I’ve been mulling over how to introduce trigonometry to my geometry students. I think I’ve finally figured out a way that is going to be conceptually deep, and will have kids see the need for the ratios.
I don’t know if all of what I’m about to throw down here will make sense upon first glance or by skimming. I have a feeling that the flow of the unit, and where each key moment of understanding lies, all comes from actually working through the problems.
But yeah, here’s the general flow of things:
Kids see that all right triangles in the world can be categorized into certain similarity classes… like a right triangle with a 32 degree angle are similar to any other right triangle with a 32 degree angle. So we can exploit that by having a book which provides us with all right triangles with various angle measures and side lengths. (A page from this book is copied on the right.) Using similarity and this book of triangles, we can answer two key questions. (1) Given an angle and a side length of a right triangle, we can find all the other side lengths. (2) Given two side lengths of a right triangle, we can find an angle.
By answering these questions (especially the second question), kids start to see how important ratios of sides are. So we convert our book of right triangles into a table of ratios of sides of right triangles. Students then solve the same problems they previously solved with the book of triangles, but using this table of values.
Finally, students are given names for these ratios — sine, cosine, and tangent. And they learn that their calculator has these table of ratios built into it. And so they can use their calculator to quickly look up what they need in the table, without having the table in front of them. Huzzah! And again, students solve the same problems they previously solved with the book of triangles and the table of values, but with their calculators.
Hopefully throughout the entire process, they are understanding the geometric understanding to trigonometry.
(My documents in .docx form are here: 2015-04-xx Similar Right Triangles 1 … 2015-04-xx Similar Right Triangles 2 … 2015-04-xx Similar Right Triangles 2.5 Do Now … 2015-04-xx Similar Right Triangles 3 … 2015-04-xx Similar Right Triangles 4)
It’s a long post, so there’s much more below the jump…
Similar Right Triangles#1: The Forward Question
I am going to begin by having students start seeing every right triangle they see as similar (or, to be technical, close to similar since these only have integer angle measures) to one of the right triangles in this book, titled the Platonic Right Triangles book. (If you want to know where Plato comes in, check out the packet…)
That’s how we will start. So they are going to use the Platonic Right Triangles book along with what they have learned about similarity in order to find missing side lengths of triangles. At the end of this, students are going to measure the height of the organ in the chapel in our school, using what they have learned.
The big idea of this first packet is getting kids to see that all right triangles are similar to this “Platonic” set of right triangles, and given a right triangle with a particular angle, they can use that fact to find missing sides of triangles with very little effort, by setting up a proportion.
Similar Right Triangles#2: The Backwards Question
This is the backwards question: Given a triangle with some side lengths, can they find the missing angle? At this point, all they have is their Platonic Right Triangles book. Mainly, the first few pages of this packet has kids try to figure out how they can use the book to find out the missing angle. They’re going to struggle at first, and have various approaches to it, some of which will be more efficient than others. But I see the first four pages of this packet, their struggle, and most importantly the group and class conversations we have based on their approaches, to be the key thing here.
Hopefully, at this point, we’ll converge upon the idea that the ratio of sides is a super awesome way to find the missing angle. If they are given the side opposite the angle and the hypotenuse, life is easy. If they are given the side adjacent the angle and the hypotenuse, life is easy. (This is because the Platonic Right Triangles book gives all the triangles with hypotenuse of 1.) But if they are given the side opposite the angle and the side adjacent to the angle, life is hard. And that’s okay. But it would be so much easier if for each of the triangles in our Platonic Right Triangles book, we had a list of all the ratios of the opposite side to the adjacent side.
The important of these ratios gets highlighted with some simple work with this simple Geogebra applet.
From here, we collapse our Platonic Right Triangles book into a simple Table of Right Triangle Ratios. An eighty-nine page book that can be represented in a two page table.
Students hopefully will finish this worksheet seeing the usefulness of ratios in a right triangles, and being able to use a table of values to help them find missing sides and angles.
Similar Right Triangles #2.5: Why Three Ratios?
This is a classroom opener. It came about when my co-teacher recognized that we never actually need all three ratios… the only reason we have all three ratios is for convenience. And so I whipped this up to get kids to recognize that fact.
Similar Right Triangles #3: Naming the Three Ratios
In this packet, students finally get names for the three ratios we’ve been using (sine, cosine, tangent), and hopefully the questions asked will require them to understand they are basically doing what they’ve been doing — but with fancy names. Specifically, I want students to leave understanding that all a calculator is doing when calculating the sine/cosine/tangent of an angle is that it is going to a super extensive internal table of values and finding the appropriate ratio of sides for that right triangle. And that all a calculator is doing when calculating the inverse sine/cosine/tangent of a ratio of sides is that it is going to a super extensive internal table of values and finding the appropriate angle for that right triangle.
Then there are some challenging conceptual questions that students will hopefully be able to answer, which require them to understand the geometric understanding of the trigonometric functions.
Similar Right Triangles #4: Special Right Triangles
I don’t think that special right triangles are all that important, but I know that I ought to expose my kids to them. So I made this…