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# Is side length & area proportional?

CCSS.Math:

## Video transcript

so let's draw a square so it's not perfectly drawn but this is a reasonable attempt at a square and of course one of the things that we know about a square is all of the sides are going to have the same length so if the length of this side is X well all of them are going to have all of them are going to have length X we also know this is a these are going to be right angles that's what makes it a square now the question I want to ask are the lengths of the side of a square proportional to its area and I encourage you to pause the video and think about it are the lengths of a side are the length of the sides of a square proportional to the squares area well to think about that let's draw let's have a little table here so side length side length and that's going to be given as X so that's going to be X and then I'll have another column I can draw that a little bit neater so let me use this tool so there you go and I'm going to make two columns two columns right over here and then over here I'm going to put area area and what's the area going to be well the area is going to be one of the sides squared it's square you could view it as the width times the height so it's going to be x times X or x squared so let's just pick some values for X and then figure out what the area is going to be so if X is going to be equal to one the area is going to be 1 times 1 which is still 1 if X is going to be equal to 2 the area is going to be 2 times 2 which is equal to 4 if X is going to be equal to 3 the area is going to be 3 times 3 which is equal to 9 and I could keep going but I think this is enough these are enough points to think about is the side length proportional to the area or is the area proportional to the side length and one of the ways you can think about proportionality is are the ratios between the side length and the corresponding area always the same so we want to look at the ratio between side length and area or area and side length as long as they're always constant air yeah and side-length it let me do this as two separate and side lengths so let's make an extra column here let's make an extra column here so there we go and let's look at that ratio the ratio of area to side to side length so here the ratio is 1 over 1 1 over 1 which equals 1 that seems reasonable for this next one the ratio of area over side length is 4 over 2 4 over 2 which equals 2 I don't even have to go to this third one I can I could say this ratio is going to be area over side length is going to be equal to 3 and notice I get a different value every time this is not equal to a constant I don't have a constant ratio between area and side length so that tells us that these two that the side length and the area are not proportional not not proportional and if you look at it it makes a lot of sense that the ratio is actually going to be equal to the actual side length that it changes depending on the side length and that's because we're squaring things it's it's literally you're literally taking x squared if you know if you have X here this is going to be x squared and the ratio is going to be x squared over X which of course is always just going to be equal this is x times X divided by X that's always going to be equal to X and we see that right over here but it's definitely not proportional