Calculus at a Fifth Grade Level

that the single aerosol and postulates can move objects with a high-energy eigenvalue multiplied by fy'y equal to minus h-bar squared Delta P Delta X is greater than or equal

calculus

is a notoriously difficult subject that students often However, when

calculus

is used to its full potential, it becomes a beautiful fundamental tool for solving real-life problems, each year almost half of the students who enter the first class of calculation receive a failing

grade

. But it does not have to be like that. The calculation is difficult because it is different. It introduces completely new concepts, such as limiting the derivative and the integral.

These are novel concepts that seem completely unintuitive and difficult to understand when students do not understand the concepts, their applications are almost impossible, so to understand calculus we must first reinforce the concepts that are fundamental to its foundation. I believe the key to understanding calculus lies in teaching these concepts, algebra and complicated mathematics that trip up students can be learned over time, but a student who never understood the fundamental ideas of calculus will never be able to 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 very big or very small, infinity has the reputation of being known as the largest number, have you heard of that before?

If it is not. You are correct if I asked them to count the numbers between one and two, a point one is slightly. bigger than one, right and it's definitely between one and two, we all agree, so right now we have a number between 1 and 2, 1 point 1, 1 is also between 1 and 2, right, it's a little bit bigger than 1 dot 1, so now we have two numbers that are definitely between 1 and 2 1 dot 1 1 1 this is also between 1 and 2, so if we keep adding 1, that's what this dot dot dot means there, that it means we can keep adding 1 over 2 and then in the summer forever and every time we add 1 to the number it gets a little bit bigger so it's a unique number it's a different number that is between 1 and 2 every time we do this and if we keep counting these numbers as many numbers are between 1 and 2, we will never finish, that is infinity, infinity is a concept and this is crucial to understanding not only infinity but also calculus, so now with that let's talk about 1 over infinity, if I have 1 over 2, we have this pizza we cut it in half and this red one here is the amount of my portion, we all agree so now let's go to 1 over 3 we have this pizza and we divide it in 3 equal portions this red portion is the amount of one slice and now, if we divide it into 4, we notice that it becomes even smaller, so we have 1 out of 5, it is a little smaller, okay, 1 out of 6, 1 over 15, let's look at 1 over 80, what would happen if we go to 1 over? infinity the question is if it's equal to 0 so remember infinity is not a number so one over infinity doesn't represent anything remember we had to have a number at the bottom of this thing we have to divide it into a certain number of slices and if we divide by infinity to me that doesn't mean anything, but what does that mean about our slice of pizza?

Well, we said it's not equal to zero, but what we can say is that one over infinity gets to zero when we increase that number at the bottom of our Snubs our portion gets smaller and smaller and smaller and smaller and if we keep doing that forever and we keep adding one to the lower number our portion gets closer and closer to being nothing but it is never equal to anything what is this 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 in the area.

I want to tell you about an interesting way to calculate area. Let's say we are trying to calculate the area of ​​a triangle, one way we could do it is by taking something whose area we know and filling our triangle with that, so let's say we have our coins stacked like this and we want to say what is not the area of ​​a triangle that has this shape. Well, let's count the quarters and say how many quarters fit in this triangle. One way to do it is to count it by just 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, add 1 plus 2 plus 3, we get 21 quarters, are shown here and we can say that our quarters we fill approximately this shape, okay and there are about 21 quarters in this triangle, but if we fill this triangle, the first thing I wanted to show you is that there is a little space here where the quarters don't touch at all and if we fill in the triangle and see that there are a lot of protrusions on the coin, then how can we make this a more accurate measurement?

Well, let's use nickels now that we have a lot more columns and what we can do is add them up. columns again and say: well, there are 36 nickels here and now, if I asked you how big this triangle is, what would you say? And again we did this by counting all the columns and now if we get the inside of the triangle, the space between the quarters is a little bit less, there's not as big a space between the quarters between the coins, which makes it a little more precise and if we draw it we say there's a little less overhang, so let's go even smaller, let's use a dime and if we count all the columns the same way we did before and there's a lot more columns, so it's a little more difficult, we get 136 times, if we put this triangle on top, we notice two things: one, this space is very small now compared to the rooms it has.

It's definitely still there, but it's definitely a smaller space and if we fill this triangle it almost looks perfect, we know there's a little bit of space on the inside that we have to deal with, but as far as the overhang goes, it's pretty much gone. It's okay, it's still there. but this is much less, so now let's compare the three triangles we just talked about, the dimes are definitely the most accurate of these three, we all agree, so we look at the columns we use. The width of these columns is as small as the width of this quarter and if we say that this nickel is half the width of this dime and this dime is a quarter 1/4 of the width of this coin, we're taking our column and we're making it. smaller and smaller if we continue forever and ever and make our coin smaller and smaller, making the width of this column smaller and smaller by using smaller coins, the precision It will keep getting better and better and if we make it infinite, our accuracy should eventually be one hundred percent, so what would that look like?

Well, let's take a look. This is a decent image of what it would look like now, obviously one over infinity is so small we can't really represent it. Well, we can't make an infinitely small column on a computer or even draw it because we can always make it smaller. Well, we can always add it to that infinity, but it could look like this and if we zoom in on this corner here. we have these columns that go up to the right and we imagine that these are the width of our infinitely small coins, if we add all these columns we would get the area of ​​our triangle and it would be one hundred percent accurate, this is one of the great concepts.

What I want to emphasize is that one over infinity can be used to calculate the area and find the area, and this is huge because this is one of the principles of calculus, this is possibly the second most important idea of ​​calculus, is that if we use infinitely small columns we can find the area of ​​anything. Well, I want to talk about another concept that is really important to calculus and that has to do with slope. Let's start by defining what slope is. When I think of slope, I think of a kind of ramp that you're writing from left to right, so, for example, if we have this guy here, he's on a skateboard, he's going up this slope, he's going from left to right, We said this is a positive slope, he is going up now. this guy, the same skater, maybe he got to the top of the hill and he, oh, low now, he's going down from left to right, so our slope is negative, it's downhill, we understand the difference between those two, okay, so let's get on with this.

To talk about apples, let's say I have ten apples and I eat five of them in a minute because I'm like a fast 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 graph we can read how many apples we have right now, so we have this line and what it looks like, it looks like a slope to the right and we can put our skater on it, then this guy goes downhill, so it's positive or negative, but what is the value of this slope?

How can we calculate it and, more importantly, what makes this slope here? So this line is different from this slope or this slope, what exactly is it? the numerical difference what is 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 during 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 going to be minus five, but what if we had a line that looks a little more complicated?

This is not a straight line that we have. a line that looked like this is not straight, it is not, it is not easy to calculate that slope and the reason is that the slope changes, let's look at a skater here, she will not go very fast as if it were a pretty rhinestone fall. right, it's a really negative slope, you agree and that let's say at that point the slope is here, well, pretty negative and if we put the same skater here, it's like riding on flat ground, he's not really going anywhere. , right, it just slides like that. that slope maybe somewhere around here, but the important thing is that this same graph, this same line has many different slopes because this is different than this, which is different in this and each point has a slightly different slope, so that we want to be able to say well.

What is the slope at a given time? How fast does our skateboard go if we follow this line at any given number? So if we look at this graph and we have it right, we take it and we cut it in half, so if you look at the bottom here, you see someone at ten and it looks like a pretty curved line, now let's say if it's one over one and let's take half of that so now we go from zero to five look at the numbers at the bottom we go from zero to five in our time okay so if we go again now we go from one to two point five now it looks like an even straighter line and if we go again one two one point two five which almost looks like an exactly straight line it's slightly different it's not a little bit straight it has a slight curve to it but it's definitely a lot better and remember that everything we did It was going from ten to five to two point five to one point two five.

We still have that number, it's fine and we saw more and more. clearer, then the question is what are we doing here? What I would say is that we start at 1 over 1, let's say we start at 1 over 1 and we have it, we get 1 over 2, so now we are in the middle of our initial graph is centered at 1 because 1 is always there, to the right, and now we're at 1 over 4, following me, we still have the length of our graph centered at 1 and we're getting this number on the right is getting closer. and getting closer to 1, what would happen if we traveled a distance of 1 over infinity?

You will be a straight line. Remember that infinity is not a number and of 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 that we recover a straight line, isn't that surprising? Remember we started on this and we said you can't measure that slope because it's different everywhere, it's different at every point on this graphically like a different slope, but if we have it more and more and we focus on one, we focus right here. , we are focusing on this time and we look at just that instant of time, it looks like a line we said before that every single point has a slope and we also said that to measure that slope we need a straight line, so if we look at an instant in the time which is essentially just a point, we better get a line because we need a line to measure the slope.

Do you agree? So this makes a lot of 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 thatIt exists and that is important. is that we know that the slope has to exist, so there must be a line there and the question is how can we look at just 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 image we know that it exists and we know that it is calculable we know that we can find it if we have the right tools I want you to stay with this idea that 1 over infinity can be used to find the slope of a curved line and this too it's crucial.

Remember I said the area was so central. One of the two central ideas of calculus. This is the other one. This is the second half of the full calculation picture. It's that we can use calculus. not only find the area of ​​a shape but also find the slope of a function of a line of a graph that otherwise we couldn't find the slope, so let's recap what we learned. today we learned the most important thing that infinity is not a infinite number 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 becomes smaller and smaller because infinitely small and goes to 0 and 1 over infinity allows us to find the way the area of ​​shapes using really small infinitely small columns, which is crazy if I give you a strange object like maybe I give you something that looks 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 want to be able to use the columns. We also show that one over infinity allows us to convert a curved line to a curved line like this. on a straight line at a specific point at a specific time, which is amazing because it allows us to find the slope at a given time of a line that we normally couldn't. Students just beginning the study of calculus always encounter concept limits. Derivatives and integrals are difficult to understand, but when these concepts are broken down and unique new ways explained, such as using coins to visualize integrals, they become infinitely easier to understand.

In fact, the unconventional methods of teaching calculus in this video enabled the students themselves. We usually hate things that are false for following these daunting concepts, so next time I hit a roadblock and want to give up because you just can't understand a concept right away, take a step back and try to approach the concept in a way new, just like we did. I did it with the use of pizza to clean the cinnamon and the skaters to explain the slope, then once you die, because you will like to fit into an amazing field of mathematics.

There is a thirst for knowledge in these videos and one day, like those students, they will change the world. you