# Mathematics and Why, Part II: How to Teach Trigonometry for Understanding

Last week, I gave some background into Mathematics and Why. This week, we are picking up where we left off: with a second alternative for a better way to teach maths in the US – a way that begins the ‘Mathematics Revolution’ from inside the walls of our curriculum rather than through changing the curriculum completely.

To begin, I must reference you to my ‘Welcome Letter’ – as I mentioned in that post, I think creating a Welcome Letter that outlines the big ‘why?’ questions for a lot of the practices we use in our classrooms on a daily basis is incredibly enlightening for a teacher and will sometimes cause us to change some of our policies for the better. However, I am also referencing the post because my own Welcome Letter explains a lot of the structural designs of my own classroom; in this post, I plan to jump far more heavily into the specific math content rather than the broad policies (like, for example, why I use a Character Point Average as 25% of the grade in my classroom).

So let’s jump into it by starting with a broad-scale discussion of the organization of content. I teach at an Expeditionary Learning School where I am the Chair of the High School Math Department for two reasons: one, my extraordinary skill in teaching mathematics, and two, because I am the only math teacher. OK, fine, I made up the first reason, but the point is that the second reason allows me some flexibility with moving content around, and here I’d like to share the results of some experiments I have done in that realm.

To begin, I changed the starting unit of Freshman year; where we used to jump straight into solving linear systems of equations and inequalities, we now begin the year with a unit called ‘Do Bees Build it Best?’ – a unit that derives from the Interactive Mathematics Program (IMP). As with all of my units, I do a lot of adaptation and innovation from the base unit, but the topic of this unit is an exploration of the honeycomb shape that bees create in their hives – thus the essential mathematical features of the unit are Geometry, Trigonometry, Simplifying Radicals, and a heavy focus on deriving equations that will work for specific scenarios.

The reason I made this change is twofold: first, students are coming off of a middle school experience where they did quite a bit of Algebra I, and secondly because this suite of topics sets students up very well for understanding the idea of deriving a ‘new’ concept. The first reason is important because, let’s face it: Algebra I is abstract. Given that many mathematics programs don’t ground the topics in real-world usefulness to begin with, students are commonly feeling like math is a mysterious set of difficult-to-memorize rules that were derived for the sole purpose of inflicting misery upon middle school students. In starting these students off with ‘an exploration of the shape honeycombs take on,’ students begin to understand that math is not about arbitrary rules… it is about observing the world around us, trying to recognize patterns within it, and eventually deriving rules about how the world works based on the fact that we observe it can’t possibly be any other way. They also get a break from the ‘abstraction’ of Algebra I – they can literally see the shapes that we are working with, and start to develop their ability to ask the next question about the shapes. For example, when looking at a honeycomb, students might ask “How much area does the honeycomb ‘hole’ cover? Obviously it has to be enough space for bees to fit through it, but not more, so how much is that?” The obvious ‘next question’ is “How do we find the area of this shape?” The answer: “Well, it would probably be easier to find if we break it up into triangles first.”

Once the hexagon of a honeycomb has been broken up into triangles, we have to ask “What is the area of each of the triangles?” Obviously this means that we have to find the base and height of the triangles, which launches us into a full-on study of triangles and trigonometry. Now, as I mentioned previously, the second thing that this unit allows me to do is set students up for success with later concepts in the class, including deriving equations as well as attending to precision and understanding that math class in high school will be a balance of big, creative style explorations (like that of the honeycomb) and skill-development. Specifically, how this happens for the Bees Unit is that I introduce (remind) students to the three main ways to express patterns in mathematics: tables, graphs, and equations. Tables are incredibly useful tools for exploring patterns when we don’t already know what pattern exists – so I have students derive the area formula for triangles using both ‘geoboards’ and a bunch of triangles of different base/altitude lengths where they use the principles they learned on the geoboards (halfing rectangles of particular sizes) to manually calculate the area of the triangles. Yes, my students already know that A=(½)bh, so forcing them to “see” the area of the triangles is a challenge, and we spend a lot of time having me force them to prove area to themselves rather than believing the formula they had been taught. Eventually, they use their tables, which have an input of base and height of the triangles they have sketched on graph paper and an output of the area, to derive anew the area formula for triangles. “Good!” I tell them, “soon enough, you will be using the same skill to derive formulas you don’t already ‘know’!”

The next question sparked by the honeycomb area is “what angles do the triangles involved in the honeycomb have.” This launches a week-and-a-half of me instructing students to learn how to be precise with measurements – both length using a ruler and angles using a protractor. I have them eventually cut out right triangles with an angle of 55 degrees; as you may imagine, these triangles are all sorts of different sizes, but they are all right with an angle of 55 degrees. We post these shapes up on the whiteboards and start measuring the side lengths and comparing the ratios of various side lengths. After recognizing that the ratios have to be the same, I ask if students think this is a trend for all triangles or just right triangles with a 55 degree angle? They eventually determine that if the triangle is right, then the ratios of the side lengths will be the same for any triangles that have the same angle measure of the non-right angles.

This leads me to do a bit of expert acting: if this is true for all right triangles, couldn’t we make a giant table of the side-length ratios for every angle possible for a right triangle, from 1 degree to 89 degrees? They say, “yeah, we could! But that would be hard.” I respond, “well, what if we start by deciding upon a few easier concerns – do we really need to know the ratios for every possible side length? Meaning, if we get the ratio for the side opposite of the angle and the hypotenuse, do we really need to do the reverse – hypotenuse over opposite? Wouldn’t it just be the converse?” Then we go into a discussion and test this theory. Students eventually decide that we really only need three ratios, and I then ask what we should call the ratios before we start doing all the work to create this giant table that may help us in the future. They begin arguing with each other, and then I bring them back to our full-class discussion and announce that I have an idea – perhaps somebody has already done the work of finding all of these ratios? I send them on a scavenger hunt where they can search through any book in the classroom (I have a large library of math and non-math books) to see if there’s already a version of the table we want to create. They go through the modern struggles of having me explain to them how books work (e.g. the use for an index vs. an appendix, etc) and eventually cheer with excitement when one of their classmates finds the table they need. It turns out, they realize, that we don’t even need to name the ratios we will use – they already have names! Etcetera, etcetera, etcetera.

The downside to this approach is that it just doesn’t work with the linear or industrial model of education. It takes FOREVER before my students are even introduced to the words Sine (NOT ‘sin’ despite the unfortunate abbreviation of the term!), Cosine, or Tangent! In the linear model, we have some serious ground to cover – there’s no time for this whole progressional charade! But think about how much more students get out of this approach than out of being told to memorize these strange and abstract terms and concepts, and then being told that they need to just use the ‘sin’ or ‘cos’ buttons their calculators to find the missing side lengths. Beyond contributing to the over-reliance on technology, students would end up seeing trigonometry as a ‘magic black-box’ process rather than understanding that the field comes from somewhere, specifically from a person’s need or desire to make predicting how things will work in the natural world easier.

It is for this reason that I never once tell my 9th-grade students about the fact that calculators have trig functions built into them. Obviously some of them eventually find out on their own, but the process of using a physical table with trig ratios has some important advantages. For one, students begin to learn that the abstract statement “sin(55)” is actually a real number – a decimal that represents the ratio of the length of the opposite side from the 55-degree angle over the hypotenuse side. This allows them to grasp both substitution and the idea that we sometimes use abstract symbols to represent numbers, when convenient.

From this point on, we obviously go into some skill-building portions of the unit, where students are completing more traditional-style “complete these 30 practice problems” sorts of assignments; however, we always transition back to the more creative problems as well:

As you can see, we go through a similar process of ‘deriving’ the Pythagorean Theorem, which then allows us to ramp up our derivation to a ‘new’ formula: what would be the area of a regular polygon with a known perimeter?

All-in-all, I feel that this approach, despite being very slow and thus limiting the amount of content we can cover in a year, allows students to actually understand mathematical thinking. I know it’s not the large-scale curriculum redesign we discussed in the first post on this topic, but what do you think? Feasible in your mathematics classroom next year? Beneficial for student understanding? Going slow to go fast, sort of like the Degeneration Effect describes? What did I miss?

-mmm