Definitions of the Trig Ratios and a Fascinating Chart

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SWBAT use the definitions of the trigonometric ratios to check their measurements of a rectangle.

Big Idea

If similar triangles have congruent angles and proportional sides, then what is the relationship between these two properties?

Opener: What is the value of an interquartile mean?

10 minutes

On the group work component of Part 2 of the Similar Triangles Project, students analyze the set of calculated ratio values from their measurments of rectangles.  This is empirical data, because it consists of the actual measurements made by students.  In order to use this empirical data to come up with predictions of what the right values should be -- on our way to theoretical values that are expressed by the trigonometric ratios -- students find the basic measures of central tendency on each data set.

In today's opener, Students are asked to calculate the mean, median and quartiles of the data set before I show them how to calculate a modified version of the inter-quartile mean. Inter-quartile mean is a term with which some of them may be unfamiliar.  In this case, outliers have a strong influence on the calculation of the mean for the entire list of numbers. The interquartile mean discards these values, and what seems like a better prediction of center is achieved.  

When I discuss this problem, I ask students to judge the "usefulness" of the mean vs. the interquartile mean.  I explain that what I mean by "usefulness" is how well a measure of central tendency might be used to predict what will actually happen.  The mean of this data set is 9.99; the IQM, as we calculate it, is 8.9.  This IQM example makes it pretty clear to students why we use this, and many of them realize the power in this tool in the context of their own data sets.

Group Work: Finish determining theoretical ratios

10 minutes

Most groups still have to finish up their work from the previous class, by finding their rectangle groups.  Using what we just did in the opener, students return to their data sets and try to agree upon an accurate theoretical value.

I set up a chart on the board where groups can record their theoretical ratio values.  This photo shows what the chart looks like when it's filled in.  When I first make this chart on the board, I leave off the angle measurement columns.  After each group has filled in their ratio values, I add the angle columns and we move on to discuss how we're going to check the accuracy of these theoretical values. 


Mini-Lesson: Using the trig ratios to figure out if you're right

20 minutes

Now it's time for the big reveal: How do we check the answers proposed by each group?

There are always students who get it already, because they understand what the learning target says: that we are defining the trigonometric ratios as relationships between acute angles and the ratios of sides in a triangle.  Others still haven't made this connection for themselves, and for these students there's a tantalizing little mystery unfolding.  My role as a teacher is to take the pulse of the class and figure out whether to go the showmanship route with a big, dramatic pulling back of the curtain, or a more conspiratorial, "hey, I know you've all already noticed that this is happening, but now we're going to use trig ratios to check your work."  Either way, kids are invested in these stakes, because they've put some effort into agreeing upon the best possible theoretical value for each ratio, and now we're going to make some calculations and see how well everyone did.

We look at the learning target again: 

Triangles 1: I can use similarity to define trigonometric ratios as properties of the angles in a triangle.  (CCS G-SRT.C.6)

I ask the class which of these words we've been working with so far.  They share what they've learned about similarity.  They say they've been calculating ratios.  They recognize that we've been talking about properties of triangles.  I talk about how they've used the angles in a geometric sense to prove that they have similar triangles on their colored rectangles, but haven't yet measured those angles.  That's our next step. 

Whatever sense students have made of this activity so far, they need to measure angles.  It's the measurement of the angle that dictates what these ratios will turn out to be.  

I add the last two columns to the chart on the board, and show students how to measure both acute angles in their right triangles.  This is the second opportunity for students to use the protractor, and it always takes some review for some students to feel confident about using these tools (MP5). Because they're already in groups, students are able to check their measurements with their colleagues.  After they've been measuring for a few minutes, disagreements may arise - especially when the diagonal through the rectangle looks like it goes in between the degree marks on the protractor.  

There are two teaching opportunities here:

  • First, I make the generalization that our tools might not be as precise as we wish.  I hold up a protractor and say, "This plastic, school-issued protractor is not a precision instrument.  It only measures to the nearest degree.  If we were trying to build anything real, we'd want to be able to be more precise than this allows us to be.  I apologize, but you might just have to your best to estimate the measurements of your angles with the tools I've provided."  
  • Second, I want to make sure students recognize that these will be complementary angles.  I ask questions to lead them to make sure that they've got this, and I tell them to use this characteristic by making sure that the measurements of the two angles add up to 90 degrees. There are a few reasons that this might prove problematic for some students (for example a diagonal that's not perfectly straight), but again, that's why it helps to do this work collaboratively.  Putting multiple eyes and minds on a mathematical task is one way to attend to precision (MP6).

As groups agree upon their angle measurements, they add them to the chart on the chalkboard. When the chart is complete, it's time to check our work.

One by one, we check student answers by calculating the sine, cosine, and tangent of each angle. Students are able to judge the accuracy of each theoretical ratio on their own, and there's great excitement here.  I leave open the question of which trig ratio goes with which theoretical ratio column for two reasons.  One is that it will become clear to students as they work; the other is that is that it depends on which angle we're talking about.  This is the first time students can see the relationship that if two angles are complementary, then the sine of one is equal to the cosine of the other.  Additionally, when they're calculating tangent, they will see that the tangent of one of the angles matches the a/b column, but the tangent of the other one is not represented in our chart.  I leave this mystery open for a little while - we'll be working to figure out how tangent works throughout the semester.

Several Sources of Error & Several Ways to Check Our Work

Although they're often quite close, it's rare for students to come up with theoretical ratios that are accurate the nearest hundredth.  The discussion of sources of error is an interesting one.  We talk about the imprecise nature of our tools.  We talk about rounding errors.  We talk about the decisions that each group made as they came to consensus on their values.  I run with this conversation as long as students are interested in it.

I point out that even if the calculator disagrees with a group's theoretical ratio, we still can't be certain where the error lies.  It might be that the group miscalculated the ratio value, but the error might also lie in the measurement of the angles.  I ask the class which is prone to a greater source of error.  I ask if anyone would prefer to use different angle measurements, or even if they'd like to try using angle measurements in-between two decimal numbers.  

I also show students that there are a few ways I confirm the accuracy of our work.  Of course the calculator is one of them.  But also, running the algebra to see that (a/e)/(b/e) = a/b is powerful stuff.  After students are satisfied with the algebra, we check to see which groups have a quotient of the first two ratios that's equal to their predicted a/b value.  This of course, lays groundwork for the relationship sin/cos = tan, which we'll continue to study throughout the semester.

A Fascinating Chart

20 minutes

In one sense, this is busy work.  But it's busy work with a purpose.  I tell students that.  I tell them that repeated calculations are one way to find structures and patterns (MP7).  I implore them to share the work, and pay close attention to repeated numbers, weird numbers, and anything else they may notice.  If they think something of the numbers in the chart, they should write it down.

I like it when students challenge me on my (cheeky, in my opinion) use of the word "Fascinating" here.  We joke around a little, and I wink and promise them that, indeed, this is a most fascinating chart.  As soon as they get started, kids are hooked on the structures that immediately become apparent in this chart, and it's fun to watch them agree with my thesis.  

What is there to notice here?  First of all, there's the way the sine and cosine columns are the same, just inverted.  Then there is the pace at which each of these columns grows - why are there so many numbers between 0.8 and 1.0?  There's the tangent column, which is the only one to go past 1.  There's that error for tan(90) - what the heck is up with that?  There's the 45 degree row of the table, and there's those couple of 0.5's sprinkled about the table, amid all the other much messier looking decimals.  I encourage students to own any of their thoughts about what's going on this table, to voice their questions and share their observations.

For many students, this chart will prove to be a valuable resource throughout the semester, one that students will want to return to as they make sense of subsequent ideas on upcoming projects.  As I circulate, I ask students how they might use this chart, rather than a calculator, to estimate the values of trig ratios for angles other the ones they see here.  They like the idea of being able to use this chart to estimate those in-between values.

The Similar Triangles Project Part 3 Overview & Work Time

15 minutes

This project has been pretty fast-paced up to this point, and some students need time to complete various parts of it.  This is ok - they'll have that time during tomorrow's class, and I expect some of this to be finished for homework.  Wherever everyone stands, with about 15 minutes left I call the class to attention to introduce the final part of the project.

I distribute the handout - it's double sided with Part 3 on the front and the rubric on the back.  I show students the checklist of everything they'll hand in, and I take questions about the short essay they will write for Part 3.

There is no formal closing today, instead I just give students time to make sure they have each item in the checklist and I tell them to make a plan for how they'll get everything done by the due date.