available, but hopefully if he comes through tonight then maybe he'll be on the bench on Monday and that'll give us something a little bit extra, something a little bit different, someone who can come off the bench and then with those three massive home games the following week, just give us that little bit of booster that we haven't had in the squad. In the squad of late, we've got quite samey players in a lot of different positions but he gives us something a little bit different. Obviously he can play on the left, can play on the right and can play up front and off the frontman and as I said I know we haven't seen the best of him this season but if he can just give us something a little bit different then that would be massively important. So hopefully he gets through tonight and you know there's no issues and he just finds that fitness that we've been looking for him to get because it just seems a very long time since we last had them in the squad, so good luck to him tonight and good luck to all the rest of the boys in the under 21s against Blackbane Roas. Moving on to the dreaded financial part of the news, hate having to talk about this but perspective ever known is triple seven of hell talks with private markets investor blue owl capital about arranging a £360 million loan tied to the club's new stadium according to reports. I stress that according to reports. There's been obviously a lot of talk about a deadline next week for 777 to repay MSP, the loan that they put into Evan, I think it's just short of £150 million. 777 apparently will be looking for a deadline on this and obviously this would help pay that and maybe help generate the funds for some of the debt or possibly to finish off the stadium. I don't know that's all conjecture at this point but that story is out there today and as I said the story about them wanting possibly an extension. There's been a lot on this hasn't there's been lots of stories filling the media in the last few days about this, about 777 being able to pay this off and what it would mean for the club. It was said a couple of weeks ago that the Premier League had given 777 for stipulations for taking over the club and this was one of the stipulations paying back MSP. I suppose if those four stipulations are met then that will give everyone a little bit more confidence that 777 can do a good job at Evan. I don't know whether to say 777 or 777, I'll say 777 here. It's been 777, a lot of people would say they've been paying the bills for the last few months they've been keeping Evan afloat when Farad Meshiri hasn't so you've got to give them the benefit of the doubt. Well maybe you do but there is a lot of stories about them out there and if they repay this next week I think it would make a lot of people feel easier and whether that comes from a loan or getting investment from Blue Owl capital I don't know but we'll have to wait and see. Also today Evan has paid about £30 million in interest charges to an opaque lender associated with a tax exile corporate record suggest that's come from the Guardian and get today as well. Again that becomes as more and more eyes get on the Evan situation and with the points being deducted and with Russian money and things like that getting mentioned more and more people are digging into Evan's story to see where the money's gone, who it's being lent by, who do we owe money to, how much we owe money to, how much is debt to Meshiri, debt to other people and how much is it tied into the stadium. A lot of people are digging into this. Two Evertonians, a lot of these stories are not new, they've been around for a very very long time but to the outside world who maybe don't quite understand the situation. There's an interest and there's a feeling that these are all shock or a story where this to us as Evertonians think we already know a lot of this so it's not much of it comes as a surprise really to us either does it. So yeah that's another one out there today and Everton have paid 10.6 million in agents fees only three promoter teams, only the three promoter teams sorry paid less and Brentford from the first of February 2023 to the first of February 2024. The highest team is Chelsea who paid 75 million pounds to agents so 10 million might seem like a huge figure but when you think it's only the other three teams who've come up and Brentford who were running the ship really really well you can see that it's not a massive figure. Of course that ties in with the amount of people we've brought in and the fact that we haven't done a lot of transfer business and we haven't brought in any huge buys either really so yeah it seems a massive figure but I don't think it really is. Just on to the football, the football yeah that thing. Miritio Pochettino has revealed that Ben Chilwell is on in the process of returning to training ahead of Monday night's game although they have injuries over Enzo Fernandez and Axel Diazzi for Monday night's games. There's also a few rumours that there's a bit of a sickness bug going on at Chelsea as well but I'm not sure how true that is. It's more about our plays isn't it and getting our plays back and as Sean Dicefield yesterday we do have a few knocks in the squad and obviously Dominic Arvent-Lewin has a little bit of an issue that he needs to sort out ahead of Monday night but let's just hope that Andrew will get through tonight's game and he'll be ready for Monday night's game. You can check that out right now if you're watching this video straight away. I think it's on Evan's YouTube channel and over on their kick channel as well. If you want to follow that game as well there you go. Make sure to check out the match preview over the weekend and some of the other great videos that we've done this week. If you want more exclusive daily live videos, check out Toffy TV Premiere. The link is in the description. The QR code will be coming up on the screen now. All of the videos we've put out are on there and all the podcasts and none of them have any advices for the price of a cup of coffee for a month's worth. Check it out. See you later.


the non-tech community that you referred to earlier, you know, which has probably taken that back during the decade that we've seen from 2010 to 20, and it's still going on, of course. You think that they are going to be the bigger participants in the in the Web 3 space, the Web 3 industry? I do. Now, the reason why I say that is because, you know, from our lens when you look at NFTs, the biggest beneficiaries of the participants in NFTs are not tech guys, they're not the ones who wrote the protocol, right? They're the creators, they're the artists, right? I mean, if you look at just what happened in the NFT and the NFT gaming space, actually, most of the people don't really have a traditional tech job, and they're not the kind of people that Google would have hired, right? So, right? And so my point is that it's given rise to a completely new group of people who could achieve something through their creative endeavors.


Everybody should be working and nobody bigger than the next nigga around here You know and Drake was when even when Drake came to the studio. It was like, you know, the Dean is here Or he was in the air again in the hallways. I don't know, but you had to you should you better act like he was Ergon and cocky because that's that's kind of only thing. I kind of remember from Drake. It all lies on me movie Oh my god.     Yeah, I always say something me and pop always had like You know words Like and I don't even know if they were true words this was words Whatever I said, shut up nitty boy.  I was super serious about this company do my video Jojo and Casey here.  You should have been there You thought he would learn this the slip of slide video.   They were calling, you know Poxay without a great gas drain. It was all kind of stuff said being said you heard you was there So, you know, what was that?  Well, he just flipped like that. Like they were kicking it on california love and they were I think he kind of felt pot. Uh pot kind of felt that dre wasn't participating with You know, the artists as much and you know, everybody should be working and nobody bigger than the next league around here You know and and dre was when even when dre came to the studio. It was like You know, the dean is here or he was like, was he arrogant in the hallways?  You I think a look at looking how he is now I mean, man. It was serious and you know, and it's in this b studio.   I don't know you had that part and straight out of come That's what I thought about when he was saying that where he came out and he was telling sugar Like they were out there messing around or whatever and he was like man He was trying to record. Was it park was in the inside of the studio Whatever you say I got parking here. Yeah, he's like y'all need to be quiet. Y'all out here messing around That movie thing crazy.    No, I think you know all eyes on me movie. Oh my god. Oh, it was crazy horrible movie Why did you like it?   That's right. I mean it kind of had me think to do with sugar which is him You know, I inherited a lot of his problems and um It's just still weird To be left out of the story like I would watch things um even with working a job and and and anticipate on sing my part Or sing what you know the little small part because that uh A lot of people would be like man, you didn't know pock that long it felt like forever Yeah, but the work that we've done and the things that we've done You couldn't it's a lot of people that won't experience that in a lifetime Do you think it's cause you from the west side of chicago and they was from l.a. And I happen them niggas ain't from l.a. Most of more than what I'm saying they be saying that and they like l.a. Come on bro. Where you from? I'm just like you I came from chicago with a vision and a dream Don't hold it against me.      So I don't know, you know what I mean? I just I guess that's why I'm blessed. I go places You respect people You respect you respect.      You see what I'm saying? Do you know how I am about the music, man? Oh, I love, listen No, you see, I'm an R&B He's rap I do it, I do it, I do it, I do it, I don't stop playing Well, I'm an R&B, I'm sorry


So, what we will do we will I will just start on assuming that you know basics about the normal random variable. What a normal random variable is what its applications are and I will start from something we call the standard normal random variable. Did he go through that? Standard normal? No, right? So, I think that is good then ok. So, you have normal random variable or the Gaussian which is the most common type of random variable has a huge amount of applicability in real life. You know normal random variable is very very good at modeling phenomena which occur in real life, various phenomena like natural phenomena, synthetic phenomena everything. Normal random variable is very good at that. The standard normal is a specific type of normal random variable as you can see from the definition is the normal random variable and instead of x now you are putting z with mean equal to 0 and standard deviation equal to 1 ok. So, normal random variables can have any mean and any standard deviation. Now, standard normal is the normal random variable with mean equal to 0 and sigma equal to that means standard deviation equal to 1. You will see that I have emphasized the because there is only one standard normal random variable ok. You cannot have two standard normal there is only a unique standard normal random variable. So, we continue with that ok. So, first you look at the functions that means the density function and the distribution function and you can of course derive them from the definitions of these same functions that we know about the normal random variable. So, for normal random variable we had this big equation and we continue with that we just put sigma equal to 1 over there mu equal to 0 over there that is the basic idea. So, now for pdf and here you will see we are using the sign the Greek letter phi ok the small phi over here this is typical for standard normal. So, whenever you see that phi you know that it is a standard normal random variable that we are talking about ok. So, phi evaluated at that z is equal to just like what you had for the normal random variable except that we are putting sigma equal to 0 and mu equal to 1 ok. Similarly, if we go ahead we can also find out the cdf the cumulative distribution function for the standard normal random variable and for that we use the capital letter capital phi which means the cdf. So, instead of f we are using the Greek letter phi here ok and it is a it is usual it is customary that instead of x we usually denote it with z ok. So, how do we get that just by integrating the pdf from minus infinity to z ok and this z over here is a dummy variable like we have used for the definition of cdf of normal random variable. So, overall the basic idea is that the standard normal random variable is a special case of normal random variable where the mean is 0 and the standard deviation is 1 of course, the variance is also equal to 1. Now, in general not only for the normal distribution, but for any continuous random variable we say that there is a standard form for that normal random variable. What is the standard form? For any given random variable x we have this standard form given by x minus mean of x divided by sigma that means standard deviation of x ok that is the standard form. So, even for normal distribution if we use the standard form we get the standard normal distribution. Similarly, if you take any other let us say uniform you can also define the standard form of a uniform random variable just by applying let us say if the random variable is u then u minus mu everything divided by sigma u that would be the standard form for a uniform normal uniform random variable ok. So, that is what we say that you can even define it for a non normal random variable not only for normal random variable, but standard normal is the most important one because we use it a lot use it to compute the values of pdf and cdf not only for the standard, but also for the regular normal random variable with mu not equal to 0 sigma not equal to 1. We will see how ok. Let us discuss this property of the standard normal random variable if we have this standard normal random variable z then you can write that cdf at minus z is equal to 1 minus cdf at z and it is very easy to see how it takes place I had the time. So, I have drawn it already. So, this is the pdf small phi let us write pdf over here of the standard normal random variable. So, it is mean is at 0 this is 0 ok and normal random variable is symmetric about the mean. So, we have a symmetry about the mean now what we have is cdf at minus z 1. So, here we go to z 1 this area beneath the pdf up to z 1 is the cdf at z 1 right. So, this is phi at minus z 1 and symmetrically opposite we have plus z 1 equidistant from the point 0. So, this area would be equal to 1 minus phi of z 1 is that ok everybody I think it is pretty easy to see how it is. So, this part the left of z 1 if you take the area of that that will be phi z 1. So, this is 1 minus phi z 1 and you can see it is symmetric. So, this area will be equal to this area that is why we can write that phi of minus z is equal to 1 minus phi of z very important property of the standard normal distribution and we use it a lot for finding out values of the cdf ok. The cdf is also treated as an operator like any mathematical operator you operate it on any value it gives you another value and you will see how we do that. So, we apply this phi on value 2.25 we get some value we apply this phi on minus 1.244 we get this 0.10675 what is this this is again this is nothing but the cdf of the standard normal random variable, but for every value of the standard normal random variable z you have a value of the cdf right and that is what we treat as an operator. So, we call it the standard normal operator and you have standard normal tables you open up any book you open up your text book by Ross or open up any book on probability and statistics which discuss which discusses about this distribution there you will see a table of standard normal distribution. Where does that table give you? That table will give you values of the cdf for standard normal variable z for different values of z you will get the value of the cdf and also software packages will have built in functions using which you can determine what is the value of the standard normal random variable at a given value of the standard normal random variable ok. You can easily obtain the cdf software packages by mean we will see some examples using Sylab. If you are familiar with spreadsheet operations like numeric spreadsheet or open office calc those things will also have some built in functions ok and they are very useful for finding out the cdf of standard normal random variable. Now, so far I have been talking why should we find out the cdf of standard normal random variable because it is useful for finding out the cdf of a normal random variable. Do you see how if we know the cdf of a standard normal random variable can you find out the cdf of a normal random variable? I leave that to you ok think about it ok. This is what basically it is cdf of any random any normal random variable x with mean equal to mu x and standard deviation is sigma x for that the cdf which we denote by f of x at x should be equal to this is the cdf of the standard normal random variable at x minus mu x by sigma x. What is this? This is the standard form of the normal random variable we already showed that z equal to x minus mu x divided by sigma x. So, f x at x will be equal to f z at z or phi at z capital phi at z ok. So, you use that property to find out the value of any normal random variable with a non-zero mean and non-one standard deviation ok. So, if we have the standard normal tables in your book and you have to find the cdf of a normal random variable with mean equal to 3 and standard deviation equal to 0.2. What we do? First we have to find out let us say we have to find out it at a value of x equal to 5. So, 5 minus 3 divided by 0.2 we find out the value of the standard form and then we go to the table find out the value of phi at that standard form very simple task. We also define the inverse normal operator or the inverse standard normal operator which is the inverse of this capital phi function. So, it is the inverse of the cdf of the standard normal operator and if you remember the previous examples these examples are exactly opposite of what we discussed then we said earlier that phi at 2.25 is equal to 0.98778 or so. Now, you turn it around you say that phi inverse at 0.98778 is equal to 2.25 and similarly the other one ok. So, what is this? This is the cdf of the standard normal z at 2.25 ok. So, we also have the inverse normal operator it has its applications I will see when you go through some examples we will see when you use the inverse of the standard normal operator and of course, one can also define the pdf's and treat them also as operators. So, this is the pdf of the standard normal this is the inverse pdf of the standard normal, but these are not used as commonly as operator as the other two the cdf ones are very very important. Now, we come to Sylab and the function that we have in Sylab to find out the cdf of the normal random variable it is very easy to remember it is cdf not finds the cdf of any normal random variable having any mu and any standard deviation ok. You have to know the mean and standard deviation of course, unless you have the mean and standard deviation we cannot define the normal random variable. So, definitely you need to know those two parameters those two moments of the distribution. Once you know this then this is the format cdf not then within quotes you write pq in capital then the value at which you are finding out the cdf of the normal random variable then the mean and then the standard deviation ok enter this function you get the value of the cdf at value x for any normal random variable we go to examples ok. So, these what we type in the Sylab prompt cdf nor within quotes pq we are trying to find out the value of the cdf at 2.25 for a normal random variable which has mean equal to 0 and sigma equal to 1 that means this is the standard normal random variable. So, using the same function cdf we can also find the cdf for a standard normal random variable. So, for that again is the previous example we can see that let me go back we found the cdf of the standard normal at 2.25. This what you are doing cdf of the standard normal at 2.25 and you get the same value this pq over here tells you was the value that we are trying to find out which is the unknown over here ok. The cdf nor function has multiple applications it can be applied to find out the cdf it can also be applied to find out the inverse of the cdf ok. The inverse normal we will see how this first argument tells you what you are trying to find out using the cdf nor command. So, that is what I said you can also use it for nonstandard normal variables putting any mu any sigma instead of 0 and 1 you can put 3 and point 2 and evaluate it at 5 and also you can use cdf nor for finding out the inverse ok. And here I am not writing capital phi, but I am writing capital F because this is not the standard normal inverse right. This is inverse for any normal it can have any mu any sigma and you can still take the inverse. Now, the format changes a little bit from the previous one the first argument is x, x is the value at which you know the cdf right. What you get? What you get if you take the inverse of the cdf you get the value at which the cdf is taken that means x. So, that is what you are trying to find out we are trying to find out x we supply the value of mean the value of standard deviation the cdf p is the cdf or probability q is also the probability this is 1 minus p for some reason sila needs both p and 1 minus p ok. On the probability axioms that we know that q is always 1 minus p, but it needs both. So, we will supply both that is ok. So, here is a program here is a command to find out the inverse of the cdf of a normal random variable. So, we are trying to find out the inverse 0 mean standard deviation equal to 1 and I took the value remember it was 0.98778 or something I just rounded it off and for q you can write 1 minus this or you can find out the value of q and write it over here either way both works you can write it recursively that is the good part of it. And this is the answer sila gives you ok you can check this what we had p inverse of 0.98778 equal to 2.25 same thing from the sila command cdf now. The important part is over here the first one x signifies that we are finding the value x for which we know the cdf. Instead of 0 and 1 we could have used any other value of mean any other value of sigma. We go to an example this what we have daily rainfall in months and months in Mumbai can be described with this normal random variable x which has a mean of 55 and a standard deviation of 15.5. These are in millimeters, but if you remember I said that for random variables we do not need to put the units, but as engineers we need to keep a track of what units are there. What is the probability that it rains more than 70 millimeter ok. So, you want to find out the value 1 minus the cdf at 70 is that ok. What is the rain is less than 70 millimeter less than equal to 70 millimeter that is the cdf at 70. So, the probability that it is more than 70 millimeter is 1 minus cdf at 70. So, you find 1 minus cdf nor we are trying to find out the cdf. So, you put pq over here at 70 for a mean 55 for a standard deviation 15.5. This is the answer we get which means 16.7 percent ok. Very simple just go to Sylab apply a function like that and you get the answer. What is the probability of a moderate rainfall and by moderate let us say we define moderate by 30 to 70 millimeter rainfall. So, that is the cdf at 70 minus cdf at 30 for the same thing and this is the answer you get 78 percent. And you can see 55 is the mean 30 to 70 is around the mean. So, there is some concentration. So, most of the data 78 percent of the data is in the range around the mean. Mean is a measure of what is a measure of central tendency of a distribution. So, this one tells you that you have more data around the mean. Now, you look at the other way what is the 84.6 percentile rainfall. For design purpose many a times it is defined that you have to design for a 90 percentile rainfall. That means it will be exceeded only 10 percent of times. Usually design guidelines tell us that way. So, let us say the design guidelines ask us that you have to design for a 84.6 percentile rainfall. So, you find the inverse now. So, again you use the cdf NOR command and you are finding out the inverse for a mean 55 sigma equal to 15.5 cdf is 8.846 that is 84.6 percentile and this is q this is p this is q equal to 1 minus p you put that you get a value 70.8 millimeters. So, now you can design with this intensity of rainfall. We go to a related random variable log normal which is kind of close to the normal and it also has a lot of application and the primary use of log normal is probably that it is very good when you are dealing with multiplication of various random variables. Then log normal becomes very useful. See the overall goal is whenever you are dealing with real life data you want to fit some known distribution to that data. Why? Because once we have that description in terms of a known distribution then it is very easy to incorporate the uncertainty the variability in a given random variable in your calculation. Because for a normal random variable we know the cdf we know the pdf we know the mean sigma everything.  So, log normal is another option most of the times we try that. How do you define it? X is a log normal random variable if its logarithm is normally distributed. So, y defined as ln x is normally distributed then x is a log normal random variable. It is valid in the range of 0 to infinity. Now if you go to the theoretical things the base of the logarithm it is not important. So, instead of the natural logarithm you can take it to base 10, base 2 whatever you want. Log normal random variable sometimes you will find they are very close to normal distribution. So, we will see some pictures. This is the definition of the pdf. Does it look similar to the pdf of a normal random variable? Does it look similar to you? It has some similarity. At least in this part you can see this half is there denominator of sigma y square and so on. y over here is the corresponding normal random variable y is log of x. We said x is a log normal random variable is if ln x that is y is a normal random variable. So, y is a normal random variable with mean equal to mu y and standard deviation equal to sigma y. Then you can write that the pdf of the log normal random variable x is this. We have not put the mean and standard deviation of the log normal random variable x in this equation. Do you see that? We do not have mu x or sigma x in that equation or in that expression for pdf. So, definitely we need to know the mean and standard deviation of the corresponding normal random variable when you are trying to find out the pdf of a standard normal random variable. I am sorry when you are trying to find the pdf of a log normal random variable. This is how it looks like. Log normal random variable pdf for different values of sigma. Sigma here is the sigma of the corresponding normal ln x or y is the normal random variable. Remember that? So, for different values of sigma and constant value of mu you can see how the log normal random variables pdf change. You can see if you go by this one, the red one which has a very low standard deviation 1 over 4 kind of looks like the normal random variable. If you go even further down sigma equal to 1 over 8, it is almost the pink one, almost the normal random distributions pdf. So, sometimes log normal is very close to normal, but not always. For example, the blue one or the green one. Now, go to the cdf. So, you want to find out the cdf of the log normal random variable x. That means capital X less than equal to the value small x. You can write that in a form that the probability of capital X being less than equal to small x is nothing but the probability that ln of x should be less than equal to ln of small x. Do you agree to this? Is ln x a non-decreasing function with x? Yes? If it is so, then you can write this. That means ln x is nothing but y. So, this is the definition of cdf of y. So, f y at y and y is a normal random distribution. If it is a normal random distribution, then its cdf can be obtained from the standard normal operator. This is the cdf of the standard normal random variable and how do you obtain using the standard form? So, y minus min of y, everything divided by standard deviation of y. u y and sigma y are not the mean as standard deviation of the log normal random variable, but they are the mean as standard deviation of the corresponding normal random variable for a log normal random variable x. This is important. So, basically what you see that we need to find out those two and here how the cdf looks like. So, again if you go to very low value of sigma like 1 over 4 and 1 over 8, the pink and the red curves, the s curves like you have seen for normal distribution and if sigma is very high like 10, the black one or the blue or the green, it is very different from the s curve, but even this black one at one point of time has to be asymptotic to cdf equal to 1 has to be. Finally, it has to reach cdf equal to 1. Maybe it takes equal to infinity, but it has to reach there. So, all the expressions of pdf and cdf is based on the mean and sigma of the normal random variable. So, we need to find out the mean and standard deviation of the corresponding normal random variable given the mean and sigma for the log normal random variable and these are the equations. You can note them down. So, if we know random variable x which is log normal, if we know the mean and variance, we can find out the mean and variance of the corresponding normal random variable which is ln x. And once we find these two, we can find out the cdf and pdf of the log normal random variable x. And again we take an example, we will continue this similar example. Again the same monsoon rainfall, instead of considering it to be a normal random variable, we consider it to be a log normal random variable with the same mean and standard deviation. And the question you ask is what is the 84.6 percentile rainfall. So, this is what we are trying to find out. We know that the cdf of the log normal random variable is 0.846. We want to find out the value at which it obtains that cdf. And we know this expression f x of x is equal to phi of ln x minus mu ln x divided by sigma ln x. I just wrote that a few slides back. This is what we wrote f x of x is equal to this y is nothing but ln x. So, we know this we are going to use this. So, we find the mu of ln x, we have the expression, we know mu of x and variance of x. So, you get the value and the variance of the corresponding normal. Again we know the mean this is 55 standard deviation is 15.5. We put those, we get the variance. If you know the variance, we can also obtain the square root which is the standard deviation of the corresponding normal random variable. Once you have these two values, then you can plug it in into the basic expression. And this is the answer that you are trying to find out at which value it obtains the cdf of 0.846. Now, does this expression look ok to you? Let me get back. So, x will be phi inverse of this thing from that we get this. Is that ok? Well, you can do it for yourself and see what answer you get. You can use standard tables to find out phi inverse. How do you use standard tables to find out phi inverse? Tables have values for given x, what is the cdf of the standard normal? You calculate it back and most of the times we do linear interpolation between two cdf values or two x values. Same thing can also be done using psi lab. So, we write the same expression, exponential of this. We are trying to find out the inverse. This is the inverse over here. So, that is what we have. This is what you get. We go to the last one to discuss today. This is gamma distribution. There is a pdf like this. It is defined only for positive x values. P and k are parameters to describe the pdf of gamma distribution. And this gamma k is called the gamma function. Are you familiar with the gamma function? So, you know what the gamma function is. I would not go into the details of it. This is the pdf. There is a specific reason why I am discussing gamma. When we go through some examples, we will see. So, the gamma function is kind of like the factorial. Even you can calculate it for non integers. Factorial is only for non integers. But, gamma function can be calculated for non integers as well. Are you familiar with this? The incomplete gamma function? Well, if not, then this is the similar expression. You have the gamma function in the denominator and the numerator is a little changed one. You can find out the differences. It is defined for parameters u and k. k is the same k as the parameter in the gamma distribution also. u is where you are evaluating it. That is why you have integration from 0 to u. The incomplete gamma function is what we use to find out the cdf of the gamma function. How do you find the cdf of the gamma function? On the basic definition, we can use the pdf. So, probability that x is between a to b, that means cdf at b minus cdf at a. So, that probability you integrate the pdf from a to b. Is that ok? I will go back to the pdf expression. This is the expression for the pdf. It is valid only from 0 upwards. This is the expression for the pdf. You can note that v and k are constants. So, of course, gamma k is also a constant. So, you can take out that v raised to k and gamma k outside the integral. We do an integration of the other part of the pdf from a to b. You can divide it up if you expand it. We take the vk inside the integral. Then, the limits change to 0 to vb and 0 to va instead of 0 to b and 0 to a. Since you have taken the v raised to k inside, we change the integration limits. What are these? If you look at closely, these are nothing but the incomplete gamma functions. These are the incomplete gamma functions evaluated at vb. Earlier, we had u over here. These are the incomplete gamma functions evaluated at va. So, you can write the probability that x is between a to b is the incomplete gamma function at b minus incomplete gamma function at a. So, the incomplete gamma function is kind of like the cdf of the gamma distribution. See, probability that x is between a to b is the cdf at b minus cdf at a, which is same as i at vb minus i at va. So, not exactly cdf, but very similar to cdf. You can easily find out tables for this i so that you can evaluate the cdf of the gamma distribution. These are the two moments. Mean of gamma is given by k over b and the variance of gamma is given by k over v square. k is called the shape parameter and v is called the scale parameter for the gamma distribution. So, what do they do? They change how the cdf looks like. They change how the cdf looks like. So, these are again descriptors. Instead of mu and sigma, you can probably use k and v. You remember similar things that we use for Poisson distribution. What is the parameter that you use for Poisson distribution? Do you remember what is the mean of a Poisson distribution? Lambda, standard deviation also lambda. There was only one parameter to describe the pdf and cdf of the Poisson distribution. That was lambda. For the gamma distribution, we have k and v, which is the shape and scale parameters. Using those, you can find out the mean and standard deviation. Now, this is why I have discussed the gamma distribution. It has a specific relation with the Poisson process. If you remember what the Poisson process is, you remember we discussed that when we discussed the discrete random variable called the Poisson distribution. What is the Poisson distribution? When do we apply Poisson distribution? We apply it to describe the occurrence of an event over time or over space. For example, for this table, if a crack is going to appear over the length, where is the crack is going to appear? That will have a Poisson distribution. Again, I gave examples like number of earthquakes of such and such magnitude over next 10 years. How many? That usually follows a Poisson distribution. We call that a Poisson process having a specific mean occurrence time, which we denoted by nu and then we had t and we had a relation with lambda, nu and t and so on. If you look back, you will find those things out. Now, we are looking at the relation of the gamma distribution, which is a continuous distribution with the Poisson process, which deals with a discrete random variable. I hope you remember those things. So, what we say that if the occurrences of an event constitute a Poisson process in time, then the time till the kth occurrence of that event follows a gamma distribution. At the first hearing, it may sound a little strange, how are we relating these two, but later on you will see it is not very complicated and you can even prove that. So, if the occurrences of an event follow a Poisson process, that means how many earthquakes over the next 10 years, then the time till the kth occurrence, let us say the second earthquake will follow a gamma distribution. Now, this time till the kth occurrence, what kind of variable is it, discrete or a continuous one? Continuous, right? So, no problem in that. It can have a gamma distribution and it will have a gamma distribution, which we can prove. Before we go into an example, again we will look at psi lab, what are the functions that we have? Again, just like normal random variable, we had CDF NOR for normal random variable. For gamma, we have CDF GAM and the expression is very similar. Again, within quotes, you put PQ, then you put the value at which you are evaluating the CDF, then you put the shape parameter and then you put the scale parameter. These are non mean and standard deviation. These are the two parameters K and V. This will give you the CDF and you can also, of course, obtain the inverse of the CDF. For that, instead of PQ, we write x over here and then the shape parameter, scale parameter, CDF and 1 minus CDF. Using that, we can find it out. Example, we do the same thing as we did for the normal random. So, CDF gamma, that means we are trying to find out the CDF using this PQ argument at 2.25 for K equal to 2 and V equal to 0.5 is this. So, instead of mean and standard deviation, we use K and V, which are equivalent. If we know K and V, we also fix mean and standard deviation. If we know K and V, we fix the PDF and we also fix the CDF. If the CDF is fixed and the PDF is fixed, you can obtain mean and standard deviation from those. So, those are also fixed. So, you put mu or mu and sigma or K or V is the same thing, but silo requires you to put K and V over here and the inverse. I think you are doing the inverse of the same thing. We obtain the CDF at 2.25, we got a value of 0.310 and we turn it around. We find the inverse for K equal to 2, V equal to 0.5 of CDF 0.310 and this is Q. Q is always 1 minus P and you get back at where you obtain this CDF of 2. So, you get the value of 2.25. Salib also has this function gamma K, using which you can obtain the gamma function for any value K, which can be integer or non-integer. Now finally, the example and with this we will stop. So, this is known an earthquake larger than M5. M is magnitude moment magnitude in Richter scale. So, larger than M5 hits Mumbai once every six years. That is known to us and you also know that it follows a positive process over time. And you are trying to find out what is the PDF of the time of occurrence of the third earthquake. And by earthquake we mean earthquake having a magnitude larger than M5. So, you can see it is following a positive process and we have a occurrence rate once every six years. So, use that information and this the PDF for the time of occurrence of the third earthquake. We said that if this follows a positive process, then the time to the third earthquake will follow a gamma distribution. So, use the gamma distribution for this. So, based on the information we know the annual occurrence rate nu is 1 over 6 once in every six years. And the occurrence time would follow a gamma distribution with K equal to 3 that is the third earthquake and V the shape parameter equal to the occurrence rate equal to 1 over 6. K is the kth event and V is the occurrence rate nu. So, the expression for the PDF. So, T 3 we put. So, time to third earthquake that is the random variable. This is V and this is V x raised to the k minus 1 this factorial or gamma function of k minus k and this is e raised to minus V x. So, that is the PDF that we wanted to obtain. What is the PDF for? This is the PDF for the time of occurrence of the third earthquake. And what is the mean for that? E of the time of occurrence of the third earthquake from now is 18 years. Where is the 18 coming from? This is K equal to 3 over V. I will get back mean or expected value of x is K over V. So, K over V, K is 3, V is 1 over 6. So, you have mean equal to 18 years. Make sense right? The occurrence of earthquake has a rate once in every six years. So, when do you expect the third one to happen on an average at 18 years from now on? That is what you have. Also, you can find out the probability that the third earthquake will occur between the 16th and the 18 years from now. So, we find out the CDF at 18 years minus CDF at 16 years. This is for T 3. We are trying to find out that T 3 occurs between 18 and 16. So, CDF of T 3 at 18 minus CDF of T 3 at 16. With K equal to 3 and V equal to 1 over 6, you get this 0.0786 as well. So, that means 7.86 percent. So, we relate the Poisson process which basically describes the number of occurrences over time or space of a discrete random variable like number of earthquakes, number of floods, number of defective pieces in a given production and so on. We are trying to find out when is it going to occur, the k-th event that will follow gamma distribution and you can find out the probability of that occurrence for any given instant. That is what we do over here. So, this is what we covered. Standard normal, standard normal operator and the inverse, the silo function for it. Example using a normal random variable. Then, we discussed quickly the log normal distribution. It is CDF and PDF and how it is related to the normal random variable. LNX is the normal random variable. How to obtain the mean and sigma for that? Then, one example with log normal random variable. Then, the gamma distribution properties relation to Poisson, the silo function and example. That is it. Thank you.


Do you think this is a good idea? It's dropping busloads of people, including last night there was a one-month-old baby on a busy street. John wasn't happy with the previous administration because the previous administration actually secured the border. Let's be honest. I think that what this is doing is this is highlighting the hypocrisy and I think in many cases fake outrage by those on the left and some in the media. The migrants in Florida, they're not arriving on the shores of Florida. They're not walking across the Florida-Georgia line. They're being flown in by the Biden administration. So somehow it's okay to fly them across the interior of the United States, everywhere across the United States, but it's wrong for the governor to then send them to one of the wealthiest communities in America. I think it's highlighting that hypocrisy. But there's a difference between doing it in a coordinated fashion and just showing up at Martha's Vineyard. Of course. Who could have guessed that the response from a Democrat state media mouthpiece would be that it's different when they do it. Here's my question. Since Martha's Vineyard, can other cities and towns now remove their illegal immigrants using the military? Or, more likely, it's just different when they do it.