Today's opening exercise helps to solidify knowledge of the Pythagorean theorem and its connection to the equation of a circle. The first two problems in Old and New Knowledge of Circles involve a radius of 10, and therefore the 6-8-10 triple. In the previous lesson, students looked at a circle with radius 5, so this opening activity extends from that, and students should be able to find all 12 lattice points.
In the next problem, we see that for r = 4 there are only four lattice points, because there is no triple with hypotenuse = 4. I like to list perfect squares on the board if students are having trouble understanding this. I ask them to find two perfect squares that add up to 16 (or whatever value of r2).
For #4, students see that it’s possible to have lattice points on a circle without integer radius, because r2 can be the sum of two perfect squares without being a perfect square itself. Again, list the perfect squares less than r2 on the board if students need help. This circle has 8 lattice points.
Slide #3 is a review slide that may or may not be necessary. I typically assign the Challenge Problem on Slide #4 as something for students to investigate on their own time. I may also post it on a wall as a problem of the week.
Teacher's Note: You don’t really want me to answer this question for you, do you? Here’s a hint: there are two solutions with radius less than 10.
Interactive Discussion: What is Pi?
The multiple choice question on slide #5 activates student knowledge of circumference, and gives me a chance to see which students go straight for the calculator. The correct answer is 8π (choice D), and to the nearest thousandth the value of that is 25.133 (choice C). The key here is to show that choice C is not “wrong”, but it’s also not as precise as it could be.
It’s best when a student asks what would happen if this question were to appear on a test. My answer is that D would be the right answer, but that I would never try to trick you like this. More specifically, if the question asked for the exact value, then D would be the right choice, but then if it says to round to a certain place, then another choice would be the right answer. This attention to wording comes up in some Delta Math modules in which students are practicing with non-integer solutions to Pythagorean Theorem problems.
Slide #6 prompts students to write their current definitions for π on post-it notes, then to stick these to a poster (see definingPi). Over the next few days, students will learn more about what π is and how it was derived (see the Defining Pi Project). It’s always amazing to me how many students feel they’ve never really learned what π actually is, and how satisfied they are when they explore it and “finally get it.”
Circumference and Area Notes
Next, as we continue to work through the presentation I will have students copy and complete the chart on Slide 7 (radius, diameter, circumference and area). Extending from the discussion of the previous two slides, my students copy and complete this chart to review circumference and area. I say to my students, "As per our earlier discussion, you should write numbers in terms of pi and radical, rather than decimals.” I want them to recognize that while this chart is bread and butter for them, there is something knew to think about with respect to writing precise answers. Students should feel like this is simple: just take the diameter and slap a π next to it to get the circumference, and they’ll be happy to notice that the r2 value shows up in each Area, even if the radius was some unsimplified radical like √19.
Formalize a Definition for pi by comparing it to the trig ratios
I didn’t originally run this part of the lesson on Powerpoint slides, but I’ve created some to help guide a discussion. I prefer to run this conversation by having students suggest and draw the equilateral triangle, the square, and the circle, then to discuss the implications of each. The key idea is that the square and equilateral triangle give us a chance to review the two special right triangles, and to reiterate that some ratios are always the same. Because these two shapes are special, there are some especially “nice” trig ratios that work out.
The general rectangle then extends the review and allows us to remember that for any triangle, we can know the ratios of its sides based on its angles. Here, I write out the words sine = opposite/hypotenuse next to a diagram.
Then when it comes to circles, we have another shape that is “always similar.” Just like similarity in triangles leads to the trig ratios, similarity in circles gives us pi. Next to the sine example, I write π = circumference/diameter, and students are thrilled when they think of π this way. Slide #18 reiterates this connection of pi to the trig ratios.
Slide #17 guides us back to the chart from slide #7, we see that this ratio holds very nicely. (Of course, this is because we use the definition of π in the circumference formula, but it’s still satisfying to see the diameter cancel out, leaving π behind.
Completing the Square: An Application
I bring this topic to the attention of the class by saying something like, "So now you know that by looking at an equation you can know where a circle is and its radius. And once you know the radius, you also know diameter, area, and perimeter...so let’s practice!" This sets the stage for some more examples, but rather than simply practice today’s lesson will take one more twist as we move into a review of Completing the Square!
I say to students, "Now that you appreciate the standard form of a circle, we should think about what we’d do it the equation were not yet in that form." I may give them this resource as a handout, but I prefer just to project it on the screen and allow students to make the best possible notes.
This Circles Check In Quiz touches on all parts of today’s lesson. It helps me to assess how students are doing with all the ideas covered today. By seeing this concise three-part prompt, students get the idea that even though today’s lesson moved quickly from idea to idea, everything is coming together.