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Linear Relationships

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okay so we're starting module for right now and module for is going to start off with the topic of linear 6 linear relationships and number relationships that make a straight line okay so let's talk about feet feet feet feet no not like feet like that I like feet like feet and inches kind of feet okay so every foot  includes 12 inches we all know that so feet and inches have a number relationship and  if you the ratio of it you would say that there are 12 inches to a foot that was a rate to be specific and if you took that rate and you made a chart when we say that for every one foot there's 12 inches for every two feet there's 24 inches and for every 3 ft there Reese X engines  and if we said that the number of feet is at Sandy number of inches is why then we could actually go ahead and put these values on a graph like you could say one foot and 12 inches into foot and 24 inches and I make this graph really tall to fit 3ft in 36 inches but you get the idea here this is a linear relationship because although cops are going to be in a straight line okay know what would the slope of this line be well doesn't it go up 12 time because you're increasing the number of inches by 12 each time so wouldn't the slow B12 over one up 12 / 1 up lower one up 12 / 1 yeah slope in a linear relationship is really the same as rate interesting and that's why we call a linear relationship linear relationship because it has the same slope it has the same rate and when the slope is the same between every two points it makes a straight line know when you have a linear relationship you can take the X and the Y and you make an equation out of it and notice here that slope is the same as rate and slope is the same as the constant of proportionality just like the Y equals KX equation form that we used before except now every y equals 12 times x and that 12 is the slope and it's the rate of change and it's the constant of proportionality so really slope and rate of change and constant of proportionality what is different names for the same concept interesting and this happens in the type of relationship do we call a linear relationship I want you to try going back to our good old quarters and dollars problem you know there are four quarters for every dollar so if you have you have four quarters right  you can go ahead and finish up this table and tell me how many quarters do you have if you have $2 $3 in $4 and then I want you to go ahead and graph those values out and then I want you to answer this question when you graph this out what do you notice about the slope of the line that you got compared to the ratio of quarters to dollars and is linear relationship  alright well speaking of linear relationship let's take a little play can talk about the Orange County Fair it's fall now but remember back in summer when you got to go eat some funnel cakes and Rob some pig races yeah the Orange County Fair Good Times  it probably cost you about $15 to get into the Orange County Fair but you don't I just get in the gates at the Orange County Fair because you want to go on the rides and stuff but to go on the rides you can't just pay $15 to get in the gate you got a bite and I don't remember exactly how much tickets cost for let's say they cost like $2 well then you going to have to pay another $2 to go on more rides and then another $2 to go on more pretty soon you paid $15 and you paid a whole bunch more cuz you bought a whole bunch tickets to go do a whole bunch of extra stuff besides just getting my day well when you go to the fair we could make an equation for the cost the cost of going to the fair is $15 but 15 players + 2 times $2 for every ticket that you buy so two times the number of tickets so one variable in this scenario what is the cost will it cost to be different depending on who you are and what you do when you get there  the other variable is how many tickets are you going to purchase you're in the fairgrounds so what we can do is whenever you have these kind of two variable equations we can make this serious linear relationship and any linear relationship you can put into a table and we can just make it table that tells us the cost Bay when the number of tickets so if you buy a certain number of tickets what's your cost going to be well if you don't buy any tickets you're just going to pay 15 bucks to get in the door you're going to water around the fair and maybe you'll have a good time and maybe you won't and maybe you'll want to buy one ticket well that one ticket means you're going to pay two more dollars so now 217 and if you buy it a second ticket two tickets are $4 plus the 15 to get in the door so that's 19 bucks and if you buy 3 tickets sets two more dollars in that so you're at $21 I think you get the pattern here is this a linear relationship yeah cuz we cannot only put that relationship in a table format we can put it on a grass and notice if you put this on a graph where money is how do I access and the number of tickets is on the x-axis then all the little dots are going to be in a nice straight line so this is Sir play a linear relationship because it makes a straight line  okay but watch what happens if we try to see if this is a proportional relationship doesn't work does it look at the cross products is 0 * 17 the same as 1 x 15 sing even though it's a linear relationship it's not a proportional relationship it's non-proportional why is it not proportional is 0 * 17 is 0 interesting even though there's a pattern here and even though there's a pattern here it's still non-proportional and the reason that is non-proportional and I'm just going to let you in on a little secret here is the reason is because it doesn't go through the or the 0 0 point on the graph okay so you can have linear relationships that are non-proportional and they would be non-proportional if they don't start off in 00 for instance with the fair it doesn't start off at zero zero because you just started 15 to just get in the gates  but if you have that inches to feet relationship 0 ft 0 inches and that's a linear relationship that also happens to be proportional because it does go through the zero PowerPoint so how do you know if a linear relationship is proportional or non-proportional well if a line does go through the origin the 0/0 then it is a proportional relationship but if a line does not go through the origin it is a known proportional relationship okay you might want to write that down because I'm going to ask you to do another practice problem right now let's go bowling shirlyn's it costs $3 per game but you got to get your bowling shoes first so that's two bucks right off the bat plus $3 for every game you bowl so it's 3 * x + 2 hopefully you see where that equation comes from $3 per game plus $2 to rent some shoes now and I want you to do is I want you to make a little table a little chart I want you to tell me what's it going to cost you to Bowl One game What's going to cost you the ball two games or three games are for games and then of course I want you to I fit out and I want you to see if the relationship between these numbers ends up being a linear relationship and if it is linear is it a personal relationship  let me know what you find out and we will talk about linear relationships and proportional relationships in class tomorrow alright is that is it Sia

Uploaded by Jeff Marbut.

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