There are many ways to teach about linear equations, right. You can (a) use the old t-table approach; (b) you can draw a line and then figure out its slope and y-intercept; or you can (c) first explain the slope-intercept formula and then explain how the equation aligns with the formula.

However I would contend that all three of those approaches leaves students somewhat baffled.

The (a) t-table approach makes it appear that points just appear because there’s some machine (the equation) that spits out points in a rather random way, and then these points (the ones on the lines) are just special points that we have found, and perhaps the only ones that are on the line.

The (b) technique gives the impression that — I don’t know —every line just has these two weird properties, slope and y-intercept, and you are simply finding them for this line. While this is true, it does not feel inherently interesting to students, in my experience, partly because slope and y-intercept seem so different from each other. Why would a line necessarily have these two properties? There’s nothing obvious about why a line would have these two properties and no other properties.

The (c) technique is little better than the (b) technique. Students are still left wondering why a line would have these two properties: slope and y-intercept. The two properties seem unrelated to each other and not necessarily essential to the nature of a line.

So is any approach better? Here’s an idea. Get kids thinking about lines by thinking about the relationship between the x- and y-values of the points on the lines.

Start out with this line: y = x. Point out that y = x is in a sense an equation that sends out a call, much as a town squire might shout out a call in a village square for certain people to assemble. While the town squire might be calling for certain men (or women) over a certain age to assemble, the equation y = x, on the other hand, sends out a call for certain pairs of points to assemble, specifically, the pairs of points whose y-values are equal to their x-values, since y = x. Ask students what kind of points those would be. So if the x-value of such a point is 3, what would its y-value have to be? Answer: it would have to be the y-value that is the same as the x-value, so the y-value would also need to be 3. So the point called to assemble would be this point: (3, 3).

Similarly, if x = -3, the associated y-value would be what? Answer: -3. So this point called to assemble would be: (-3, -3). And if x = 0, the associated y-value would be what? Answer: 0. So this point would be: (0, 0). So when these three points are assembled, you would have these points standing bravely on the cold, coordinate plane: (3, 3), (-3, -3), (0, 0). Tell students to graph these three points. When they do so carefully, using a ruler, ask them to name some other points that appear to stand on this same line. They should report that this line also contains points such as (–5, -5), (6, 6), (9 9), (5,5), even (0.5, 0.5), etc.

Students should gradually come to see that when they graph y = x, they wind up graphing the infinitely many points whose y-values are exactly equal to their x-values. And thus they should see that this is NOT a coincidence. They should make the connection that the line, y = x, calls forth the ordered pairs whose y-values are equal to their x-values! This insight is the fundamental concept behind this lesson.

What is so amazing is how many students do NOT realize this most fundamental fact. I have been tutoring students in algebra for 25 years, and I find that when I explain this rather fundamental fact to students, 9 times out of 10 students are amazed.

So I cannot state how important it is that we help students make this connection: there is a relationship between an equation and the coordinate points that the equation generates.

Once we help students see this for y = x, we should move forward in small steps. I will show how I do this in the next post.