Circled Triangles

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SWBAT explain and apply the connections between right triangles, the Pythagorean Theorem and the unit circle

Big Idea

Connections are Key! Make a clear connection between right triangle trigonometry and the unit circle with this lesson

Set the Stage

8 minutes

I begin class by asking students to sketch a right triangle then to label one of the non-right angles as theta (θ) and the sides as opposite, adjacent and hypotenuse.

(I walk around while the students are making their sketches to catch any who are still confused about labeling the sides.) When all students are done, I ask them to jot down the trig ratios on the same paper and explain that we’ll be looking at these ratios and right triangles in a new way today.  (Again, I do a quick check to make sure we’re all on the same page.  Refer to the first page of the unit circle resource for a sample sketch and ratios.) 

I wrap up this section by asking students to think of any circumstances where they’ve used a right triangle to help solve a problem.  I usually don’t get much response to this other than examples using the Pythagorean, but sometimes a student will mention the distance formula.  If so, I use that to segue to the main activity for the lesson. 

Put it Into Action

40 minutes

Whole Class Discussion (15-20 min): I've included a resource titled "Unit Circle" that shows all the work you will be doing on the board and that you're expecting your students to create.  This can be used with your students or strictly as a teacher resource at your choice.  If nobody mentioned the distance formula in our beginning discussion, then I bring it up by drawing a X-Y coordinate plane. (You can also project an image of a coordinate plane on the whiteboard or use an image under the document camera.) I label a point on the plane and ask how we can find the distance from the origin to that point.  Most students recall the distance formula, or at least understand that they can use the horizontal and vertical distances and the Pythagorean to find the missing value.  If not, I add the horizontal and vertical components to prompt that understanding. 

Once we clear the distance formula hurdle, I ask students to add an X and Y axis to their sketch paper, along with labels of a, b, and c for the sides of their triangle and the corners labeled (0,0), (X2,0) and (X2, Y2).  (Some students will struggle with aligning the axes because of how they drew their original triangle. (MP6)  I encourage them to either change their triangle or to change alignment of the X and Y axes.  You can see an example of this labeling on the second page of the unit circle resource.)   Students should now have a triangle that has its legs along the X and Y axes and angle θ at the origin.  The sides should be labeled with hypotenuse, opposite and adjacent and also with a (adjacent), b (opposite) and c (hypotenuse). 

Instead of checking each student myself at this point, I have them share their sketch with at least three other people for review.  I encourage them to move around and talk through any disagreements about the accuracy of each other’s sketch. (MP3) Students are not allowed to remain seated and must initial each other’s work as they review and approve it or recommend changes.  (This encourages student ownership and also gives them a chance to move around a bit after sitting for a while.)  When all students have successfully completed the review process I ask them to return to their seats and prepare for the specialty of the day – circled triangles. 

Referring to the triangle I drew on the coordinate plane on my whiteboard, I add a circle that has the origin at its center and just touches the outermost corner of the triangle (X2, Y2).  (Because I’m a pretty terrible artist, I use educreations to make a short video presentation of this and the link is in my resources as “circled triangle”.)  I have students to add the same circle to their papers then ask them if they can identify any of the parts of the circle.  Most students will pick out the center as being at the origin, and if you’re lucky, at least one will say that the hypotenuse of the triangle is also the radius of the circle. (MP2) At this point I tell the students that when the hypotenuse is exactly 1, the circle is called a “unit circle”.  (I usually have to elaborate on the connection between 1 and unit.  Although they should have learned about the unit circle in earlier classes, I'm trying to make the connection between unit circle and right triangle explicit rather than implicit.)  Their sketch should now look like the triangle on the third page of the Unit Circle resource, including the labels they’ve added along the way. 

I ask them to look carefully at their sketch and find the parts of the triangle we started with that now have more than one label.  I have them pair-share what they find, then ask for a few to summarize for the class. (MP2) I ask whether the trig ratios have changed because we’ve added a circle and placed it all on a coordinate plane.  (That may elicit good discussion/debate or not, depending on your students.)

Team work (15-20 min): I ask students to work in teams of 2-3 for this next part.  I propose the following challenge for each team: Can you re-write the trig ratios in a simpler way, using the information from your sketches? (MP1, MP7) (This can be frustrating for some students, so I try to set up my teams carefully for this activity.  I make sure that at least one student in each group is fairly confident and comfortable taking risks/perservering.)  I allow at least 5 minutes for this discussion, depending on the success and effort of the teams.  I also use this time to walk around and monitor teams that are struggling and individual students that have “checked out” for one reason or another and are not engaged at all.  For these teams/kids, I try to give a gentle nudge toward changing the hypotenuse to 1 in the ratios and seeing where that goes.  I pull the class back together when all the teams have either accomplished the goal or have stopped working.  I have one student act as scribe to record information on the board while teams share what they found.  After all teams have had an opportunity to share, we edit the work on the board and finalize our trig ratios. This should include the following:

sin θ = opposite/hypotenuse        sin θ = opposite/1= opposite         

sin θ = Y2

cos θ = adjacent/hypotenuse   cos θ = adjacent/1= adjacent  

cos θ = X2           

tan θ = opposite/adjacent                   tan θ = Y2/ X2 

I have students add this last piece to their original sketches.  (The final page of the unit circles handout has a summary of all the mathematics.)


Wrap it Up

5 minutes

For the closing piece, I give each student a notecard and ask them to write one thing they think they really understand better now on one side of the card.  I then ask them to write one question they have about today’s lesson on the back of the card. This lets me know what areas of this lesson are still confusing to my students and also lets me know what they think they've "got".  If these questions are too general for you, you can try using the questions below instead (or in addition).

  • Explain how a right triangle is related to a circle.
  • Explain or demonstrate how the trigonometric ratios change from a right triangle to a unit circle.
  • Explain or demonstrate that the trigonometric ratios are equivalent for a right triangle and a unit circle.