okay we want to go ahead and look at

inductive and deductive reasoning for some folks this can be a little bit

confusing but the main thing is we’re looking for patterns with inductive

bracing we’re looking at specific examples trying to see what is the

pattern what’s changing here what’s the next thing so in this particular problem

we’re looking at basically left side right side and asking ourselves what’s

changing how is it going from one line to the next on the left hand side we

start with one plus two the next one goes one two and then three okay then

the next line is one two three four so it looks like we’re just kind of adding

the next number what about the next line sure enough one two three four five

makes sense then that the next line would be you guessed it one plus two

plus three plus four plus five plus six okay now what’s it going to equal now we

look at the right hand side and look for the pattern over there two times three

over two okay so we had a fraction next one also a fraction over two divided by

two but this time it instead of two times 3 we have 3 times 4 so we kind of

like 1 up 1 what about the next line still divided by 2 but instead of 3

times 4 we go to 4 times 5 again up by 1 and then from 4 times 5 over 2 we go up

to 5 times 6 divided by 2 so sure enough what do you think the next numerator is

going to be that’s right 6 times 7 divided by 2 right well this is beautiful patterns but does it make sense mathematically let’s just check a

couple of them the first one 3 plus 5 sorry 1 plus 2 is 3 and then 2 times 3

is 6 and we divide by 2 and sure enough I think that’s actually 3 sorry about

that and then 1 plus 2 Plus 3 1 plus 2 is 3 plus 3 is 6 3 times 4 12 over 2 yes

again that works let’s jump down to the one we just

made up 1 plus 2 is 3 plus 3 is 6 plus 4 is 10 plus 5 would be 15 plus 6 would be

21 over here what’s 6 times 7 oh that’s 42 and then if we divide that

by 2 sure enough it is 21 so it really works it’s not only a

cool pattern it works now let’s try another one

see if you get this a little bit different um we’re doing 1 times 8 plus

1 yes that would equal 9 because 8 plus 1 is 9 and then 12 times 8 that’s 96 96

plus 2 sure enough that equals 98 and then they just keep going down and

building this pattern like a pyramid okay so take a look at what’s going on

over here on the left and see if you can figure out what the next line would be

what’s that next number gonna be notice the x part at the x 8 stays the same

every single line but then there is also a change right here what’s the pattern

what’s going on from one line to the next can you predict the left-hand side

of the next line okay so what we’re doing we’re taking that first summer

we’re just adding the next consecutive digit so we’re gonna have 123 456 123456

it’s still gonna be like all the others times 8 but this time what are we gonna

add the next consecutive number would be 6 okay so what about the right-hand side

well we started out with nine then it goes to it two two-digit number then a

three digit then a 4 then a 5 so we’re getting bigger numbers but notice how

we’re getting them it goes from 9 then 9 8 10 9 8 7 then 9 8 7 6 looks like we’re

going down what would you predict the right-hand side of this next line would

be that’s right nine hundred eighty seven thousand six

54 you can trust me on this one if you don’t go ahead and get your calculator

it actually works pretty cool stuff all right let’s take a little bit different

twist this time we want to do one of these prompts like you may have seen on

Facebook looks like somebody has this magical and you know knowledge or

something it’s math that’s all this behind this alright so let’s look at two

examples this first one we’re going to multiply a number by 4 add 8 divided by

2 and subtract 4 and then we’re going to figure out what our answer r is we’re

going to look for a pattern and then prove that it works no matter what

number we start with alright so the first one we want to use this 2 so 2

times let me get a marker here and I’m going to use blue so 2 times 4 it’s

going to be 8 and then we’re going to see 8 plus 8 16 and then it says divide

that by 2 so that’s gonna be 8 and then 8 minus 4 is loops 4 alright so my

answer here was 4 now it says try 5 all right so we’re gonna do the same thing

go through all these steps again so 5 times 4 and then add 8 and then we’re

going to divide that 28 by 2 and get 14 subtract 4 so final answer is going to

be 10 right what about the next one well uh says 8 is my starting point so 8

times 4 again we’re going to start there says add 8 to that so that’s going to

take me up to 40 and then divide that by 2 and get 20 oops and what happened

there sorry about that 20 and then 20 minus 4 would give me 16 okay so

started with two got four start with 5.10 start with eight got sixteen are

you seeing a pattern well the next step says write a conjecture look for the

generalization that relates the original that the result to the original number

so two going to four five going to ten eight going to sixteen how what could

you do in each of those cases to get that answer put it in terms of n what do

you get well if two goes to four or five goes to ten eight goes 16 I’m looking

at doubling so the way I would write that when n is 2 times n I’m doubling

it and multiplying by 2 okay so here’s where we start using some math again but

this time we’re going to use deductive reasoning we’re going to use general

principles of mathematics to prove that this actually works no matter what the

starting number was okay so we’re gonna select your number call it in we’re

gonna multiply by 4 so how do you represent that algebraically or

mathematically you put 4 n how do you add 8

you put + 8 so far so good right now divide this by 2 this is where it gets a

little bit tricky for some of us so take a minute and see if you can divide by 2

and simplify your answer the important thing to remember is that you have to

divide all of it so it’s the 4 n divided by the – it’s the 8 divided about it –

and when you simplify the result you get the 2 n plus 4 okay now from that I want

to subtract 4 ok so I take the whole thing I subtract 4 and my answer is

going to be move it down a little bit further there we go -2n plus 4- the

4s cancel and I’m left with sure enough the 2n which proves my conjecture all

right let’s try another one it’s a little bit different

similar idea but I will see how well you can do it

this time we’re going to add three double the result add four divided by

two subtract the original number that’s used so go ahead and start with five and

then nine and two and see what your answers are okay so what do I get if I

do this five plus three and then double this 16 then sixteen plus four is 20

divided by 2 is 10 subtracting the original number I get five what if I

start with nine well nine plus three is going to be twelve doubled is 24 adding

4 is 28 divided by 2 is 14 and subtracting the original number is high

huh so I start with five I got five I thought maybe I get back my number but

here I start with ninety then I get back five again try third one so two plus

three is five sorry my pen switching on me five we go and then we’re going to

double the double it 10 add four and divide by two and subtract the original

number well seven minus two is sure enough five so it seems like the

original number I start with doesn’t even matter my answers gonna be five

that’s what I would suspect my conjecture is so I’m gonna say if my

original numbers in doesn’t matter my answer is gonna be five all right so

let’s go to proving it let me see if you’ve got this down all right so how

could you go through this procedure first off selecting the number that’s

basic that’s n adding a number you can do that means plus three right now think

about doubling that number and simplify it the important thing to remember is

that we’re going to double the entire amount so I write 2 times the quantity

and remember that I distribute so that I get 2n plus 6 now if I want to add 4 I’m

just going to go ahead and add 4 and then combine some like terms and get 2n plus 10 and what about dividing that value by two what does that look like

what I’ve simplified it that’s right we have to do it in parts we have to do the

whole thing we have to divide the entire amount so the two end and the 10 both of

them get reduced and get n plus 5 so then my final thing is to subtract the

original number well the original number was n so when I subtract it sure enough

I get the 5 I suspected