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?


Sneak in Solving Linear Equations — SolveMe Mobiles

As part of my day to day warm up series in my grade 9 applied class we are solving  Solve Me Mobiles. Like what VisualPatterns does for my students and learning and discovering linear relations — Solve Me Mobiles is having students solve equations without really knowing it.

Puzzles are presented with minimal distraction and with clarity. Puzzles require no explanation. Students know exactly what its asking for.

Today we started on Puzzle 12 and completed up to puzzle 14 (first 15 minutes of class).

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As students explain their strategies to the class I translate their words into small equations…. All with the goal in mind of sneaking in equation solving.

Jill easily solved a 1-step equation on the left side…and then used pictures to help solve the 2-step equation on the right.

Onto Puzzle 13,

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After Carl explained his strategy I used the opportunity to discuss opposite operations.


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Here’s what one student wrote to solve this one…


And we translated that answer into this one.


All this took about 15 minutes of class time….and then we were onto something else!

Work it in! —- SolveMe Mobiles

Other Warm Up Posts:

Sneaking in Factoring

I started a series of new warm ups for my MPM2D class today. My goal is to sneak in factoring as warmups throughout the semester. By the time we need to learn it (like when we need to factor to solve equations) we will have mastered it already. I also previously snuck in multiplying binomials when we tackled quadratic patterns as Mary Bourassa did in her 2D class.

So today I gave them this slide and said I want you to solve a puzzle!

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They broke out their iPads and used the Algebra Tile app to put together the rectangle. The kids worked away and you could see them trying to put tiles in a way to make the rectangle

….and they soon found out that they had to fit a certain way!! 
On take up we made sure everyone had either my rectangle or a rotated version.

Then we did this one…..

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After we were done I asked the class: “If the combination of squares and rectangles makes up the area, what are the dimensions of the rectangle?” They had a little bit of a hard time here, but finally could see the x + 4 and the x + 2 as the length and the width. I then wrote …

 And then I heard some “aaah”s. We had previously seen both versions of the quadratic expressions and discussed why the factored form helped us out quite a bit if we wanted to find the x-intercepts.

We stopped there….It only took us 15 minutes. Tomorrow we will do a few more…..always writing the factored form after. I will also try to get students to notice efficient strategies to make the rectangles.

  • Why did you put 4 x terms along the width and 2 x terms along the length?
  • How does that relate to the number of singles?

Where I hope to go with these warm ups is to factor all types of trinomials:

  • Perfect Squares


    This time…..make a square

… and get this…



  • Trinomials of the Type ax^2 +bx + c


  • Completing the square too!!!!

This time…make a square

We’ll be definitely working our way out of the app and onto paper with area diagrams…





Completing the square


Completing the square

I think working with these puzzles for the next few weeks first will give us a strong base when it’s time to factor to help solve equations and then complete the square. I think I’ll track all the warm ups we do like this and I’ll post them all!

Lesson Study & Big Ideas

For the last few years my school has been apart of the SSI (School Support Initiative). In small teacher groups we discussed learning gaps we see in our students and then implemented teaching strategies to shorten those gaps.

Last year we identified perseverance as a learning gap. We felt that our grade  9 and 10 applied level students gave up too easily. If an answer didn’t come to them right away they “packed it in” and didn’t really try. We also felt that, especially in math this “packing it in” was due to lack of confidence in their ability. We spent the year focussing on giving great feedback and using growth mindset language. We spent our release time money on mostly bringing in supply teachers while we conferences with students. We talked about where they were and gave them specific feedback to help them get better. We always talked in terms of constant improvement.

This year we are a cross curricular group: 3 math teachers, 1 science and 1 geography. We had a great brainstorming session on learning gaps. Although I still think confidence in math is a huge deterrent in producing quality work we decided our learning gap would be

“Student understanding of main ideas and the big picture”

Our group felt that students sometimes were missing the main idea during and after a lesson. Students might be able to get by memorizing what we were doing in class but missed the big idea.

In my class I wanted to see if this was a problem…. So we set out to get a pre-assessment. We needed to see who in our classes was getting the big idea and who wasn’t. I decided to use Andrew Stadel’s Filing Cabinet problem and not tell them in advance we were learning about surface area. I wanted to see if they could see past the filing cabinet and see that we are solving problems with surface area.

Act 1: The video

If you’re not familiar with this problem read about here or grab the full lesson on

Have a look at this solution….there is a 936 post it note answer in there somewhere!


After we solved the problem I presented them with this slide

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Here are some of their responses.

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Our next step is Lesson Study! Every time we meet we will plan together an upcoming lesson for one teacher. Then we will all go and watch to see the student reactions and thinking during this lesson. This will be my first time going through this process and I’m pretty excited to plan a lesson with 4 other teachers. I am interested to go through this process to improve my teaching and improve student learning around the big ideas!