Important Ideas in Tutoring Sample Lesson
In the first group lesson, we cover foundational ideas we will continually reinforce throughout our High School Admissions program in weekly classes for our 7th graders in preparations for admissions to high school through the TACHS, HSPT, SHSAT, ISEE, and/or SSAT. These ideas are important for the tests and also everyday life. Those who are not using them are missing out. After the initial lessons to get everyone on the same page and set the tone, students focus on their specific tests and topics most beneficial to them, while taking multiple practice tests. This video is a slightly abridged sample of what our students typically encounter in a first online or in person lesson with us. Many of the ideas motivate students to try to improve a little bit every day. We think some of the most important are the following:
- Look at problems from multiple perspectives and angles; Multiple methods should yield the same correct result.
- See the similarities between ideas learned in school (and life) and apply them to other domains.
- Learn the foundational ideas and build your way up.
- Every 7th grade math problem can be solved backwards to check.
- Break problems up into easier ones you can solve.
- Not let ego get in the way of growing and asking questions.
- The most common mistake on the HSPT Language section, which is the hardest for most students
- The most powerful force in the universe. Those who apply it get to benefit, but those who don’t, pay it to everyone else.
- Some fun history of scientists and their accomplishments
- Many more ideas and test taking strategies.
This is only the first lesson! Most seventh graders attend 20-50 lessons and grow tremendously over that time. Grades in school typically increase, which also helps for admissions. Sixth graders show even more improvement.
More Info on Our TACHS, HSPT, SHSAT, ISEE, SSAT High School Admissions Prep Program
Transcipt of Tutoring Sample Lesson
Let’s start with divisibility rules. You probably know a lot of these. There’s one in particular that I want to go over that you probably don’t know, but maybe you do. How do you know if something’s divisible by two?
It’s an even number.
Even number. Great! Is there another way to express that also because it’s important to kind of look at things from different perspectives?
I think probably if it ends in a two, four, six, eight, or a zero?
Exactly!
So, those are both the same idea, and I think it’s important: in school you’re going to learn many different things that are really the same idea. You want to see the connections between them. Instead of learning the same thing five different times, you want to learn it well once, and then realize, ‘Oh wow, this other thing they’re teaching us is exactly like this, only this is the difference this time, or this is the situation we apply it to.’ All right, so even numbers or ends in two, four, six, eight, or zero. How about divisible by three? How do you know if something is divisible by three?
An odd number, right?
Not necessarily, because 7 is an odd number, and that’s not divisible by three, and six is an even number, and that is divisible by 3. So not just because it’s odd, that won’t work. Half of the multiples of three are odd and half of the multiples of three are even. So, there is another way though. I know that a number like 111 is divisible by three just by looking at it. How do I know?
I think it’s because if like you add up all the numbers and if it’s divisible with by three, it’s divisible by three.
Exactly! So, if you add up the digits and that number is divisible by three then the original number is also divisible by three, which is really bizarre, but this is true. This is a very useful skill, so I’ll write that out. You might want to write this one down. You don’t have to write down divisible by two. I think you guys all know that one, but this is one you might want to write down.
So, if you add up the digits and that’s divisible by three, then the original number is. So something like 712, is that divisible by three?
No.
No, because seven plus one is eight plus two is ten. Ten is not divisible by 3, so 712 is not. What about 711? Is that divisible by three?
Yes.
Yes, seven plus one plus one is nine. Nine is divisible by three; therefore, 711 is. Cool? Any questions about that?
How do you know if something is divisible by five?
If it ends in a zero or a five.
Yup, zero or five. And how do you know if something’s divisible by ten?
Ends in a zero.
Ends in a zero, all right okay. So the hardest one here is the three. The other ones you probably know, and this is really all we need to know. We can work off this.
I like learning the building block ideas, the basic ideas, the definitions, and then always working your way back from there. If you do that a lot of times, eventually those will just become subconscious, and you’ll know them.
The more times you can actively consciously go back to the definition and work your way back up, the easier it will be to learn the mechanics of a lot of these things. That’s what I used to do all the time at your age whenever I got stuck, so that’s a good first line of defense whenever you get stuck with anything: go back to the definition. Go back to the building block idea.
So, I mentioned the fundamental theorem of arithmetic before. What is that? What am I talking about? What is this crazy guy talking about? Fundamental theorem of arithmetic… if I could spell arithmetic.
Let me do an example and then you tell me if you’ve seen this before. So, 54, we can write as 2 times 27, which is the same as three times nine, which is the same as three times three. 3 is a prime number and 2 is a prime number so we can write this as two times three to the third power. 54 is equal to 2 times 3 to the third power. The same exact thing; they’re equivalent. Three to the third power is 27 times 2 is 54. Two times three is six times three is eighteen times three is fifty-four. The order of that doesn’t matter no matter how we do it. It’s commutative. Well, what am I talking about? Where have you seen this before? The fundamental theorem of arithmetic is what?
Distribution like the distributive property?
Close, it’s related to that and that’s an idea we’re going to talk about very soon. We’re going to use that a lot in mental math, the distributive property. I’ll write that on the side. Not quite the same. It’s related, but I think there’s something else you’ve done in school that’s a little bit closer. That’s a good guess, and remember, don’t worry about looking silly in front of other people, especially in front of me. I don’t care at all. I make mistakes on things all the time, and I’m sure you guys do, too. As you get older, you guys will make mistakes too, and that’s okay. The idea is to learn from them as much as you can. I don’t think any of you know each other in this group, so really don’t worry about feeling silly in front of them at all.
Ask as many questions as you want. Your goal is to learn this stuff as well as you possibly can, so that you know it for the rest of your life, and that you’ll always know, and you can always apply it.
This stuff makes your life a lot easier. I think the people that say they didn’t learn anything in school, or they don’t use anything in school on a daily basis are kind of missing out. They don’t see the things and where they can apply these things.
Even when you’re a billionaire on Shark Tank, you’ll need this stuff. All the time they’re doing mental math in their head, and if someone comes in and they say all right I want $500,000 for 10% of my company, and then Mr. Wonderful goes “Well, why is your company worth five million dollars?” He’s doing the valuation of his company (of the company) in his head right there. So, it’s still going to be important to be able to do that. So, any questions at all on the stuff feel free to ask. Have you ever seen…if I say the word, I think you guys will all know it… you guys have all done factor trees before? Yes. So what I did for 54 was: I did a factor tree and then I just circled the prime numbers, and set it equal to the multiplication of those prime numbers. So the fundamental theorem of arithmetic means that you can write any number as a multiplication of prime numbers. Any…let’s say integer, positive integer…. we’ll talk about those, too…. can be written as a multiplication of prime numbers. Can anyone put that in their own words for me? What do all those words mean together? Any positive integer can be written as a multiplication of prime numbers. Can anyone interpret that if you were trying to explain that idea to a fourth grader, or maybe a fifth grader? I don’t think a fourth grade would understand this. How would you explain it to them?
Well, how about this? Let me pick someone. What’s a prime number? I’m going to pick a random person from the group. What’s a prime number?
A prime number is something that can only be divided by itself and one, right?
Good! So, a prime number is: the only factors…only unique factors, only distinct factors are one and itself. I’m adding those words distinct, unique because one. The factors of one are just one and one, right? So that doesn’t count as a prime number. That’s why I’m adding this word. Otherwise, you might think that one is a prime number, but it’s not. It seems like it would be, but one is not a prime number. Does anyone know what the smallest prime number is? What is the smallest prime number?
Two?
Two! Excellent! So the only factors of two are one and itself. Why is two the only even prime number? Why can’t you have another even prime number after two?
It can always be multiplied by two and another number?
Exactly. Because 2 always goes into it, 2 is always a factor. It can always be multiplied by two and another number. Exactly! So, any time we’re looking for prime numbers, if we see that it’s even, it’s automatically not prime. Definitely not. So, we never want to test an even number to be a prime number.
I have a question.
Sure.
So, for this factor tree, we only take the prime numbers, so it couldn’t be like a two, a three, a three and a four?
A four couldn’t be in the answer, but when you’re breaking it down, you could divide by four first if you wanted to. Not in this problem you couldn’t. But let’s say you want to divide by nine first and you wanted to say that 54 is 9 times 6, you can do that. There’s no real wrong way to do the factor tree. You’re always going to come up with the same answer at the end. So, this is three times three and six is two times three.
Okay and that’s one thing I do like about math: there’s more than one way to do it. I could give you a number to do a factor tree and everyone can have a different answer for it. That’s definitely possible. What I want to do right now is go around the room and count prime numbers.
So, we’re going to start with two. Someone just said 2 is the smallest prime number. What’s the next prime number after two.
Three.
Three, good! What’s the next prime number after three?
Four?
No, unfortunately four is not a prime number, because it’s two times two, and we don’t have to test any even numbers. There can be no even prime numbers after two. So, anytime you get to an even number automatically skip it. So, not four. What about the next number after that?
Five.
Five is a prime number. Are there any ways to multiply to five except by multiplying one and five? No, right? Not with integers. If we get into fractions, if we say two times two and a half, yes, but that’s not really what we’re talking about. These numbers have to be integers. They have to be whole numbers. Let’s say they should more be, instead of integer, that should really be a whole number. I said positive integer. We’re good. That’s a whole number. All right what’s the next prime number after five?
Uh, six?
Unfortunately, six is not, because six is two times three, so it’s not a prime number.
Sorry.
No problem. I make mistakes. You guys will make mistakes in the future. It’s all right. The goal is to learn from them.
Seven.
Exactly! The only multiple of seven is one and seven, right? There’s no other way to multiply to seven. We can’t say two times anything is seven, three times anything, but there’s nothing that makes that seven. Four times anything. There’s nothing that makes that seven, right? So, that’s a prime number. What’s the next prime number after seven?
Eleven?
Eleven. Very good! You skipped eight and ten, because they’re even and nine is not prime because it’s three times three. Very good. So, 11. What’s your next prime number after 11?
13?
13, very good and the next prime number after 13.
17?
Very good. Why did you skip 15 correctly?
Because it has multiples of 5 and 3.
Exactly! Three times five. Very good. Next prime number after 17?
19?
19. Very good. Next prime number after 19?
23?
23, good and you skipped 21 because 3 times 7. Two plus one adds up to three; therefore, 21 is a multiple of three.
Next?
27? Not 27 because 3 times 9. And then two plus seven is nine. That’s a multiple of three; therefore, 27 is a multiple of three.
29?
29, good! And they’re going to start getting a lot harder. But yeah, 29. There’s no way to make 29 except 1 times 29. What’s your next prime number?
31?
31. And also, just as an idea, this is a very similar, you guys know in colors the primary colors? This is a really cool idea for the idea of prime numbers I think. You all know primary colors, right? What are primary colors?
Any artists here? I’m not an artist. I’m not a very good one at least. With a computer, I’m much better.
They make any other colors?
Yeah. Colors that what?
Colors that can’t be made by any two other colors.
There you go. Yeah, they can’t be made up by any two other colors. They’re the building block, fundamental color. They’re primary. That’s what prime means, right? Even pre as root word p-r-e means before, so this this root word means first, beginning, fundamental, building block and the idea is: so,
2 is the prime number, 3 is a prime number, and 5 is a prime number, right? That’s the same thing as red, yellow, and blue. If you combine red and yellow together, you’ll get orange and that’s the same thing as six, right? If you combine two and three together, you’ll get six. If you combine red and blue together, you’ll get purple, and that’s the same idea as 10. If you combine blue and yellow together, you get green, and that’s the same thing as combining three and five together and getting 15.
There are more combinations with numbers than what colors. What’s your next prime number after 31?
37?
Good, 37. You skipped 33, 3 times 11. You skipped 35, 7 times 5. 37, very good. All right, next prime number after 37?
Sorry, um, 41?
41, good. Not 39 (3 times 13), but 41. All right, next prime number 41?
43?
43, great! Next prime number after 43?
47?
47, good. Not 45. 9 times 5. 47. The next prime number for 47?
51?
Nope. Five plus one is six, so therefore 3 goes into 51. It’s 3 times 17 is 51.
53?
Good and you skipped 49, 7 times 7. That’s a good one. 53.
After 53?
57?
5 plus 7 is?
59? Good, 59. Five plus seven is twelve, and that’s going to be 3 times 19. We could even say 3 times 20 is 60, right? So, 3 times 20 minus 1 (or 19) is 57, and if we do that, that IS the distributive property. And we’re going to use that for mental math a little bit more with addition. I’ll show you when we do doubles. I’ll show you how to do that: what I’m talking about. But we said 59 is the next one, right?
All right, next prime number after 59?
61.
Yep, 61. Next prime number after 61?
67.
Good, not 65. It ends in five. What’s your next prime number after 67?
71?
Good. Next prime number after 71?
73?
73, good. It doesn’t end in a five. Seven plus three is not three. Maybe, there’s some other hidden number as we get bigger and bigger, but it doesn’t seem like it. All we really need to test, all I’m really asking you guys to do, is test two, the three, and the five. If it doesn’t go into any of those, it’s very likely to be a prime number, and then if it’s not, hopefully I’ll catch it and let you know, but I think we’re good. All right, next prime number after 73?
Um 79?
79, good! Not 75. Not 77. 7 times 11. 79.
All right, great! Next prime number after 79?
83?
83, yeah. Not 81. 9 times 9 and 3 times 27. 83, good!
Next prime number after 83?
87
8 plus 7 is?
Oh, uh, 89?
89. Yeah, 89 almost feels like there are numbers that go into it.
What I want you to do is write 432 as a multiplication of prime numbers. So, 432 as a multiplication of prime numbers.
And remember, there’s no wrong way to do this. You can put whatever number you want into this first. I would probably just keep dividing by two until I couldn’t anymore instead of trying to figure out two big numbers that went into this. That’s going to take some time. I think it’s pretty easy to just divide by two, divide by two, divide by two until you get down…until you can’t anymore, and then pick another prime number and start dividing by that.
You could have divided by three from the start because you know four plus three plus two is nine; therefore, three goes into it. I think it’s a little bit tougher to do. It’s harder to divide by three in your head, especially for bigger numbers. I think if it’s a little smaller, it’s a little bit easier, but three is fine. We could have also done six. You could have also done 18 if you wanted to, but that’s crazy. Who would see that 18 goes into this number? If you saw that somehow, yeah, you could do that. There’s no wrong way to do this. As long as these two numbers always multiply to the number above it, you can’t be wrong. And another thing I like about math besides that you can do it multiple ways: anything you could do forwards in math, you can do backwards. So, when you get your answer, you should be able to multiply that together and get 432. If it does not equal 432, something’s wrong. All right, so some of you guys said you divided by three. Did a lot of you guys divide by two or what numbers did you guys divide by to start?
Three.
You did three? Three. Okay. A lot of people did three?
I did eight.
Eight? How did you know eight goes into this? But, okay that works.
Because I just looked at the 54 at the side for the second and then I multiplied it by 8 and it equals 432. Clever! Okay, all right let’s do three. I rarely do it with three. I usually do it with two. Let’s do three. So, it’s going to be three times 144. Is that correct?
Yeah, and another way we can check that: three times 100 plus 40. I’ll write it as a three multiplication. Three times a hundred is three hundred, right? 3 times 40 is 120. And 3 times 4 is 12. Now, if we add all those pieces in our head…it’s a little bit easier if we have two pieces…. but we’ll get 432. 300 plus 120 is 420 plus 12 is 432. All right and then what’d you guys divide that by? Did you do 3 or 12 and 12.
I did three.
Three again?
12 and 12.
Yeah, that works too so if you did it times three, this would be 3 times 48. Is that correct?
Yeah.
Because I know 3 times 50 is 150, right? So, this should be 3 times 50 minus 2 to get six less because 3 times 2 is 6. So, I’m using the distributive property to do this in my head, but if you haven’t done things like this before, this is a really good way: any time you’re stuck with anything. Anytime you’re trying to do a harder problem, break it up into easier problems that you can do a lot more simply. So, you can do 3 times 50 in your head, you can do three times two in your head, and then just add that extra step of subtracting them or adding them.
Okay, 48. 48, that’s one of your times tables. You said that’s like six times eight or two times twenty-four. I’ll just write it as six times eight. It’s probably how most of you guys did that. And then six is two times two, sorry, two times three. Eight is two times two times two or two times four. I’ll do it in steps, but I have to make a bigger picture. So, this is going to be 2 times 2. And you all have a tree that looks something like this? It doesn’t have to look exactly, but it should all end in one, two, three, four twos and then one, two, three threes. So, how can we write this? We can write this as two to what power times three to what power?
Two to what power times 3 to what power or just give me the first one? Two to what power?
2 to the fourth power?
2 to the fourth power. Great!
And three to what power?
Second power?
I think I see three threes though.
Third power?
To the third power, no problem.
That’s all right I think that’s more of a ‘not seeing this because it looks like a two’. That’s my handwriting than anything else. So, two to the fourth times three to the third. Great. And then we can check this. Two to the fourth power is what?
16.
16. Great. 2 times 2 is 4 times 2 is 8 times 2 is 16. And what’s 3 to the third power?
27?
27, is that what you said? I think I that’s what you said. 27. Yeah, so 3 times 3 is 9, and then 9 times 3 is 27. The definition of this exponent means times itself that many times, so when you’re stuck with anything go back to the definition and just work it out. And these are going to be pretty common exponents. 3 to the third is 27. 2 to the fourth is 16. Working off the two exponents like 2 squared, 2 to the third, two to the fourth, two the fifth, and for three and four, those are going to be pretty important. At least you’re going to see them multiple times, so it wouldn’t be awful to memorize them or just not even memorize them. You’re going to see them so often that you probably will have them memorized after the end of this year. You don’t have to try to memorize them.
And that’s where a lot of the stuff, if you practice it a lot, you don’t really have to memorize it. It won’t feel like work if you do a little bit every day from now until November. It’ll feel very painless. You’ll learn a lot. You pretty much make memories every night in your sleep, so it’s very beneficial to do it a little bit every day rather than just to pack it all in the night before you do something. If you usually study the night before a test and then take your test, try to take that time and just space it out a few days before your test, and you’ll have much better long-term results. Even if you’re doing well now, you’ll do better if you do that. And if you space that out over a week or more, that would be even better. You learn a lot more over time rather than trying to cram everything together.
And also, that brings up the idea: a tree that grows with that wind collapses under its own weight, which brings me to the idea of would you rather have one million dollars at the end of a month or a penny that doubles every day for a month?
And what I was saying before: a tree that grows with that wind collapses under its own weight. Trees, humans, biological systems, they get stronger when we stress them. If you just sat on the couch and watch TV all day, your muscles are going to go to mush, your brain’s going to go to mush, and they’re not going to be there when you need them. But if you train them and stress them a little bit to a point, they’re going to get stronger, and they’re going to be better when you need them, and way better than you can ever possibly imagine, which is hopefully what we’re going to prove here. So, you can take a little vote what would you rather have: a million dollars at the end of the month or you get a penny on day one and it doubles every day for a month and then you get whatever it is at the end of the month.
Is it a million dollars a month or just that one month?
Just that one month.
The penny because by the end of the month you probably have more than a million dollars.
Okay, possibly. We’re going to find out soon. We’re going to do the math to see. Does anyone think it’ll be more than a million or do you think it’ll be less than a million?
I think it would be less.
One more one, and we have one for less. I’m not sure who the other voice was, but anyone else want to vote?
I think I’ll be the less.
So, we have two for less, one for more. Which one do you think it would be?
Can you show it again? Yeah, would you rather have a million dollars at the end of the month or a penny that doubles every day for a month and whatever you get then?
I think I’ll take a penny.
All right, split jury. Let’s find out, so at the end of one day (after one day), we’re obviously going to have a penny. After two days, we’re going to have two pennies. I’m gonna give this to you at some point. This is going to be a doubling exercise. So, obviously after three days, 4. Four days would have eight. After five days, we’d have 16. After six days, we have 32. For seven days, we’d have 64. After eight days, we’d have a dollar twenty-eight. After nine days, we would have two dollars and fifty six cents. After 10 days, we’d have five dollars and twelve cents. Okay, it seems like there’s no way this is going to be over a million at this point, right?
Okay, all right, after 11 days, what’s double five dollars and twelve cents?
Five dollars, no ten dollars and twenty four cents.
Ten dollars and twenty four cents. Excellent!
All right, double ten dollars and twenty-four cents.
Um, twenty dollars and forty-eight cents.
Twenty dollars and forty-eight cents. Great!
Double twenty dollars and 48 cents? And the way we can do this we can break it up into double twenty dollars and double 48 cents. Twenty dollars you can do in your head very easily. What’s double twenty dollars?
40.
And double 48 cents?
If it was 50 cents, 50 cents times two would be a dollar, right? Yeah. So, 2 times 48 cents has to be a little bit less. How much less?
Is it 96?
96, good. So it’s gonna be 40.96. $40.96. All right, great. Double 40.96?
Uh, say like 81 dollars and something cents?
Yes, 81 dollars and something cents. Can you be a little bit more specific?
Hey, I’m trying to figure that out. Sure, that’s a very good estimate. 81 dollars and 19, no 90, yeah, 92 cents.
Very good and a way we can do this: we can do the distributive property with a subtraction. We could say 40.96 is 41 minus four cents, right? So, if we’re going to double that, we can double this and say 2 times 41 is 82 and 2 times 4 is 8 cents, so it’s going to be 82 minus 8 cents. That’s a good way to do it. You don’t have to do it like that. The more you practice this, the more you’ll know what numbers to break it up into or the more you know what’s easiest for you. We could have also done it as 2 times 40 plus 96 like we did before. So, 2 times 40 is 80 and then 2 times 96 is a dollar ninety-two. That’s a little bit tougher. But either of these work. You can do it multiple ways and get the same answer. All right.
81.92 times 2? Round this one. Let’s round this one down just so we know at the end, our answer is actually going to be bigger than this if we round it down. So, let’s round this down to 81. Double 81?
162?
162. Good. Double 80 is 160. Double one is two. 162. All right, double 162?
324.
324. Great. Double 324?
648?
648. Great. Double 648?
1000, no wait, wait, one thousand two hundred and fifty-six.
A little bit bigger than that because double 650 would be 1300. The six is right.
One thousand two hundred and something. 1298.
96 I’m getting and let’s check it just to make sure. Maybe I’m wrong. Yeah, 96. So, if we did it as 650 minus two, it would be 2 times 650 which is 1300 minus 2 times 2, which is four, so it’s thirteen hundred minus 4. That’s the easiest way for me to do it. I think that’s the easiest way. We could have broken that up into 1200 plus 96. Sorry, 600 plus 48. 600 times 2 is 1200; 48 times 2 is 96. That’s not bad, but you have to have some experience doing 48 times two. I think you might have done it as four days is 96 hours. That might be where you’ve seen that combination before. Okay, any questions about this? All right
1296 times two?
2592.
Two thousand five hundred and ninety-two. Very good. 1300 times 2 would be twenty-six hundred and this has to be eight smaller than that because this is four smaller than thirteen hundred. That’s pretty much the distributive property we just did. Great. Double 25.92?
5 thousand one hundred eighty-six
Yeah…wait a second. 84. I always confuse my 4s and 6s. Because you were subtracting 16 from it, right?
Oh, I thought it was 86. This is going to be 2600 minus eight. If we double both of those, we’re going to get 5200 minus 16. So, this has to be a four.
Oh, good
All right, 5184 times two?
Is that 10,368?
10,368, very good.
And 10,368 times two?
Twenty thousand seven hundred and thirty-six
That’s correct, right? Let’s check it. So, I broke this up into 10,350 plus 18. Yeah it’s 36. 10,350 is twenty thousand seven hundred plus your eighteen times two which is your 36. Great!
So, this is about 22 days. Do you think this is going to be over a million or under a million so far?
I still think it’s going to be over a million.
Yeah, it’s still very, very far away, right? We’re still missing a lot of money here to get to a million and we’re almost done. We only have a few more days left. The first question everyone should have asked is ‘Is this February? Is this a leap year? Is it January?’ It’s gonna be a big difference at the end whether it’s February or January, but let’s keep going. Let’s round this down again. This number’s getting a little bit unwieldy, so let’s round this to twenty thousand five hundred. Say it’s about twenty thousand five hundred.
Well, I already got an answer for twenty thousand seven hundred thirty six times two.
Sounds good, what is it?
I got 41,472. 41,472?
Yeah, sounds good. Twenty thousand times two is forty thousand. Seven hundred times two is fourteen hundred. Thirty-six times two is a seventy-two. Yep, all right.
Do you have the next one or you want me to round?
82,940…um…four
I believe so. What was this digit? Two, I think, wait I think it was two.
Yeah, okay, I thought you said something different. Let’s check it to make sure. These are getting pretty tough to do in your head. I think these are a little bit too tough for anything you can see on the test, but that’s okay. This is really good practice. One of the things that’s actually proven to work for teaching is preparing students for things that will be more difficult than what they’ll actually see and then those things actually do work. And it’s kind of like lifting weights. If you lift a heavier weight, the lighter weight feels easier and you do better on it. Same idea. We can just do 82,000 at this point.
164,000.
164,000 you said? Yeah, all right good, so we can round this down to 82,000 double that and this is 164,000. Double 164,000?
328,000.
Great! Double 328,000?
656 hundred thousand six hundred fifty six…that’s okay. I kind of felt like that 100 should be there though when you said it. Kind of wondering why it isn’t because that would be a bigger number. This is just 656,000.
All right 656,000 times two? What’s that? You can do the honors. I’m just writing the work down.
One million three hundred and twelve thousand. Great!
So, now it’s over a million after just 28 days, right? Isn’t that amazing? So, it started with a penny after only 28 days we’re already over a million, and then for 29 days we’d have 2.6 million.
And 30 days: 5.2
31 days: 10.4 million if it was January or a 31-day month. So, I think this is a good example of that humans are not really good at seeing exponential growth. Einstein said it’s the most powerful force in the universe. Those who understand it get to benefit from it. Those who don’t have to pay it to everybody else. And it goes off like this.
It’s kind of like the tortoise and the hare idea. You’d rather be an average student of average intelligence who works very hard and gets better a little bit every day, improves a little bit every day, than someone who’s a genius that doesn’t do anything. You’re much better off in the first situation. The genius isn’t improving. You’ll eventually surpass the genius. And I’m not a finished product. You’re not a finished product. When we’re 70 we’re not gonna be finished products. We want to constantly do this for anything we want to get better at our entire lives. And anything at your age now that you want to get better at that you want to be one of the best in the world at, you start doing it now a little bit every day, you’ll have a chance to be one of the best in the world. If not one of the best in the world, you’ll be way better than the vast majority of people that do it if you start now and just improve a little bit every day. That’s for anything: for shooting free throws, for hitting a baseball, for doing math, for reading, for anything. So, it’s a long way to go. None of us are finished products. If we just keep improving a little bit every day, we’ll be way better than we can ever possibly imagine. Any questions about this stuff at all? Is that a cool idea?
That was fun. I really enjoyed it.
Awesome, thank you. All right, let’s switch and go to English. There’s a very important idea in English that almost everyone confuses initially. Do you guys know the difference between it’s and its? Its and it apostrophe s.
Isn’t its referring to a certain thing and i-t apostrophe s is referring to… I can’t find the word.
Can you use it in a sentence? Possession?
All right, good. Which one of these is possession and which one of these is it is. That’s pretty much the question, and I think a lot of people confuse this from the start; it’s a little counter-intuitive because we’ve all learned that apostrophes make things possessive, right? But that’s not true for pronouns, only for nouns, so this one on the right over here, this is ‘it is’. This one on the left over here without the apostrophe, this is the possessive, which is a little counter intuitive.
So, this one is the contraction. If there’s an apostrophe on a pronoun, you say it like a contraction. This is ‘it is’.
Can anyone think of any other words that are similar to this that follow the same rule? There’s no opposite rule to this. This is the rule for all of these, but can you think of any other words that are that are very similar to this?
How about whose? What would that be as a contraction? How would you spell that? How would you say who is?
W h o apostrophe s?
Exactly, this is ‘who is’. This is probably one of the most common grammar errors to look for. Sometimes they’ll use it correctly. Sometimes they’ll use it incorrectly, and you want to be able to pick it. The grammar on the HSPT, I think, is one of the hardest things. The one for the Regis test, the scholarship test for Xaverian, Xavier, Seton Hall, St. Peter’s, Fordham, many schools now, Loyola, St Edmund’s. So, I think that’s one of the hardest parts, and this is one of the biggest things you’re looking for. And also we have ‘they’re’ and this is they apostrophe r e ‘they are’, so everything on the right here, these are all contractions. Anytime you see an apostrophe on a pronoun like this, like ‘it’s’, say it as two words ‘it is’, ‘who is’, ‘they are’, and it will jump out a little bit more for you. It’s a little bit harder the other way, but these are all possessives. If it doesn’t have an apostrophe, it’s possessive, just like the other possessive pronouns that you guys know: his, hers, ours, yours. None of those have apostrophes, right?
So, no pronouns that have apostrophes are possessive. All possessive pronouns do not have apostrophes. There are no apostrophes in anything on the left here. I think that’s a good way to remember it. Any questions about this?
All right
All right, which one should be there? It’s cold outside.
The contraction?
The contraction, good, so this should be ‘it is’ cold outside. Sorry, if I could spell cold right. All right
How about “Who’s at the door?” Which one is that? Who’s at the door?
Who’s, the contraction?
Good! Who apostrophe s. Who is at the door? Very good!
Just the first one here. ‘They’re over there with their friends.’ Just the first one.
Contraction? They are.
Good! So that should be ‘they are’ over and they’re over there. Just this one. How would you spell this one?
It’s a little bit of a trick question, so if you think it’s a little bit of a trick question, you’re right.
Um, the possessive? No, it’s not possessive and it’s definitely not the contraction, right? You pretty much eliminated the contraction. You know it’s not that one, so you’re picking the other one there, but unfortunately there’s a secret option here. There’s another ‘there’. What’s the other there?
T-h-e-r-e like here and there. The other words don’t really have an analogous relationship, but there’s just a separate there here, so this was a little bit of a trick question. So, that’s this one. They’re over ‘there’ with their friends. What’s the other one here? ‘With their friends’, which is that? Which would that be?
That should be the possessive one.
Good!
Okay. I think this is a good time to bring up that we’re going to have homework on this, and let’s pick our group name. Okay, so the groups that are left. So, basically I’m going to have weekly homework. It’s going to be posted at one of these links. If you try to search for it on Google, hopefully this page should come up and you should remember your name, and then you can just click the link if you do forget this. But I’m also going to post the group link, the meeting link there, and everything will be there in one place: the book, everything. So if you lose anything, it’s all right there. There’s also some extra information about the SATs, chemistry, and schools you might be interested in like Sea, Loyola, Sacred Heart, so there’s some cool stuff you can look through there. But you get to decide what your name is, and I’m going to go through each one and describe/tell you what they’re about. And then you can pretty much write it in chat which one you want. She was, they were all about radioactivity. It was a husband and wife. Madame Curie is probably more famous, so I usually reserve this for a mostly female group, but you can definitely pick this. It’s a very cool group. Galileo. He invented telescopes. He also labeled the dark surfaces on the moon ‘mares’ because he thought they were oceans. ‘Mare’ like calamari, frutti di mari, the Seattle Mariners, maritime, marine. Those have something to do with seas or oceans. He thought there were oceans on the moon. He was a genius. He helped advance Humanity a lot and he was very, very wrong about that. There are definitely no seas on the moon, so I guess my point from here is don’t worry about being wrong. Whatever the evidence points to, go with that and then if you have different evidence in some way or if you have another reason to think something, feel free to be able to change your mind. And be like “All right, well now I think, based on what I know now, this should really be this answer” and that’s okay. You’re allowed to do that. That’s showing growth as a person. Gibbs. This is very important. It’s a very important calculation in chemistry to determine whether a reaction will actually occur. It’s named after a person called Gibbs, but this is a pretty cool calculation to learn. You’re going to learn all about that in probably AP Chemistry. Joule. That’s a unit of energy. You’ll learn about that in chemistry and physics. You probably know calories. 4.18 joules are one calorie. Those calories in food that you see at the back: those are units of energy. That’s how much energy it would give you if you could break it down perfectly.
There’s such thing as kilojoules, right?
Yes, a kilojoule is a thousand joules, just like, if you have a meter a kilometer is a thousand meters. Kilo just means a thousand.
That’s a really good question. We’re going to talk about that. We’re going to have to be able to do unit conversions for the SHSAT and for the HSPT. They’re going to have a lot of those. And you’re going to have to know kilograms. The units of kilo-, milli-, and centi- are going to be very important. I think those are the only three you really need to know.
Nash. He was the movie, A Beautiful Mind. He was all about getting/optimizing moving things from one place to another, but Nash is taken so you guys can’t pick that. Newton is also taken. You guys are not allowed to pick this one unfortunately. He invented calculus. Pretty much, he was just sitting around doing physics, and he was like ‘I need better math for physics’ so he invented calculus, which is amazing. He also has an awesome paraphrase: if you think I’ve accomplished so much or if you think I’m so smart, or you think I’m so great, it’s because I’m standing on the shoulders of giants. So, basically he’s saying that everything that everyone who came before him helped pave the way and helped him. He couldn’t have done any of this without all the people that have made advances before him. So basically, same idea now. We/you wouldn’t know some of the math or some of the science that you’re going to learn over the next few years unless Newton, Curie, Galileo, Gibbs, Joule, Pascal, Salk and Turing. Unless they came before and showed that way before, we probably wouldn’t have known these certain things. And then we’re going to learn those things and be able to do some things that they invented. Nitrogen is 80 of the air. Air is about 20 oxygen, 80 nitrogen. We don’t really react with it. We take it in, but we don’t really use it…in the air at least, but in food it’s very important. It makes up amino acids, which make up proteins, so if you didn’t have nitrogen, you would not be able to make muscles. Pascal. You probably know him from the Pascal triangle/Pascal candle. Probably heard the name Pascal from the Pascal candle in church. He’s also done the Pascal triangle you’re going to learn in algebra one. The standard unit of pressure is named Pascal. That’s pretty important, which is a Newton per square meter. So, pretty much Pascal couldn’t have done what he did unless Newton came before him and talked about forces, because pressure is all about forces per square meter. Salk is a really cool one right now also. He invented one of the first vaccines. About the third or fourth one I believe: for polio in the 1950s. Before 70 years ago, we couldn’t invent vaccines. We didn’t have many vaccines and after this we learned how to make all sorts of other vaccines based on this method. Before this, it was only the smallpox vaccine, rabies with Louis Pasteur, and I think one other. Turing. He invented computers. If you’ve ever done a captcha, like which one of these has a crosswalk, which one of these has a light, which one of these is a fire hydrant, that’s a Turing test. They’re making sure you’re not a computer, so that you can go in. If you are a computer, if you are an automated system, they don’t want to let you in. They’re making sure you’re a person, so that’s a Turing test. All right, so any questions about any of that stuff?
Okay, why don’t you guys vote for just one of whichever one is your favorite group. Joule, Nash, and Newton are already taken, so it cannot be any of those. If that person just voted for any of these, they can vote again, but it just can’t be one of those three groups. You can take any other group you want. Vote once and then if there are any overlaps or if there are any ties, we’ll work from there. Sounds like everyone voted. All right, Turing it is. We have three votes for Turing, so it’s going to be at masterofchemistry.com/turing. I’ll type that in right now and that’s where your homework is going to be. It’s going to have something to do with the divisibility rules. Actually, your homework for now, for this week is just doing practice test corrections. I want you to go through your test. Anything that you can easily do that you know how to do that was kind of a silly mistake, just write an explanation of how you can get it right in the future to show that you actually know what you’re talking about. Anything that you’re uncertain about, ask about in class. So, go through your entire practice test. Anything you got wrong, write a correction for how you can get it right in the future. It only has to be a couple of words or a sentence. It doesn’t have to be long and tedious. Just a short, succinct way. For example, you could say that i t apostrophe s is a contraction not possessive. That’s it. That’s all you have to say. Something like that. So, for any question you got wrong for reading comprehension, write the line where you can find your answer. For vocabulary, just write some synonyms. For math, write out the work. For abilities, write the rule. You probably don’t know how to do that yet. We’ll talk about that next week, but any questions that you guys got wrong, write corrections for, and then email them to me. All right, sounds good. I will see you guys next week, and then next week if you want, this is going to run into an SHSAT class for the students that didn’t take the real SHSAT yet. It still got pushed back, so you can see what they’re doing if you want to stay a little bit extra next time. All right, have a good week, I’ll see you next time.
Bye! Bye!
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