After reviewing example 1 on page 2 of student notes, we have a great opportunity to have students practice a host of great CCSS skills, like transformations and similarity. These relate directly to standards in congruence, specifically HSG-CO.A.4 and HSG-CO.A.5, which ask students to use transformations and manipulatives to show completed transformations. This standard is directly linked with the key standard for this lesson, HSG-C.A.1, where we prove that all circles are similar. This lesson provides an opportunity for students to make connections between three big ideas in Geometry: circles, similarity and congruence. Connection of larger concepts like this directly links to mathematical practice #2, since students are asked to connect these important abstract concepts (mentioned above) with concrete circle and transformation examples. To get a sense of student understanding, I usually ask that examples 2 and 3 be done by students in partners or independently.
I always like to keep my students on their toes, and I usually will look for people who may be unfocused or off-task as “volunteers.” Many times to do this, I will write student names on the board, and ask them to come up and write their answer. I almost always will write student names in pairs so that they can check in with each other before writing their answer publicly. Many students want to know if their answers are right or wrong before putting their work on the board. I encourage students to just try and even if they are right or wrong, our whole class will learn or to check in with their assigned partner. This helps to create a classroom environment where one mistakes are seen as an opportunity rather than a chance to criticize, and I find promotes cooperation between different students. This links directly with mathematical practice #3 where students are asked to critique each other’s work.
When teachers review example 2 and 3, this is a great chance to really dig into depth about our definition of similarity as it relates to transformations. Students should be expected to write each specific transformation and then describe how each circle maps to cover, completely the other circle.
We then will then jump into a critical topic in geometry, tangent lines! In this quick exploration on the top of page 3, students can practice identifying tangent lines, and using a protractor to identify angles that they form. I make a big deal about tangent lines forming … what kind of angles? I ask my students again and again to repeat tangent lines = right angles! It can become a chant, if you have time, and the kids have fun with it! It’s important to discuss that questions #5, is the opposite true that when given a line perpendicular to a radius at an endpoint, we have a tangent line. If teachers have time, they could quickly sketch this out for students, and students will be able to see that if we have one point touching a radius that is perpendicular, this satisfies our definition of tangent line.
Applications of Tangent Lines:
After reviewing these important theorems, teachers can then jump into applications of tangent line problems, which can be tricky for students. Example 4 is a great problem because it requires students to first apply their knowledge of tangent lines and then also to remember the Pythagorean triplet, 5, 12 and 13. This again connects another important Geometry concept to circles, and also, gives the class an opportunity to review Pythagorean Theorem. Teachers can ask students to plug into Pythagorean Theorem or can ask students for “short-cut” way to find the missing leg. All students love short-cuts and when students remember their triplets, it’s a great connection to a prior topic.
The remaining examples (ex 5) to (ex 6) ask students to apply the idea of tangent lines in different ways and also apply the idea that segments which are tangent from a point to the same circle are congruent. I would suggest that teachers talk students through (ex 5) particularly because it requires students to multiply polynomials and is also tricky to set-up. Then, (ex 6) may become a question that students can work on in pairs. Most of my students with partner help or a little hint, can find the value of the radius, however, I make a BIG deal if they forget to find the distance from the golf ball to the center of the cup. I always get worked up, and ask the class, what do we really want to know in this question, the radius or a distance between two objects? Then, I usually ask them to think about someone writing the SATs, which we have already discussed at length is a sneaky test where the authors try to trick students. What question will be one the SATs, finding the radius or finding the distance? It helps to remind them to go back and read, and to think more critically about a question and hopefully dig deeper into the purpose of why we’re learning these concepts.
Practice for Lesson
I find that at the end of class, when time is running out a great way to introduce the proof or bigger idea behind a theorem is through a video. In the Investigation on page 5, if teachers have time, students can derive the proof for why the two tangent lines (connected to an exterior point) are congruent. However, I’ve also included a video link (http://www.youtube.com/watch?v=R1PBIyuFvAQ ) that provides a brief overview in the first 1-2 minutes of how to show that the two tangent lines are congruent using congruent triangles. If there is time, teachers may want to get into the details of how and why we have congruent triangles in this diagram; more specifically we can use right triangle properties and hypotenuse-leg congruence. Teachers can pause at any point in the video, maybe after the teacher on the video draws two triangles, and ask students to predict the next step. If you watch the entire video, you’ll see a full written proof, but I think that a brief overview is enough to stick with the students and then apply this idea to problems.
Lastly, for (ex 7) and (ex 8), I will ask students to continue to “popcorn” read, and then we will complete (ex 7) together. This example asks not just for the value of x but for the length of GH. Many times, I will find x and then pause and pretend like I’m done with this example. I will then wait and see if anyone noticed that we didn’t find GH yet. As I do this often in the year, students will frequently be clued into this pause and realize that they have not done something. This is great and hopefully will allow students to build-in their own pause time to make sure they have completed the question completely. Students can then work in partners on (ex 8) and a volunteer can put their answer for (ex 8) on the board.
To wrap up class, teachers can give students a chance to practice in small groups on the last page of scaffolded notes which are practice questions. These questions will reinforce the applied concepts learned in class and can be done in small groups or pairs.
Before we do the exit ticket for this class, I typically write student names on the board to answer the practice question. If time is short, which it was when I last taught this, I asked one student to explain their answer to a more challenging question. We reviewed #3 as it required students to remember the ideas for reflective and transitive properties. Teachers can also ask for volunteers to read the answers, and students can check their work. I like doing this if I have a sense that all students are “on-board” with the lesson’s concepts. However, I find that doing just 1 question together really helps to solidify a key concept, particularly a challenging one, before students leave for the day.
At the end of class, I put the exit ticket on the board and asked students to complete this question. The exit tickets are a chance for a quick-check in for both myself and my students. In this exit tickets, students are asked to review how to identify key parts of the circle. This is the main objective for me in this lesson – students will really struggle in future circle lessons if they can’t identify these key parts, particularly the new words like tangent and secant. One way to discuss the mathematical difference between these words is that a tangent hits 1 point on the circle, while a secant goes completely through the circle.
Further, I like to make the exit tickets quick and basic; my goal is to see if students got the main idea of class and if they can answer this correctly on their own. Once students complete these exit tickets, usually on post-it notes, they then stick their answers on part of the wall in the back of my room. At the end of the day or the next morning, I look to see who got these ideas right and who got them wrong. My overall goal is not to “grade” students but rather to adjust my next lesson, if needed. If I find that more than a third of my students are not able to answer the exit ticket questions, I will often adjust the Do Now for the next lesson.