Tree Height #3ActMath #MPM2D #MFM2P

Here is a tree height 3 act math activity I do for right angled trigonometry with both my 2D & 2P classes. The screenshots below were taken from my 2P class this semester.

Act 1: Setup


Some noticings:IMG_2298

Some wonderings:IMG_2299

We do some turn & talk guesses for “too low” & “too high” then we go back to Pear Deck for our best estimate:IMG_2300

Act 2: Measure & Solve

Students downloaded a clinometer app onto one of the phones in their group.

Here are photos of last year’s group out measuring:

Up to the “vertical non-permanent surfaces”¬†to solve in their “visibly random groups”:

This slideshow requires JavaScript.

Act 3: Consolidation

This is one activity I don’t have a true act 3 for – I don’t know the actual height of this tree ūüė¶ I led a class discussion going over the solutions from various groups. We discussed the fact that trig would not find the whole tree height & that groups needed to add the height of the person up to eye level to their value found using trig. I sent groups back to their boards to adjust their solution for this (final photos above).

The whole activity, including the Pear Deck file, can be found here.

‚Äď Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

Problems We Solved in #MFM2P

My plan was to blog about every problem-based learning activity I did this year. I did not succeed; I think I blogged about two from the my MFM2P course? So as a runner up to a full blog post reflection on each, you’re getting one post with a summary image of each activity or problem & a link to my materials for it.
I’ll group them by strand here, but they are not listed in the order that we did the activities. If you’d like to see the progression of activities I used, you can see that here.

Linear Relations

26 Squares: This one I did manage to blog about.Summary (11).jpg

Banquet Hall: 2016.04.22 2P summary.png

Phone Charging: 2016.05.19 2p.JPG

Phone Plans: 2016.04.27 2p.png

Gummy Bears: I did blog about this one here.Summary 2016.02.29 2P.jpg

Measurement & Trigonometry

Lamppost: w/ shadows 2016.04.15 2P Summary.jpg

School Height: w/ mirrors 2016.05.25 (1).JPG

Tree Height: w/ clinometer 2016.05.11 2p summary.jpg

Wheelchair ramp: 2016.06.08 2p (1).JPG

Filing cabinet post-its: Summary Filing Cabinet 3-Act.jpg

Pyramid Post-its: 2016.05.13 2p (1).JPG

Quadratic Relations

26 Squares: I did blog about this one. 2d 2016.02.08 (1).JPG

Visual Pattern: 2016.04.12 2p summary (1).JPG

Not every lesson we did was problem-based. Sometimes I need to do some direct teaching right from the get go, like with expanding & factoring. Other times we explore & investigate by drawing & cutting out shapes, like with similar triangles & trigonometry. But in any case, maybe someone new to the MFM2P course (or not so new to it) will find these activities useful!

‚Äď Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

Trig Practice ‚Äď missing angles #MFM2P #3ActMath

This is the follow up to a post I wrote last week in response to¬†a blog post by Kate Nowak titled In Defense of Unsexy¬†about posting our regular “boring” functional lessons, and not just the cool, innovative & “sexy” ones we think will impress others.

Act 1: Prompt, notice & wonder


Missing Side Length

Act 1: Estimate
CaptureAct 2 & 3: Solve & Discuss strategies & errors as a class
IMG_0918.JPGSince I wasn’t originally planning to blog about this lesson, I only took a photo of one group’s board; the one we voted as the most correct & nicely communicated. The class especially liked that they had solved with 2 different formulas to check their work.

Missing angle w/ Trig

Act 1: EstimateCapture

Act 2 & 3: Solve & Discuss strategies & errors as a classIMG_0919We use a trig table in order to find values, but during act 3 I did some direct teaching about how to use their calculator buttons for sin, cos & tan as they needed to do so in order to get angle measures to the nearest hundredth in their homework on Khan Academy.

Remaining missing angle

Finally we used the PearDeck drawing slide in order to solve for the 3rd angle (this was done individually at their desks as opposed to in our small groups at the vertical boards).Capture (2).JPG

Homework was this set on Khan Academy.

As always, here is the full folder for this activity.

‚Äď Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

Trig Practice – missing sides #MFM2P #3ActMath

A few days ago I read a blog post by Kate Nowak titled In Defense of Unsexy. She argues that we like to blog about the “sexy”, awesome, new, innovative activities & lessons we’ve tried in our classes. But that there is a lack of sharing the plain old “normal” lessons we still use. I’ve been trying to blog more this semester about the activities we do in MFM2P (grade 10 applied), but am guilty of posting mostly¬†the ones I think are more “sexy”. So I wasn’t going to post what we did the last 2 days as the problems chosen were very basic trig ratio practice problems. But Kate has convinced me that this, too, is worth sharing and reflecting on.

So the problems & the prompts are very basic here. But I adapt them & try to make it engaging by layering the 3 act math model onto them.


Act 1: Prompt, notice, wonder, estimate

using_trig_ratios_to_solve_triangles_sides_img1 (1).pngCaptureCaptureCaptureAt first we had a bunch of estimates larger than 8cm. So I asked the students what they thought of the estimates made. A few said that b could not be longer than 8 cm, judging by¬†the diagram which was drawn to scale. So I allowed them to adjust their estimates to what you see above. I still had 2 students that left estimates longer than 8 cm. So does that mean they weren’t listening to our discussion? That they didn’t understand our discussion? That they just didn’t bother to change their estimates? I’m not sure.

Act 2: Solve

IMG_0912.JPGAll the groups solved this, but I only photographed 1 board (the clearest, nicest looking one) as I wasn’t originally going to blog about this activity.

Act 3:

When the problem is an “unsexy” one like this, I consider act 3 to be the class discussion surrounding the strategies used to solve, the clarity of work on the boards, etc.


We then repeated the process for the other missing side.

Act 1:

Act 2:
Half the groups solved using trig, but they often make mistakes when the variable is in the denominator. Another group tried to divide both sides by 8 at that point, and while they got the right final answer, their work did not support it (now I wish I’d photographed each board! lesson learned). This group shown here originally multiplied 8 x 0.9063 to get an answer smaller than 8. They knew their answer was wrong, so I wrote the expression out in green at the top right for them to work with.IMG_0913The other half of the groups used Pythagorean Theorem.IMG_0914

They then started on this Khan Academy homework.

All materials for this lesson are here.

‚Äď Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

TV Antenna Problem #MFM2P #3ActMath

Tuesday’s MFM2P class was presented with this image:TVantenna (1)It is a problem out of a sample textbook I have, but I created my own image for it. Students¬†were asked “What Mathematical questions could we ask about this scenario?”. Their ideas:CaptureSome good ones in there … we talked about the misconception that the wire is 15.4 feet long. Reflecting now I wish I’d taken off the bottom measurement before asking them for questions. I believe in Dan Meyer’s theory of stripping a question down in order to “be less helpful”. And I think I was a bit too helpful here. Although I like that it led to questions about converting between feet and metres as that’s part of our curriculum this year.

Students were then told the angle of elevation of the guy wire is 56¬ļ and to estimate:¬†Capture

I then sent students to their boards (VNPS) to solve.¬†I went to my desk & took attendance & checked homework. I listened to conversations trying to discern whether or not groups were too stuck. Up until today we learned about the trig ratios and could use the ratios and a trig table to solve for missing angles. We had not yet learned how to find missing sides at all (which is what this question requires). Most groups decided fairly early on to label the sides as opposite, adjacent, and hypotenuse; whether by themselves or by looking at another group’s board. But they weren’t sure what to do next. One group figured out they could use the trig ratios. Originally they had written down $latex¬†sin\theta =\frac{opp}{adj}$. I questioned them about whether or not this matched their formula sheet, and then listened to them reason out whether they wanted the sin formula or the tan formula. They settled on tan & proceeded to solve:IMG_0231.JPG

Another group saw this group’s work and tried to do the same:IMG_0232.JPGThey used the top angle as their theta. But did not adjust their opposite & adjacent sides accordingly. This¬†gave me a great teaching moment when we were done – you can see my writing in green & blue. I showed how their work would have been different & that also allowed me to show how to handle the scenario where the missing side winds up on the bottom of the proportion.

A note here: I try not to teach the “trick” of cross-multiplying. But I wonder if flipping the proportion is simply another trick I’m teaching them without them really understanding WHY we can do that. Something to think more about.

Here are the other groups’ work:IMG_0233.JPGThis group seems to have misread the trig table & gotten the wrong tan value.IMG_0234.JPGThis group is dividing angles and also writes the Pythagorean Theorem formula down.IMG_0235.JPGThis group is getting some ideas off the other boards but not too sure where to go with it.

I revealed the correct answer (which the group on the 1st board had) and did the “teaching” you saw written on the 2nd board.

I then gave them the period to work on these practice problems.

The entire folder of materials is here.

‚Äď Laura Wheeler (Teacher @ Ridgemont High School, OCDSB; Ottawa, ON)

VNPSs to the rescue!

This month I have a student teacher teaching my two grade 10 classes each morning. He’s been doing a great job trying out the¬†spiralled curriculum & activity-based teaching approach that I use. He’s also continued using the visibly random groups (VRGs) & vertical non-permanent surfaces (VNPSs) that I have set up in my classes. Today we had a moment that really cemented for us why the VNPSs are so powerful:

A bit of background first. This year I’m teaching the primary trig ratios using trig trainers¬†& a trig table. The trig trainer provides the sine & cosine values for a right triangle with a hypotenuse of 1. Students then use similar triangles to solve for missing information like this:

Screenshot 2015-04-15 at 10.50.27 AM

So far we had covered how to find missing sides, but not yet how to find missing angles using this method. The students had all the knowledge they needed to do so, there was nothing new to teach them besides how to apply their knowledge in a way to find a missing angle.

So yesterday my student teacher started his lesson by putting this problem on the board:Screenshot 2015-04-15 at 10.46.12 AM

He asked the class questions about how they used the trig trainer to solve for missing sides (activating prior knowledge) to elicit ideas about similar triangles and scale factors. He then asked them how they might use the same ideas in order to solve this problem.



No answer.

There were a few awkward minutes while he waited for them to figure out how to apply their prior knowledge to this new example type. He tried rephrasing his question but they weren’t giving him anything. They weren’t willing to venture a guess out loud. He was hoping they would suggest to him the method to solve for the missing angle & he would solve it on the board for them (direct teaching).

But I suspected that if asked them to, most of the students could solve the problem based on what they’ve learned so far, even if they couldn’t verbalize how to do so (or weren’t willing to verbalize it). So from the back of the room I piped up & suggested sending the groups to their assigned vertical surface (each group has a blackboard or whiteboard space assigned to them). My student teacher obliged & sent them to their boards.

Within one or two minutes a couple of the groups were¬†solving the problem – using the exact method that my student teacher hoped they would explain to him in the earlier discussion. The groups that didn’t figure it out right away looked at the boards of those groups that had & quickly caught on to the idea and started solving themselves also. Here is the solution from one group:IMG_8438

Once most groups had solved it, my student teacher asked them to return to their desks & consolidated their learning with the whole group and then assigned some practice problems.

This experience really drove it home how beneficial the vertical surfaces are. When asked to explain orally how to solve the problem, students were not able. But working on the problem at their boards, most groups solved without having to be taught how to do this specific type of problem. And those that didn’t get to the final answer were still able to see the full solution presented, and done so in multiple ways by different group.

So powerful!