proudly we hail New York City where the American stage begins here is another program with a cast of outstanding players public service time has been made available by this station for your army and your Air Force to bring you this story as proudly we hail the pioneers of air safety is entitled a medal for Joe a story of the exciting early days of flying and of the men who risk their lives for safety our first act curtain will rise in just a moment but first America looks up to her men in the sky yes our country looks up to the young men thundering their way to new glory in the skies theirs is the task held in high esteem by the entire nation you are needed to swell their ranks and can do so by enrolling now in the aviation cadet training program of the United States Air Force if you're between the ages of 19 and 26 and a half have had two or more years of college and are otherwise qualified visit your nearest United States Army and United States Air Force Recruiting Station and ask about the aviation cadet training program do it today and now your army and your Air Force present the proudly we hail production a medal for Joe a cup of coffee and I'd explode gee that was a wonderful dinner Julie you're even a better cook than I remember well it's been a long time since I had a chance to show off in the kitchen for you yeah it has been a long time hasn't it and yet it seems like just a couple of hours ago since we were all stationed in Texas and I was barging in on you and Bob now I must have eaten every dinner at your house two years ago two years ago awful lot has happened since then sure has Bob and I were on our feet then instead of in the air remember how we both signed up for the air service on the same day remember I'll never forget it a date was August 27 1919 anniversary I was afraid then it what's there to be afraid of most accidents happen in the home people slip in the bathtub or fall down sometimes I can't help but be a little worried haven't you heard anything from the field no I called about five just before you got here said the flight was delayed ran into some bad weather well I don't think there's anything to worry about if something was wrong you'd be getting a call from the field those boys are pretty good about keeping everybody up to date yeah I know that now let's just forget about any silly worries let's just think about how surprised Bob will be when he gets home and finds that ad hoc old major Hoffman that is right here in his own quarters in McCook field you know by the way I don't think it's exactly right for me to just move in like oh nonsense we've got plenty of room and the kids love having it too when some good bachelor quarters come up you can move out providing it isn't too far away for you to come and have dinner with us every night well with this dinner as an inducement I think I'd be able to make it a couple of hundred miles for the same thing alright here let me help you with the dishes while we're waiting no no I certainly won't let you help you just go into the living room and read the paper won't take me a minute Bob will be here before I finish might as well have blindfolds on let's try to see through this bug you may got anything Chuck your soup Bob your soup hey can you take it down any think we must be way beyond those mountain ranges by now if our calculations are right we're not far from home it's times like this I wish I were patrolling the Mexican border again on foot yeah you'll sing a different tune when you walk into your front hall and give Julie a big kiss nice close your eyes let's go down a little see what we can see it's fine up Bob look there's the field they got the flares up for us that's fine up pull her up she's stuck Chuck I can't pull her out Julie stumble over something no the glass just fell out of my hand yeah something's wrong I don't know what it is but something's happened to Bob let's sit here in the sun and tell lies to each other you're like a dope having to be waited on like this wait till I get out of this cast and I'll throw you over my left shoulder with one hand just to get even with you for having to carry me out here like a baby every afternoon I stole lemonade coming up oh it's wonderful having your home all the time darling now there's true love for you she works like a dog both of you do cutting me around waiting on me hand and foot and you say it's wonderful having me on the ground well it is now and don't laugh at me I wish you could stay on the ground always Bob look Julie if they told me I couldn't fly I don't know what I do so let's just make up our minds to that soon as I'm out of this plaster cast I'm gonna be at the ready room waiting for my orders you know I've been doing a lot of thinking in the past five weeks since that night we were sitting here waiting for you to come in Bob I don't tell me you're on Julie's side look at you are the one who got the idea of transferring to the air service of this US Army but I was more than five seconds behind you but remember all the arguments you had stored away to answer Julie oh sure sure I know don't think I transferred any other branch no the thinking I was doing was a little different well what Ed if it's anything to keep you out of the air you two lunatics I'm all for it no no that's not it Julie we'll fly gosh we were born to fly it's just that not enough work is being done on making flying safe well you're crazy they're doing everything they can what happened that night couldn't have been avoided it wasn't anybody's fault but you could have done something Bob you and Chuck both done something what we did everything that was humanly possible you can't blame that wreck on me or Chuck or anybody else wait a minute calm down fella I'm not blaming anybody but look at it this way you weren't chained in your pilot seat were you of course now what are you getting at and Chuck could have moved around to couldn't he sure but I don't see what you both could have climbed out of the cockpit couldn't you sure and then we could have soared right down to work and let the plane crash without us Bob I'm serious after your accident while Julie and I were here alone and when well all right let's be honest when I thought you were gonna die I stayed up alone night after night thinking about something I saw once that might have prevented all this nothing could have prevented it Chuck and I did everything we be quiet now let it explain once I was at a county fair and a guy went up in a balloon I suppose he was crazy to do it but he jumped out in a parachute he could have killed himself but he didn't parachute what's that well they've been doing a lot more experimenting on them in Europe than they have here it's sort of like an umbrella a silk umbrella the wind slows it down enough so that the guy strapped underneath it just floats down to earth sounds like a dream sure it's a dream the Wright brothers had a dream just eight miles from this field and we're flying in the product of that dream every day well why hasn't more been done about it oh I don't know lots of reasons there isn't a lot of money around to spend on aviation yet and what there is is being spent to make better planes to make those planes safer besides that there aren't enough people that are convinced it could ever work are you convinced yes I am yes I'm completely convinced there's an awful lot of work in experimenting to be done and maybe somebody will get killed trying the things out but parachutes will work if we work on them hard enough I don't think I get it entirely I wasn't at that county well it's not hard to explain it's look you see that milkweed pod there in the garden that white fluffy thing that's floating down to earth well sure we've got dozens of them out here all during the summer all right now that is the perfect parachute see how it opens out and just like a silk umbrella it floats the pod down to the plants you think that man could build something like that for men to parachute to earth yes I do if you got the right cloth and the right pulleys and the right well the right everything you see that's where the trial and error experimenting comes in but it could be done it could work it could have worked that night for you and Chuck you're a dreamer Ed like I said oh yeah and I think I'll be called worse than that before I'm through and this is a paint section major here's where we've been doing some research on paint that won't rust or chip off yes I see core pro looks like it's in fairly good shape and we've been making some headway sir and that room in there what's that oh that's just sort of a supply and storage room I'm afraid it's not as neat as we'd like it sir well if you've known that the new material chief would be such an eager beaver you probably would have had everything shining huh well sir we are pretty short-handed but we can get it in shape right away well let's go in and take a look anyway we've got a lot of junk in here most of it is just that I guess major but I don't like to throw anything out until they tell me to thought you might want to see it all and then we can throw up whatever you say what's over here in this box boy it's dusty enough all those those are a couple of what do you call them parachutes somebody I can't remember his name some fellow who's attached to this section a couple of years ago he was always trying them out wait a parachute you mean there's been some experimenting on them here at McCookfield nothing much from what I hear I wasn't around then but this fellow I was telling you about was hit on the subject kept weighing them down and dropping them out of planes never got anywhere with them though everybody said he wouldn't but he kept trying until he was transferred somewhere else so they've just stayed here gathering dust should I throw them out major no sir read dust them off I want to inspect them thoroughly this afternoon and corporal get out anything else that that fellow used in his experiments I want everything he used out and ready for me to look at right after Chow yes sir all right gentlemen anything else how about you Hoffman you're getting around all right finding out about your new section yes Colonel it all looks to be in good order there's just one thing sir well go ahead staff meetings the time to bring up any problems you got that's what we have these sessions for well I have a list of requisitions here Colonel well you can just put those through channels can't you normal requisitions can just be ordered in the usual fashion that's all you've got and we might as well break this up no oh wait Colonel this is a little different this is something I think I'll need staff approval on you see it's a list of requisitions for a new section a new section well that's different here let me see that one sewing machine 12 volts of silk hey what is this are you planning on opening a dress making shop for the wives of the post well no sir it's well I was hoping to get staff approval to start some experimenting on parachutes parachutes now look Hoffman let's get this straight you're in charge of materials your jobs to work on paint and metals and fabrics anything else it will improve our air service we don't want any crazy experimenting on silk umbrellas going on around here I haven't got the money or the personnel of the time I'm not going to risk the lives of any of my men Colonel they're risking their lives now every time they go up without any protection I know that just as well as you do but my job is to improve what we've got to eliminate the hazards in the equipment that exists your job is to put your section to work on the improved research now let's not consider any hair brain schemes well Colonel if I work on them in my spare time would that be all right for sure there's nothing I can say about how you spend your time off me deffy idea you want to carry out on your time off your business but my business right now to throw this requisition in the wastebasket is missed you see Colonel I honestly think it will work now not right away I know it's going to take a long time probably but look at it this way if we don't try we'll never know if it'll work or not can't we just try when you've dug my footsteps for four months now every time I look up there you are with a requisition in your hand I'm getting so I I look under my bed at night to see if you're there well I'm sorry Colonel I know I've been a nuisance but if you just put through this requisition I promise you won't see hide in her hair of me for months just let me try and see if I can make a parachute work you know I'm not just starting from scratch a lot of guys in Europe and here in the States have been working I've got something to go on all right all right you've worn me down get a bowl of silk and fool around with it if you must not get out of my sight oh and and Hoffman yes sir good luck you are listening to the proudly we hail production of a medal for Joe will return to our story in just a moment young men of America your country is building a mighty Air Force to maintain the security of our nation this means that there is a job for you the chance to do an important job with one of the finest organizations of its kind in the world today the United States Air Force if you're between the ages of 19 and 26 and a half have had two or more years of college and are otherwise qualified the Air Force needs you as part of the aviation cadet training program for complete details visit your nearest United States Army and United States Air Force recruiting station today you are listening to proudly we hail and now we present the second act of a medal for Joe here corporal give me a hand with one of these crates sure thing major gosh these things are heavy what's in them sir sewing machines here let's put this one over in the middle of the room so we can open it up easily right sewing machines and you really got the stuff you asked the Colonel for you're really gonna make some of those parachutes mm-hmm that's right and go ahead and laugh if you want to oh sir I don't think I will you don't think you're what laugh what do you mean I've convinced you after all these months not really convinced me but well I'll put it this way if I were up in a plane instead of here on the ground I'd like everything I could get my hands on to make sure I'd get back on the ground who knows this thing this idea of yours might work I say it's worth trying well what do you know corporal I've got my first convert yes come in yes you gentlemen wanted to see me your major Hoffman that's right I'm guy boy my name's Floyd Smith oh you don't have to tell me who you guys are I've seen you make parachute jumps over at right field glad to meet your bow the same here go on go on sit down both of you well what brings you men over here well we heard you wanted to hire some men to help out on your experiments with parachutes I sure do but you're the first ones to come in here since I placed ads all over the country weeks ago I mean you really want job yeah we sure do we've been both risking our next jumping out of planes affairs all over the barnstorm and country we heard you're trying to perfect parachutes and if there's one thing Floyd night like it's a perfect parachute well I've never been so glad to see two people in my life you two guys probably know more about parachuting than anybody in the country look I'm afraid we can't offer you the kind of money you get for trying to break your neck there's major majors it's like guy says we'd like to get some good parachutes that's all that's why we're here yeah both of us not to mention their wives we're a little sick of believe in every jump is gonna be our last we'd like to work with you but you're hired when do you want us to go to work right now listen you can tell me more this afternoon that I could learn in three years of using weight what do you want to know I'll just start from the beginning what kind of shoots did you use what were the police like what about the winds just tell me everything you know and when you finish we'll figure out what we should do to get you to that perfect parachute and there there he is a hundred and seventy pounds and the best looking guy I've ever seen good morning half man Smith morning colonel hello colonel what's this pretty large for a puppet colonel I'd like you to meet Joe dummy Joe well what's his assignment around here he's gonna make all our experimental jumps colonel you see we found out that 50 pound weights weren't working right we discovered that if we wanted to make a good parachute for a man we'd have to have the next best thing to a real man that's right so we built dummy Joe here you see colonel his arms and legs are weighted to put the stress on the cords the way a man would do it well anyway almost the way a man would do it and we've even got a time fuse in his right hand so Joe can pull the rip cord for himself pretty ingenious looks like you're making progress here well so we we've been getting along how about materials you got what you need here seems to me I've been signing enough requisition I think we've got enough for Joe's first trial anyway sir we finally got just the right kind of silk had to send all the way to Japan for it but it should be just right you see the wind goes through it it's porous enough for that but enough wind should stay in it to billow a man Joe that is down to earth just as nice as you please well it sounds as if you're getting along pretty well frankly you're doing better than I thought you would I'll have to hand it to you Hoffman you know how to stick to a job well thanks colonel it's all up to dummy Joe now well he's not going to prove everything in one jump is he no no sir not by a long shot it'll probably be more like a thousand before we're ready to test the shoot with live personnel well don't go using any men on those things until you're sure and Hoffman yes sir let me know when dummy Joe completes a thousand jumps I think by that time we ought to have a medal pinned on him and we got a rig up that time few some other way and you're right dummy Joe's got a delay before the ripcord's pulled otherwise you'll get all tangled in the tail surfaces of the plane all right let's fix it right now now this is the way I've worked it out a load of 400 pounds released from a plane flying in 120 miles an hour will put just as much strain on the shoot if not more as a delayed drop of 16 seconds yeah okay I'll take your word for it let's go find something for dummy Joe to carry on his next jump come up with anything to figure out why those rigging lines are breaking yeah I think so this thing looks complicated and it is sort of but you see it should record the amount of strain on each individual cord then we'll know what's pulling too much in the drop look at you what's the matter your company got a parade well can I put on my dress uniform once in a while sure sure I just haven't seen you in anything but fatigue for so long it came a sort of a shot well you and Guy and me are invited over to Julie's and Bob's tonight so go find yourself a white shirt and the tie she's expecting us in about half an hour we really should have you know this should actually be his celebration that is he performed some MacDee on the call of duty can certainly say that tomorrow is going to be jump number one thousand I'd like to make that bail out instead of dummy Joe tomorrow why do you look so shocked why do you think we've been making all these experiments we've been working on these parachutes for men to use not for dummy well sure of course I know that but it's well I guess I never really thought about you or Floyd or guy or Bob parachuting from a plane Julie you confuse me nothing would satisfy you but Ed had to have everyone believe in him and help him in his experiments now that he's got it ready for a man to try out you're sitting there mumbling that a man shouldn't do it I know it's silly but I just wish somebody I didn't know would try it well it's not gonna be someone you don't know Julie gonna be someone you've known for almost ten years me look at I know how you feel I've been in on a lot of this I know just as much about that shoot as you do and furthermore I've had actual experience I've jumped in shoots a lot worse than the ones you started with I'll jump tomorrow and now wait wait a minute how about me I've chucked up just as many jumps as you I've got a better idea than any of you after all it was my crash that got you started on this let me have the first try at the thing you thought about because of me I think you're all crazy there's only one way to settle this we'll do it the way we did when we were kids now here I'll break these toothpicks don't watch you'll draw straw stop crumbling just draw one edge you go first Floyd guy and Bob thanks I hate to be selfish but I sort of hope you're out I am I think you stacked them oh let's see the rest of them of all the rotten luck okay Floyd it's dummy Joe shoot for you tomorrow I'll go along in the plane with them all right I'll check from the ground Julie you and Bob come on out to the field with us tomorrow you were in on the beginning I want you to see our dream come true I almost think it'll be easier up there in the plane than down here and right it would be planes coming in for the jump now cross your fingers and pray there he goes not opening the shoes not opening oh it'll open it's got to there it is it's open oh just as pretty as you please and not a tree in sight for him to land in and no wind to blow him anywhere but right down here at our feet I almost didn't make it I wouldn't miss this for anything Colonel I well I thought you were off the base I was darn near broke my neck getting back here wouldn't have missed dummy Joe's jump for anything 1,000 today isn't it I've been keeping pretty close track of that little guy Colonel this is a little difficult to explain you see you went around what are you mumbling about Hoffman well I'm just trying to say that well that isn't dummy Joe sir not dummy Joe well then who Hoffman you don't mean to say there's a real live human being in that contraption it's Floyd Smith Colonel we drew straws and he won Colonel look how beautifully it's working why it's just like a child in the swing he's down let's go out and give him a hand well how'd she work Floyd perfect perfect not a thing wrong you ought to order enough of these for every man Colonel what I ought to do is court Marshal you Hoffman Colonel think what it would have meant to the men like Bob who crashed if they'd had parachutes think what it will mean to all the men who are flying and will fly in the future and besides Colonel all you said was not to let a man jump until the parachute was safe well you just saw it work sir yeah I guess that's what I said well I'm a man of my word and I said something else too so let's go where Colonel into the hangar you've had your glory for today I'll give another guy a chance going to pin a medal on dummy Joe America is depending on her leaders in the air if you're a young man between the ages of 19 and 26 and a half have had two or more years of college and are otherwise qualified you are eligible to join the ranks of America's leaders in the air you can become an aviation cadet the defense of our nation hangs heavily on our air strength the finest in the world but we cannot relax our efforts the Air Force still needs pilots and aircraft observers if you have the primary qualifications visit your nearest United States Army and United States Air Force recruiting station today ask about the aviation cadet training program do it now this has been another program on proudly we hail presented transcribed in cooperation with this station proudly we hail is produced by the recruiting publicity bureau for the United States Army and United States Air Force recruiting service this is Kenneth Banghart speaking and inviting you to tune in the same station next week for another interesting story on proudly we hail


Welcome back to our lecture series math 4220 abstract algebra one for students at Southern Utah University As usual, I'll be your professor today. Dr. Angie Misseldine Lecture for we are still in section 1.2. It turns out there's a lot of stuff to say about sets and such Although in this lecture, we're not gonna on these will be sets, but we're really gonna focus on the equivalence relationship Side of the section header there. And so what is an equivalence relationship? Well, one answer is this is one of my very favorite topics As as one who studies algebraic convictors partitions is something that is dear and true to my heart They they come up with sure rings and a lot of other stuff. I do my own personal research So we're gonna talk about equivalence relationships in this lecture and there's a lot that can be said here This one mostly will present the definition and some examples of equivalence relationships So remember before we had to find what a relationship is right? What is a relationship? say that we have Two sets like say a and b and so a relationship are was some subset of the direct product of two sets And so then whenever the element a comma B is inside the inside the relationship are we would say that a Rb things like that. Well, it turns out using our as your relationship symbols kind of corny Typically, we'll use a different symbol like in this case a little twiddle will say like a twiddle B or something like that So that's the general definition of a relationship We then proceeded after we define this to define what a function relationship is In this lecture, we're gonna talk about a different relationship Which is often referred to as an equivalence relation so an equivalence relation is You often use something that in some regard looks like an equal sign. So you don't typically use equal itself because that's typically reserve for equality of sets But we could use things like a congruence symbol like when you do modular arithmetic the triple equal sign there or congruence like maybe if you do geometry or Approximately or just a single twiddle or twiddle line There's a couple of different options one could do here But you often use a symbol that in some regard resembles an equal sign here now an equivalence relationship is going to be a relationship on a single set x so that means is it's a subset of x cross x Equivalence relationship always relates things from the same set to another element of that set and so it's a relationship on a set with itself That satisfies three properties, which we could call the axioms of an equivalence relation That if something's an equivalence relation the following three properties must hold and without one of them It's not an equivalence relationship. The first one is referred to as the Reflective property the reflective property says that every element is related to itself So if you take any element x inside the set x will be Related to x or as this is an equivalence relationship. We'll say that x is equivalent to itself and when one thinks of an Equivalence relationship you often want to think of With the following idea that we're trying to generalize the notion of equality So when it comes to equality of like numbers, right x is equal to x. That's a property we want for general equivalence relationships The second property is referred to as the symmetric property Which says that if x is related to y then y is related to x the the order doesn't matter Who's on the left who's on the right? It doesn't matter whatsoever And this is again a property we see with equality that if x is equal to y then y is equal to x That that's a property of equality equivalence relationships are generalizing this principle and lastly For an equivalence relationship. We require the transitivity property the transitive property This tells us that if x is related to y and y is related to z then x is related to z as well And we see a similar property for equality if x equals y and y equals z then It must be that x equals z So as equivalence relationships are trying to generalize the notion of equality You see that these three axioms of an equivalence relationship are exactly these three properties of quality that we demand when We Equations and such so when one has an equivalence relationship some related terms that we're going to find here Is if you have a typical element x inside of the set that has an equivalence relationship Then you often draw something like x Inside the box there so bracket x bracket sometimes people write things like x bar Or something like that what this is described as will be called the equivalence class This is going to be the set of all things to kind of wrap around to the other side there all things We want all elements of x which are related to x there now x of course is going to be inside of this set Because of the reflexive property Equivalence classes are never empty. This is the main reason why we have property one here x always belongs to the equivalence The the x always belongs to its own equivalence class But then you'll also contain any other member of this set that's equivalent to x and Because of the symmetric property and the transitive property It doesn't matter which element you use inside the set here an element that belongs to an equivalence class We call a representative and if you interchange x with anyone it's equivalent to that as you pick a different representative It'll describe the same equivalence class. It doesn't matter which representative you choose Oftentimes there's a natural selection on who you choose here and let's see some quick examples of this So let's take as our set x the set of ordered pairs of integers So a and b are both integers such that b is not zero So we take the set of ordered pairs of integers so that the second Coordinate is never zero and we define an equivalence relationship on those elements. They're ordered pairs So we have two integers here, right? We say that the pair a b is related to cd if the product ad equals bc All right, so we're gonna take the first term times this So we take in the first in the first pair we take the first term and in the second pair We take the second term we multiply those together. It's ad that should equal The product of bc Which rides you take the second term from the first one in the first term of the second one So if you always take the product of the two numbers in different positions We want that to be the same and if that happens we say the two Pairs are equivalent now. We want to show that this forms an equivalence relationship. How does one do that? Well, we have to check the three axioms. We would begin by showing that The operation sorry that this relationship is reflexive. So to show the reflexive property We have to take a generic element. So we would say something like let a comma b be inside of our set x So it's just a generic element the only thing we can suppose about this element a comma b is that it's inside of x Which means that a and b are both integers and b is not zero. All right, so then notice the following then The product a b is equal to a b That is a fact of equality like we mentioned earlier a b equals a b and therefore We get that a comma b is Related to a comma b because that's what's expected right if you take a times b. That's equal to b times a so we get Equality there and so therefore we get that a sorry a b is Related to a b that proves the reflexive property So that's how one always shows the reflexive property you're going to just take a generic element of the set and argue by the definition of the Relationship why that generic element is related to itself. That's generally a fairly easy thing to do Symmetry, how does one do symmetry? So to prove the symmetry The symmetry axiom what you do is the form you're gonna select two elements So we're gonna say a and b and then c and d. These are elements in x such that a Comma b is related to c comma d So to prove symmetry you always have to assume a Relationship that two elements are related to each other and then from there you want to argue that I can reverse the relationship So by assumption we did we assume there's a relation there to get started because symmetry is it is a conditional statement if Then type statement, so we assume the if part and we're gonna prove the then part of based upon that Hypothesis we're assuming so what do we do next? Well, we now assume that these things are related so unravel the definition so This tells us that a a d Whoops a d equals b c And so then we might have to you know twist some things around a little bit thus We see that See b Equals d a so you kind of have to turn that thing around a little bit But these are just properties of equality and so therefore we get that Cd is related to a b Like so and so notice looking at the definition of this object right here if a b is related to cd That means ad equals bc. So using properties of equality. I was able to take this equation and switch it around I had to use the commutative property multiplication and I use the symmetric property of equality But these two these two equalities are equal to each other this equality right here is the one associated to this relationship and This equality right here is the one associated to this relationship It's a subtle thing, but that's what one would do to show the symmetric property here And so for the symmetry property you assume One pair is related and then you have to show that the reverse relationship is also present All right the transitivity so transitivity requires we assume we got three elements in the set that are related So we'll say something like the following let a comma b C comma d and e comma f be inside of x such that a b is related to cd and We want that cd is related to e f So we assume that we have two Relationships here and so then we want to infer from this why a b is is related to e f here now The next thing to do is to unravel the definition. What does it mean for a b to be related to cd? So then this tells us that a ad excuse me is equal to bc And that's what the first relationship tells you this one right here. The second relationship right here tells us that cf is going to equal De Assuming I did all those correctly right ad equals bc and cf equals de Hmm now this one's a little bit trickier than the previous ones like how do we how do we relate these things together? Interesting well with this one. We're I'm gonna try to combine all of these things together. So we might say something like the following. Okay, hence We see that adf is Equal to the following. So adf Adf let's see. Well ad is equal to bc. That's great So adf is equal to bc f That's the first part and then cf is related to de so that will equal bde and So we have that these things are equal to each other via the transitive property of equality You're probably gonna use that since equality is coming up here And then the next thing to notice is that well, we're looking for and this is a strategy I can't overemphasize when one does something like this whenever you're working on a proof if you ever get stuck Sometimes it's a good idea to go to the end of the proof and work backwards. So we say something like therefore a comma b is related to e comma f That's what we're trying to prove and it's important that we pay attention to what is that we're trying to prove here But what does a b relate to e f even mean that would mean something like thus A f is equal to b e so this is the statement. We're trying to show a f equals b e We're kind of there. We have a df equals b df. That's exactly what we need except for we have this extra D How do we get rid of the D? Well, we can say something like the following. Okay, well, couldn't I just like? divide by D Well, one has to be careful. You can't just divide by any integer. You could only divide by integers that are not Zero oh Yeah, well D's not zero right because D was the second coordinate and by definition that second component can't be zero Aha, we see why that's part of the definition now This is something you should also pay attention to that when we have a definition There's usually a reason why those things are put as part of the definition. It's not just some arbitrary thing It's like we only will accept purple numbers. Why purple, right? There's a reason and same thing with a proof Right if you're trying to prove if this then that there's probably a reason we listed all of the assumptions We did if there's an assumption We didn't use in the proof then it means we didn't need it and we can make a stronger statement by removing it So the fact that we had to have the second component be non-zero means we probably needed eventually so we could say something like by Canceling oh Boy, are there two L's in canceling you're now going to see why I am not having a PhD in spelling here by canceling D which does not equal zero we then get I'm going to erase this thus We get that AF equals BE and we finish the proof showing the transitivity property And so this is not the most polished proof you ever going to see here If you want to look in the lecture notes, you can see a better version of these proofs I'm trying to write this out in real time so you can kind of see as your as yourself Need to become a mathematical proof writer I want you kind of see the thought process that goes into writing a proof believe it or not writing a proof is a Creative process some people often think that oh math is so so objective There's always a right there's always a wrong and it's very clear-cut what that is And then people like oh art is so subjective, right? There's these subtleties nuances, you know, it takes a real talent to become a great artist It turns out that mathematics is both Objective and subjective there are some things that are very very objective right clear right clear wrong But it turns out that mathematics is also a creative process writing a proof is like creating an art form of some kind It's very difficult to teach proof writing because it's a creative process How do you teach someone to be creative students be creative? You know, I can say things like that, but this creative endeavor takes a lot of practice And I can try to guide you along the way, but it's very difficult. So give yourself some patience here And so that's that's this is an example of how one writes a proof. It's a creative process this now shows that this set of Pares of integers is an equivalence relationship and we actually call the set of equivalence relationships the rational numbers a rational number is just a pair of Fractions, right? I should say a pair of integers, which we call a fraction We often denote a column B instead by the symbol a divided by B That's what we usually do But we say that two fractions are equivalent to each other exactly in that context one half. Oh boy one half equals three 36 yeah, I can do that because One times six is equal to three times two This is the typical relationship on the rational numbers and so by placing on a relationship on Pares of integers we can then create the set of rational numbers that we know and love


Welcome to BCH Technologies. Today I'm going to show you how to set up this printer to print as much as those super expensive Canon printers. First thing to do is unpacking and add some paper. And lastly, we need to get the ink system ready. The ink system is really simple. You just open it up. I got too much ink. So if you get this too low, the ink will come here. So when you remove this plug, the ink will come out. So the best way is actually do not just put it on the floor all the way. What you have to do is I have this nice holder. So when I open it, I rest out this one. And you can see there's no ink here. So I can now remove the plug. And then you just close it. That's all you need to do. Just chomping ahead of time. So when you use up the ink, you will need to add ink. So you just open it. It's the same thing. You rest your thing on this. And this will be perfect level. So you just remove this refill hole. Just add ink to it. Because most people know how to set up the printer with a USB cable. And we're going to just put how to set it up at the end of this video. So I need to talk a little bit more about the ink levels. There are two ink levels. One ink level is actual ink level. And this is actual ink level, which is totally... I mean, there is about 20, 30 times more than what you need. But the printer has an ink monitor. It doesn't know you have this big tank. So the printer's ink level monitor is going to run from full to empty as your print. So no matter what you do, you have to let that ink monitor go empty. I'm talking about the printer's ink monitor. So let me print enough pages. So the printer's ink monitor is going to turn itself off. When it turns off, it's going to give you an air and ask you to turn it off. And there's nothing you can do. There's nothing you can do to make it full. You have to run it to empty and then turn it off. And here you can see this is the printer's ink monitor. And I already think the black is about 80%.  So what we're going to do is we're just going to let this cartridge to let this ink monitor to completely run to the empty. And we're going to turn it off. Another thing I want to show you is those two lights. Those two lights are going to light up steady yellow when you only have about 20% of ink left in the printer. And then when you turn it off, it's going to have a flashing yellow. So now I will just let the printer run and print the 500 pages. We can see the ink level. We can see the ink level monitor on the computer is really low. This is action level. So this is what the printer think the ink level is. And when it's below 20%, you can see the indicator light is permanently yellow or called amber. So the printer's ink monitor will be reset pretty soon. I just keep printing until it reaches that point. So finally the printer thinks the ink is run out. So I give you this error and the support code is 1688. What we want to do is do not close this window. So come here and you can see the black cartridge is flashing. So what you do is you just hold the stop button for at least five seconds. And then you'll hear a click. So the printer is back to working. And what you can see here is where we set the black cartridge. So the black cartridge doesn't have an excellent mission mark anymore.  So the black cartridge can work forever from now. And we're going to do one more time for the color cartridge. Again, you cannot force the printer to turn the ink monitor off. You have to wait until you print plenty of pages. Then you see that the error window pops up. Then you can do it. After you've done it, you can print forever without cartridge. So right now we just keep printing until we get the color cartridge ready to reset. And here you can see the color cartridge is still steady yellow. And the black, because we turned it off, is blinking. We can keep printing until the ink reaches the critical level, which is right here. So when we add ink, we want to have something to hold the position. So we put this white block right here. And you can see what we wanted. We wanted this to be leveled with a table. And then we just gently remove the refill plug. And then we just pour it in. And then we use a paper towel to wipe out the excessive ink. And then we put the refill plug back and gently push the tanks back. So here's when the color ink runs out. It looks similar to the black ink runs out. Again, we do not close this window. Just keep it open. And we go back here. And you can see the color is flashing, which is hold for five seconds. And you can see the window disappear. And you're ready to print again. I hope you enjoyed this video. Stay tuned if you don't know how to set up your printer with a USB cable. Canyon showed you how to set it up for wireless connection and how to install. Actually, 90% of the problem they have is the user cannot follow their complicated procedures. And so what we do is we included a USB cable to make the installation much easier. So we're going to plug the USB cable into the computer and make this installation much easier. Another thing is we do not have a CD drive. Most people right now, their laptop doesn't have a CD drive. So what we're going to do is we go to the Canon website, which we're going to post in the comments below. So you can click and go to the real Canon site. I'm going to search Canon TS 3122. And you can see the first one is actually Canon. Okay, yeah, it is this printer. So we're going to go to drivers and download. Then we're going to download a full driver with the software. Download is complete. So we can double click. And now we can start set up. Okay, it really took a while. So here it is. United States agree. Yes, agree. So the firewall, of course, we want the active block. So we plug the power on. And so here we choose the wireless connection. And they asked us to turn the power on. So we connected the USB cable. And then we turn on the power. And then we click next. So this printer actually is another printer I set up earlier. So this is not this printer that we want to set it up. What you should see is that there's no printer here. So you want to set up the network connection according to the guide and do your next. And it's going to open up this website. And just ignore that website and go back to your installation guide. And then what you want to do is click alternative method. And then you select set up wireless connection using a USB cable to the next. And then I'll tell you the connection completed. It will ask you to a printhead alignment. Normally I just skip this step. It's really not really necessary. And you can start to a test print. One is like show you how to print from smartphone. And you can click instructions and they will bring to your website. And show you how to do it. And printer registration and I'll skip. But you can register your printer here. And finish. So after you finish, you're going to have this getting more. We just show you how to print from cloud. And how to print from Google cloud and the air print, which you can see it. You're going to print it from iPhone iPad. And again, if you have any question and probably with those functions, please contact Canon instead of BCH because it's not in related issue.


Welcome everyone. We're glad that we have several people joining us today for this discussion about the recertification process of genetic courses. The way we're going to structure it today is we're going to start with a panel presentation just to give a general overview of the process and the expectations, but we would like to devote a lot of time to your questions. So we have posted a message in chat. Please feel free to type any questions that you have as they come up during the presentation or we're a small group so also if you're more comfortable unmuting your microphone just asking questions that's fine as well if you would prefer to do that. So we are joined today by Maggie Slattery and Jodi Bender with that Richardson and with the Spankler. So these four people will be our panel who will be answering your questions and talk with people full process. I think Maggie is going to start us off with a slide presentation that gives a general overview. Thank you very much. I apologize for my voice. This is remarkably good for where I was at yesterday. So I'm using some slides that were provided for the Shire conference by Michelle Duffy, who's the chair of curricular affairs. And so we'll go through them, many of them rather quickly because I'll provide them to you. You can use them as a resource after the fact. I do love, you know, questions are probably much more useful than any of these slides. So we'll leave a lot of time for that. There is a video of this presentation from the Shire conference on the Shire Institutes website. If you ever feel like for another hour and lots of questions, there's lots of Q&A there. And while I'm not currently in chair of curricular affairs and I'm not a voice of curricular affairs, I had, I previously spent two years serving as chair of curricular affairs. I have a sense of what typically happens at curricular affairs. That hasn't changed too much since I was chair, but you know, there is a little bit of nuance depending on who the chair is. So I'm not speaking on her behalf today. Okay. So we're going to talk about just a little bit of information about what the process is like because many people don't know the process or don't use the process frequently. So just a refresher there, some tips on preparing the proposal and preparing responses to the general education costs. So we're primarily talking about recertifying general education courses. Earlier, while we were getting ready for this meeting, I found that there are 64 EMS courses that need, that were on the books prior to the beginning of recertification. So that's not an insignificant number. And I think that's why we've been asked to have this session today. So you can go ahead and slide. There are a fair number of resources on the Gen Ed website. It was the under faculty and staff. There's a job aid, which is a word doc of all the possible prompts on a curricular proposal. There's a sample integrative studies proposal. There's a sample syllabus that the curricular fairs is looking for. So curricular fairs is not looking for a syllabi that detail every graded assignment and attendance point in a course. They're looking for the durable content that makes up a course. So there's many, many sections that you would include on a good regular syllabus. You do not need to include on this, don't need the disability statement, don't need all of that. So there's a sample syllabus template of key sections that we're looking for, or what they're looking for. So they're really looking for what makes this course distinct from any other Penn State course, and what is the content that would be covered in this course to make it this course. So following the idea of the 80-20 rule, which is that of course needs to be 80% similar amongst different operating. There's another webinar link there, something that was more focused on the integrative studies courses, which is not necessarily recertification, but it is all getting walked into one. And then on the far left are some websites, the Gen Ed website, the Senate website, and then the sites.phu.edu la, that's the liberal arts curriculum website. They put together a whole list of resources that I think other units will see as useful. And it was all put together by Susie Lin, who's currently one of the vice chairs of curricular fairs. She's been able to put more samples, more sample proposals up there. So this is a really busy slide, but it's helpful to look at the overall flow of the curriculum process, and just to kind of get a sense of where all the different approvals are. It's kind of unique in that every unit has a different process. So between one and two and three can be quite different depending on where you're coming from. But in general, faculty proposal course or course change, the AS for consultation, it goes to their units for approval. And then there are three signatories that are required on a curricular proposal, or your curricular affairs rep, a department head, and the dean's representative for EMS. So for EMS, I think that's a bet. Once you have all of that, then it goes to curricular affairs. At campuses, that stuff often includes a stop at their own Senate. Departmental, I'm sorry, a college level committee in some colleges. I don't believe that occurs in EMS. Perhaps at some point in the future that might happen here. So there's a little bit of flexibility depending on where you're coming from for one, for two, three, or four. And then once it's at curricular affairs, it needs to go to a subcommittee. So the subcommittees are Gen Ed subcommittee. Any of the special designations is a USIL subcommittee right across the curriculum. First year, not that first. Anyways, so any of the special designations on a course have a subcommittee with you. So that means that you need a minimum of two weeks prior to any deadline for it to go through a subcommittee there. Again, this is a lot of small print, so you can look at it at your leisure afterwards. But the flow is generally there. At the end of curricular affairs approves it. I've been getting questions about what if things don't get approved and very rarely do things not get approved. They may have to go through several iterations and there might be some compromise and those are typically more programs than in courses. But usually eventually things get approved. There does need to be a provost and then there used to be a technical beauty approval now it's a very official rubber stamp. No question. All right, you can go for it. This is a set of the dates. This was from this academic year. And the proposal due date on the far left column is the date that you would want the proposal to be at the Senate office for the meeting in the far right column. But for recertification it needs to go through at least one subcommittee so you need it two weeks prior to that deadline. So I put notes on that on the Gen Ed website, but it's worthwhile knowing that that's an absolute drop dead date due date. That is not the day to get into the curriculum system or anything like that. Here, Susie's actually said three weeks prior. Technically it's two weeks. So if a course does not, if I have a Gen Ed course and I don't do any of this, what happens to that course then. Oh, and a year and a half you'll get notice that says you really intend to not recertify this course. And if the answer is yes, you just want to drop the Gen Ed designation, then the curricular affairs is planned is to look at if that impacts any other units. If it's just truly a Gen Ed course that's not required by any major anything that's not a big deal. And then it will just get dropped. The designation will get dropped. If it's a designation on a course that a lot of people use as a Gen Ed, I don't know, like I'll be ridiculous. They'll take the GN off of 10-1-10. Absolutely. That's unrealistic, but you know that would have to go through lots of deliberations. That course is so widely used in so many majors as a Gen Ed and lots of other things. I have a related question. If a University Park department is no longer actively offering Gen Ed course, but another campus may be offering it. Is University Park obligated to initiate recertification? No, that campus can do the recertification. So that example has come up especially around the STS courses where for lots of reasons they were largely created and administratively put under UP colleges, but are largely offered exclusively at campuses now. So in most cases, they just said, you know, if you've been offering it, initiate it. Yeah, there's another example. In education, there was a course that was developed here and I was used only at Abbey 10 or somewhere else. Are there any current Gen Ed courses that for some reason would be exempt from needing to go through this process? No. So we have used, and for lots of reasons, we have used expedited processes for many previous recertifications of other things. So I, for a separate project, looked through all of the old proposals that we have on probably two of our most popular courses. It's like 100. There have been many updates in expedited forms that course, but the original proposals are so old and the expedited forms required such little content. We do not have learning objectives for any of those courses. So then, yeah, that leads to a whole series of concerns. Now, I'm sure that individual faculty members may and even disciplinary communities do, but we do not have anything on books. So if somebody was to argue the ED20 rule, there's just nothing there. So if we have all that, we're doing changes. Do we have to do consultation? If you already have that on the books. Right, we're just doing a recertification. So it's a time to make sure all those fields are complete and yes. And so you got to go through recertification.  Yes. We have a course. Because it impacts everybody who's teaching that course. The recertification? Well, it wouldn't impact. Basically you're recertifying so there's no impact, right? Well, but it's the alignment with the new learning objectives. It's the alignment with the revised domain criteria. It's, so it's all getting lumped under recertification, but it's cleaning up lots of things. So we have a course in Geoscience, Geoscience 20. It's taught as a continuing ED course. We in the department need to recertify that, even though it's not taught by any of the faculty members in our department. We don't know what's taught at the campuses. It's just not taught here. Because it's only taught as continuing ED. At UP, it's only taught as continuing ED. It's offered at the campuses as an actual. And there's a faculty member that's been teaching the course, but they're not. So in the reality of things, it doesn't matter where the recertification comes from. So if there's a campus who really. But it needs to come from an academic unit. It needs to come from an academic unit and continuing ED is not an academic unit. Okay, you can go ahead. I started. I mentioned most of the next couple slides already, but. Recertification collaborative amongst the disciplinary communities. So I guess this kind of goes a little bit with David's question. Especially in courses where we're changing anything. Whole groups of faculty teaching that course, and they can't be left out of the process of society. Is it changing? How is it changing? Or even if it's. They just ultimately just need to be notified that things are changing. So consultation amongst disciplinary communities, disciplinary community. Think of it as people who actively teach in earth. Or Geo side. And the person, there's no person like that at a campus and that's fine. But if there is, then we should be trying to include that. What if there is not a regular faculty. It's always adjunct. And not. So if they're adjuncts. There's, you know, that's all the campuses are, or less hesitant to require, or they're hesitant to require consultation. Mostly because contracts are temporary. And that, but if it's saying. If it's not a tenure track faculty member, but it's a member who's been there teaching for a long time. You know, campuses haven't been giving, been given tenure track positions, even Phil. So. To just say because they're not tenure track is not. No, that was the question. But adjuncts. Yes. But even for short time. Yeah. Right. So that's not possible. And now, and actually the campuses don't want us to involve them because that involve, you know, implies a lot more commitment. But is there somebody else at that campus? Yes. Yeah. The ADs usually, because they want to know the changes. So that when they hire somebody on that f t two, they can say, Oh yeah, there's a new proposal on that. Yeah. Yeah. Yeah. So. Division has a good strategy for the. Five campus colleges.  It's a different usage of the term, but because they break it out as humanities. They have different groupings. Yeah. So actually. Most of those are also on the liberal arts website. And I, I'll put, I'll put it on the gen ed website too. Okay. We'll move down. To the next slide. Yeah. Right. So. The job aid on the gen ed website and the knowledge base document for the curriculum system. But it's all online at curriculum. And then you'll profit different series of questions depending on your answers. So which domain, whether or not it's going to be integrative studies or all of that. Go to the next slide. So this is a kind of a generic, what makes a complete course proposal. The. The general. Evidence of adequate and complete consultation. So on genetic courses, if you only talk to university. That's going to be a flag. Just in general, because it's gen ed. And you can full stir your consultation response. There is a field about consultation. So it helps that you put in there kind of your rationale for how you picked the songs. Like. So like one is CAMHS, like CAMHS courses, this is a little interesting here, CAMHS courses generally are offered outside of the University front. So CAMHS is, you know, from say, of all of our 35 direct courses, how many they have, you know, there's a history of a couple being taught at this campus, so we consulted at this campus, we also consulted with, you know, the primary historian at these two campuses. So you could say something like, Earth courses have a history of being taught X, Y, and Z, we consulted with those locations, or if you're doing something like a 400 level course, you know, this is a unique 400 level general education targeted to a subpopulation of students that is only half University but it helps if you put your rationale in there.  Something that faculty have commented on when completing proposals is that there seems to be a lot of redundancy in the field. How do we best address that? It's going to be, it's going, yes, and it's going to be addressed in the new curriculum system. No one wants to feel left out of new systems. We're delaying, we're delaying, I'm trying to get it delayed as long as possible. I know why the university purchased it. Oh, it's off the shelf. Yay. Is it related to the curriculum project? It is. The same software? Yeah, so they were, so they're, so that they work together, we've got the curriculum system that is off the top. So will that replace what's in my bag? Will that replace what's in mine? The formatting for the catalog is. The bulletin, the bulletin will, the bulletin is going to replace the course catalog. It will not replace the course schedule, right? Right. And the fields, so this system that we purchased, which is called Course Leap, and then this is the Kim. They work with campus solutions, so they, they know campus solutions and they have, they have their own version of campus solutions that they test everything for. Will it replace my bag? That cattle, you answered it. Okay. That catalog part that is just a total mess. Yes. That listing, when you go to a subject and there are 15 versions of the same course, you don't know which one to click on. Yes, that is going to be better with the bulletin. Now what's not that better is the course entries and descriptions. So when things merged from Isis to Leipa, several fields just got pushed and that's the technical term. So we're, they're looking at ways of cleaning up that data, but, and what's actually prompting that is it looks, it looks bad now, but it just kind of also is in a really bad format. So you don't know what part is worse. But when you're in a good format, like it's going to look like in the bulletin, the bad content jumps out at you even worse and even more dramatically. So they're looking at ways of cleaning that up, but you can't clean up the bulletin without cleaning up line path because it's a direct need. So you need to clean up the source data. So you can skip to the next slide. Oh, skip. Well, so this, I mentioned the sample syllabus, some of the areas that the curricular affairs is really looking at. There is the sample template on the genetic website. But then again, large sections of what you would normally include is the syllabus that are not there. That's either very instructor-specific or in spoiler play content. This is, so these are some of the main heading areas under the genet section where you need to address places where there's been hiccups is in describing how a knowledge domain or a learning objective is being accomplished in the course rather than just saying, oh, curricular affairs gets a lot of, well, this is a history course. Of course, it's whatever course. And it needs to be a bit cleaner articulation of how the course is maybe the learning objective or the knowledge domain. Actually, at the end of this PowerPoint, there's a way that I mapped out, like if you have well-written course objectives, how you can map your course objectives to the learning objectives and the domain criteria. So that presumes well-written course objectives, but many faculty have those. So that you probably already have a good sense of if you can do that mapping, how your course is doing this in a pretty straightforward way. You can skip to the next slide, move to the next slide. So these are some points that were put together by Michelle and Susanna. Anytime the proposal asks for assessment, there are examples, but a cleaner layout of, you know, just an explanation of how a specific assessment would be able to measure a learning objective. I have found this is much more straightforward for online courses because they've been developed in a more structured way. The next slide is the Gen Ed domain criteria. So this is how course is either arts, humanities, social science, things like that. But it's basically you just need to do the same thing. So consultation. This is some of the standard recommendations for consultation. There's actually a senate report that David wrote on consultation. It was a committee, but it's the standard by which the consultation recommendations are made. You know, some places have, what it really gets down to is content. So finding discipline. So finding persons with the right area of background. And that no college owns a course. It's a university course. No college owns a domain. That's been attempted recently. And and so that it's just really looking out for, this is ways of helping to improve problems with curricular redundancy and just in general making sure people are informed. Although it's not an information system. It's not like spam everybody with curricular consultation. That's not the strategy either. It's really about getting informed input from somebody who could possibly be teaching the course or has something constructed. So really looking at where courses have been taught in the last several years and the curriculum just to provide you with that. Anywhere that's offering programs with similar content, especially if there's a major at other campuses. If you're changing a pre-rec, the units that offer that pre-rec. So if you're going to remove a math pre-rec or add a math correct and math community. And then there's these other ideas that kind of bridge large portions of the university. Entrepreneurship, innovation, sustainability. Like nobody owns those words. Don't forget librarians. They're a really helpful resource. I was just going to say too, this is just something I've learned over the years, that if you're going to ask somebody for consultation, I always send them an email outside and just say, hey, I'm going to be sending a course your way. Are you the right person to talk to? And our response rate for consultation is almost 100 percent because we do that. Sometimes they say, yep, send it my way. Sometimes they say a better person is a person. Sometimes they say, I'm on sabbatical. I'm not, you know, I'm not doing any of that. And so, you know, just kind of letting everybody know kind of outside the system that I mean, they get emails. The system sends them emails, but that way you get to the right person and they know it's coming and they know you're watching, right? So you don't get these default acceptance things. All right. So who is doing what? Earlier this academic year in, I don't know if it was August or September, all of the AQ deans got a list of general courses. Yeah, that was the one I referenced earlier. Yeah. And Michelle Duffy's been putting, she'll probably, if she hasn't, does she send anyone for that? And they've sent you. Yeah. And now, I have my update. Okay. Michelle Duffy has intended to share an updated list every month or so, but so big disciplinary communities are difficult to coordinate. So the Office for General Education and the Learning Outpost Assessment Office are doing work to try and encourage them to coordinate. So did some things with the psych community as a model, we're trying some other things with sociology. But if you have a big community that you want to try and get together, we'll do like that. And then when you're asked to consult, please consult, but be collegial about it. So, you know, so collegial feedback from people, especially when I saw some feedback from a campus, I was like, oh, his office isn't two doors down from hers, that's not so nice. It's okay to differ. And so even okay to say no, I disagree because of X, Y, and Z, but we all know how difficult that is. You know, one of the things I think hinders consultation is people are fearful of this negative feedback. And not because they're not willing to take constructive input. It's just if it's not being done in a cultural way that people kind of defer to avoiding it. Okay. So the last three slides are all on integrated studies courses. That's not really recertification. Could be if you're taking an existing course and making it those questions. And you mentioned earlier that there's now an H possibility on the course. Yes. My understanding from the Senate office is they have decided to double up on two suffixes. They will have Ns for inter-domain, Zs for length, and you want to add an H to those. They're hoping that there's not many examples of that. If it's either of those, then they could use the T and F. I think our time is just about up unless there's any other questions in the room. I don't see any questions online. We will post the recording on our Dutton Food for Thought series. And we'll put that as well at that spot as well. But we very much appreciate you coming and spending your time.


Root me in the face. See you Friday with Bob Rossi! Great. That was really great. We're gonna add bongo soon. So, this is about tape. A really good tape tip from Jeff Turner. He's a handleist. Jeff Turner Art. He's become a great friend of us. Jeff Turner Photo. Sorry, Jeff. He gave us this little piece. So nice. So, this is called Tuck Tape. Can you see on the side there? Tuck Tape. It's used in, you can find it in the hardware stores for in-between walls. You put insulation in the plastic sheet. You use this for the seam. This tape is waterproof. Super sticky. So, you put it on the side. The resin runs off. You can even fold it. The drip just runs off because it's waterproof. It rips up nicely. Stuff like painters tape and duct tape. They work great. But the resin soaks through a bit. So, sometimes it's hard to get to peel off. So, Tuck Tape. T-U-C-K-T-A-P-E. And we'll see you next Friday. See you tomorrow.