Mathematics

7.RP.A.2d

This is Common Core State Standards Support Video in mathematics. The standard is 7.RP.A.2d. This standard states: Recognize and represent proportional relationships between quantities, and specifically, Part 2d states: Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

First, let’s look at some related standards to 7.RP.A.2d. In the previous grade level, in Grade 6, there’s standard 6.RP.A.3a that talks about plotting the pairs of values on the coordinate plane, and this connects back to the idea of a graph on this 7th grade standard. Students should have experience with the idea of a graph based on this standard back in sixth grade. Also in this same standard, the idea of unit rate in the seventh grade standard connects to this idea of equivalent ratios in 6.RP.A.3a. Now, this standard, 2d, is the last of the sequence of very similar standards at the seventh grade level. The first in this sequence is 7.RP.A.2a that talks about deciding whether two quantities are in a proportional relationship. Then 2b talks about identifying the constant of proportionality in tables, graphs, and so forth. And then 2c talks about representing proportional relationships by equations.

This seventh grade standard sets the foundation for a standard at the next grade level, in Grade 8. Standard 8.EE.B.5 talks about graphing proportional relationships and interpreting the unit rate as the slope of the graph. Now, at the eighth grade level, students will be graphing proportional relationships, but that’s not quite the expectation here in seventh grade. But they are supposed to look at graphs and determine proportional relationships based on the situation. Now, in seventh grade, we’re talking about the unit rate. This lays the foundation for this eighth grade standard that talks about interpreting the unit rate as the slope of the graph.

This eighth grade standard is the first standard where the term slope appears as part of the standard. However, the term slope does appear in Grade 7 in the introduction, and in the introduction it states: Students graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line called the slope. And again, that introduction ties to what’s coming up in eighth grade as far as graphing proportional relationships and interpreting the unit rate as the slope.

Now, let’s look a little bit closer at how this is stated. In eighth grade, okay, no problem: interpreting the unit rate as the slope of the graph. But there’s a bit of an issue with the seventh grade introduction, this whole idea of understanding the unit rate informally as a measure of the steepness of the related line. Well, what is it about this informal interpretation being a measure of the steepness of the related line? If you were to ask students to give a real-life example of slope, in all likelihood, what’s going to happen is we’re going to get examples like this—a hill or a mountain, a roof, a ramp, something dealing with the steepness of something. Now, what happens here is that this leads to the standard English interpretation of slope, and it takes away from the mathematical meaning. Well, as far as a real-life example of slope, well, what about something like $3.15 a gallon of gas or 60 miles an hour?

Now, note that these examples of slope are mathematics, and in this case, they’re also unit rates because we’re comparing to one. We’re comparing to 1 gallon of gas and to 1 hour. If we look at this idea of rate and we add “of change,” well, there’s your definition of slope. Slope is your rate of change—how fast one variable is changing compared to how fast the other variable is changing. If we focus on rate again, but this time let’s look at unit rate, and going back to rate of change being the slope, notice that the idea of rate is what we have in common. So what we have here really is—unit rate is a special case of slope where a comparison is being made to one.

Let’s look at this context. Irene earned $100 in 5 hours. Our rate of change, or the slope, is going to be $100 compared to 5 hours. So, in other words, for every change of 5 hours we have a change of $100. Now, our unit rate in this context would be comparing to 1 hour instead of 5. So what we have to do is simplify this, and in simplified form, the unit rate would be $20 every 1 hour. If we were to plot this context, here are our numbers, and $20 an hour would be our unit rate. Let’s fill in the rest of the values, and now let’s take those values and plot them on a coordinate plane. So here’s our point (1, 20), the point (2, 40), the point (3, 60), and the rest. We note that they are in a linear pattern, so we connect the dots, and we have this line for our graph.

Let’s look at the first part of the standard: explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation. Well, here, any point (x, y) on this line is going to be a solution to how much money is earned at a unit rate of $20 per hour. So, for example, if I worked 3 hours I would get $60. What about the point (0, 0)? Oh, it’s not in our table, so let’s include that also because that is where this line begins. Let’s take this same idea, but let’s use some different values. In this case we have points such as (1,15), X being 2, Y being 30, and so forth, and if we plot these, we plot our point (0, 0), the point (1, 15), the point (2, 30), (3, 45), and the rest. Connect these two, and here’s our linear graph.

Now, here’s the graph of the previous context; same type of idea, hours worked and money earned. It was just a different unit rate. Now, the standard says to pay special attention to the point (0, 0). Okay, so we have these two contexts, but what if we had some others? So here’s an additional example. Here’s another. But we do see a pattern. It seems like they all start at the point (0, 0). So, a conclusion here is that graphs of all proportional relationships will include the point (0, 0), also referred to as the origin.

Now let’s look at the point (0, 0) in terms of tables of values. Notice here that in all of these contexts, all of these tables, the point (0, 0) is something that’s in all of them. So, again, all proportional relationships will include the point (0, 0). Let’s look again at this point (x, y). This is the first context that we looked at where someone is making $20 per hour. Now, if we put this in fractional form, y over x, that, as we already figured out, is our slope. Now, of course, we can’t include the point (0, 0) because we cannot have 0 in the denominator. Let’s take our values, the first being 20 over 1, then 40 over 2, and then the next one, x is 3, and y is 60. Then we also have the point (4, 80) with 80 being on top for the y value, 4 being on the bottom for the denominator, our x value, and then 100 over 5, and 120 over 6. Notice that these are all equivalent fractions.

Let’s look at that other context. Here, the initial point, 15 for our y value, 1 for our x value, that’s our unit rate in this context. Then we have all the other points, and again, notice that these are all equivalent fractions. So, in all of these contexts, y over x was our slope with the unit rate being something that was always compared to one. So, in the first context, it was $20 an hour for the unit rate. The second was $15 per hour for the unit rate, and the third example was $10 per hour as our unit rate.

Now, the standard talks about r being the unit rate in that point (1, r). A unit rate is typically defined as the ratio of two measurements where the second term is 1. Now, notice we have a ratio of two measurements, so there’s some dissonance here because the single letter r is supposed to represent the ratio of two measurements. The problem here is that a variable typically represents one number, but a unit rate is actually a ratio, again, of two measurements. Well, let’s see. The slope of a proportional relationship would be y over x, and the unit rate of a proportional relationship would be y over 1, where again, the x value is 1 and then you have whatever the y value is. But again, a unit rate is a situation where we’re always comparing to one. So, the point (1, y) is going to represent the unit rate of a proportional relationship. So, if r is a variable that represents one number, then there’s a little bit of a problem with how the standard is stated.

When we look at this standard where it says the point (1, r) where r is the unit rate, it might be a little bit better for students to really understand this as saying, well you have the point (0, 0) and where the point (1, y) determines the unit rate. This might be a little bit better interpretation as far as students understanding what the standard says and not getting confused with this idea of just r being the unit rate. Again, what happens here is that if r is the unit rate, for the above to hold true, well r has to be interpreted as being a ratio, not a variable that represents a single number.

Now let’s go back and look at graphs and this idea of unit rate. For this first example, notice that we have the point (1, 1). So that is the slope, but it’s also the unit rate since the denominator is 1. The second example—the slope here is 3 over 1. This third example, the slope is 5 over 1. If the slope is y over x, then the unit rate has to be compared to one, so the unit rate would be whatever the y value is compared to one. So in all of these cases, we actually do have the unit rate represented because all of our denominators are one. And again, notice for each of these three, the x value was one and then you had whatever the corresponding y value would have been.

Let’s look at this example where the slope is 5 over 4. Now, the unit rate would have to be where our x value is 1, but we have a little bit of a problem here. Well, any number can be expressed as something over 1, so we could take the slope of 5/4 and just put that all over 1. So our unit rate would be 5/4 all over 1, which would correspond to the point (1, 5/4). It might be a little bit clearer to students if we express this as a decimal. So, here the unit rate would be 1.25 to 1.

Let’s look at this example where the slope is 4 over 7. Our unit rate has to be something compared to one. If we convert 4/7 to a decimal, that’s .571428, which is okay. It’s a little bit of a long decimal, but we can see that it’s a little bit more than 1/2 compared to 1. Again, we can take any number and put it all over one, so here the slope could also be expressed as 4/7 all over 1, and this is okay although again, it’s a little bit of a mess because we have a complex fraction. But again, our unit rate, the x value is going to be one. In this case the y value turns out to be a fraction, 4/7. But in a real-life context, the unit rate is actually a lot more complicated. So, it seems like in some contexts, using the slope itself may be better than using the unit rate because the unit rate might turn out to be a pretty sloppy number.

Let’s look at our standards for mathematical practice. The first four: if we do some of these activities, students would be reasoning abstractly and quantitatively. If we had them working in groups, they would construct viable arguments and critique the reasoning of others. Looking at the last four, students would be attending to precision, and they would look for and express regularity in repeated reasoning.