Can the Dog Reach the Bone? Determine whether a Point Lies within a Circle
Lesson 1 of 10
Objective: SWBAT describe the connection between the distance formula and square areas, by rediscovering the Pythagorean Theorem. Given the coordinates of the center of a circle and its radius, SWBAT determine whether a point lies on a circle, in a circle or outside of a circle.
This lesson is going to throw students for a loop. They have gotten used to doing math with numbers, graphs, and equations. This Warm-Up does not look like that.
In my class, there are always students who come in to this lesson and say, “Yay! I loved geometry!” There are also students who come in to this lesson and say, “Oh, we are doing geometry now? I never got this stuff, I’m not doing it.” To all students, I say:
We are going to start looking at the overlap between geometry and algebra and functions. If you loved geometry, that’s great. If you hated geometry, don’t worry because you don’t need to remember ANYTHING from that class in order to be successful. We will start from basic big ideas and work our way up, so as long as you can THINK and ask questions, you will be fine.
I find that most often the students who shut down are people who didn’t really understand geometry and come into this class scared. They are afraid that they are missing some secret and important knowledge so they give up before getting started. The upcoming lessons are set up in such a way that even if students know almost nothing about geometry, they can work through the lessons to develop the ideas. On the other hand, for students who remember a lot from geometry, the lessons require more justification and a greater depth of understanding than they might already have, and they also have very challenging extensions built in.
SO—what is today’s warm-up all about?
The big idea is that if students can find the areas of these squares, then they can find the lengths of their sides using a square root. This seems like a simple idea, but it is in fact the idea at the root of the Pythagorean Theorem.
I love this warm-up because it starts with such a simple question: “Is this a square?” From this simple starting point, students to develop the Pythagorean Theorem. As long as you don’t say the words “Pythagorean Theorem,” even students who can already rattle off “” probably won’t make the connection right away. So, ask all students to work through the warm-up to ensure they fully understand.
For advanced students, the explanation questions are essential, though it isn’t essential that students write comprehensive answers. These are the questions to really focus on when you circulate, especially with students who seem to be whizzing through this. Consider this sample dialogue:
T: “How do you know that those are squares?”
S: “The sides are all the same.”
T: “How do you know the sides are the same? Is that enough information to prove that they are squares?”
S: “The sides look the same… yes it is enough information.”
This student does not yet understand the big idea here. First, how do we know all the sides are the same? The goal is for students to identify that the sides of the square are all hypotenuses of congruent right triangles. They might lack the language to explain this, so I will probe their thinking and say something like: “How could you use triangles to prove that this is a square?”
For students who rush through this warm-up, my goal is to push them to justify their thinking using precise language (MP3, MP6). For other students who have struggled in the past with the geometry, the purpose of the warm-up is really to just help them make sense of the ideas, even if that takes a little more time (MP1).
I want everyone to understand the problems on the first page of the warm-up. And, in tomorrow’s follow-up warm-up, I want all students to fully understand problems (4) and (5) as well. In order to further this agenda, I constantly circulating asking: “Why? How do you know? How would you convince somebody who doesn’t agree with you?”
The purpose of this investigation is two-fold:
- First, the real-world context helps illuminate for students the definition of a circle. Obviously, the dog can reach points that are at most 5 feet from the stake in any direction. It seems visually clear to students that the dog will run in the shape of a circle if it keeps the leash taught.
- Second, many of the points are chosen so that students will have to actually test distances to determine the distance between the point and the stake. The idea is that once students have created a working definition of a circle, they will then have to figure out how to apply the ideas from the warm-up to answering this question.
During this lesson, I do not say anything about the Pythagorean Theorem or the distance formula. If and when students bring up those topics, I ask them, “Why do you think that might relate to this problem? How could that help you with this problem?” It’s great to praise the fact that the students are thinking using these concepts by saying something like, “Wow, that’s an interesting connection. Do you think it applies to this problem?” But, I think it is important that in praising students, I don’t tell them whether or not it relates to the problem. The point of this is that by not validating the students' ideas directly, the students have to convince themselves and each other. I believe that this is where deeper thinking comes in (MP3).
Many students will jump to making a graph of the situation, which is awesome, but the challenge with the graph is knowing how to draw the circle precisely. If students ask for tools like rulers and compasses, I make them available, but I don’t tell students to use them. Students should be the ones thinking about the tools they need (MP5). Some of the points given in the problem were deliberately chosen to make it difficult to just “eye-ball” the answer. If students seem to get stuck, here are a few questions I plan to ask to help them move forward:
- What is the furthest the dog can get from the stake? How does this fact help you answer the question?
- What is the connection between this activity and the warm-up?
If students get stuck, I say, “I would suggest finding a way to make a connection between the key ideas of the warm-up and this problem.” This is a pretty huge hint, so I give them time to think about this before giving them any more guidance.
I have lots of dot paper available throughout this lesson. I just make a bunch of copies and put a pile on each table. This is another tool that students can think about how to use (MP5).
The ultimate purpose of this investigation is for students to apply the Pythagorean theorem or the Distance Formula. If they accomplish this in solving this problem, they will eventually develop the equation for a circle. It is important to not rush students. At the same time, I stay on task with my students to make sure that they are making progress in this direction. In other words, if students are trying to eye-ball every problem, I want to start asking them questions like, “How could you get a precise answer to this question if it is hard to tell from the graph?”
Apply this New Learning
Many of my students may not have time to start the Is this Point on the Circle? task today. It is available for students who quickly master the ideas of the previous problem: Can they extend these ideas to a more abstract situation? By the time students get to this activity, they should have some idea about how to determine whether or a not a point lies on a circle without using a graph. The purpose of this activity is for them to articulate this method in an organized way.
Once a student or a group of students accomplishes the “Dog and the Bone Problem,” I give them this task. I explain to them that the point of the task is not to solve the problems, but to articulate a method for solving the problems. This distinction is sometimes a hard one for my students to understand, because they have developed the idea that math means solving problems. So, this task is a good example of a Common Core shift to incorporate MP6 and MP2.
I find that students are best able to engage in these types of generalize the method tasks when I take time to explicitly explain my intentions. I say something like:
The point of asking you to develop your own method is to make this task a higher order thinking task—I want to push you to articulate your reasoning and use clear and precise vocabulary. Also, I think it is better to do a smaller amount of more challenging and thought-provoking work, than a lot of tedious and repetitive work. So more than just solving problems, I want you to think about how you are solving the problems.
It may help students to include a word list on the board, but I like to post this once students have already had time to think about how to explain their method, or to encourage students who have had trouble getting started to use the word list as a scaffold:
- Pythagorean theorem
- x-coordinate, y-coordinate
Over the course of the next few days, I will ask all of my students to tackle this challenge. So, even if some students don’t get the chance to start working on it today, I make sure that I explain the challenge to all students by the end of class so that they know what they are working towards.