# Digging for deeper mathematical thinking

I recently wrote about using an electronic graphing tool to speed up the mechanics of graphing so that students could make deeper connections about how the content we were covering. The activities that followed were a totally serendipitous foray into some very rich math discussions. Here’s the story…

One of the requirements of the math framework in the school where I teach is that we assess students’ ability to investigate patterns, describe them a general rules and verify/prove that they are true. So, I had students investigate how the values of m and b affect the shape of linear graphs (y = mx + b). Students used an electronic graphing tool in order to generate a variety of graphs quickly in order to make observations. If you’re interested in that part of the story, you can read about it here. If you’re interested in using electronic graphing tools with your class, you should also check out this one, which I only just discovered and is much better than what I used with my class.

In general, students did a great job of seeing that the value of m affects the slope and the value of b affects the y-intercept. They didn’t yet have the language to describe it that way, but they could say things like, “the larger the value of m, the steeper the angle of the graph” and “when you increase the value of b, the line shifts up”. Students were also able to verify their general rule by predicting how a new value of m (or b) would change the shape of the graph and then making the change to confirm their prediction.

Most students, however, struggled to justify why m and b affected the shape of the graph that way that they did. What was is about multiplying the input number by m that made the graph steeper, while adding b made it move up or down? This is a tough question to answer, and requires some pretty sophisticated thinking.

Since my class was a couple of lessons ahead of the other grade 8 class, I decided to go on a bit of detour that would allow students to grapple with some abstract math questions, provoke some debate and (hopefully) get students offering compelling justifications for their opinions. The next lesson, I had students working in small groups to determine whether these statements were true or false:

• Lines with the same slope are always parallel.
• Lines that intersect must have different slopes.
• Lines that have different slopes must intersect
• Lines that intersect must have different y-intercepts.

The statements that were false were easy to disprove. For example, students could disprove the last statement by giving an example of two lines with the same y-intercept and showing that they do indeed intersect (at the y-intercept, specifically). Statements that were true were much harder to prove. Students had to make connections between what they new about slope and y-intercept and what it means for lines to be parallel or intersecting. The result was a very rich conversation that engaged all students in some deep thinking about what they had observed and learned.

What was even more exciting is how their deep thinking continued when we started talking about parallel lines intersected by a transversal. Because students had thought carefully about the nature of parallel and intersecting lines, they were able to determine (after a bit of debate amongst themselves) that parallel lines can be treated as equivalent to each other with respect to the angles formed when intersected by a transversal. While they didn’t know the words for “alternate interior” or “co-interior” or “corresponding angles”, they could explain why alternate interior angles are congruent and co-interior angles are supplementary. All I had to do was give them the vocabulary for what they had already shown.