Touching Tangents

You will want to have copies of the “calculating tangent values” handout ready as soon as you’re done with this discussion.  The goal of this part of the lesson is for students to recognize that while sine, cosine, and tangent functions are all trigonometric ratios, sine and cosine values can be read directly from the unit circle, while tangent values come from the ratio of sine to cosine.  I begin this class by asking my students to reflect silently for a moment about the similarities and differences between the sine, cosine and tangent functions. (MP2)  I give the further directive that when they’re ready to share their ideas, I will know because they will put all writing materials down and be looking quietly at me.  Even my seniors have fun with this because they like the idea of all staring at the teacher at the same time…trying to get a response out of me.  I keep it deadpan until the last kid is ready, then feign surprise to see so many eyes all focused on me.  I sometimes make a goofy comment about my features, like “Am I glowing now?” or “I must be having a really bad hair day today.” Other times I just ask for volunteers or draw popsicle sticks (see my strategies folder – “Calling on kids”). 

When everyone is ready I have students share their ideas by either posting them on the board individually or by having a scribe post for the class.  I like to have students get up and moving because it seems to keep them more engaged, even if it’s just to write on the board, but it takes more time, so I have to weigh the value for each class and lesson.  I’ve included two versions of the format I use for recording this kind of information, a venn diagram and a compare/contrast table.  You can hand a copy of one of these out before you ask the students to reflect if you want to give them more guidance, which can be especially helpful for students with language or learning differences. 

I always hope that at least a few students will remember how we generated the sine and cosine graph using the idea of unit circle, but don’t be too frustrated if you only get responses about the trigonometric ratios they learned in geometry.  I just take whatever the students give me and then guide them toward the unit circle and graphing sine and cosine functions.  Your students may more easily see that both sine and cosine use the hypotenuse while tangent doesn’t.  You can ask what the value of the hypotenuse is in a unit circle and move from there to the tangent being sine/cosine.

10-15 minutes

There is a video narrative in my resources that further discusses the pedagogy for this section.

• The goal of this part of the lesson is for students to recognize that the tangent function has some values that are undefined and to identify the asymptotes associated with these values.  Before I hand out the Calculating Tangent Values worksheet I tell my students that they will be working in pairs to complete this activity and ask them to pair with their left shoulder partner. (see my strategies folder for grouping ideas)  I also say that I want students to use “the calculator they were born with” rather than their graphing calculators or any other electronic devices for this activity.  I walk them through the first two lines of the table which give the sine and cosine values for 0 rad and π/4 rad.  I ask how we know that the sine of 0 is 0 and that the cosine of 0 is 1.  After brief discussion, I then ask what the tangent of 0 is.  Because we just finished talking about tan = sin/cos in the previous activity, they generally remember that and can apply it fairly easily.  When I ask about how we know that sin and cos of π/4 are both √2/2, students tend to struggle a bit.  If no one can recall the values, I have them look at the unit circle chart in the back of their textbook.  If your textbook doesn’t have this, I suggest giving the students a copy or having them create their own copy for reference. If your students have never had the opportunity to derive the values for the unit circle, you might consider taking a day to do that with them…it’s a great way to make the connections between right triangle trig, the unit circle and trig functions. I have included nine points for the first part of the table, which should show my students a clear pattern. (MP7) The values and open spaces on the lower section of the table are for putting in additional points to fill out the tangent curves.  Once I’m sure that most of the teams understand how to complete the table, I walk around the room helping as needed while they work.  Within a few minutes I usually have a team tell me that one of the values is “wrong” or “not possible” because they’ve come to the π/2 and get a number divided by zero.  I advise them to simply note it in their chart and try to find the remaining values.  As teams complete the top nine rows, I encourage them to try the remaining rows and even to generate their own angle values.   This allows you to differentiate for your faster/brighter students.  When all the teams have completed the first nine rows, I bring the class back together for discussion of what we’ve found.  At this time I ask if anyone remembers what we say about division by zero.  If there are no responses I prompt them with the idea of the slope of a vertical line (y/x where x is 0).  I don’t spend too much time on this because I want to focus on what the tangent graph looks like, so if they still don’t remember, I tell them it is “undefined”.  We then look at what the tangent values look like graphed on a coordinate plane with the x-axis labled 0 through 2π and the y-axis labeled -1 through 1.  Since we can’t plot “undefined”, there are two places on our graph where the function is not continuous, π/2 and 3π/2. (MP2, MP4)  This leads directly in to a discussion of asymptotes as the lines that the tangent function approaches but never actually touches.

15-20 minutes

• The goal of this part of the lesson is for students to become familiar with graphing the tangent function using their graphing calculators and to recognize the effects of transformations.  I begin by asking my students if they’d rather graph several transformations of the tangent function on graph paper or using their calculators.  Those who opt for graph paper usually switch within the first few minutes of this activity when they see how much quicker and easier the calculators make the task.  (MP4, MP5) I tell the students that for this activity they will be working independently.  I ask each student (including those who want to use graph paper for the rest of the activity) to graph the tangent function on their calculator and make sure that their window is set correctly (x_min = 0, x_max = 2π,  y_min = -1, y_max = 1)  Once everyone has the graph I ask for a volunteer to graph it on the board.  Almost always, the student will include the asymptotes as part of the function.  To avoid putting my volunteer on the spot, I simply ask if everyone agrees with what is drawn.  I generally get universal agreement, although occasionally a student will recognize that the asymptote should not be connected to the tangent curves.  Either way, I remind the class that there are points on the tangent graph that are undefined, with asymptotes.  I point out that on most calculators these appear as one continuous line, but in reality they’re not.  I have them note where the asymptotes cross the x-axis so that they can see that it corresponds to the same points we looked at earlier when we were calculating values. (MP7)  I then tell them they will be looking at transformations of the tangent function, like we did with the sine and cosine functions.  The challenge will be to identify what changes in the parent function result in what changes on the graph.  I hand out the Transforming Tangents worksheet and tell them their objective is to figure out the patterns in tangent transformations.  While the students are working, I walk around checking to make sure they’re on track and helping them with calculator issues. Some struggle with setting up the window, others have a hard time seeing the differences.  For those who still can’t get the window right, I offer time outside of class to help them because I want them to put their energies into the tangent activity now.  For those who aren’t seeing the differences, I try to refer them back to the parent function, and will limit the window even more than one period so that they can see more clearly.  I’ve also found that using the “trace” and/or “table” function sometimes gives these students a numeric connection to the graph which makes it easier for them to understand.  When most of the students are done, I bring the class back for discussion and wrap up.

To close this lesson, I ask students to share and compare their summaries with their table with the goal of coming to a consensus about the effects of different transformations on the graph of the tangent function. (MP3)  I then ask one member from each table to share the consensus of his/her table with the class.  We discuss any discrepancies to reach a class consensus, with direction from me as needed.  The final piece of this lesson is for each student to update her/his summary to reflect the class consensus and then complete the Challenge problem sheet as homework.