Problem Solving with Isosceles Triangles and Circles
Lesson 4 of 17
Objective: Students will be able to apply their knowledge of circles and right triangle trig ratios to solve for the unknowns in an isosceles triangle.
This Opening_Exercise refreshes students' memories of a few key ideas that we’ve covered regarding circles. I try to push them to work quickly here, and to practice the habit of referring to previous notes if they’re having trouble recalling any of this.
The point (2,1) does not lie on the circle, but it’s close, and because the radius of the circle is 4, students should recall that only four lattice points can be found. As much as possible, I encourage students to discuss their answers and show solutions on the board; if all goes well, students will recall enough that I won’t have to say much here in terms of speaking to whole class. As they work, I circulate and make sure that at the very least, every student has the equation of the circle written accurately in their notes.
Following the opener, the task on Slide 3 of Problem Solving Slides helps us review isosceles triangles and how we can use trig ratios to solve for unknowns. The big idea here is that, because isosceles triangles have a pair of congruent angles and sides, we can connect this to the 30/60/90 triangle and its derivation as half of an equilateral.
We can find both the height and the base of this triangle by using sine and cosine. I let students guide this, and let them decide what we find first. I am aware that we could use the Pythagorean theorem to find the third side, but…
- That would be less precise
- It’s actually a little bit more computation intensive…
- …especially when we move on toward generalizing later
We’ll be able to reference this work later. It lays groundwork for deriving a formula for the area of a triangle (G.SRT.9) and establishing the law of cosines (G.SRT.10 and 11).
Problem Set, Part 1 Work time
Here, I give students a chance to practice a few problems (Problem Set Isosceles and Circles) like the one we just did in the mini-lesson. Depending on how that mini-lesson progressed, students will more or less have a “recipe” to use on the first four problems. The first two problems are as straightforward as the example; the third is the same but with a much bigger obtuse angle, and the fourth gives an odd number for the angle measurement, which will result in decimal values for the other two angles. The fifth and sixth problems provide measurements for different parts of the isosceles triangle, just to make sure students can be flexible.
The key here is the generalization! After running through six exercises, I want students to formalize the procedure they’ve been using into some concise algebra. When they can do it successfully, they’ll have provided themselves with a great tool for the upcoming Defining Pi Project. It’s important to remember, as we make these moves, that an algebraic expression is itself a tool, which is a message I try to repeat throughout my curriculum!
The generalization in 7b might be a little "harder” than 7c, because it requires us to consider that the measurement we find using trig is half the base of this triangle, and we have to account for that in our expression. That’s ok, I don’t believe in strictly sequencing problems in order of difficulty – that’s not how problems appear in life!
While students are working on these problems, I encourage, but do no require them to work together. When I see one student developing expertise I may refer other students to him or her for help. When I notice that neighboring students have different solutions, I’ll ask them to discuss what they did. When students are developing generalizations on #7, I might ask them to write another problem similar to #’s 1-4 and see if their expressions work.
Problem Set, Part 2 Work Time
These problems are designed to get students thinking about some of the salient aspects of our continuing study of circles. For the first problem, they sketch circles and make estimations for non-lattice point values on the circles. #1e helps them to recall that the domain of the circle is strictly bounded by the edges of the circle. Problem #2 is a chance to notice that sometimes describing a circle with an equation is far more convenient than drawing a graph, and we must be able to read these equations and imagine the circles they describe. Problem #3 is a chance to review the area formula for circles, and Problem #4 is about the intersection of two circles – one of these intersection points is a lattice point and one is not.
Problem #5 is the most interesting, because it lays some groundwork of our upcoming study of the unit circle, without really referencing that structure in any way. Students are asked to consider the idea moving around the circle, and after warming them up with the easily conceivable trip of ¼ of the way around a circle, I ask them to consider the coordinates of points 1/8 of the way around. By reaching back to what we know about 45/45/90 triangles, it’s possible to find the exact coordinates of these 1/8 and 3/8 points on the circle, even though they’re not lattice points.
A Pair of Exit Tasks
In summary, this lesson opens with a problem about circles, then moves into some exercises regarding isosceles triangles, continues with less routine problems about circles, and finally here comes back to two non-routine questions that will help us teachers get a quick picture of what students know and can apply about the unique properties of isosceles triangles.
The first Exit Task asks students to explain whether they can use what we know about isosceles triangles on a more general scalene triangle, given the measurements of just one angle and one side. I like this question because it requires students to activate their knowledge of why we’re able to make the moves we make on isosceles triangles by explaining an example of when we cannot.
The second task is really a preview of the Defining Pi Project, which begins in the next lesson. In that project, we borrow from the technique of Archimedes, by inscribing successively larger regular polygons inside of a circle. Here, students are given the distance from a vertex to the center of the pentagon, and asked to determine the perimeter of the shape. I’m looking to see how quickly my students notice the five congruent isosceles triangles that can be found inside of this shape, and how well they can now use isosceles triangles as a tool in this calculation. It’s great if they can do it, and it’s fine if they cannot, because that lesson is coming right up. I’d like to know how carefully I’ll have to describe this process when we get there.
One additional note on this second task. Some students might need a quick reminder about how to find the measurement of the angles in this pentagon. I’ll use my judgment in the moment to decide whether to:
a) Just tell them that each angle measures 108 degrees
b) Remind them of the formula , where n is the number of sides
c) Or just allow them to work together on this problem because some students know where to start and some do not.
I have chosen not to include this information explicitly, because I want my students to have the experience of seeing a non-routine problem and drawing on this knowledge from their previous mathematical experiences. I’ll just be at the ready to swoop in an give a nudge if they need it.
Both of these problems follow an important principle for exit tasks: although they may take a little time for students to get their thoughts on the page, they’re actually very quick for me to assess. A quick glance at an answer to the first question is all I need to know whether a student understands what make isosceles triangles unique. A moment looking at the solution to the pentagon problem is all I need to see how successful a student can be at applying their knowledge of isosceles triangles to this new context, and how ready they are for tomorrow’s lesson.