Shadowing Practice: ALL OF MATH explained in 14 minutes - Learn English Speaking with Video
Creating lesson...
1
Hi, you're on a rocket ship, flying to the edge of the universe.
2
Wait, how big is the universe?
3
It's 1 plus 2 times 99 times e plus 100 to the power of- Stop!
4
What the heck is going on here?
5
The universe is infinite.
6
Infinity can consist of numbers which are symbols that represent a value.
7
We can add these values, we can subtract them, and we can also do some strange things like this.
8
But, we'll get to that later.
9
Besides numbers, there's also these things representing instructions, relationships, operations, and concepts called signs and symbols.
10
Some of these signs are inverses, meaning that putting them together just make them want to undo each other, or, as the nerds call it, cancel each other.
11
This little fixed number right here is known as a constant, and then there's also these unknown mysterious letters called variables.
12
One of them by themself don't really tell us that much, but put them together, and now we have an equation.
13
Like, you know, this one for example.
14
The goal of every equation is to get solved.
15
We do it by rearranging it in a way that isolates the variables from the constants.
16
Think of it like this.
17
If we would like to solve this equation, we would use the inverse operation to cancel out the constant on this side.
18
By doing that, we've isolated the variable and thus gotten its value.
19
Now, there are also different types of equations.
20
There's the quadratic equation, which always has one variable raised by no more than two.
21
And then there's also the polynomial equation, which can have an unlimited amount of numbers their variables can be raised by.
22
Meaning, they can look like this sometimes.
23
These two are basically more hardcore versions of the linear equation, which can't be raised by anything more than one.
24
A key thing to remember is that the variables in a quadratic
25
and polynomial equation can be equal to multiple values at the same time.
26
To determine how many possible solutions an equation has, we look at the largest number of variables in the equation
27
is raised by x cubed minus 3x equals zero for example will have these three solutions.
28
Okay, now that we know about these equations, we need to, you know, actually be able to solve them.
29
These four formulas plus some extra methods pretty much explain step by step how.
30
Looking at a quadratic equation, we will see that it's basically the same as the first formula.
31
If we then plug the number in the formula and then simplify, we will then find out that the value x1 equals negative 1 and x2 equals negative 5.
32
This is pretty neat, but that's too complicated for us.
33
The second, third, and fourth formula tells us that we can rewrite these equations by factoring them into smaller parts.
34
We can then put each of them equal to zero and then solve them separately.
35
There is a twist here though.
36
Not every equation will be this simple.
37
They can also look like this, which are close to impossible to factorize, meaning if you're not smart enough to come up with your own formula, you're gonna have to use this one instead.
38
Now, when it comes to the polynomial equation, things can get.. weird.
39
Except if you're four steps ahead and see that we actually can factorize the equation, and again solve each of them separately.
40
Oh yeah, did I mention that there's also equations that have a constant, or as they say, a base raised by a variable?
41
They're called exponential equations, and tell us how a starting value grows or declines exponentially at a certain rate.
42
The base of the formula tells us the rate of growth or decline, the constant tells us the initial value, and the exponent tells us for how long it grows.
43
All right, enough about equations.
44
Let's get ourselves out of the atmosphere.
45
If you've ever been to space, you'd know that the Earth looks like this.
46
Oh, sorry, wrong picture.
47
You would know that the Earth looks like this.
48
That, my friend, is a shape, aka a sphere.
49
There are two-dimensional shapes.
50
Fuse some of them together, and now you got a three-dimensional shape.
51
Now combine some of these and you have yourself this thing.
52
You see, everything around you is just a bunch of shapes combined to make objects.
53
Now, let's say you wanted to know the size of this thing's surface, or as most people call it, the area.
54
First stretch the surfaces out, which gives us a rectangle.
55
Then measure the base and height, and apply this formula.
56
Area is measured in square units, and every shape uses the base formula B times H.
57
Actually, that was kind of a lie.
58
The circle uses its radius squared times pi, but you get the idea.
59
Now, if we instead wanted to know how much we could stuff inside of an object, we would need something called volume.
60
Calculating how much candy you could fit inside of your cylindrical shaped bucket on Halloween could be pretty useful, so in that case, you would use this formula.
61
Similar to area, every shape pretty much uses the same base formula B, which is the surface area of the base times the height, with a few adjustments for each shape.
62
Everyone except for, you guessed it right, the sphere again, which now times pi by the radius squared.
63
Hey, wanna see something cool?
64
Said a man named Pythagoras almost 2,500 years ago as he formalized the Pythagorean theorem.
65
It tells us that this side, called a hypotenuse and a right-angled triangle, equals the sum of the squares of the other two sides.
66
But that's freaking whack, said this guy who 300 years later gave us a way
67
to calculate the crazy relationship a triangle's angle has with its sides, aka trigonometry.
68
If we look at a triangle, we will see this angle.
69
In trigonometry, it's called theta.
70
These sides have a particular ratio to one another based on the value of the angle theta and are labeled hypotenuse, opposite, and adjacent.
71
There are three functions that pretty much sum up all of trigonometry.
72
Actually, there's one more thing too but we'll get to that later.
73
The sin function tells us the ratio between the opposite and the hypotenuse.
74
The cos function tells us the ratio between adjacent and the hypotenuse.
75
And the tan function tells us the ratio between opposite and the adjacent.
76
Imagine a triangle with these values.
77
By using the sin function and then calculating it as an equation, we can then find the value for x, which equals 6.88.
78
The other key thing to remember is this little circle divided into 4 quadrants with 12 points called a unit circle.
79
The points in the circle are defined by various measurements like, for example, degrees.
80
But the most common measurement, however, is radians, which basically is the radius of the circle stretched out over its side, creating an arc.
81
6.283 radians make up a whole circle, which is equal to the circumference of the circle, 2 pi radius.
82
But, since we only use radians as the measurement and not the radius, we can rename 2 pi radius to 2 pi radians.
83
Yeah, I know, they literally sound the same.
84
Deal with it.
85
Congratulations!
86
Now that we know the circle in total is 2 pi radians, we can also say that the points in our circle become various fractions of pi over radians.
87
Now, here is when things get kind of freaky.
88
If we draw a line from the center of the circle, this line according to trigonometry will have the unit 1.
89
One what, you may ask?
90
Just one.
91
Yeah.
92
You see, since we're dealing with ratios between lengths, they'll cancel each other out, therefore we just use 1.
93
If we know that this length is one unit long, we can also figure out that the end of each axis have these coordinates.
94
Now let's place this angle theta in the circle and draw one line alongside the x-axis, and then another one from the point where the angle intercepts the circle.
95
Wait a minute, that's the same triangle we drew earlier!
96
This side, aka the ipotenuse, will have the unit 1.
97
If we look at the functions sin, cos, and tan, we can then plug our values into one of them and then simplify it.
98
Do you see what's going on here?
99
We're marking out the coordinates for the angle theta.
100
In simple terms, we're figuring out the lengths of the triangle's sides.
101
Knowing this, we can then say that the coordinates for the angle theta are cos theta, sin theta.
102
Now, to most normal people, this tells us about as much as this kid trying to explain his dream.
103
So, we're gonna need some actual numbers.
104
The three key numbers to remember are square 3 over 2, square 2 over 2, and square 1 over 2, which is simplified to 1 over 2.
105
Remember those dots in the circle representing the fractions of 2 pi radians?
106
Turns out, if we draw a right triangle with a 30 degree angle to pi over 6 radians, then draw a mirror image triangle below it, we can then see that the length of the original line,
107
aka the wide coordinate, aka sin theta equals 1 over 2.
108
You see, there's a pattern here.
109
If we instead draw a triangle with a 45 degree angle to pi over 4 radians, we will see indeed that sin theta equals square 2 over 2.
110
Doing the same with a triangle with the angle 60 degrees
111
to pi over 3 radians will give us the conclusion that sin theta equals to square 3 over 2.
112
Long line big number, medium line medium number, short line small number, and the same pattern applies to cos theta.
113
Fun fact!
114
We can actually draw out the function of cos and sin in a graph.
115
It shows us the variation of cos and sin with respect to the angle theta.
116
You can think of cos theta as the x-axis.
117
When theta increases, so does the cos theta line, until theta goes over the second quadrant, which sits on the negative side of cos theta, making the line slope down.
118
Sin theta?
119
Pretty much does the same thing, just in reverse.
120
All that talk about graphs and functions, I think it's about time we talk about calculus.
121
Using a function, we can understand the change in motion in whatever your imagination can come up with.
122
This is f of x, and this is an equation.
123
Put them together, and then they become a function.
124
A function tells us that for a certain input x, we get a certain output y.
125
To join the special club of functions, their inputs and outputs need to either have a one-to-one relationship or a many-to-one relationship.
126
If we then plug all the input values
127
and output values into a graph and then connect all the values with a line, we will then get a longer line.
128
This line can tell us multiple things.
129
Like, for example, this kid's heart rate when he realizes someone threw his diamond sword into lava.
130
Functions come in multiple variants.
131
Hey look, those are the equations we looked at earlier.
132
Depending on the equation, the function fx equals, you will get a certain type of line.
133
A linear function gives us a straight line where the constant
134
tells us how much the output y increases in relation to the input x.
135
These lines can also get very bendy, and can look like this, or even this.
136
One thing many people confuse is that while these three lines might look similar, they are vastly different.
137
The function of a quadratic equation will always have one turning point where it either turns up or down, whereas the function of a polynomial equation will have multiple turning points.
138
Fun fact!
139
The more variables you have in a function, the more of these points you'll have.
140
Not so fun fact, this means a line can look like this.
141
Okay, but this graph, doesn't it also have a turning point right there?
142
No, that is an exponential function which, as you know, grows or declines exponentially, meaning it won't at any point turn around.
143
Speaking of points, limits, describe the behavior of a function as it approaches a specific point.
144
If you're not confused by that definition, you're probably like, four steps ahead.
145
But, for the rest of us, think of it as a point on a graph that, when the line gets closer and closer to it, it can start acting weird.
146
For example, why don't you try to evaluate this function?
147
You can't because it equals zero, which means we can't find out what the line does at x equals two.
148
What we do instead is that we find out what it does as it approaches two, aka the limit.
149
In this case, we can then plug in a number that is just over two, which then gives us 4.1, and to prove we're right, we use a limit which gives us the number 4.
150
Now, when we have a function that equals zero, it becomes discontinuous.
151
Continuous functions are functions that can be drawn without lifting the pencil of the paper, aka, they won't have any holes or gaps that break its flow.
152
Discontinuous functions, which have the three states, whole, jump, and infinite, are the total opposite.
153
They couldn't think of anything worse than to stay in flow.
154
If you thought this was the limit when it comes to the usage of limits, you were wrong.
155
They were also used to point out the holy grail of calculus, or at least the second holy grail, derivatives.
156
Derivatives are used to calculate the rate of change in a function at a certain point.
157
If we zoom in on a function, we can see that it's basically just a straight line.
158
Now let's draw a line right here.
159
This is called a tangent, and is the derivative of the original function.
160
Now, I'm sorry to disappoint you, but getting the derivative isn't just zooming in on one line and drawing another.
161
Let's say you have the function f of x equals 5x minus 4.
162
You would then have to plug everything into this formula by first replacing x with x plus h, and then divide everything with h, simplify, to then get the derivative 5.
163
Nah, screw that, said this guy who basically saved every future calculus student from getting an f by formalizing the power rule.
164
It's basically an idiot's guide to derivatives, and tells us that the derivative of x raised to some power is equal to
165
that number times x raised to the power of that number minus 1.
166
For example, x cubed when derived becomes 3x squared, and just x when derived becomes 1, and a constant derived is just completely cancelled out.
167
Now, the crazy thing here is
168
that you can actually derive the derivative of a function to get a function for the derivative, and then you can derive
169
that function to get a function of the derivative of a function of the derivative of a function of a function.
170
Yeah.
171
Okay, that's cool and all, but what if we have an exponential function where a constant is raised by a variable?
172
Well, then you would derive using the exponential rule.
173
You see, there is also these things called the differentiation rules, which gives us the formulas for how to derive much more complex functions.
174
Like, you know, this one.
175
Okay, now let's say you would like to know the area of the surface under the function.
176
This is where we use the second holy grail of calculus.
177
Integrals.
178
By slicing up the area under the line into an infinite amount of slices, you can then calculate the area of each slice, add them up, and then get the whole area under the function.
179
Now, I know what you're thinking.
180
Infinite rectangles.
181
Isn't that like impossible?
182
Technically, yeah.
183
But not if we use an integral, which gives us a really specific estimate of the area under the function as we approach an infinite number of squares.
184
This is the formula which is used by first deciding the intervals we want to calculate the area of, and then we do something that literally can't stand derivatives so much,
185
they've named themselves antiderivatives, which are basically derivatives in reverse.
186
3x squared, for example, is x cubed.
187
6x antiderived is 3x squared.
188
Eh, yeah, there's actually a problem here, said some mathematicians.
189
Since the derivative of a constant equals zero, we wouldn't know if the anti-derivative has a constant.
190
Now, here is when an absolute genius came along and said, what if we make the constants subtract each other out by adding this part of the integral formula, and then, boom, we get the area.
191
Hey, wanna play a game?
192
If you were on a train station and this weird guy came up to you with two pieces of paper, what would be the chance of you being able to flip this little piece of paper on your first try?
193
Well, assuming that the game isn't completely rigged, you would firstly have to calculate the amount of preferred outcomes, then divide it by the total number of possible outcomes, and then you get the number 0.5.
194
This means that there's a chance of 50%.
195
Now, there are two other key rules that you probably should remember.
196
The first one tells us that the probability of any event must fall somewhere between 0 and 1,
197
with 0 being something totally impossible and 1 being something 100% certain to happen.
198
The second rule tells us that if you were to add up all the probabilities of a certain event, you would get the number 1, and the last one tells us
199
that the probability of something happening plus the probability of it not happening is also equal to 1.
200
Anyways, if you've been listening really well, you should at this point be able to solve this mathematical problem.
201
Now, rumor goes that first person to comment the right answer gets a shoutout in the next video. So, subscribe.
Why practice speaking with this video?
This engaging video, titled "ALL OF MATH explained in 14 minutes," provides a unique opportunity for english speaking practice within a scientific context. As you immerse yourself in the explanations of mathematical concepts, you will not only enhance your comprehension of English but also familiarize yourself with technical vocabulary. Watching educational content like this can significantly help you learn english with youtube, allowing you to pick up new phrases and expressions as well as understand complex ideas in a simplified manner.
Through the shadow speak method, you can repeat phrases and sentences directly after they are spoken, helping to reinforce your language acquisition. This technique encourages active participation, ensuring you are getting the most out of your learning experience.
Grammar & Expressions in Context
Throughout the video, several key grammatical structures and expressions stand out, enriching your vocabulary and understanding. Here are a few examples:
- Questioning Techniques: The speaker often starts with a question, such as "Wait, how big is the universe?" This technique invites engagement from the audience and can be a useful conversational strategy.
- Using Comparisons: Expressions like "Think of it like this" help clarify complex ideas. By adopting similar methods in your own speech, you can make your points more relatable.
- Defining Concepts: The speaker explains terms like "variables" and "constants" in simple language. Learning to define terms succinctly can aid in your own explanations and enhance your clarity when communicating in English.
These structures provide a solid framework for improving both your speaking skills and your overall command of the English language while discussing specialized topics.
Common Pronunciation Traps
While watching this video, pay attention to certain words and phrases that might be challenging to pronounce. For instance, the term "exponential" can be tricky due to its length and the placement of vowels. Similarly, phrases like "solve them separately" might pose a challenge with the blending of sounds. To improve english pronunciation, try repeating these words aloud several times, focusing on the articulation of each syllable.
The speaker's accent might also influence pronunciation. Listening closely to how they enunciate may provide valuable insights into speaking clearly and effectively. By actively practicing alongside the video, you can employ the shadowing technique to mimic the speaker’s rhythm and intonation, further solidifying your learning experience.
What is the Shadowing Technique?
Shadowing is a science-backed language learning technique originally developed for professional interpreter training and popularized by polyglot Dr. Alexander Arguelles. The method is simple but powerful: you listen to native English audio and immediately repeat it out loud — like a shadow following the speaker with just a 1–2 second delay. Unlike passive listening or grammar drills, shadowing forces your brain and mouth muscles to simultaneously process and reproduce real speech patterns. Research shows it significantly improves pronunciation accuracy, intonation, rhythm, connected speech, listening comprehension, and speaking fluency — making it one of the most effective methods for IELTS Speaking preparation and real-world English communication.