Determining if slope or y-intercept is reasonable
The narrator says 100 pages times 0.2 gives two extra centimetres (it would be 20), and reads the intercept as 15 cm but the answer choice says five centimetres. Interventions avoid testing the specific slope/intercept numbers.
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A linear model links the number of pages Luca writes to the length of his pencil. Using common sense, the video checks that the y-intercept (the starting pencil length) is reasonable but the positive slope is not, because a pencil should get shorter, not longer, as he writes.
You'll be able to
- interpret the y-intercept of a linear model as the value of y when x is 0.
- interpret the sign of the slope as whether y goes up or down as x increases.
- use context to decide whether a slope or y-intercept is reasonable.
Key words
- linear model
- A straight-line equation used to describe how one variable changes with another.
- slope
- How much y changes for each increase of 1 in x.
- y-intercept
- The value of y when x equals 0.
- negative slope
- A slope where y goes down as x goes up.
Transcript
- [Instructor] We're told a linear model for the relationship between the number of pages Luca writes x and the length of his pencil y, in centimeters, is all of this. Are both the slope and the y-intercept of the line reasonable in this context? Pause this video, have a go at that before we do this together.
And you can look at the choices. So let's first just think about what it's telling us. It's saying, well, common sense, is there a relationship between the number of pages you write and the length of his pencil? Well, assuming he's using that pencil to write the pages, which I think is a safe assumption, the more he writes, the more that he's going to use up the graphite in the pencil, the more that he's gonna have to sharpen that pencil and the pencil should get shorter.
So the pencil should start long and as x increases, the pencil should get shorter and shorter. So let's see what's happening here. So this line right over here, if you think about it, when x is zero, so before he's written any pages, we have a length of 15 centimeters.
That sounds reasonable. I'm literally with my fingers saying, "Oh yeah, for about 15 centimeters, that seems like a reasonable length for a pencil." So the y-intercept actually does seem reasonable. Now, what's strange though here is as x increases, y is increasing by a little bit.
Let's say x, let's say Luca were to write 100 pages. You multiply 100 pages times 0.2, that would imply two extra centimeters. So somehow the pencil is growing as he writes more, and that does not make sense to me. So the y-intercept makes sense but the slope does not.
You should be shrinking. So this should be a negative slope. So I'll go immediately to this does not make sense. No, because the slope suggests that Luca's pencil grows longer as he writes more pages, even though the intercept of five centimeters at x = 0 is reasonable.
Yep, that's exactly what we just talked about.
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