Background: Before this week, I introduced function notation as we worked with linear equations. On Monday, I introduced the definition of a function, and we applied it to graphs, tables and other situations to see if they fit the bill. Tuesday, we sketched and analyzed graphs — very little precision, just graphing the general shape of my hair over time, or the number of students in school over time, or a family road trip. Wednesday, we did Graphing Stories. Thursday, I introduced a bunch of vocabulary — linear vs non-linear, maximum, minimum, increasing, decreasing, etc — and applied those to a bunch of different piecewise functions. My goal was to give my students lots and lots of examples of what functions are, and make connections between the concrete and the abstract.
The lesson: Friday, we did Function Carnival. If you haven’t tried it, go check it out right now, or try a walkthrough of what I used: go to student.desmos.com, and type in the code 67nx.
Cannon Man: The task is simple: watch this guy get shot out of a cannon, go up, go down, and then parachute the last few (feet?) to the ground. Then, graph his height vs time.
This was surprisingly hard for students. If I did this lesson again, I would change the order to put this a bit later. The general idea — he goes up, then he goes down — wasn’t too complicated, but the fact that the first part of his flight is non-linear, and then it is linear at a much slower rate as he parachutes down, is not intuitive for 8th graders. Then, they got the general idea, but it is a hard function to graph correctly, and it started the lesson off with more frustration than I think was necessary. Here’s a screenshot of the teacher view from the lesson.
Most kids got it, but in part because after a few got very close and most of the rest of kids had the general idea, I used the teacher view to show the “very precise” graphs to start a quick discussion and get kids over the hump and onto the next graph.
Bumper Cars: This is where Function Carnival really makes its money. A bumper car moves on a circuitous path but at a constant speed until it hits another car, and then stops. Students graph distance vs time.
There are so many misconceptions here, and I think that these few minutes of graphing and sharing were the most important of the whole lesson. There is the misconception that the graph looks like this
where students are confronted with the fact that suddenly three cars appear. This is the best illustration I’ve seen of why the definition of a function is important — why it’s worth remembering for students. Here’s what student work looked like as they converged on the correct graph.
After most kids had the right idea, I showed some model graphs again to move them along.
Ferris Wheel: This is another good one for the definition of a function. A ferris wheel moves through one full rotation, and students graph height vs time. Few students fell for the trap and drew a circle, which I was pretty happy about. After that, it was just a question of working through some trial and error to get the best graph. Here’s what the student graphs looked like.
Roller Coaster: The last graph we discussed as a class was the roller coaster. Watch a car run through this roller coaster, and graph it’s height vs time.
This reiterates the function misconception, and also gets at a key idea of graphing something vs time — it’s not how far the car travels at a height, it’s how long it spends there. Here are some student graphs:
What I learned:
- If you do this lesson, create a “Custom Carnival”. The default gives students cannon man, bumper cars, and the ferris wheel. I added the roller coaster, a more complicated ride called the zipper, and a graph of the roller coaster’s speed vs time. While few students would’ve finished the first three working at their own pace, in order to facilitate discussion, I moved many kids along faster — and some kids graphed 5 rides successfully in one class period. Extending the carnival minimized the work I had to do to keep kids productive.
- Discuss! Desmos embeds questions into lesson to push student thinking, but a quick glance at the teacher view showed that many students skipped the questions if I didn’t prompt them to go back, and their answers missed the key points I wanted to get at. While it was a bit of a pain to tear students away from their graphs for a moment, having students articulate the meaning of a function, why certain parts of a graph were linear, non-linear, or constant, and to explain how they found their graph made a huge difference for the whole class’s learning.
- Showing students the teacher view is key. I don’t think it’s beneficial for kids to struggle with one graph for 20 minutes — instead, show them what their peers are doing to scaffold the questions and give them some footholds. Showing all of the graphs at once doesn’t give the whole thing away, but it minimizes the number of students who fall far behind spinning their wheels.
- Desmos has filters in the teacher view for “holes”, “multiple values”, “very precise”, “needs help”, and “points”. I used “very precise” to quickly show the class what the ideal graph should look like while shouting out those kids, and I also used “points” to show how some students plotted a series of points to create a skeleton for the graph. The other filters weren’t as useful — “holes” usually just meant the student wasn’t finished,and students who were basically correct often fell under “multiple values” because they missed erasing a small piece.
- This lesson is awesome. Kids loved it, and there were some great learning moments. I’m looking forward to Water Line coming up next!