Teaching Conics in Algebra 2

I really like opening the day with an open question. They’ve been kind of easy to think of so far. But, what about conics? What’s a good group of open questions that can be used with conic section lessons? Before I could think of that, I really had to look at the new standards for Conics. In doing that, I realized I really hadn’t examined exactly what the kids are supposed to learn. So, I had to research and think about that for a while first.

I did a little research about conic section topics and standards that need to be covered in Algebra 2. I checked this publication on the California Dept. of Education website: California Common Core State Standards for Mathematics. You’ve probably all seen it, if you’ve been working in California.

Here’s the conics standard for California – yes, there’s only one, but it’s loaded (p. 83):

3.1 Given a quadratic equation of the form  ax² + bx + cy² + dy + e = 0, use the method for completing the square to put the equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola and graph the equation. [In Algebra II, this standard addresses only circles and parabolas.] CA

Um… okay. Let me think about that. first of all, is this the same or different for what we’ve been doing at my school for teaching conics. Do we need to address the directrix and focus or foci? Because, I talk about those whenever I talk about conics. Even when introducing them in Geometry.

Here are the Geometry standards (p. 74):

Translate between the geometric description and the equation for a conic section.

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

2. Derive the equation of a parabola given a focus and directrix.

In the past, I taught about the foci of an ellipse and parabola and hyperbola. Last year, we didn’t test on graphing hyperbolas. According to the Algebra 2 standard above, it seems like it’s all four, except for that part in the brackets. Is that only for other States? I needed to find out. After all, I’m serving on a County wide committee to discuss teaching this class and we should really know what those apparently contradictory statements mean. The bold is supposed to be California.

So…. here’s what I found out… Go to this website IXL, scroll down to whatever standard in which you are interested, hold your cursor over the standard and a sample questions will pop up. Wow. Great stuff. No foci/directrix stuff until Pre-Calculus. Okay – I guess they’ll handle that in Pre-Calc. Looks like just graphing parabolas, those that open horizontally or vertically, and graphing circles. For circles, be able to find the center. I think they still need to be able to tell whether the conic is a circle, ellipse, parabola or hyperbola from the equation, though.  Please make a comment below if you understand this all to be different than what I’m writing here. This seems much less than what I’ve taught in the past. So, maybe that’s a good thing. 🙂

Next, what will the lessons be? Then, I checked the NCTM website, Teachers Pay Teachers the NRICH websites for ideas related to those standards. And, of course, Desmos. Well, on the day I had to start the topic, I didn’t have a good ‘open question’ opener. I just asked kids about the equation of a parabola, in vertex form. I asked about the equation of a circle. I put it on the board. I asked if they’d seen that before. I asked them to notice that there’s a x-squared and a y-squared term. I asked if that meant it’s a function. So, it was a weak start compared to what I would have liked. But, it was the day before break and my goal was to introduce conic sections. I had them watch this Conics video from YouTube This was pretty much a vocabulary lesson with a graphic that was pretty good for getting them to understand the basic concept of what conic sections are.

Then, the fun began…  as they started to use the Desmos activity, Polygraphs: Conics, found here. Now, I’m figuring out my unit plan. I have the week off. I plan to find open questions, interesting activities and relevant homework for them. Something that spirals old stuff, too. I plan to write more about it, too….  Ideas?

 

 

 

 

Asking Open Questions, why it matters for math

I’ve been asking open questions and I think you should too. Maybe you already are. If so, please reply to this blog with some of yours so we can share! 🙂

The other day, I started my Algebra 2 class by asking students to think of 10 numbers. I know I’m not a mind reader, but if what they say is true and 90% of communication is non-verbal, then I was 90% reading their minds…

I knew they had easily thought of their numbers as I continued to give instructions, “draw a number line and try to think of numbers that would represent the number line… .” Their faces fell as they had to rethink this. They were once again happy and satisfied when I demonstrated and started to draw my number line and scaled it -5 to 5 and started to plot points above the number line at -3, 1, etc. I asked them to think of maybe some negative values, maybe a decimal or a fraction, maybe a mixed number… I asked them to try to think of numbers that represent numbers they’ve learned about over the years.

They all started to write down new numbers. So, at this point, one of the great things about open questions was that everyone was involved, not just the kids who could ‘figure it out’. Everyone could think of numbers. Everyone understood a number line and negatives and fractions and decimals – this was an algebra 2 class, after all. This was feeling easy for them. They great thing is that they could all have different numbers and all be ‘right.’ Satisfying.

So, even better, they were hooked. I asked if there were any other interesting values they could think of. One student said, “Pi!” That was perfect. I marked it on my number line on the board. Many kids marked theirs. I said what about 2π? or -π? or 1.5π? Any others? Wait…. nothing, that’s okay, I’d leave radicals for another conversation.

Then I asked, “What is my number line missing? What am I not including?” They had to think. I asked about my domain. They said it was very small. I agreed that I had only included small numbers. Then I asked, who picked the largest number? Hands went up. The largest ended up being only 100, the smallest -100. I asked if we should have larger numbers or infinity symbols somewhere.

Anyway that wasn’t the point, but it was fun. So, then I marked a new number on my number line, at about 2.7. I asked if anyone new what the value was. They shouted out guesses, finally I heard 2.7!. I said, yes, that’s very close! Then I wrote the expression lim as x→∞ of (1+1/x)^x on the board, over the point. From there, we talked about limits, what happens as x gets larger, we made a table of values and tried larger and larger numbers for x, only to see that y was changing less and less and moving towards, 2.718···. Then I said this is a special value, just like π, and it’s called the natural number, e

I’ve introduced and derived e with my students many times over the years. The difference this week, was that they were all hooked, all involved, they all had money on the table (intellectually speaking). Everyone was writing and thinking. You could see it from the expressions on their faces.

In Algebra 2, asking open questions feels so important to me. It’s a tough class to teach. There’s a lot of material and a short amount of time. There’s a wide range of students in the room – varied grades, varied backgrounds and varied attitudes towards learning math. Open questions allow for many entry points which generate stronger feelings of success and inclusion.

The new standards seem to want us to go deeper than we have in the past and students need to think more critically. Engagement needs to be high. Asking open questions really helps them engage and think and stay tuned to see what happens next. Even if what happens next is a bit of traditional lecture and on to some problem solving. They enter it with more curiosity and more confidence.

I’ve been doing more open questions this month and I’m seeing a change in the culture in the room. I have to thank the great educators and researchers who introduced these ideas to me at CMC-North this past December. Steve Leinwald in particular, Dan Meyers and Michael Fenton and the dream team at Desmos. My teaching has transformed (see It All starts to Gel…  http://wp.me/p73p86-2)

Resources for me have been:

  • NRich website
  • The book: More Good Questions, Great Ways to Differentiate Secondary Mathematics Instruction, by Marian Small and Amy Lin  link to amazon
  • The book: Styles and Strategies for Teaching High School Mathematics, by Thomas, et al., 2010 link to amazon

 

It all starts to gel…

I’ve been reading books, articles and blogs. I go to training meetings and conferences and see presentations from experts with great ides. I am always inspired to try new things. So, I do. But, they don’t stick. Finally, I’ve found the right combo, the right approach, the right attitude.

I am exploding with ideas. But, I’ve finally found a formula – that I don’t intent to stick to all of the time, because that will cause problems too, in terms of engagement.

I am starting the day with an open question. To get kids to enter where they are. For example, yesterday’s question was “How many solutions are there to the equation  x+1 = x^2-1  ?” They worked in pairs and also had to answer, “How do you know?”

These questions allowed for multiple entry points, and was an effective and engaging differentiation strategy. Some students used guess and check and found one or two solutions. Some students combined like terms across the equation and either factored or analyzed the discriminant, much to my delight. No one graphed. I knew what everyone had done because I had time to circulate, discuss methods and ask questions that moved kids into engaging more deeply with the problem.

For the students who analyzed the discriminant, I asked them if they could then find the solutions. This was puzzling for them at first. For those using guess and check, I asked, “how do you know when you have them all?” They went back to the drawing board and asked other students what they did. It was great to watch.

Using a “You, We, I” strategy, thanks to CMC-North conference and Steve Leinwald’s presentation, I then brought the class together and tell them I saw three great strategies used. We discussed the merits of each. Then decided which might be best for that particular problem. This was the ‘We’ part. We started with the ‘You’ part, where they generated their own ideas and tried to articulate their reasoning.

Then came the ‘I’ part. I asked them what they would do if the functions were higher degree. I asked them which of the strategies would still work. I also asked them what they would do if the expressions weren’t factorable. I asked them if they’d like to learn a method that would work every time, for any two expressions, from an isolated constant term to a higher degree polynomial or other more complicated function. Of course, they were hooked and interested.

Thank you, Desmos! I graphed and displayed the functions on the overhead. They could easily see the intersection of the linear expression, x+1, and the quadratic expression, x^2-1.

Here’s the handout to students, Desmos Intro and Parabolas.

From there we went into the lesson. More You, We, I. Then into a short homework assignment. This is the 2-4-2 idea (not perfectly executed by me) that I learned about from the same Steve Leinwald presentation. I also asked for feedback. Today, when the kids came in and had to  turn in the sheet, one student let me know she really appreciated it. I can wait to read their feedback. I would have done it then and there, but, y’know, lots going on in a room of 28 Algebra 2 students.

The lesson itself was really a summary of the structure of both the standard and vertex forms of the equation of a parabola. It was meant to introduce them to Desmos and better demonstrated what a, b, c, h and k, do in the equations. I wish I had known about teacher.desmos.com and the calculator with so many pre-existing awesome tools. If you are not aware of Desmos and it’s many wonderful capabilities, you have got to yourself educated on Desmos. It’s easy to learn. The team there seems to be a great group of people, thinking up wonderful activities. Next week, the kids will use Marbleslides for rational functions to review graphs of rationals. I can’t wait.