I do not mention anything the term "trigonometry" as we start this exercise. My Geometry students have briefly learned about right triangle trigonometry in their Integrated Algebra class; I have observed over the years, however, that very few students really achieved any understanding of the concept, due to the pacing of that course. I therefore intentionally leave it a bit of a mystery as to where we are heading in this lesson.
I divide the students into groups of three (if the number of students does not divide evenly into groups of 3, a group or two of 2 students works fine). I give each member of a group the two handouts (the same sized quarter circles, and a table), a protractor, and an additional straightedge (this seemed to help when measuring the angles).
The table contains nine different angle measures. I ask the group to split these angle measures up between the group members, so that each student is responsible for three angle measures. I explain that they are to use the protractor to mark off their angle measures on their quarter circle. Once they have marked off their angle measures, they draw three right triangles, as shown in the example. I provide an example of this on the board, just to make sure that everyone gets off to a good start.
Next we start the measurement phase. I briefly discuss with the students the meaning of the words “opposite”, “adjacent”, and “hypotenuse” with respect to right triangles. When all seem clear on these concepts, I explain that each student is responsible for measuring and recording the lengths of the opposite and adjacent sides of all three of his or her triangles, accurate to one decimal place. The adjacent side of the triangles is measured easily using the horizontal ruler on the diagram; using their straightedge or protractor, the students can extend the opposite side of the triangle horizontally, connecting to the vertical ruler on the diagram. We also discuss the length of the hypotenuses, which will always be equal to the radius of their quarter-circle.
Once each member of the group has recorded their own data in their table, they share their data with the other members of the group, so that each group member ends the measurement phase with opp, adj, and hyp filled in for all nine angle measurements.
At this point, with opp, adj, and hyp filled in for each angle, my students are ready to find and fill in the ratios. I specified that I wanted the ratios rounded to 4 decimal places, though I must confess I don’t exactly know why I chose ten-thousandths, other than trig tables are often given to this level of accuracy.
When everyone’s tables are complete, we end the data gathering stage, and are ready for discussion.
I begin our class discussion of the Data by asking the students to observe general trends in the columns of their tables.
What happens to the value of opp/hyp as the angle measures increase?
Is the same true for adj/hyp?
Then I ask each group for their values of opp/hyp for 30° and 45°. As the students begin to notice that their values are all very similar, I ask them why they think this is true, hoping that they will begin to see and discuss similar triangles.
At some point in the discussion, we assign the names sine, cosine, and tangent to our columns. My students were introduced to trig in their previous year of math, so these terms are at least somewhat familiar to them and they may advance this idea. We also talk about the use of SOHCAHTOA as aid to remembering the appropriate ratios.
After the discussion, we will next we put our trig tables to use. I draw a triangle on the board with a 60 degree angle and the length of the hypotenuse indicated, and ask the students to find the length of one of the legs of the triangle using a ratio. We talk about the process of deciding which trig function to use, and how to solve a proportion.
I then hand out a trig table that I have photocopied from a textbook. The students are usually unaware that trig tables even exist, and are able to compare the accuracy of their trig values to the values on the table. I ask the students how this table might have been constructed, hoping that they will talk about the role of similar triangles.
We use the photocopied trig table on a problem, and then move onto locating and using the trig functions on the calculator to find the lengths of sides in a couple other problems.
I put a sample problem on the SMART board and pose a series of questions:
1. What should our first step be when faced with a problem like this?
I am looking for the answer, “Read the problem.” A lot of students will begin a task like this – what they perceive as a word problem – by panicking and claiming that they “have no idea what to do.” After a brief discussion of this, I ask the students to read the problem silently to themselves.
2. What should our second step be?
Here I am looking for the answer, “Fill in the diagram.” I ask a student to read aloud the first sentence of the problem, and then ask the students, “What is significant in this sentence? Where does the 200 feet belong in our diagram?” After the students respond to the question, I fill this value in on the diagram.
I ask another student to read aloud the second sentence, and ask, “What is significant in this sentence? What the heck is angle of elevation?” After the students respond and we discuss angle of elevation, I fill the 40o in on the diagram.
I ask a student to read aloud the first part of the third sentence, stopping at the comma. Again I ask the students the significance of this statement and fill in the little’s boy’s height on the diagram.
I have a student read the final part of the problem, and we discuss where x should go, and what exactly x represents. I do not at this time make any explicit remarks about what the question asks for or about the addition of the boy’s height.
3. Can we solve this problem using trig?
Here I focus on our diagram and the question, “Is there a right angle in the diagram?” It is key that the students understand at this point in the unit that they must have a right angle in order to use trigonometry.
When I feel that the students understand the question fully, I ask them to set up a proportion and to solve the problem using their calculators to generate the trig value.
4. What does the problem ask for?
I have purposely not directed attention to the boy’s height – I am interested in seeing how many students consider this when they solve the problem. Some will add the 4 feet at the end, some will not. This provides a great opportunity to discuss the importance of reading and rereading a problem, in order to make sure that a question is answered fully and completely, and to ensure that the answer makes sense.
To begin our closing activity, I hand out the problem set entitled Lengths of Segmentsand I ask my students work in their groups to answer the questions. In this problem set I include the diagram for each problem. The next lesson will require the students to provide their own diagrams. I also did not specify what level of rounding I wanted for the answers. This is something that the students and I discuss when they get to it – what level of precision is appropriate in the problem? Usually we agree to round these answers to the nearest whole number.
Problem 7 asks that the students solve for both missing segments. I do not specify whether to use trigonometry or the Pythagorean Theorem, and I encourage discussion among the students, in order to ensure that the students understand that both methods work and provide similarly precise answers.
Problem 10 involves angle of depression. I discuss this with students as we encounter this, and compare it to angle of elevation. This is an opportunity to revisit parallel lines and alternate interior angles.
Problem 12 always generates interesting discussion about the height of the girl and its impact on the problem. This could be an interesting extension of the problem; since the water balloon sails beyond the girl – what would her height have to be for the balloon to hit her head?
I realize that some of these problems may seem slightly violent. The students usually find humor in these (as well as the goofy diagrams) and this helps to make the problems more engaging.
Any problems not completed in class today will be assigned for homework.