Calculus at a Fifth Grade Level

that the single spray and postulates can move objects with a high high energy eigenvalue times fy'y equal to minus h-bar squared Delta P Delta X is greater than or equal calculus is a notoriously difficult subject students often only see as the class that they must get out of the way in order to graduate however when calculus is used to its full potential it becomes a beautiful tool that is central of solving real-life problems still every year nearly half the students who enter the first calculus class receive a failing grade but it doesn't have to be

this way calculus is hard because it is different it introduces completely new concepts such as to limit the derivative and the integral these are novel concepts that appear completely unintuitive and hard to grasp when students don't understand the concepts their applications are next to impossible so to understand calculus we first must reinforce the concepts that are fundamental to its foundation I believe the key to understanding calculus lies and teaching these concepts the algebra and complicated math that trip students up can be learned in time but a student who never grasped the fundamental ideas of calculus

can never succeed so let's take a step back from everything we know about math and try to learn calculus in a whole new way so infinity is really cool because it allows us to talk about things that are either really big or really small infinity has this reputation of being known as the biggest number have you guys heard of that before yes it's not yes you're correct if I asked you guys to count the numbers between one and two one point one is slightly bigger than one right and it definitely is between one and two

we all agree so right now we have one number between 1 and 2 1 point 1 1 is also between 1 & 2 right it's a little bit bigger than 1 point 1 right so now we have two numbers that are definitely between 1 & 2 1 point 1 1 1 this is also between 1 & 2 so if we keep adding 1 so that's what this dot dot dot means over there that means we can just keep adding 1 on 2 then in the summer forever and every single time we add 1 on the

number gets a little bit bigger right so it's it's a unique number it's a different number that's between 1 & 2 every single time we do this and if we keep counting these numbers as in how many numbers are between 1 & 2 we'll never end that is infinity infinity is a concept and this is crucial to understanding not just infinity but also for calculus so now with that let's talk about 1 over infinity if I have 1 over 2 we have this one pizza we cut it in half and this red right here is

the amount of my one slice we all agree so now let's go to 1 over 3 we have this one pizza and we divide into 3 equal slices this red slice is the amount of one slice and now if we divide into 4 what do we notice gets even smaller right so we have 1 over 5 it's a little bit smaller okay 1 over 6 1 over 15 let's look at 1 over 80 what would happen if we go to 1 over infinity the question is is it equal to 0 so remember that infinity is

not a number so one over infinity doesn't represent anything remember we had to have a number on the bottom of this thing right we have to divide into a certain number of slices and if we divided by infinity to me that means nothing but what does that mean about our pizza slice well we said it's not equal to zero but what we can say is that one over infinity goes to zero when we increase that number on the bottom our slights our slice gets smaller and smaller and smaller and if we keep doing that forever

and we keep adding one to that bottom number our slice is getting closer and closer and closer to being nothing but it's never equal to nothing what this is called and this is important this is infinitely small 1 over infinity is an infinitely small number just like infinity is an infinitely large number let's move on to something that I know you're familiar with area I want to talk to you guys about an interesting way to take area so let's say we're trying to calculate the area of a triangle a way that we might be able

to do it is by taking something whose area that we do know and filling our triangle with it so let's say we have our quarters stacked up like this and we want to say what isn't it the area of a triangle that has this shape well let's count the quarters and say how many quarters fit into this triangle so one way we can do this we can count it just by going 1 2 3 4 but we can do that but the way I want to talk about doing it is to count all the columns

if we count all the quarters we add 1 plus 2 plus 3 we get 21 quarters are shown right here and we can say that our quarters roughly fill this shape ok and that there's about 21 quarters in this triangle but if we if we fill this triangle the first thing I wanted to show you guys that there's a little space in here where the quarters don't quite touch and if we fill the triangle and we see that there's a lot of there's like overhang on the quarter so how can we make this a more

accurate measurement well let's use nickels now now we have a lot more columns right and what we can do is we can add up these columns again and say okay well there's 36 nickels here and now if I ask you how big is this triangle what would you say and again we did this by counting up all the columns and now if we get the inside the triangle the space in between the quarters are a little bit less there's not as big of a gap between the quarters between the coins making it slightly more accurate

and if we draw it we say that there's a little less overhang right so let's go even smaller let's use a dime and if we count up all the columns the same way that we did before and there's a lot more columns so it's a little bit harder we get 136 times if we put this triangle over we notice two things one this space is really small now compared to the quarters it's still definitely there but it's definitely smaller space and if we fill this triangle up it almost looks perfect we know that there's a

little bit space inside that we have to deal with but as far as overhang is concerned it's pretty much gone okay it's still there right but this there's a lot less so now let's compare the three triangles we just talked about the dimes is definitely the most accurate out of these three we all agree so we look at the columns that we used the width of these column is only as small as the width of this quarter or the coin and so if we say this nickel has half the width of this quarter and at

this dime has a quarter 1/4 of the width of this coin we're taking our column and we're making it smaller and smaller and smaller if we keep going on forever and ever and ever and making our coin smaller and smaller and smaller making that making the width of this column smaller and smaller by using smaller coins the accuracy is gonna keep getting better and better and if we make it infinity our accuracy should be a hundred percent eventually so what would that look like well let's take a look this is a decent picture of what

that might look like now obviously one over infinity is so small that we can't really represent it right we can't make an infinitely small column on a computer or even draw it because we can always make it smaller right we can always add it onto that infinity but it might look something like this and if we zoom in to this corner over here we have these columns that go up right and imagine that these are the width of our infinitely small coins if we add up all these columns we would get the area of our

triangle and it would be a hundred percent accurate this is one of the big concepts that I want to drive home is that one over infinity can be used to calculate area to find the area and this this is huge because this is one of the principles of calculus this is the second most arguably first most important idea of calculus is that if we use infinitely small columns we can find the area of anything okay I want to talk about another concept that's really important to calculus and that has to do with slope let's start

by defining what slope is so when I think about slope what I think of is the in kind of a ramp that you're writing from left to right so for example if we have this guy over here he's on a skateboard he's going up this incline he's going left to right we said this is a positive slope he's going up now this guy same skateboarder maybe he got to the top of the hill and he oh I go down now now he's going down this from left to right so our slope is negative it's downhill

we understand the difference between those two okay so with this let's move on to talking about apples let's say I have ten apples and I eat I eat five of them in one minute because I'm like a speed apple eater and now I have data points I have two numbers two groups of numbers we can put this on a graph this line tells us that if we go to any point on this on this graph we can read how many apples we have at this minute so we have this this line and what does this

look like looks like a slope right and we can put our skateboarder on it so this guy's going downhill so it's a positive or negative negative but what is the value of this slope how can we calculate it and more importantly what makes this slope right here so this line different from this slope or this slope what exactly is the numerical difference what's the difference in the actual slope between these two what in these three what we can do is say that the slope is equal to the number of apples that I ate over the

time that I ate them if we have this line we start at 10 go to five how many apples do we eat five and how long did it take one minute so our slope is gonna be negative five but what if we had a line that looks a little more complicated this isn't a straight line we have a line that looked something like like this it's not straight it's not it's not easy to calculate that slope and the reason is is because the slope changes let's look at a skateboarder here she's not gonna go really

fast like that's a pretty strass t'k drop right it's a really negative slope you agree and that lets say at that that point that slope is right here okay pretty negative and if we put the same skateboarder over here he's like riding a flat ground he's not really going anywhere right he's just he's coasting so that slope is maybe somewhere over here but what's important is that this same graph this same line has many different slopes because this is different than this which is different in this and every single point is slightly different slope so

we want to be able to say well what is the slope at any given time right how fast is our skateboard are going if we followed this line at any given number so if we look at this graph and we sort of have it okay we take it and we cut it in half so if you look on the bottom here go someone to ten and it looks like a pretty curvy line now let's say if that's one over one and let's take a half of that so now we go from zero to five look

at the numbers on the bottom go from zero to five in our time okay so if we go again now we go from one to two point five now looks like an even more straight line and if we go again one two one point two five that almost looks like an exactly straight line it's slightly different it's slightly not straight it has a slight curve to it but it's definitely a lot better and remember all we did is we went from ten to five to two point five to one point two five we kept having

that number okay and it looked more and more straight so the question is what are we doing here what I would say is that we started at 1 over 1 let's say we start at 1 over 1 and we have it we get the 1 over 2 so now we're at half of our initial graph a centered about 1 because 1 is always there right and we have that now we're at 1 over 4 following me we keep having the length of our graph centered about 1 and we're getting this number on the right is

getting closer and closer and closer to 1 right so what would happen if we go a distance of 1 over infinity you'll be a straight line remember infinity is not a number and from these a concept and 1 over infinity is infinitely small so we go from 1 to 1 plus 1 over infinity and what we see is we recover a straight line doesn't that blow your mind remember we started out at this and we said you can't measure that slope because it's different everywhere it's different every single point on this graph as a different

slope but if we have it more and more and we focus in on one we focus in right here we're focusing on this time and we look at only that instant of time it looks like a line we said before that every single point has a slope and we also said that in order to measure that slope we need a straight line so if we look at an instant in time that is essentially just one point we better get a line because we need a line to measure the slope there do you agree so this

makes complete sense if we look at a time from 1 to something just after one infinitely close to 1 it better look like a line because we want to be able to measure that slope because we know it exists and that's important is that we know the slope has to exist so there must be a line there and the question is how we have to just be able to look only at that point in time to find that line and this is very important because we can measure the slope of this line based on this

picture we know that it exists and we know that it's calculable we know that we can find it if we have the right tools I want you to leave with this idea that 1 over infinity can be used to find the slope of a curved line and this is also crucial this is remember I said the area was like this central one of the two central ideas of calculus this is the other one this is the second half of the complete picture of calculus is that we can use calculus to not just find the area

of a shape but to find the slope of a function of a line of a graph that otherwise we wouldn't be able to find the slope of all right so let's just recap what we learned today we learned most importantly that infinity is not a number infini is a concept we also learned that 1 over infinity is not equal to 0 it leads to 0 it goes to 0 if we look at 1 over infinity it gets smaller and smaller and smaller because infinitely small and goes to 0 and 1 over infinity allows us to

find the shape the area of shapes using really small infinitely small columns which is crazy if I give you a weird object like like maybe I give you something that looks like like this how do you find the area of that thing that's pretty hard right you won't have a formula for that we have we want to be able to use the columns we also showed that one over infinity allows us to turn a curvy line a curvy line like this into a straight line at a specific point at a specific time which is amazing

because it allows us to find the slope at any given time of a line that we normally wouldn't be able to the students just beginning the study calculus always finding concepts limits derivatives and integrals hard to understand but when these concepts are broken down and explaining the new unique ways such as by using coins to visualize integrals they become infinitely easier to understand in fact the rather unconventional methods for teaching calculus use in this video allowed the same students usually hate things that were false to follow these daunting concepts so next time did a roadblock

and want to give up because you just can't grasp a concept right away take a step back and try to tackle the concept in a new way just like we did with using pizza to clean cinnamon and skateboarders to explain slope then once you die because an awesome field of mathematics you will just like to fit there's in this videos [Music] have a thirst for knowledge and one day just like those students go on to change the world you