Introducing Trig through Slope

Here is our lesson today to introduce trigonometry for the first time. We had spent a few days with solving problems with similar triangles. We are spiralling and have done  lots of work recently using slope and the distance formula to classify triangles. I wanted to capitalize on that familiarity with slope to introduce the tangent ratio for the first time.

We started with this….again

Most students like last time chose A and their reason was it was less steep. So I asked “How much less?” “How do we measure that?”……SLOPE was the response and they calculated the slopes to verify.

Next I had them do this…
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I stressed supreme accuracy and added “Try to create a size of triangle you think no one else will make”……I had them measure their rise and run and enter them in this table on the board.


I also kept a running table in Desmos…


As more students added their triangles I could hear them say, “I bet all the slopes should be the same” , “They’re all similar triangles” We took a moment to discuss similarities and make it clear we all have similar triangles and that the ratio between the rise and the run should all be the same. We also discussed why some of our triangles did not have a slope of 1.7. I had them repeat the process with an angle of 45 degrees.


I said out loud that MY slope ratio was 1….and I could see all their heads bobbing up and down….”Yep, we got 1 too”.


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I asked them again to create an angle/triangle (Had them keep the same orientation of the triangle as I did in my diagram) that no one else would.

Measure the rise and the run, then calculate your slope. Keep your triangle and slope hidden, especially from ME.

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Keeping their angles and ratios hidden from me I said…”When I point to you tell me your angle….and I’ll magically tell you your slope” Cue the Oooohs and aaaahs.

I played up the magic bit. I held my calculator up to shield the screen from them.

I pointed at one student they told me “34 degrees”. I punched on my calculator mysteriously and said…”0.67.” The student yelled out….”Hey that’s right”. I went around the room pointing at students and telling them their slopes (ratios). I could see it on their faces, they wanted to know how I was doing this……Boom Let’s talk about Trigonometry.

So I said:

“In math we have these things called functions….they’re like black boxes that take an input and do some number crunching and spit out an output. One function you have used already is the square root function. You give the function 9 and it spits out 3. We math people use a symbol for this function so we all know what is going on. There is another function that will calculate the slope of a right triangle if you give it the angle. So we could write something like this “(I used one of the students angles).


“This is what I was doing when you gave me your angles….I was using the function to calculate your ratio between rise and run. But we don’t usually use the term slope when we talk about right triangles. We use fancy words.” I had them draw a right triangle in their notes and we labeled it with Hypotenuse, opposite and adjacent. Screen Shot 2015-10-08 at 5.07.34 PM

“Instead of using a slope function…..we use the word TANGENT. And instead of using the word rise we use the word OPPOSITE and instead of run we use ADJACENT. So we can write this tangent function equal to the rise/run = opp/adj.”

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“And we math people don’t like to write too much so we really use this version.”

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Then we practiced using the tangent button on our calculators. They pretended to be the magicians and checked each others ratios. We practiced using the inverse tangent button to find angles.

Once we were comfortable we moved into writing the ratio and finding the angle out. We also used this example to write the tangent ratio of the other angle.Screen Shot 2015-10-08 at 2.21.05 PM

and then one more for lengths:

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Since we are spiralling I gave them the homework set (Mary Bourassa Style) to work on….here.

Tomorrow I’ll introduce the Sine and Cosine function.

Using slope here to introduce trig allows us to take something familiar and make something new. Students could see the progression happen and not have trig just thrown at them.

Would to love to hear your thoughts on this. How do you introduce trig?


Solve, Crumple, Toss – Trashketball!

Instead of doing the same old worksheet in class on finding exact values of trigonometric ratios……change it up! I saw the blog post “Solve Crumple Toss” on Kate Nowak’s site and decided to give this a try.


I Took the worksheet and cut up all the questions. I put each question on a half piece of paper.


I called it Trashketball


  1. Take one question at a time and solve/complete it.
  2. Get it checked by the teacher:
  3.     if correct —put your name on it, crumple it up and put in the basket. Get another!
  4.     if incorrect — try again!

Bonus Shot

  1. If you were correct on your first attempt ….. then take another slip of paper, put your name on that,  crumple that up and shoot! If it goes in… stays in the basket. Otherwise it goes in the recycle bin.


At the end of the time the teacher randomly selects one crumpled “ball” the name chosen is the winner!


Just a little different that doing the same old worksheet!



Grade 10 Principles of Mathematics at JMSS will be involved in an ongoing mock crime scene investigation. Throughout the semester clues will be discovered and evidence will be analyzed using Linear Systems, Analytic Geometry, Quadratic Relations, and Trigonometry. A full list of evidence and An introductory video revealing some evidence can be seen below. Statements from the 4 witnesses/suspects can also be downloaded below. As a Grade 10 math student it is your job to discover what happened and who the murderer is! Good Luck.

1. Car tire mark found on the road. 5.  Video evidence that may be related. (REVEALED) – (Need QuickTIME)
2. Picture of suspect and shadow caught on camera. (REVEALED) 6. GPS locations of all four suspects. Statements about locations have been given. (REVEALED)
3. Blood spatter found at the scene of the crime. Re-created in classroom lab. 7. Victim: Mrs. Curl – an ex JMSS math teacher. Also the ex-math department head.
4. Suspects found on location immediately after time of death

  • Mr. Roesch
  • Mr. Orr
  • Ms. Koiminek
  • Mr. Coates
The following are statements taken from each of the suspects found at the scene

  • Mr. Roesch: “I never liked that Mrs. Curl. I never forgave her for making me use that darn Smart Board in my classroom. I didn’t kill her though”
  • Mr. Coates:   “I always like Mrs. Curl. I can’t see why anyone would want to hurt her. I definitely didn’t kill her.”
  • Mr. Orr:         “When she left the school last year she it left ME with double the work!!! I’LL NEVER FORGIVE HER FOR THAT!”
  • Ms. Kominek:“I never really knew Mrs. Curl, but I know I didn’t like her”