# Inductive & Deductive Reasoning 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
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
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
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
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