Solving Quadratics Using Misconceptions

8:33am

I’ve just finished a conversation with a colleague to frantically finish/start planning how the hell I was going to teach solving quadratics to my Year 10s and 11s at 8.40am.

I posted this photo on Twitter the night before the lesson, probably while procrastinating from actually planning the next morning’s maths lesson:

This is a question that I ask myself before I plan any lesson. Why? It helps me to be more critical about how I’m presenting the maths to the students in my class. If I can’t answer this question, my students will struggle to see any purpose in learning maths apart from needing it for a test (an issue that one blog post is incapable of containing). So I started jotting down ways that I could tap into what they’re comfortable with and push it. Then push it some more. Let it slide back into a comfortable state, before pushing it again. Here’s a photo of my notes I took for “Future John”, in case he knew how to make it into a lesson. Thank god he did, even if it was 7 minutes before the lesson.

*bonus points if you can spot an error
**Some Dan fans might notice a familiar problem at the bottom. I pinched this one from his post where he discussed the need for efficiency in computation (read it here).

8.41am

“Yesss, we’re not late! As long as we beat John to class, we’re on time!” Students shout while walking in like they’ve won Gold in the steeplechase.

It was time. I took an extra minute to wait for my low-tech PowerPoint presentation to save as Presentation 1.pptx… This is what I brought to class:

Me: Write down your full working out to solve this problem. What is x? If you can do it in your head, write down what the working out would be… Please don’t call out the answer.

The reason that I wanted students to show their working was to address any misconceptions that they brought to the lesson about using the square root. This is why…

Me: Once you have an answer, place your pen on your paper and fold your arms – so I have an idea about how everyone is going.

Vincent: That was too easy, the answer is 3.

Me: Sure, that’s an answer, but is it the only correct answer? What else could it be? Have a couple minutes to think about another number that might work.

Students looked at me like I had absolutely no idea what I was talking about. Some students tried 4, 5, 6, and 7. Others tried 0, 1, and 2. Not one single student could figure it out. I then showed them my working out and I only had to get this far before I heard the biggest “ahhhh” from the class I’ve  heard in a long time.

I didn’t even need to tell them what the other solution apart from 3 could have been. I wouldn’t have been able to shout over the students explaining to their friends that -3 was the other possible solution. So I gave them some relief by showing them this one:

Students gained confidence by telling me that this one in fact had two solutions: -7 and 7. They told me the answer like they had always known that there could have been two solutions. I now knew they were ready for the next one:

Me: If you’d like to figure this one out in your head, feel free to. Just make you sure that you write down your answer. Arms folded once you’re done.

Atilla: It’s 2.

José: Thanks, Atilla! I didn’t want to figure out the answer for myself anyway…!

Me: Ok, well thanks to Atilla, you might have guessed the answer is 2. What else could it be?

Boom. Maths fight round two had started. In the blue corner are the students who think that if the teacher asks any question at all about an answer, they must be wrong. In the red corner are Atilla and his friends, who think that this is all just a cover up because Atilla yelled out the only correct answer. So I let the fight go for as long as it needed to – when someone asked the question, “What about -1?”

Me: That sounds a bit random, doesn’t it? Try it out, guys, see if it works…

Of course, it did. Students realised that the solution could not only have two possible solutions, but they could be two completely different numbers. I slipped this one in because I didn’t want students to just assume the other solution would always just be the negative version of the first root they found. I had another chance to catch students still doing this in the next one:

Me: Alright, for this one, same process as the previous one – just don’t yell out the answer please.

Atilla: Yeah guys, be considerate!

Once everyone’s arms were folded, I asked for the solutions that people had. The three answers that were given were: 3, 0, and -3. So, I substituted each one in to see if it checked out. First the 3, then the 0. By the time I got to the -3, which many students felt was incorrect, I got them to type it into their TI-84 Calculators. They got 0.

Thomas: I think my calculator is set to radians.

Me: It could be? But that shouldn’t matter. Why is your calculator wrong? IS your calculator wrong?

After letting them argue about which calculator was better, Casio or TI, I explained why they got 0. The calculator was reading -3^2 as -(3^2) or -1*3^2. On a TI, the students would have to enter it as (-3)^2. A great opportunity to place further emphasis on using brackets!

Then I asked them to pick a number from 1 to 10 and had a class poll to find out that most students love the number 7. They then had to find out what their number made this equation equal if it were the x-value:

Atilla: (the only student who picked 9) Easy, 9.

Atilla wrote his response at the top of the board. The others followed with the result below:

After asking them what the other value would normally be, if one was “x”, I wrote up x and y at the top of each column. We then worked from 1 to 10, plotting it out one by one on Desmos. The vibe was intense. Here’s what it looked like:

The main question on the students mind was, “are all of these points correct? It kinda looks like a quadratic…” So, I asked them how we would be able to check all of them at the same time and they begged me to graph the function. This is what it looked like:

From here, we talked about why the equation can have two different x-values for each y-value until the vertex. These conversations were so much better because the students had an invested interest into understanding why people who chose 1 and 6 had the same result.

9.40am

I had essentially used misconceptions to teach students some really important aspects of quadratics that I usually don’t spend much time (if any) addressing. I would really love to hear your thoughts about this lesson.