One of my favorite lessons to teach middle school students is how to use a graph to relate two quantities. This is usually the first time students have created a graph from scratch to represent a real-life scenario. Many of them have prior experience creating graphs from other math classes, but my experience has been that they have only used graphs within a math context. This lesson seeks to expose students to the reality that characteristics of graphs can have deep and significant meanings when context is added.
The class starts by watching a Dan Meyer video where he swings on a swing set for a few seconds and then jumps off and walks away. The goal is for students to graph the height of his waist over time. The video plays first at full speed and then at half speed to allow students the opportunity to fine-tune their graphs. I play the video for students (they do not play it on their devices because the solution is at the end of the video) as many times as they would like. Once they are satisfied with their graphs, I ask them to give me their pencils. I do this because nearly every student wants to alter his or her graph to match the solution once it is revealed. This would rob us of opportunities for reflection and learning, though, so I take their pencils. I then play the end of the video to reveal the solution.
Most students have constructed a graph that has similarities to the solution. There are always a number of differences, however, and those differences are what lead to a hilarious review. I walk around the classroom and choose some examples to display to the class. Before I do, I preface the conversation by noting that every student had errors in their graphs in some way. It is ok to make errors in mathematics, as long as we learn from them and improve in our craft. Success in mathematics is often built on a mountain of past failures. I tell the students that their shared misconceptions often generate reflection in their peers, which leads to an enhanced understanding of the content. Once they understand that, they are ready to share. Here are some examples of some common misconceptions I have seen over the years:
The “Missing in Time” Graph
This graph is usually created by the students who are the most organized and meticulous. Their approach is to look for the waist location at each second interval since that is what is displayed on the timer on the video. The result is a discrete graph rather than a continuous one. This result leads to a great discussion about what lines and curves truly mean. Lines/curves are a collection of an infinite amount of points, each representing the location of his waist for a split-second of time. The graph must be a continuous line/curve because his waist is always present in the video. If the graph above were true, we would not see him for a vast amount of the video.
The “Charlie Brown” Graph
This graph is the most common. It is very close to being accurate with two exceptions. The first is that there are no curves at the maximums and minimums on the graph. When I ask students why they drew sharp angles, their reasoning is usually that there is one moment in time where he changes direction. I ask them if there is also one moment where the direction changes on the solution graph and after some close inspection they agree that there is. I then ask what the difference is between lines and a curve before and after that change in direction. After further discussion students arrive at the concept that speed is represented by the steepness of the line at any given point. The trouble with the lines is that the speed remains the same both before and after the change in direction. I play the video again for students and they note that he actually slows down (and hovers for a split-second) before changing direction. This results in the curve on the solution graph. I tell my students that if the sharp angle changes in direction were represented in the video, he would have had to have been hit by the Incredible Hulk to get a change in direction that fast!
The second misconception here is the height of the waist after he jumps off the swing. His waist is really at a height of three feet instead of two. I usually joke that the force of the landing must have been enough to cause him to shrink.
The “Most Painful Swing Set Ever” Graph
You would not think this misconception would occur, yet every year I have a student or two draw something similar. The shift downward from the solution graph, I feel, is the result of their past experience with graphs. Just about every graph students are exposed to begins with a y-value of zero. They took the low point of the swing and shifted it to be at zero, not making the connection that zero in this case meant the ground instead of the minimum height of the swing. If they do go below the x-axis, they usually only do it once, normally as a result of drawing their graph too quickly. It is a perfect moment to talk about the math practice of attending to precision. At this point in the class discussion, simply showing the graph starts to elicit a chorus of laughter and groans. It appears the entire swing set has been lowered three feet at the start of the video. Students quickly deduce that the points in quadrant four of the graph represent moments where his waist would have been underground, making this the most painful swinging experience ever! My principal walked in for an informal observation during my display of this graph once and commented how cool it was that our students were finding entertainment value from a graph. I agreed.
The “Thrill is Gone” Graph
I find this graph intriguing. Some students have the misconception that every graph begins at the origin. Using this context makes it easy to show students that this would mean he started by sitting on the ground. The compressed graph is the result of students trying to show that he was swinging above ground, but not being precise as to where the minimums were.
My favorite part is the end. There are two misconceptions here. One is that the waist having a height of zero would mean that he must have fallen out of the swing and is now lying unconscious on the ground. The other is more subtle. I circle the vertical line segment at x = 8 and ask students about its meaning. They say it represents his waist falling quickly. I ask them where exactly was his waist at the eighth second of the video. After a video review, they begin to realize that vertical lines represent instantaneous action. Also, since lines are composed of an infinite series of points, that segment represents that his waist appeared at every height between zero and seven feet above the ground for one split-second of time. That would be a sight to see.
If you are using Discovery Education’s Math Techbook, this lesson ties in nicely with this apply problem. The task features graphing relationships while planning for an extended bike ride over changing elevation. Students will examine relationships between elevation and time, speed and time, and distance and time. Students can use their whiteboard tools to create graphs to present to the class.
Do you have a creative approach to introducing the concept of graphs representing a relationship between two quantities? If so, tell us about it in the comments below!