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2.1.2 - CPM CCA | Robert Anderson | YouTubeToText
YouTube Transcript: 2.1.2 - CPM CCA
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Video Summary
Summary
Core Theme
This lesson focuses on understanding and calculating the slope of a line, exploring how changes in variables affect steepness, and relating this to the concept of growth rate and starting values in functions.
hey everyone it's Mr Anderson coming um
to you again um we are in our second
section of section 2.1 Chapter
2.1 um chapter 2 first lesson in our CPM
Common Core algebra book and our goal
today is to do the Deep dive on slope
and we are going to talk about what
makes slope change right how do I
measure that steepness and um you know
what what might have an impact um on
that so what things can we manipulate
and toggle in order to make a line
steeper a line less steep Etc um so
previously we
determined um the the the growth rate
and starting value using those tiles in
our last lesson if you need to back up
and look at look at 2.1.1 again go ahead
and do so uh and we made a table and an
equation in this lesson we're going to
talk about um manipulation of that slope
so uh our our questions that we want to
focus on today what makes a line steeper
what makes a line less
steep all right how is growth related to
steepness and then where do we find that
starting value um where's it located uh
on a line where's it located in a table
etc etc so we're going to look at those
things today and we've got in um 21 this
is our first core problem that we're
going to start today um we've got any uh
a table that's representing a uh a function
function
um um in in in down here on the right
hand side and we're going to go ahead
and write the equation of that and I
need to know two things I need to know a
starting value and I need to know a
slope and you saw this if you watch the
function video you saw me set up things
using um maybe something you've seen
before our slope intercept form our y mx
plus b format and the way that it's
easiest to do this is maybe
consider your M and your b as blanks and
these are things that we're going to
have to fill in and the information is
given away in the table uh or on the
graph and and we'll look at graphs today
as well and what I'm looking for in this
particular case is how is this growing
now the first thing I'm checking is my
figure number my dependent variable uh
I'm sorry my independent variable my x's
those particular things are growing by
one and that's good because if that's
the case if I go from 0 to 1 1 to two 2
to 3 etc etc I'm counting by ones it
also means that my denominator my slope
is going to be um one and it means my my
slope uh can maybe very easily be
evaluated as a whole number and we'll
we'll get to this uh a little in a
little greater detail as we look at
graphs in a moment but um take special
note that we're counting by ones in the
figure number here next thing that we're
going to go ahead and do is we're going
to look at the yv values the number of
tiles that we might have and here
because I'm counting by ones is where
I'm looking for that change so the
question that I would ask is what am I
going up by there I count from two all
the way up to seven what am I what am I
climbing by and is that the same as I
count from 7 to 12 and 12 to 17 and 17
to 22 and the answer is yes and and what
is that we're growing by five we adding
five each
time now here's where I said hey we're
we're counting by ones here right um so
this is maybe adding one each time some
of you guys remember that your slope is
your rise over your run your um change
in X over change in y so if you want go
ahead and write that here right 5 over
one is the same thing as five and I'll
just make a little note hey that's the
exact same thing I'm trying to star that
there that's the exact same thing as 5
over one I just wrote it as a um an
integer instead of a rational expression
but then the next thing I need is the
starting value this is back up to that
idea like in figure zero okay in figure
Z how many tiles did we start with and
that's answer will by looking at the
table everybody check this out with me
figure Z make my highlighter a little
thicker figure zero is right here and
how many tiles do we have to get
started and that answer is two now you
might be questioning you might say but
Anderson I don't know I don't know what
this looks like I don't know what this
this this tile pattern looks like and
honestly that's okay right what we're
going to learn pretty quick is if if
we're going to consider these as tiles
it doesn't really matter what it looks
like as long as we understand how we're
growing and in this case we start at two
we're growing by five each time we've
got a pretty unique function here and we
can represent that using the table we
can represent it using the growth um in
or I'm sorry the the tiles we can also
represent it using this thing right here
called an equation and that's going to
work out pretty well and note I just
referenced in Prior learning something
we probably saw in pre-algebra
and that is that yal MX plus b format
and I plugged in m and b okay in this
particular case you might already know
the answer to this but it's saying hey
does it appear to be a function if so
write this equation in function notation
if not explain why well two things does
every single input have an output and
the answer is yes in this particular
table and that's all I can see right now
in this particular table grab a
different highlighter
here in this particular table I can see
that every every single one of these X
values is sent to a yvalue and that's
good that means we are onto right that
means we are
covering all right we're covering those
X's but here's the other thing is every
single x value sent to a unique yvalue
and and for sure I don't have um X's
being sent here to um multiple I'm sorry
multiple X is being sent to the same y
so in that particular case based on the
evidence we see in the table we do have
a function so I'm just going to go down
here and I'm going to say yes this is a
function All Right Now function notation
is going to look very very similar to
yal MX plus b notation so we've got yal
mx plus b we already discussed in the
prior problem that's yal 5x + 2 now
here's what function notation looks like
slight difference all right I'm going to
get rid of the Y and I'm going to write
F and then in
parentheses x and what that means ladies
and gents that right there that little
bit of you know symbolism that means
this particular equation is a
function of X so when you see that we
call that F ofx and we're we're we're
abbreviating that is a function of X Now
function notation has its advantages
function notation tells
people hey I can put something in and I
can very very quickly
evaluate and see what I'm going to get
out okay okay here's here's an example I
can say for
example I can put in two instead of uh
putting in an X here I can put in two
and and what is that going to look like
in this particular problem well I'm
two and I'll do a little bit of math 5 *
2 is 10 two I'm sorry 10 + 2 is 12 so in
this particular case when I put in 2 I
get out 12 now does that check out sure
it does check out back up here when I
put in two
I got out 12 and there's an advantage of
function notation that we don't
necessarily have as cleanly in slope
intercept form or y mx plus b notation
let's keep on keeping on we're going to
go ahead check out a couple graphs down
here in 2-13 all right describe how the
pattern grows and how many tiles are in
figure zero h x represents the figure
number and Y represents the number of
tiles write an equation that relates X
and Y and then decide if it's a function
here's what I'm going to go and do I'm
going to do a and I'll let you guys
pause and Tackle B and C okay so let's
check out a together and I'll zoom in if
you have this um technology available to
you um go ahead and use it otherwise
sketch this in quickly on a a sheet of
graph paper uh it's going to work pretty
pretty easily there but in this
particular case as I'm checking out a I
find it helpful with a graph this is not
an equation anymore not yet anyway I I
find it helpful to to draw in something
called a slope triangle and a slope
triangle is simply just going to
identify two dots on your line two dots
that cross through nice grided points
like the ones they identified here and
talk about how you move from one dot
usually the leftmost dot up and over to
the the rightmost dot and in this
particular case it's clear that I had to
count up two boxes and I had to count
over one box now that's pretty telling
because that means for me my slope is 2
over one and we've got this thing we'll
develop here in a moment called you know
rise over run changeing y over changeing
X but my slope is two over one up two
and over one okay the other particular
thing I'll need is my starting value and
my starting value I'm going to choose
just a different color for this my
starting value is where I cross the Y
AIS and that point is specific because
that's when X is zero right that's my
zero figure right so when X is zero
where am I on the the y- axis and in
value is three right because on this
line right
here this line crosses right here at
three and that really tells me
everything I need to know y equal blank
x plus blank oops not blank that was
there we go y equals blank x plus blank
and now I can fill these items in y
equal my slope is 2 over 1 that's
2 and plus 3 here's the other thing all
right I'm going to have in this
particular case a function and some of
you might be questioning hey Anderson
why do you got a function I'm going to
do my darnest to just tack in a vertical
line here there's my vertical line and
I'm going to throw out there I'm going
to claim that I can use the vertical
line test and I can take that line and
now I can
drag this particular vertical line
everywhere I want over the graph line
not the axes but the graph line and does
it ever touch the graph line in more
than one spot or in two or more spots
the answer is no so we would say yes
this thing is a function for sure yes
and if you were ever
asked to to to prove that or validate
that you would say something like it
passes the VLT or the vertical line test
that's how you could you could justify
that claim okay go ahead and pause this
video now and try B and C but ladies and
gents if you're ready to keep on keeping
on with me you can go ahead and uh we
can talk about one more uh question here
in uh in 214 um we got a similar uh very
very similar in fact um uh problem we're
going to be using a slope triangle to
investigate um um you know that growth
rate and we're going to then go ahead
figure out how we can take this down
into a unit rate um I don't really care
about the equation in this case I don't
I can't see the starting value but I can
see that slope triangle so let's let's
go ahead and and and and check this one
out um that that slope trial triangle
like I said we we'd start here and we'
go here we'd move from that left point
to that right point and it looks like
here we go over three right that's this
plus three and you know the scale might
confuse you a little bit but we're told
we go up
27 very very important we put the rise
on the top the rise over the run now um
the slope in this particular case we
could identify as 27 or up 27 plus 27
over 3 and if you left your slope that
way it'd be fine um you know for every
three you go over you go up 27 or for
every up up every 27 you go up you over
three but most of you guys are going to
realize that 27 is splitable into three
pieces and so you can also write your
slope in a reduced
form like either one of these things
right you could say Okay 27 divid by 3
hey that's nine right calculator will
validate that for you or you could
reduce it um by taking a factor of three
out of top and bottom and rewriting as 9
over one and that tells you that in this
slope triangle there's going to be three
other little itty bitty slope triangles
where you go over one and of nine okay
and the slope triangle with a base of
one right answers this particular
question right here how many tiles are
added each time the figure is increased by
by
one you now have a unit rate right and
that's that's going to become very very
important as as we get um later on in
our our chapter 2 but that's a that's a
pretty important skill and if you're not
um 100% convinced how I got that just go
ahead and fire 27 divid by 3 and
calculator and check that out it's not
always going to be a whole number but in
this particular case it is right um last
thing I'm going to ask you guys to try
uh and goof around here goof around with
is going to be something like this all
right in 215 we've got a a tool that's
uh in this problem and I'm just going to
kind of expose you to this tool and then
I'm going to talk about review preview
real quick um the the tool requires
something called Desmos and if you go
into our
ebook our ebook has this in here
and it's got a tool that's built in here
and it allows me to adjust Alpha and
beta now Alpha and beta are simply you know
know
um uh what's it called uh changes in y
and changes in X right um nope I'm sorry
not changes there we go okay and they're
they're really going to allow you to
manipulate the the the steepness of the
the line and and check out some of these
things that's happening first of all
it's always going to give you the slope
which is kind of a big deal um but see
if you can use that to answer these
questions here about this particular
graph and and maybe you don't even need
that maybe you can just do some some
counting there right to answer or tackle
2-15 all right real quick um because I'm
running out of time let's talk about our
review preview here our review preview
is uh right here at the bottom 19 22 and
24 uh folks thanks for watching have a
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