- All right,
we're going to start off. First of all,
this is a little warm up problem.
So I want you to read this question and answer it.
It says which costs more per pound according to the graph,
peas or cabbage.
And don't just give me a one-word answer,
but also I want you to explain how you know which one it is.
So use just a few seconds to do that and then we're going to go over it.
You can just add that to your previous bell work.
Very good, very good.
- Now I see a several different answers,
but that's okay because I want you to be able to explain
why you said what you said.
How many think it's the cabbage that costs more per pound? It's okay,
don't change your answers, it's okay.
And why did you say that, Parker?
Why did you say you thought the cabbage was more per pound?
- Because it's sloped more towards the cab bage than the peas on pounds.
- So it's sloped more towards the pounds is what you said? Okay.
Someone said that the peas cost more. Tran,
why did you say the peas cost more?
- I said the peas cost more because no matter where you are on the graph,
the slope is higher on the cost range.
- Okay, what do you mean by the slope?
The slope was higher.
- How diagonal the line is because if you go to the very tip,
the pea slope is higher than the cabbages.
- Okay. It's actually it is the peas.
It is the peas because here, what axis is this?
- X-axis.
- The X-axis. So the X-axis is talking about pounds.
As pound is increasing, what's happening to the cost?
It's also increasing on both of these,
which means if the pounds and the cost for the cabbage
is here at that same poundage,
the cost for the peas is what?
- Higher.
- It's going to be higher, okay?
Everybody understand that? Okay.
Yesterday we talked about graphing linear equations.
Who can remember what were some of the ways
that we can use graphing linear equations?
What did we do? We just start graphing?
What did we do to graph linear equations? Ryan,
what did we do to graph linear equations?
- We use some tables.
- Okay, we use tables,
function tables. What else could we use on that?
- We use ordered pairs.
- Ordered pairs. What does the ordered pairs tell us,
Valen?
- It tells us like the number of the X
like what number it's going to be on the X and Y-axis.
- The X and the Y-axis, very good.
So we use the order pairs from the tables to graph that.
What else did we talk about?
- Fractions.
- Fractions, okay.
Did we use fractions to graph though?
What did we talk about with the fractions?
- They were slope like slope is a fraction.
- Slope can be used as a fraction.
Why can it be used as a fraction?
- Because slope is a number over a number and so it's rise over run.
- Okay, so it's a ratio,
which is a fraction. Okay,
very good.
So today we're going to elaborate on that a little bit more,
and we're actually going to use a task
to investigate right of change in slopes
so we're going to take it just a little bit further
and we're actually going to use triangles,
similar triangles. What do I mean by similar?
What do I mean by similar? Tran.
- They're proportional.
- They're proportional. What does proportional mean?
- That if you enlarged it, it would be the same as the other triangle.
If you shrank it, it'd be the same as the other triangle.
- Okay, very good explanation.
So we're go ing to see why similar triangles is going to help us explain
-why slope is the same between
any two distinct points on a non-vertical line.
Show me with your hands what does verti cal look like.
What does vertical look like?
Show me with your arms. Is it like this or is it like this?
What's vertical? So it's straight up and down.
So anything that's non-vertical,
which means it's not going to be straight up and down,
we can see how the distance between
any two points on that non-vertical line is going to be the same.
- Then from that,
we're going to derive the equation
Y equals MX for a line that goes through the origin.
Who can tell me where the origin is? Where is the origin?
Taylor, where is the origin?
- Right where the X and Y-axis meet right in the middle.
- So that order pair is going to be what?
Zero-zero.
- Zero-zero, very good.
So at the origin it's just going to be Y equals MX
and for a line that's not through the origin,
it's going to be Y equals MX plus B.
How many like music? How many like music?
I like music.
What are some ways or places that we use music or see music or hear music?
What are some different places that we would have music?
- In your car.
- In your car, just riding around listening in your car.
What's another place?
- In an elevator.
- In an elevator, when you're sitting in an elevator,
you hear doodoodoodoodoodoo. Taylor,
what's another one?
- At home.
- Just at home, okay.
What were you going to say?
- At a dance.
- At a dance. Sometimes we go to school dances.
- When you go shopping, a lot of stores play music.
- Very good. At stores,
a lot of times they'll have music. Well,
we're not going to talk directly about music,
but how many of you all buy CDs sometimes?
Anyone buy CDs,
okay? So we're going to look at how,
on a graph, how we can relate the price of CDs,
and they're not always the same price,
but we're going to just give an average price of a CD
versus how many CDs you may purchase or you may have.
So we're going to use that to do a little investigation,
and in our investigation today we're going to have
several different segments.
First when I pass out the task that we're going to work on,
I'm going to give you a few minutes of private think time.
That just means that for the first few minutes,
for five minutes, I just want you to kind of look at the sheet,
answer the first few questions,
and just kind of see what you can think about on your own.
- And then after we have that private think time,
we're going to then get in our small groups that you're already in
and you're all going to discuss what you've already found on your own,
and then you're going to further finish that worksheet.
And then, and I don't have this on here,
but from that small group work,
we're going to add ten more minutes
and that's when I'm going to give you a chance
to put your findings on the larger posters
that you're going to share with the whole group class
at the end that we'll discuss in the last ten minutes,
okay.
- Are there any questions right now
as far as what we're going to be doing today?
Okay.
First, I'm going to go ahead and pass this out to the groups
and I just want you to go ahead and put your name and date on this.
Just make sure everybody in the group gets one.
All right, on this task is called Music is a Must.
It says the data shown on the graph below
reflects average cost of a music CD.
So you're going to take this information that's on the graph,
just whatever you gather from it,
and you're going to answer the first few questions.
- All I want you to do right now in your private think time,
I want you to actually just work on our answer letters A through D.
So right now in these first few minutes
you're just doing letters A through D.
But when you're finished with A through D,
go ahead and just to yourself read the other few questions
just so that you already have an idea
of what we're going to work on as a group.
So, but for the first five minutes just doing A through D right now.
- What's it mean by what?
Unit rate.
Unit rate? By saying that if I find the rate of something
and I want to know what the unit cost is.
How many am I trying to figure out?
For how many?
For per pound.
Per pound or per item, exactly. So unit just means one, exactly.
Okay, if you think it's that much,
what makes you from the graph figure that?
Because you said you think it's that much per CD?
What from that gives you ,
what information from the graph makes you think that it's $15 per CD.
So explain that.
- Yes, the cost on the left side that is in dollar amounts.
So the cost is in dollars and that is on your Y-axis.
- Would it be that or would it be each
would be fifteen dollars because two would be thirty?
- Okay, so if you say two is thirty then how do you get the unit price?
Like either show or explain that. You're on the right track.
You're on the right track. But isn't that the same way,
all the way up?
Because you chose 2 and 30, but could you have chosen 105 and 5?
I mean 7 or could you have chosen 120 and 8?
You see what I'm saying?
Yes sir?
- How do you find the unit rate? Like do you do like 30 divided by 2?
- Okay, if you're finding the unit rate and I said that I wanted to see,
if I had some apples,
I want to know how much it was per pound.
That means I want to know how much for how many pounds?
- One.
- For one. So the unit would be how many?
- One.
- One so if, but you're saying 2 and 30, could I have chosen 75 and 5?
Or could I have chosen 120 and 8? Now you're saying yes.
So how would I know that?
What tells me that yes I could choose any of those prices
or any of those amounts?
- How much one's equal to? The unit rate for each of them?
- Okay, the unit rate is what you're finding.
Okay. I like how you put several ratios on there.
That's good. That's good.
All right you have about a minute and 20 seconds
just on your personal think.
If you have any questions or stuck on something,
I'll be happy to help you.
- I don't understand unit rate.
- Okay, let's say that I were buying some apples
and I wanted to know how much it is per pound,
that's the unit, right?
So what am I trying to find out? How much it is for how may pounds?
- One pound?
- For one pound. So if I'm trying to find the unit rate of CDs,
I'm trying to find out how much it is for how many CDs?
- One.
- For one, exactly.
- Oh. Okay.
- But how can you derive that or come up with that
based on this whole? Okay.
You're on the right track. Because for instance,
if I look at this graph, how much is 4 CDs?
60 dollars.
Okay 60. How much is 6 CDs?
- Ninety.
- How'd you know that because there's not a 90 on here?
- Because it is in between 80 and 100.
- Okay, so you went from 60,
then to 75 and then 90. So how did you come up with that?
You looked at the chart, but what's happening each time?
- It's going up by 15.
- Okay. All right,
now what I want you to
because I'm seeing some really good information on there.
There are a few questions and that's okay.
You are all on the right track that I want you to do in your small groups,
first of all discuss A through D that you just did on your own
just to make sure that you're all on the right track.
If someone found it a different way does it mean that they're wrong? No,
not necessarily.
So just kind of come up or discuss what your different findings were
and see if you still came up with the same solution with the same answers,
but then I want you to go ahead and take it a little further
and look at the next several questions on that same sheet. Okay?