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Welcome, students! Today, we're diving into Chapter 1.3, where we'll explore some fundamental properties of whole numbers, particularly focusing on positive integers. These concepts form the building blocks of mathematics, and understanding them will enhance your problem-solving skills.
Chapters
00:00 Introduction
00:27 Introducing the properties
00:47 The Commutative property
04:04 The Associative property
05:59 Additive Identity
08:23 Multiplicative Identity
08:53 Multiplicative Property of Zero
10:43 Is Division of whole numbers Associative
16:25 Applying the properties ( examples)
20:23 Simplifying Algebraic expressions
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0:00
anyway so I'm recalling that so we're
0:02
going to talk about on chapter 1.3 which
0:04
is properties right and then properties
0:07
are pretty fun again we are still
0:09
working on like the simple stuff so it's
0:11
not really that complicated so
0:13
properties really deal with like in this
0:16
case like uh integers and more most
0:19
likely like positive integers okay so
0:22
we're going to talk about a community
0:23
property the associative and also we
0:25
going to talk about the additive
0:27
identity multiplicative identity and
0:29
multipli the multiplicative property of
0:32
zero all right these are pretty easy
0:34
stuff it's just the terminology that
0:37
could be a little bit confusing
0:39
sometimes so now who knows about a
0:41
communative property without looking at
0:43
the board can you tell me what it is
0:45
without looking have you heard about a
0:47
commutative properties without looking
0:49
at the board yeah um I'm not sure uh but
0:54
I think that means like multiple
0:56
different kind of numbers can add up to
0:58
the same answer like different kind of
1:01
numbers can add up to the same answer is
1:03
that quite right uh
1:05
Danielle Abigail yeah what do you think
1:09
it's
1:11
like um if you add something like or if
1:15
you switch it around or if it's like
1:17
just different numbers and it adds that
1:20
as the same switch Yeah okay switch it
1:22
around all right okay that's what I
1:25
meant all right a what do you
1:27
think the community property without
1:30
looking at the board and he answers what
1:31
what does he mean the community
1:44
property you be the same answer right so
1:47
yes Lee um May I write down the
1:49
definitions of the different types of
1:51
properties yeah I am very prone to
1:53
forget them yeah that's fine you can
1:55
write it own am I dealing with yeah you
1:57
you can write it so and this is going to
1:59
be a so you can always go back and watch
2:01
it right so to like she say now the
2:04
communative property it only applies to
2:06
the addition and the multiplication it
2:09
does not apply to anything else but
2:10
addition and multiplication so
2:13
basically the order in which numbers are
2:17
added or multiplied does not change the
2:20
sum or the product you guys TR in some
2:23
ways to explain that okay so what I'm
2:26
saying is this symbolically It Means A
2:28
plus b is the same as B plus a right if
2:32
I say for example Alina plus uh Abigail
2:37
and abalina am I changing anything it's
2:41
the same thing right if I say ab and
2:44
Caleb and Caleb and ab it's still the
2:46
same thing cuz just because I switch the
2:49
way I'm ordering them doesn't mean that
2:52
it's not the same people it's still the
2:54
same people right so a plus b is the
2:57
same as B plus a all right that's the
3:00
community property Community right and
3:02
then it applies to
3:04
multiplication in what ways a * B is the
3:08
same as B * a so 2 * 3 is 6 and 3 * 2 is
3:15
6 right so that does not change anything
3:18
so if you it whether you multiply uh two
3:21
numbers in a in a certain order it
3:24
doesn't change it if you switch it
3:26
around right so a * B is equal to B * a
3:30
and then a + b is equal b + a when it
3:33
comes to addition right and
3:35
multiplication this is also true so we
3:37
have the same thing here 2 + 3 is the
3:40
same as saying 3 + 2 all right 5 + 4 is
3:45
the same as saying 4 + 5 so those are
3:48
that's called a communative property the
3:50
communative
3:59
then the next one that we're going to
4:01
discuss is called the associative right
4:04
the associative property so the
4:05
associative is also an interesting one
4:09
because by definition it means that the
4:11
order in which numbers are grouped when
4:14
added or multiplied does not change the
4:18
sum or the product right so if I have a
4:21
sequence of three numbers right and I
4:24
put a plus b plus C in
4:27
parenthesis I'm going to get the same
4:29
thing if I go a plus b in parentheses
4:31
plus C in other words if I do 2 + 3 + 4
4:37
like this as an example is the same as 2
4:39
+ 3 +
4:41
4 right it doesn't change it just
4:45
because I put a parenthesis here or I
4:47
bracket these two together does not
4:48
change the
4:49
outcome right the way you group them
4:51
doesn't matter right cuz you're going to
4:53
get you going to get the same result so
4:56
again the associative law works not only
5:00
for the addition but it also works for
5:03
the multiplication okay so basically if
5:06
I do two times and I bracket three and
5:09
four together I'm going to get the same
5:11
thing if I group two and three together
5:13
and then times them by four it's still
5:14
going to give me
5:17
24 does that make sense guys all right
5:20
so the associative and the communative
5:23
property they both they apply both for
5:25
the addition and the
5:27
multiplication okay so they apply for
5:46
to and then the next thing we're going
5:48
to discuss
5:49
is there's like three more properties
5:52
that apply both for the addition and the
5:54
modification right
5:56
so the additive identity
6:00
or if you want to shorten
6:03
AI what does AI also stand for who knows
6:07
intelligence artificial intelligence
6:09
right ai ai so but in this chapter AI
6:13
stands for what artive
6:16
identity right and what does it mean how
6:19
would you define this uh let's
6:22
see uh Kinsley you haven't talked this
6:25
one yet so how would you define the
6:26
additive identity based on this syb here
6:30
how would you word to
6:34
somebody how would you like Define this
6:36
let's say you were work with a brother
6:39
like your younger brother if you have
6:40
one right and he goes I don't understand
6:42
the ative identity how would you define
6:46
isal what is this so how would you work
6:51
that um I don't
6:54
know so think about some things all
6:59
right what about you what do you think
7:01
how would you define that in
7:03
words say it
7:06
again say you switch what though what
7:08
number is being added to the
7:10
eight what number is being added to the
7:13
eight what's
7:15
that zero right zero does that look like
7:19
anything else you thought it was
7:22
o
7:24
yeah so how would you word that to
7:26
somebody
7:28
um maybe
7:30
if you add any number to zero it doesn't
7:33
add anything like say I have three
7:36
fingers and then I add zero fingers I
7:39
still have three fingers that's exactly
7:40
right yes this is a little bit off topic
7:43
but if do the weekends count as days
7:46
missing no so there only days in school
7:50
that it counts as right technically em
7:53
so anyway so the addtive identity is
7:55
this if you add zero to any number you
7:58
get the same number right it doesn't
8:01
change it if I have $20 and I add
8:05
nothing to my $20 how much do I
8:10
have $20 $20 right so if you add zero to
8:14
some a number it doesn't change it it's
8:15
still the same number so that's what it
8:17
means right now the multiplicative
8:22
identity is MI right in this case if you
8:25
multiply any number by one what do you
8:27
get
8:30
any number by one yes number one by one
8:34
the same thing right a * 1 is = 1 * is
8:37
equal 8 right 3 * 1 is 3 1 * 3 is 3 if
8:42
you take any number you times it by one
8:44
you're going to get the same thing so
8:46
that's called a multiplicative identity
8:49
okay and then the last one is
8:52
npz is called the multiplicative
8:55
property of zero which means if you
8:57
times any number by Zer you always going
8:59
to get zero so that's pretty much what
9:03
this is okay so now uh the next thing
9:06
we're going to do is we're going to talk
9:08
about
9:10
um division here and I want to ask you
9:13
guys a question let's see if this this
9:15
work
9:19
so can I eras this no no no all right
9:23
can I erase this yes no no no you can
9:26
erase that one can I eras the first one
9:28
all right
9:31
all
9:33
right now notice that all these rules
9:36
only apply for what that we just
9:39
mentioned what kind of operation have
9:42
been all these rules that we just
9:44
mentioned they applied only for what
9:48
multiplication add right they apply for
9:51
addition and multiplication so notice I
9:53
didn't include Division and that did not
9:56
include subtraction right is three - 2
9:59
the same as 2 - 3 no no so the uh
10:03
subtraction cannot be commutative you
10:06
can't do that
10:08
right but now let's do this
10:11
here
10:14
is we are discussing again whole numbers
10:17
is
10:20
division of whole
10:25
numbers associated
10:38
right is division of whole numbers
10:40
Associated what am I asking here
10:43
no is the question making sense is the
10:45
thing is division of whole numbers
10:47
associative no no now
10:50
remember when you say no what do you
10:52
have to
10:54
do yeah explain prove it explain it
10:58
right can anybody here let me go off
11:00
track for a second can you prove that
11:02
God
11:04
exists can you yeah how for the Miracles
11:08
he done in the Bible oh he said because
11:11
of what the Miracles that he's done for
11:13
us in the Bible the miracle that he's
11:15
done for us in the Bible right now
11:17
listen to that now what if I say to
11:20
you that's your Bible I don't believe
11:24
what the Bible says so can you prove
11:27
outside of the Bible that God exists
11:31
yes I can prove how because he changed
11:35
my life okay now listen to
11:38
this I've been given two examples right
11:41
and listen carefully she said because of
11:43
what the Bible said that he did Miracles
11:46
right and then she says because he's
11:47
changed her life that's your life not my
11:50
life so how can you prove objectively
11:54
that God exist yes like the just nature
11:59
like how honeycomb is perfectly slanted
12:02
so the honey can stay in the hexagons
12:04
okay so what if I say happened by
12:10
chance what what if I say that's just
12:12
chance okay that doesn't prove that God
12:15
existed that's just honeycomb and
12:17
whatever fing you're stubborn H finger
12:20
stubborn oh now she call me stubbornly I
12:23
guess now you see this right now how can
12:25
you prove that now here's your homework
12:28
this is just outside you guys going to
12:30
go and
12:31
prove that God exists I don't want you
12:33
to use the Bible I don't want you to use
12:37
I just want you to show me how can you
12:39
prove without an objection that God
12:42
exists that's going to be a homework
12:44
that you guys could do on your spare
12:45
time it's not you don't have to turn it
12:46
in I'm just saying yes yeah you don't I
12:50
mean if you turn in that's fine you
12:52
might get some extra credit but I want
12:53
you to prove to me that God exists yes
12:56
on paper yeah you can write it you can
12:58
you can write that and tell me God exist
13:00
because of again you don't have to do it
13:02
if you feel like you feel like want to
13:03
take on the challenge I want you to
13:05
prove to me that God exist suppose I'm
13:07
an atheist right I don't believe that
13:10
God exists can you disprove the fact
13:13
that God does not
13:15
exist yeah how many paragraphs I don't
13:19
care paragraphs it could be one sentence
13:21
it could be two sentence I just want
13:23
show me that exist anyway so I'm saying
13:27
all this to say that this right is
13:29
division of whole numbers associative
13:31
when you say that no you got to give me
13:34
what something that we call a counter
13:37
example right A C example is how can you
13:40
prove that division of four numbers is
13:42
not associative then you have to prove
13:43
it to me right so basically you can take
13:46
a number right
13:48
I8 right divided
13:52
by
13:55
4 ID by 2 right so my my job is to try
14:00
to prove that this is not equal to what
14:03
8 right /
14:07
by 4 / two because this is the
14:11
associative law if this was to be true
14:13
these two would be what the same right
14:15
but now I'm going to try to prove that
14:17
because this and these are not the same
14:19
therefore division cannot be Associated
14:21
because if it's true for one thing it
14:23
has to be true for everything right and
14:25
if it's false for one thing that means
14:27
it does not hold does that make
14:30
all right so now let's try to prove this
14:33
how would I how how would I do this here
14:35
8 4 2 um four all right so we're going
14:40
to use pmas right so I'm going to go
14:43
from left to right so let's try to do
14:45
this 8 / 4 / two so what's 8id 4 two and
14:53
then what's 2 ID two one now let's try
14:56
the next one right we're going to do
14:58
eight
15:00
/ by 4 / two right so what do we do
15:05
first 4 4 by two right we get what two
15:09
and then that's 8 ID by two that is four
15:13
look at the result different answer yeah
15:15
different answer so therefore we
15:16
conclude that division is not
15:20
associative right so you cannot use this
15:22
for you can use division you can use the
15:24
associative law for division it only
15:26
applies to what multiplication and
15:29
addition right now let me ask you a
15:31
couple questions here I want you to name
15:33
the property that's being shown by each
15:35
statement okay so let's make let's do
15:37
some
15:57
examples for
16:32
so your job is to identify the property
16:35
that is being used here okay so let's
16:38
try and do this
16:39
together see if you guys remember see if
16:41
you took not then you should be able to
16:43
have to know now you can use like uh
16:45
initials if you don't want to write the
16:47
entire thing so the first one here 4
16:51
plus parenthesis a + 3 = a + 3 parth + 4
16:56
what law is being used there
17:00
what property am I using
17:02
here think about it for a
17:05
second
17:08
yes everybody agrees associative
17:11
property is that right you agree
17:14
nor associative all right
17:20
associative we can just call it a right
17:23
to make it easy now the next one 1 * 3x
17:28
=
17:29
3x yes
17:33
keny multiplica identity everybody
17:35
agrees y multiplicative identity right
17:38
multiplicative identity m m i or
17:44
Mi all right Next One D + 0 =
17:49
D you raising your
17:52
hand
17:54
like so what is it yes
17:59
identity that's
18:00
correct AI right Mr yes should we be
18:05
writing down the equations and what they
18:07
mean yeah yeah you should write that cuz
18:10
you want to retain the information yeah
18:12
on paper or um whatever you choose to do
18:16
and then you all know this is available
18:18
so I can send you the link so if you
18:20
feel like I don't want to write this I
18:21
just going to go watch the video later
18:22
that's that's up to you yeah are you
18:24
going to send on goog classroom or yeah
18:26
your CL yeah you
18:29
have 8 * 1 =
18:33
8 uh I want the boys to speak today uh
18:37
Caleb what's that you can just say the
18:39
initial you don't have to say the whole
18:40
thing what is this 8 * 1 =
18:47
8 m i thank you mi I there you go good
18:52
job and then last one 5 * 7 * 2 = 7 * 2
18:59
*
19:08
5 yes
19:10
anybody uh yes
19:13
asso associative um I would say probably
19:18
not but you were Clos what do you
19:23
think community community community
19:27
right
19:30
M because you see here I did not group
19:33
them you see like for it to be
19:35
Associated they have to be group a
19:37
certain way with parentheses there's no
19:38
parentheses so all I did here was just
19:40
switch the positioning you see so this
19:42
is the communative right so you can call
19:44
commu C you can use the letter c for
19:46
that all right
19:51
so let's keep
19:56
moving all so now we're going to learn
19:58
how to
19:59
simplify algebraic expressions
20:03
okay so we're going to learn how to
20:06
simplify algebric expressions
20:30
all right so we're going to learn how to
20:31
simplify stuff like
20:47
this I want you all to try it on your
20:49
own first and then we're going to go
20:51
once you you do it I want to come check
20:53
your stuff and then we can go from
20:55
there so how would you simplify when I
20:57
say simplify what am I saying when I say
20:59
simplify solve it solve it right make it
21:03
more simple this one is a little bit
21:05
more complex but I have 3 + x + 7 and we
21:09
talk about algebraic expressions it has
21:11
numbers and variables right so now I
21:14
want you to simplify this instead of
21:16
instead of leaving it as like 3 + x + 7
21:19
I want you to make it more
21:22
simple don't know what x is you don't
21:24
need to know what x is right you don't
21:26
need to know what x is to make to to
21:28
simplify it yeah you're just trying to
21:29
make it more simple it's still a complex
21:31
operation I have 3 + x + 7 I want to
21:34
make it look simpler yeah 10 10 plus X
21:39
that's 10 plus X right you jump ahead of
21:42
me so how did you get the 10 plus x what
21:44
did you do she said 10 + x so how did
21:47
she do it yeah cuz 7 * 7 + 3 is 10 and
21:52
then you don't know what x is you don't
21:53
know the x is right so what do we
21:55
combine we combine what these are called
21:57
what kind of terms are is like terms
21:59
like terms it's the word like terms
22:02
thank you very much right so basically
22:04
we are using our we are rearranging it
22:06
so this is 3 + 7 + x so these are called
22:10
like terms right these two here are like
22:14
terms right like terms because they look
22:17
the
22:18
same right like terms so 3 + 7 that is
22:23
10 + x now can I combine 10 and x no I
22:26
can't because they're not like terms
22:28
it's just X okay so I'm going to leave
22:31
it as X now if I give you a value for x
22:33
now you can find it but here X is not
22:36
known X is unknown so this is just going
22:38
to be 10 + x does that make
22:42
sense okay so how how would you do the
22:44
next one 8 * x * 5 yes I've already it
22:48
what do you have so I first did 8 * 5 40
22:52
okay so then my answer would be 40 * x
22:54
40 * X right cuz again these are like
22:58
terms right so
23:01
40x 40x okay 40x or 40 *
23:12
X all right so now with that say we're
23:15
going to go to the book and then work on
23:16
some problems
#Mathematics