Using Special Right Triangles and Spotting Some Patterns

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SWBAT use special right triangles to determine geometrically the sine, cosine, and tangent of 30, 45, and 60 degrees.

Big Idea

What does it really mean to memorize something? To recognize the underlying structures and patterns, and to use them to make sense of and remember, conceptually, what is happening.

Opener: The Height of an Equilateral Triangle

10 minutes

Take look at today's Prezi to see the notes for this opener and for today's guided lecture notes.


Class begins with a problem asking students to find the height of an equilateral triangle.  The first thing students must notice here is that a line representing the height of the triangle will also be the line of symmetry that we discussed in the previous class.  From there, some students will want to use trig ratios: they will see a right triangle with a hypotenuse of length 10 and they have a 60-degree angle to use.  I allow students to go this route, but I pay attention to who is choosing it, because I want to make sure they see how this problem can be solved without trig ratios.

Other students will discover the method that I want to highlight.  Looking at the right triangle that forms when we split an equilateral triangle in half, we see that the shorter side is half of the length of the original side.  Once we know the lengths of two sides of right triangle, of course, we can use the Pythagorean Theorem. The reason I want to highlight this method is that it gives us an access point for a discussion of exact values and simplest radical form.

I hope to find one volunteer to present each method.  By employing the trig ratio sin(60), we'll find the height to be 8.66...; by employing the Pythagorean Theorem, we'll get √75, which has exactly the same decimal value.  It's a really important moment when students see that these values are exactly the same thing.  I wouldn't say they're surprised, but I think there's a little amazement here.  Up to this point, many of my students have felt like the values a calculator spits out for the values of trig ratios are mysterious, inexplicable things.  This example gives them some grounding, and adds to their growing body of evidence that the ratios are something real.  

I chose 10 for the length of the sides so we can look at the Fascinating Chart and notice a number that looks suspiciously like 8.66025403784...

I don't yet worry about simplest radical form.  That's coming up in a few minutes.  I do make the point that √75 is a more exact value than 8.66... because it's not rounded, and that will prove useful in applications where we want to avoid rounding error. 

This is a problem that we will revisit in the next unit, when we inscribe a hexagon in a circle.

Guided Lecture Notes: Side Lengths of Special Triangles

15 minutes

Notes & More Problems

To continue, we engage in some repeated exercises.  I jump from the 30-60-90 opener to squares, just to keep my students nimble and making connections/distinctions between both kinds of special triangles.

We begin by reviewing today's learning targets, SLT Triangles 4 (F-TF.A.3) and Mathematical Habit 7 (MP7).  I ask students if they can see any connections between these two learning targets, and I take a minute or two to elicit the idea that when we look for patterns, we can learn more about the behavior of the trig ratios.  I then frame our next two tasks by saying that we're going to do a few repreated exercises now, and as we work, I'd like for everyone to look for patterns in their work.  Spotting patterns, as we'll see today, is a step in memorization.

Five Squares

The next problem is simply to draw five squares, each with a different side length, and then find the length of the diagonal in each one.  My role is just to make sure that students can see the triangles that are formed when they draw a diagonal through a square, and to make sure that can describe these as special 45-45-90 triangles.  For most students, this is not too much trouble, especially as they discuss the work with their groups.

My expectation here is that students will use the Pythagorean Theorem to find the length of each hypotenuse in radical form.  They may also express these values in decimal form, and some may choose to verify their results by using sin(45) or cos(45) -- which, once again, we notice have the same value.  What each student will end up with is a set of radical values ready to be simplified later in the lesson, when we spend some time on simplest radical form.

Five 30-60-90 Triangles

For any students who didn't notice it in the opener, I make it explicit here that half of an equilateral triangle is a 30-60-90 triangle.  Based on this idea, what do we know about these triangles.  By now, I expect all students to be able to see that the shortest side of a 30-60-90 triangle is half the length of the hypotenuse, because the hypotenuse is one "original side" of the equilateral triangle, and the short side is half of another.  

On a side note, it can be a little confusing that the short side is opposite the 30 degree angle and the longest side is opposite the 90 degree angle - so when we're talking about "half the length," we're talking about the lengths of sides.  If we want to talk about one angle that is half the measure of another, those are the 30 and the 60 degree angles.  The proportions of similar triangles relate to their sides; there is not a directly proportional relationship between angle measures and side lenghts.

Once again, when we know the lengths of two sides, we can apply the Pythagorean Theorem, and just as we did in the previous example, we can make a set of five values in radical form, ready to be simplified.


Interlude: Simplest Radical Form

10 minutes

Now we've got a nice set of radicals that need simplifying.  This is a key skill in an Algebra 2 & Trig curriculum, and although it's not the primary focus of today's lesson, this is a great context for practice.  Of course, there are only two possible radicands here: √2 and √3.  That's a good thing, because it helps students to solidify the concept.  They will have time to practice this skill more generally on today's Delta Math assignment. 

Look at the 45-45-90 radicals photo for an example of how I set up the notes for simplifying a set of radicals.  I want students to see the structure and pattern here, and for many of them, seeing this context rather than a random set of exercises helps them make sense of this concept.  

After guiding students through this example, I ask them to repeat for the radical values in their 30-60-90 triangles.  "What would this look like for the examples from the 30-60-90 triangles?" I ask.  Many of them feel like there should still be a √2 in the result, but with minimal frustration, they see that that won't work: none of these values are the product of 2 and a perfect square.  So what are they?  By testing out a few perfect squares on their first example, they see the √3 that remains, and if they're not wondering it on their own, I'll ask, "Do you think that √3 might repeat?"

Guided Lecture Notes: Trig Ratios for Special Triangles

15 minutes

I ask everyone to find their Similar Triangles Projects, and to take a look at the "Fascinating Chart".  (Actually, it's likely that I'm handing back these graded projects today.)  I ask, "What are some really nice values in your Fascinating Chart?"  Students have been wondering about the 1's, 0's and 0.5's for a while now, because all are a stark contrast to the sea of messy decimal numbers that fill the rest of the chart.  I say that for today, we're going to ignore the first and last rows of the chart, but that still, there are a couple of numbers we should look at in the middle of the chart.  By now, everyone in the room is remarking that they can see the 1 and the 0.5, and, lo and behold, they're in the 30, 45, and 60 rows of the chart.  I jokingly ask if they think this is some magical coincidence.

"Let's look at the 1 first," I say.  "Why is the tangent of 45 equal to 1?" or if they need further prodding, "Which of our shapes provides an example here?"  I want to highlight the way similarity appears in each of our triangles.  I ask students to look at their set of squares or 45-45-90 triangles (I use these terms interchangably to highlight the connection between them).  I ask, "What does it means for all of these to be similar?"  I'm hoping that, with the Similar Triangles Project fresh in their minds, students will remember that trig ratios allow us to see the relationship between angles and ratios of sides, both of which are properties of similar triangles.  With all of this mind, an explanation of why sin(30) and cos(60) are equal to 0.5 is pretty straightfoward.

Now, with our radical values simplified, we can look at some of the "messier" trig ratios.  Taking a set of 30-60-90 ratios as our example, I write the ratios of the longer leg and the hypotenuse for each on the board, and I tell everyone to simplify each of these.  Each ratio simplifies to √(3)/2, so I ask everyone to enter this on their calculator.  I love the moment when students see the .8660254, and their initial shock that it matches one of the values in their charts.  I explain that the exact value of sin(60) and cos(30) is √(3)/2, and from now on, I'd like everyone to memorize this fact.

Work Time: Problem Set & Delta Math

25 minutes

If there is time remaining, I give students time to work together on Problem Set 4 or to take a look at the newest Delta Math Assignment, Radicals and Special Triangles.  This Delta Math assignment gives students a chance to practice simplifying radicals - both as an isolated skill and in the context of the Pythagorean Theorem.  It is also the second assignment of the unit that includes a timed practice module for calculating perfect squares.  Students have found this one very useful, especially once they see the role that the perfect squares play in simplying radicals.  This is another case of "memorization" being more possible when it makes sense: it's one thing to memorize the perfect squares out of context, but it's another thing entirely when one (a) sees a valuable use in the exercise, and (b) has a fun, timed challenge to complete in competition with the friend sitting next to them.