Comparing proportional and nonproportional linear relationships

0:00- [Instructor] We're told that a stack of four cups is 13 centimeters tall and a stack of eight cups is 21 centimeters tall. The graph shows the height of the stack for different numbers of cups. And then they ask us some questions. What is the rate of change for each cup after the first, and is the height of the stack proportional to the total number of cups?

0:18So pause the video. Have a go at that. Okay, so they did graph some of this for us. So we know that a stack of four cups is 13 centimeters tall. And so we have four cups right over here, and that is 13 centimeters tall. This is the 0.4, 13. They also tell us that a stack of eight cups is 21 centimeters tall.

0:45So let's see, we have eight cups here, and it is 21 centimeters tall. So this is the point right over here, 8, 21. And as we can see, if we had zero cups, well, we could actually, if we want, actually graph. We could start here. Zero cups would have zero height, but as soon as we add that first cup, it looks like we get to seven centimeters.

1:08So if I want, I could draw a line like this. And in reality, we wouldn't be talking about fractions of cups, so we would probably just focus on the whole numbers. But let's just go with this graph for now because this graph actually accounts for even non-whole numbers.

1:26But let's think about these different questions. What is the rate of change for each cup after the first? So the first cup gets us to seven centimeters, so I would argue that that is a seven-centimeter-tall cup. So that first cup right over here, that is seven centimeters.

1:44But each additional cup is not going to add seven centimeters because they're going to stack on top of each other. The next cup is going to look something like this if it was transparent. It's going to look something like this. So we want to think about every other cup.

1:57How much is it going to add? What is this height right over here? What is this height? Well, if we look at this right over, if we look at the graph, we could just try to figure out the slope of the line after the total number of cups is one or for total number of cups greater than or equal to one.

2:15And so we could figure out the slope right over here by just saying, okay, what is our change in height? So we're going from 13 to 21. So we could say 21 minus 13 over our change in number of cups. That is going to be eight minus four, eight minus four. 21 minus 13 is eight, and eight minus four is four.

2:38Eight over four is two. So that means that every cup is adding two centimeters. And we could see that here. If we add one cup, it added two centimeters. If we add one cup, it added two centimeters. So this height right over here is two centimeters. So what is the rate of change for each cup after the first?

2:58So it would be two centimeters per cup. Then they say, "Is the height of the stack proportional to the total number of cups?" Well, for it to be proportional, if you were to double the number of cups, it would double the height. And we could see very clearly that that is not the case.

3:18Here, we are doubling the number of cups. We're going from four to eight, but we're not going from 13 to 26. We're going from 13 to 21, so we are not proportional. Another way to think about it is if we were proportional, this original line right over here, it would have to intersect the x-axis without having a different slope below one like this.

3:44So either way, we can see that it is not proportional.