Circle Lattice Points

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Objective

SWBAT lattice points on a circle given the center and radius of the circle.

Big Idea

Instead of telling students the equation for a circle, developing it themselves enables students to make the connection between circles and the Pythagorean Theorem.

Warm-up

30 minutes

Investigation and New Learning

30 minutes

First of all, today’s investigation should begin with students taking time to finish developing their “methods” from yesterday. Yesterday’s lesson focused on testing whether or not a given point lies on a circle—the task of today’s lesson is to generate points that do lie on a circle. I frame the transition this way: if I don’t give you points, how you can find points that will lie right on the edge of the circle?

These lesson transitions can be a bit difficult, because I find that my students are not ready to transition at the same time. I know that teachers are trained often to have all students transition to new activities at the same time, but I have found that the learning experience is much more fluid for students if they transition when they are actually ready to transition. In the beginning when you and students are just getting used to this, it can feel a little bit like you are running around like a chicken with its head cut off.

At the beginning of this lesson, once I feel confident that most students understand the first three problems on the warm-up, I tell students:

The challenge today is to finish developing your method from yesterday, and then transition to a different task that asks you to actually find points on the circle. Let me know when you think you have produced a high quality explanation of your method and then I will get you started on the next task.

This puts the onus of directing the learning onto the students: they need to understand the purpose of both tasks, they need to be able to assess their own understanding and the quality of their product, and they need to decide when they are ready to transition. At its core, this way of teaching by putting the onus of directing the learning on the students, incorporates MP1 more authentically: they can only direct their learning if it makes sense to them.

My guiding vision for today’s task is this: the less I tell students, the more they need to figure out on their own. They may need feedback and scaffolding to help them figure stuff out, but my goal is for them to succeed with as little feedback and as little scaffolding as possible.

In order to accomplish this objective, my students will need to make some connections:

  • To find lattice points, treat the radius of the circle as the hypotenuse of a right triangle
  • If the square of the radius is the sum of two square numbers, the roots of these numbers can be the sides of the right triangles
  • Adding or subtracting the lengths of these sides from the coordinates of the center point will yield lattice points on the circle

I will ask students the questions below to help them make these connections.   

  • How can you use right triangles to help you with this?
  • Which part of the right triangle is the radius?
  • How can you find possible side-lengths of the triangle?
  • How do the side-lengths of the triangle help you find the coordinates of lattice points?
  • Does the center of the triangle matter?

My process for using these kinds of questions is that I ask one or two of these questions to a pair or group of students, and then I tell them:

Think about those questions and see what you can figure out. If you are still confused, then I will try to explain it better the next time I come around.

This is another way that I put the onus of learning on the students: I am giving them some help and I am also communicating to them that I hold them accountable for thinking. I have learned over the years that students are much more likely to take risks (i.e., try out ideas that they aren’t sure about) if they trust that more help will be available to them once they extend themselves. Also, on my next round of circulation, if students haven’t made progress, I can always say, “I need you to show me some evidence that you thought about the last hint I gave you, or if you aren’t sure, then you need to tell exactly what does and doesn’t make sense to you.” It is important that to insist on this, because otherwise students are going to just wait until you give them more help, which is the exact opposite of perseverance.

Closing

10 minutes
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