0:02 hey everyone it's Mr Anderson coming um
0:05 to you again um we are in our second
0:09 section of section 2.1 Chapter
0:12 2.1 um chapter 2 first lesson in our CPM
0:14 Common Core algebra book and our goal
0:17 today is to do the Deep dive on slope
0:19 and we are going to talk about what
0:22 makes slope change right how do I
0:26 measure that steepness and um you know
0:29 what what might have an impact um on
0:31 that so what things can we manipulate
0:33 and toggle in order to make a line
0:37 steeper a line less steep Etc um so
0:38 previously we
0:41 determined um the the the growth rate
0:43 and starting value using those tiles in
0:45 our last lesson if you need to back up
0:48 and look at look at 2.1.1 again go ahead
0:50 and do so uh and we made a table and an
0:53 equation in this lesson we're going to
0:56 talk about um manipulation of that slope
0:58 so uh our our questions that we want to
1:01 focus on today what makes a line steeper
1:03 what makes a line less
1:08 steep all right how is growth related to
1:11 steepness and then where do we find that
1:15 starting value um where's it located uh
1:17 on a line where's it located in a table
1:18 etc etc so we're going to look at those
1:22 things today and we've got in um 21 this
1:23 is our first core problem that we're
1:26 going to start today um we've got any uh
1:29 a table that's representing a uh a function
1:30 function
1:34 um um in in in down here on the right
1:35 hand side and we're going to go ahead
1:37 and write the equation of that and I
1:39 need to know two things I need to know a
1:41 starting value and I need to know a
1:42 slope and you saw this if you watch the
1:46 function video you saw me set up things
1:49 using um maybe something you've seen
1:51 before our slope intercept form our y mx
1:53 plus b format and the way that it's
1:56 easiest to do this is maybe
1:59 consider your M and your b as blanks and
2:00 these are things that we're going to
2:02 have to fill in and the information is
2:05 given away in the table uh or on the
2:06 graph and and we'll look at graphs today
2:09 as well and what I'm looking for in this
2:12 particular case is how is this growing
2:14 now the first thing I'm checking is my
2:17 figure number my dependent variable uh
2:19 I'm sorry my independent variable my x's
2:21 those particular things are growing by
2:24 one and that's good because if that's
2:26 the case if I go from 0 to 1 1 to two 2
2:29 to 3 etc etc I'm counting by ones it
2:32 also means that my denominator my slope
2:36 is going to be um one and it means my my
2:38 slope uh can maybe very easily be
2:40 evaluated as a whole number and we'll
2:42 we'll get to this uh a little in a
2:44 little greater detail as we look at
2:46 graphs in a moment but um take special
2:48 note that we're counting by ones in the
2:50 figure number here next thing that we're
2:52 going to go ahead and do is we're going
2:54 to look at the yv values the number of
2:56 tiles that we might have and here
2:57 because I'm counting by ones is where
2:59 I'm looking for that change so the
3:01 question that I would ask is what am I
3:03 going up by there I count from two all
3:06 the way up to seven what am I what am I
3:08 climbing by and is that the same as I
3:12 count from 7 to 12 and 12 to 17 and 17
3:14 to 22 and the answer is yes and and what
3:19 is that we're growing by five we adding
3:21 five each
3:24 time now here's where I said hey we're
3:27 we're counting by ones here right um so
3:30 this is maybe adding one each time some
3:32 of you guys remember that your slope is
3:35 your rise over your run your um change
3:38 in X over change in y so if you want go
3:41 ahead and write that here right 5 over
3:42 one is the same thing as five and I'll
3:44 just make a little note hey that's the
3:46 exact same thing I'm trying to star that
3:48 there that's the exact same thing as 5
3:50 over one I just wrote it as a um an
3:52 integer instead of a rational expression
3:54 but then the next thing I need is the
3:56 starting value this is back up to that
3:59 idea like in figure zero okay in figure
4:02 Z how many tiles did we start with and
4:03 that's answer will by looking at the
4:05 table everybody check this out with me
4:08 figure Z make my highlighter a little
4:11 thicker figure zero is right here and
4:14 how many tiles do we have to get
4:17 started and that answer is two now you
4:19 might be questioning you might say but
4:21 Anderson I don't know I don't know what
4:23 this looks like I don't know what this
4:25 this this tile pattern looks like and
4:27 honestly that's okay right what we're
4:29 going to learn pretty quick is if if
4:30 we're going to consider these as tiles
4:32 it doesn't really matter what it looks
4:34 like as long as we understand how we're
4:37 growing and in this case we start at two
4:39 we're growing by five each time we've
4:41 got a pretty unique function here and we
4:44 can represent that using the table we
4:46 can represent it using the growth um in
4:48 or I'm sorry the the tiles we can also
4:51 represent it using this thing right here
4:52 called an equation and that's going to
4:56 work out pretty well and note I just
4:57 referenced in Prior learning something
5:00 we probably saw in pre-algebra
5:02 and that is that yal MX plus b format
5:04 and I plugged in m and b okay in this
5:06 particular case you might already know
5:07 the answer to this but it's saying hey
5:10 does it appear to be a function if so
5:12 write this equation in function notation
5:16 if not explain why well two things does
5:18 every single input have an output and
5:20 the answer is yes in this particular
5:22 table and that's all I can see right now
5:25 in this particular table grab a
5:26 different highlighter
5:29 here in this particular table I can see
5:30 that every every single one of these X
5:34 values is sent to a yvalue and that's
5:37 good that means we are onto right that
5:38 means we are
5:40 covering all right we're covering those
5:43 X's but here's the other thing is every
5:46 single x value sent to a unique yvalue
5:50 and and for sure I don't have um X's
5:53 being sent here to um multiple I'm sorry
5:55 multiple X is being sent to the same y
5:57 so in that particular case based on the
5:59 evidence we see in the table we do have
6:01 a function so I'm just going to go down
6:03 here and I'm going to say yes this is a
6:06 function All Right Now function notation
6:08 is going to look very very similar to
6:12 yal MX plus b notation so we've got yal
6:13 mx plus b we already discussed in the
6:16 prior problem that's yal 5x + 2 now
6:18 here's what function notation looks like
6:21 slight difference all right I'm going to
6:24 get rid of the Y and I'm going to write
6:25 F and then in
6:28 parentheses x and what that means ladies
6:30 and gents that right there that little
6:33 bit of you know symbolism that means
6:37 this particular equation is a
6:42 function of X so when you see that we
6:44 call that F ofx and we're we're we're
6:46 abbreviating that is a function of X Now
6:48 function notation has its advantages
6:51 function notation tells
6:54 people hey I can put something in and I
6:57 can very very quickly
6:58 evaluate and see what I'm going to get
7:01 out okay okay here's here's an example I
7:02 can say for
7:05 example I can put in two instead of uh
7:07 putting in an X here I can put in two
7:08 and and what is that going to look like
7:10 in this particular problem well I'm
7:16 two and I'll do a little bit of math 5 *
7:20 2 is 10 two I'm sorry 10 + 2 is 12 so in
7:24 this particular case when I put in 2 I
7:26 get out 12 now does that check out sure
7:28 it does check out back up here when I
7:30 put in two
7:32 I got out 12 and there's an advantage of
7:34 function notation that we don't
7:37 necessarily have as cleanly in slope
7:40 intercept form or y mx plus b notation
7:42 let's keep on keeping on we're going to
7:44 go ahead check out a couple graphs down
7:46 here in 2-13 all right describe how the
7:48 pattern grows and how many tiles are in
7:51 figure zero h x represents the figure
7:53 number and Y represents the number of
7:56 tiles write an equation that relates X
7:58 and Y and then decide if it's a function
8:00 here's what I'm going to go and do I'm
8:02 going to do a and I'll let you guys
8:06 pause and Tackle B and C okay so let's
8:08 check out a together and I'll zoom in if
8:11 you have this um technology available to
8:13 you um go ahead and use it otherwise
8:15 sketch this in quickly on a a sheet of
8:17 graph paper uh it's going to work pretty
8:19 pretty easily there but in this
8:22 particular case as I'm checking out a I
8:24 find it helpful with a graph this is not
8:27 an equation anymore not yet anyway I I
8:29 find it helpful to to draw in something
8:31 called a slope triangle and a slope
8:32 triangle is simply just going to
8:34 identify two dots on your line two dots
8:37 that cross through nice grided points
8:39 like the ones they identified here and
8:41 talk about how you move from one dot
8:44 usually the leftmost dot up and over to
8:46 the the rightmost dot and in this
8:48 particular case it's clear that I had to
8:51 count up two boxes and I had to count
8:54 over one box now that's pretty telling
9:00 because that means for me my slope is 2
9:02 over one and we've got this thing we'll
9:04 develop here in a moment called you know
9:06 rise over run changeing y over changeing
9:08 X but my slope is two over one up two
9:11 and over one okay the other particular
9:12 thing I'll need is my starting value and
9:14 my starting value I'm going to choose
9:15 just a different color for this my
9:18 starting value is where I cross the Y
9:21 AIS and that point is specific because
9:23 that's when X is zero right that's my
9:25 zero figure right so when X is zero
9:28 where am I on the the y- axis and in
9:36 value is three right because on this
9:37 line right
9:40 here this line crosses right here at
9:42 three and that really tells me
9:44 everything I need to know y equal blank
9:49 x plus blank oops not blank that was
9:51 there we go y equals blank x plus blank
9:53 and now I can fill these items in y
9:56 equal my slope is 2 over 1 that's
10:01 2 and plus 3 here's the other thing all
10:03 right I'm going to have in this
10:06 particular case a function and some of
10:07 you might be questioning hey Anderson
10:08 why do you got a function I'm going to
10:11 do my darnest to just tack in a vertical
10:14 line here there's my vertical line and
10:16 I'm going to throw out there I'm going
10:19 to claim that I can use the vertical
10:23 line test and I can take that line and
10:24 now I can
10:28 drag this particular vertical line
10:31 everywhere I want over the graph line
10:33 not the axes but the graph line and does
10:36 it ever touch the graph line in more
10:39 than one spot or in two or more spots
10:41 the answer is no so we would say yes
10:44 this thing is a function for sure yes
10:46 and if you were ever
10:50 asked to to to prove that or validate
10:53 that you would say something like it
10:56 passes the VLT or the vertical line test
10:58 that's how you could you could justify
11:01 that claim okay go ahead and pause this
11:04 video now and try B and C but ladies and
11:06 gents if you're ready to keep on keeping
11:09 on with me you can go ahead and uh we
11:12 can talk about one more uh question here
11:17 in uh in 214 um we got a similar uh very
11:20 very similar in fact um uh problem we're
11:22 going to be using a slope triangle to
11:26 investigate um um you know that growth
11:29 rate and we're going to then go ahead
11:32 figure out how we can take this down
11:35 into a unit rate um I don't really care
11:37 about the equation in this case I don't
11:39 I can't see the starting value but I can
11:41 see that slope triangle so let's let's
11:43 go ahead and and and and check this one
11:45 out um that that slope trial triangle
11:47 like I said we we'd start here and we'
11:49 go here we'd move from that left point
11:51 to that right point and it looks like
11:54 here we go over three right that's this
11:57 plus three and you know the scale might
11:58 confuse you a little bit but we're told
12:00 we go up
12:05 27 very very important we put the rise
12:08 on the top the rise over the run now um
12:10 the slope in this particular case we
12:14 could identify as 27 or up 27 plus 27
12:16 over 3 and if you left your slope that
12:19 way it'd be fine um you know for every
12:21 three you go over you go up 27 or for
12:24 every up up every 27 you go up you over
12:27 three but most of you guys are going to
12:30 realize that 27 is splitable into three
12:34 pieces and so you can also write your
12:36 slope in a reduced
12:39 form like either one of these things
12:41 right you could say Okay 27 divid by 3
12:43 hey that's nine right calculator will
12:45 validate that for you or you could
12:47 reduce it um by taking a factor of three
12:49 out of top and bottom and rewriting as 9
12:53 over one and that tells you that in this
12:56 slope triangle there's going to be three
12:58 other little itty bitty slope triangles
13:01 where you go over one and of nine okay
13:04 and the slope triangle with a base of
13:07 one right answers this particular
13:09 question right here how many tiles are
13:11 added each time the figure is increased by
13:12 by
13:14 one you now have a unit rate right and
13:16 that's that's going to become very very
13:19 important as as we get um later on in
13:21 our our chapter 2 but that's a that's a
13:24 pretty important skill and if you're not
13:27 um 100% convinced how I got that just go
13:29 ahead and fire 27 divid by 3 and
13:30 calculator and check that out it's not
13:31 always going to be a whole number but in
13:34 this particular case it is right um last
13:36 thing I'm going to ask you guys to try
13:38 uh and goof around here goof around with
13:41 is going to be something like this all
13:45 right in 215 we've got a a tool that's
13:47 uh in this problem and I'm just going to
13:49 kind of expose you to this tool and then
13:50 I'm going to talk about review preview
13:53 real quick um the the tool requires
13:55 something called Desmos and if you go
13:56 into our
13:59 ebook our ebook has this in here
14:02 and it's got a tool that's built in here
14:05 and it allows me to adjust Alpha and
14:07 beta now Alpha and beta are simply you know
14:09 know
14:12 um uh what's it called uh changes in y
14:17 and changes in X right um nope I'm sorry
14:19 not changes there we go okay and they're
14:21 they're really going to allow you to
14:25 manipulate the the the steepness of the
14:27 the line and and check out some of these
14:29 things that's happening first of all
14:30 it's always going to give you the slope
14:33 which is kind of a big deal um but see
14:36 if you can use that to answer these
14:38 questions here about this particular
14:40 graph and and maybe you don't even need
14:42 that maybe you can just do some some
14:44 counting there right to answer or tackle
14:47 2-15 all right real quick um because I'm
14:49 running out of time let's talk about our
14:51 review preview here our review preview
14:55 is uh right here at the bottom 19 22 and
14:58 24 uh folks thanks for watching have a
