e. Graph proportional relationships in the form y = kx and describe characteristics of the graphs.
In the previous post, we analyzed the ways two quantities that are in a proportional relationship using a table. Today, we want to look at the idea of analyzing proportional relationships - and identifying features that are unique to proportional relationships - using graphs. So, let's consider one of the relationship we talked about last time:
When you graph this set of data, it will look like this:
Since these quantities are both continuous quantities, we can actually connect the data points using a line (actually a ray):
Let's compare this graph to graphs of three other situations. The first one is the siblings' ages.
The second situation is the candle situation: the length of candles burned and the length of the remaining candle.
The last case is the length and the width of rectangles with a fixed area measurement.
Since the first situation involves the discrete quantities - and since I don't know how to graph the curve for the last one, I am just going to plot these data points.
When you compare these graphs to the graphs of the proportional relationship from earlier, you immediately notice that the graph of the inverse proportional situation isn't a straight line. However, the other three situations seem to result in straight lines. Although it is not really appropriate to use a line to represent the siblings' ages data with a straight line, I'm going to do so to illustrate the similarities and differences - and I'm showing all three lines on the same coordinates.
From these graphs, we noticed that one difference seems to be that the graph of the proportional situation goes through the origin, but not the other two. As it turns out this is indeed unique to proportional situations. The other two cases, constant sum and constant difference situations, result in a straight line. One commonality among the three situations is that the rate of change is constant. Thus, the characteristic of the data sets that are represented in straight lines. I think this might be an idea that is worth discussing explicitly in Grades 7 and 8 when linear equations/functions are studied more formally.
By the way, the fact that the graphs of proportional relationships go through the origin relates to the fact that double number lines we use to represent multiplication and division situations are "hinged" at 0 - in other words, both quantities will go to 0 at the same time. In fact, proportional relationships are assumed in all multiplication and division situations. In middle grades, that fact should become explicit instead of being an implicit assumption.