MPM2D - Day 36: Triangle Centres

Now that my students have had the chance to practice finding equations of medians, altitudes and perpendicular bisectors, it was time to look at triangle centres. I started by giving them a piece of bright orange paper (it is the day before Halloween, after all) with a triangle on it, which I asked them to cut out.

Then I asked them to balance the triangle on the eraser end of a pencil and take note of the coordinates of the balance point.

Then I asked them how they could calculate the coordinates of the balance point and had them discuss ideas in their groups.

It was really interesting to see the groups that had thought to find the intersection of the medians react to the alternate, and much simpler, approach of finding the average of the x-coordinates and the average of the y-coordinates.

I told them that this point is called the centroid and is one of the triangle centres. Then I had them go to GeoGebra on the Chromebooks and we started constructing and investigating.

I asked them for conjectures on why the orthocentre is sometimes outside the triangle and they did a good job of figuring it out. Some needed a little hint so I added angle measures to my triangle.

There seemed to be the most confusion around perpendicular bisectors, but I think we cleared that up. 

I showed them that I could construct a circle through all three vertices whose centre was the circumcentre of the triangle. Then I constructed line segments from the centre to each vertex and asked what was special about those segments.

I think they saw that this could be useful when they heard some examples.

Then we started the one example for the day, but only got this far.

So I asked if anyone would be upset if I held onto the homework until Monday after we finish this example. No one objected so they can work on their parabolic art over the weekend. Here is today's handout.

MPM2D - Day 35: Equations of Medians, Altitudes & Perpendicular Bisectors

Given that my students had not yet practiced finding equations of medians, altitudes and perpendicular bisectors beyond the examples we did in class yesterday, they got a work period today to do just that. Here is the handout.

MPM2D - Day 34: Median, Altitude & Perpendicular Bisector

We moved on from quadratics today, back into analytic geometry. We started with a quick review of finding the equation of a line, in a slightly different way. None of my students has found the equation of a line using anything other than y = mx + b, so I showed them how to use point-slope form: y = m(x - x1) + y1 and explained why I prefer it.

I asked what they noticed when I wrote:

While we were having this discussion I had a huge coughing fit and had to leave the room as I could neither talk nor breathe very well! A little water and a cough drop helped me continue, but I'm not really sure what I said beyond pointing out the similarities of these two equations. After the coughing fit I did take it one step further and showed them that the equation of any function can be written in this form - in other words, it's not going away. However, I also told them that if they would like to continue using y = mx + b to find the equation of a line, they are more than welcome to do so.

On to the meat of today's lesson:

We worked through an example of each, with emphasis on making a plan before diving in. I also had them draw each one and we checked the reasonableness of our answers at each step against the diagram.

Here is today's handout. I did not give homework as I want them to spend time working on their parabolic art.

MPM2D - Day 33: A Little Algebra with Quadratics

As my students get more comfortable working with quadratics, some have figured out how to calculate the 'a' value when determining the equation given a graph. I decided that it was time for everyone to see how to do this. My plan was to work through four examples together, which you can find here. I would like to say that I thought about where these would take us, but I didn't. I love how things turned out and need to make more time in my planning to ensure that I don't miss going down a path when it presents itself.

We started by writing down the three forms of a quadratic equation on the whiteboard (I wrote, but they told me what to write), and needed to clarify that the vertex was at (h,k). Then we worked on the first example, beginning with "Which form of the equation should you choose?":

This one was next:

At this point they had a fairly solid understanding of how to find the equation and they impressed me when they worked on example 3 - many students didn't need any prompting to find the x-value of the y-intercept, which is traditionally a common issue.

I asked them the do example 4 and, if they finished quickly, to do it a different way.

And then it just seemed natural to ask them to...

I love being able to connect to work that we have already done like this. I also know from checking homework that many students have had trouble correctly expanding and simplifying from vertex form. This gave me an opportunity to see what errors they were making and they all were able to experience success with this type of question.

I then asked what they knew about the graph from the equation in standard form. I wrote down all their responses without comment and then we went to Desmos to see the graph and adjust the answers.

I really like how it all came together. I am trying to ensure that my students don't simply learn a collection of skills, but rather that they understand how they fit together. I think that several pieces of the (quadratics) puzzle clicked today.

Here is today's homework set.

MPM2D - Day 32: Parabolic Art

Having worked on quadratic transformations for the past week, today my students started a little project I love called Parabolic Art. You can read more in my blog post here, but the idea is that each student has to create a piece of "art work" using only parabolas. We went over what they need to know in terms of Desmos - signing in, sharing, adding a domain restriction - before I showed them samples from previous years (some are included in the blog post previously referenced). Here is one of my favourites:

Then they got to work creating their sketches. They had to draw on graph paper and when they were happy (or time was up), they showed me their sketch and I made a note of their subject and added axes to their drawing. This is my answer to the multitude of parabolic art graphs that are already out there - even if they took someone else's design (which they would have a hard time doing as I make them create their sketch in class), they would have to move all the equations to fit the axes I forced on them.

I did not give them a homework set today as I want them to spend some time working on their parabolic art. It is due 1 week from today.

MPM2D - Day 31: Quadratic Transformations

Before letting my class have some time to practice what they have been learning about this week, we went over the different forms of quadratic equations. Although they have seen all three, we had not really looked at them all together.

Next, we went over how to find the equation of a quadratic from its graph. Many were not making the connection between horizontal and vertical translations and the vertex of the parabola. I made that a little more explicit on the whiteboard before working through these examples, with randomly chosen students explaining their process to the class.

When asked how they felt about finding equations given a graph, thumbs were solidly to the side or up. I can attest that many would have been down at the beginning of class, given the homework they handed in.

After that, they worked on the handout I prepared for today, found here. And here is homework set 24.

MPM2D - Day 30: Quadratic Transformations

Today was solidify-this-tenuous-grasp-you-have-on-quadratic-transformations day. I bounced between the Desmos activities and the whiteboard to go over the material, asking questions to random students (chosen via Popsicle sticks) along the way. Vertical translations were okay and horizontal ones were equally not okay! I went over why they seem to go the wrong direction and, although I know some students followed along, others were not making connections at all. For now, they can use the pattern they have observed to graph. We went over the graphs from the first side of the handout from a couple of days ago before moving on to looking at the a value.

They were pretty quick to tell me that the parabola opened up if a was positive and opened down if a was negative. I told them the correct vocabulary for this transformation and showed them the sinusoidal transformation activity I used for my grade 12 class. I think this helped them understand why we refer to the transformations the way we do. It was good for them to see that the transformations have the same effect on any type of graph. Then we looked at the effect of the value of a, really the absolute value of a, but since they haven't seen absolute values yet I omitted that part. We looked at everything in terms of patterns and they seemed to grasp how to graph parabolas with vertical stretches and compressions.

We stopped there today. I often ask them to show me thumbs up/to the side/down to gauge how they confident they feel about the material and the many thumbs down coming into class had moved to sideways if not up by the end of the class. They are definitely still shaky on a lot of this, but they are starting to put it together. I asked them if they would like a work period tomorrow to give them a chance to get a more solid grasp on all of this and they enthusiastically supported that idea.

Here is today's homework set.

MPM2D - Day 29: DAB Quadratic Transformations Part II

I had planned on doing consolidation of the roles of 'h' and 'k' in transformation today, followed by some work on test corrections (yes, I finally got all their tests from Friday/Monday marked last night). It was a good plan, however the Chromebooks that I have borrowed need to go elsewhere in the school tomorrow so I changed my plan and went ahead with part II of the quadratic transformations activity today. While they got going, I checked homework, which was generally pretty well done.

I don't know if I said it yesterday, but I really love running Desmos custom activities. My students are all engaged, collaborating and talking math. Another positive is that students who are absent can work through it from home.

I confess that I wasn't watching what they were doing quite as carefully today as I also returned the tests by individually talking to students about what they had done well and what they needed to work on a little more.

Here are my observations about today's activity:

• Students did really well with the pattern of y = x² and did a great job predicting what y = -x² would look like. They also seemed to have a much better grasp on 'input' and 'output'.
• Their predictions for y = 2x² were fairly good. Here is a sample - certainly not all correct, but there is some strong understanding of what is changing in the graph represented. 
  
• The vocabulary they used for the previous prediction and for y = 0.5x² is interesting. They are clearly used to describing lines as more and less steep. Here is a sample of what they wrote:
  
• They did a good job determine the coordinates of a point on the new curve.
• I need to work on screen 13 as they did not know how to fill in the blank. I was expecting 'y-value' but most wrote 'point' or 'value'.
• Of the only 12 students who got to screen 16, 7 chose blue, 2 chose red, and 3 chose purple (the correct answer). We will need to look at this one tomorrow. I think part of the issue is that the graphs are very close together and many of my students were working on their phones. This screen needs some work...
  
• Only 11 students answered screen 17. It is very informative that only 1 of those students answered correctly (yikes!).
  
• Only 5 student actually entered equations on screen 18 where they had to match the graphs. There were some technical issues that made it more difficult - hopefully Desmos will figure out a better way of hiding equations soon.

Tomorrow we will consolidate both days' work. How things go tomorrow will determine what I do on Friday...

Here is the link to the activity. Here is the homework for today. 

Sinusoidal Transformations

I created a Desmos activity, using Activity Builder, for my grade 12 Advanced Functions class. I tweeted out the link, requesting feedback, and based on the results (thank you!), I thought I needed to set the stage for the activity a little more than I could in 140 characters.

My students have all done transformations of functions. They did them in grade 11, including trig functions, only they were all written in terms of degrees. They have also looked at transformations of polynomial functions this year already. To me, this activity is really making sure that they remember the vocabulary around trig transformations and it is giving them the opportunity to work in radians - that is the only new thing for them. So what may seem like a big leap should not be for my gang. I ran it with them on Monday and they did well. That said, I love some of the suggestions I have gotten and will add them to the activity ASAP. I will also put fewer graphs to match on each screen as it was very challenging for them to sort out the graphs out on a phone.

Here is the activity, if you are curious to take a look.

MPM2D - Day 28: DAB Quadratic Transformations Part I

(DAB = Desmos Activity Builder)

About a week and a half ago, I created by first Desmos activity using Activity Builder. You can read more about it here. Today, I actually got to run the activity with my students!

They were investigating horizontal and vertical translations of y = x². It was weird yet informative to spend a fair amount of time sitting looking at my computer screen as they moved through the activity. Here are some of my observations.

• Having only really worked in depth with linear equations, some students wrote about the slope of the parabola. They didn't have the vocabulary they needed to accurately describe what was happening.
• The immediate feedback I received was fantastic. For example, all students got the question on screen 8 correct. I now know that I don't need to spend extra time on that concept.
  
• The answers on screen 10 were very interesting! Having looked at vertical translations, I asked students to predict what the graph of y = (x + 4)² would look like. Here is a sample of the responses - there is a lot to dig into here:
• Despite my best efforts, they still don't understand horizontal translations. Some students saw that it is counter-intuitive, and can use that, but even they don't seem to understand why. Screens 12 & 13 made this clear to me.
• The value of h in (x - h)² also was a source of confusion. No surprise there.
• However, almost all students got this question correct. Then many got the next one wrong when they were told the transformations and had to write the equation. They are definitely on shaky ground here!
  
• My last two screens talked about 'input' and 'output' which also caused confusion. These students have not been introduced to the idea of a function, so I am not surprised that they had no idea what I was talking about. When I explained what I meant, they did a decent job answering the questions. However, not that many students got this far.

I stopped them with 10 minutes to go so that we could do a little consolidation before the end of class using this handout (which also has content for tomorrow's class). 

I am not sure how long I will need to spend working on horizontal translations tomorrow, but that is where we will start. Here is the link to part I of the activity in case you want to check it out. Here is homework set 21.

MPM2D - Day 27: Test, Day 2

While I was home sick on Friday, my students wrote the first part of the cycle 1 test. Today was part II, focusing on similar triangles and trig. Although I am still sick and really did not enjoy working in a building with no heat (it was -6 Celsius outside this morning), it was nice to be back.

After the actual test questions, I threw this at my students:

I want them to start thinking about how to connect different parts of the curriculum and what makes a good question. This will culminate in their summative task at the end of the semester which is generally very open-ended - given a scenario, students must come up with (good) questions and answer them. I will be curious to see what questions they thought of.

MPM2D - Days 25 & 26

Today was more or less the same as yesterday, and tomorrow will be part I of the test. I will try to update this post with the test sometime next week.

MPM2D - Day 24: Cycle 1 Review

Today was the first of two days of review before the cycle 1 test. I corrected and returned homework set 20 while they got to work. I suggested that they go over any homework with which they had struggled with before starting on this review package. I spent the rest of the class giving one-on-one help as needed.

Since this is such a short post, here's a cute joke that someone tweeted out yesterday.

MPM2D - Day 23: Trig Word Problems

I can't say how many times I have changed my mind on how to teach trig in this first cycle. At some point I had thought I would start with word problems to give students the context around what we are doing. I have started with slope angle which turns into the tangent ratio before, but didn't do that. I still need to take some time to reflect on whether I chose the best path and to make notes of what might work better next time.

For our third day of trig, we worked through word problems. We started by talking about the angle of elevation and the angle of depression (I have no idea where this handout came from so I cannot give credit. If you know, please let me know.) 

At this point I realized that my entire class seemed to still be in a post-turkey haze so I made up an example on the whiteboard and got them going. We talked about how you could find an angle inside the triangle given the angle of depression and came up with a couple of options.

I had them all solve using whichever angle they preferred. This turned out to be a really good thing. I wrote both solutions on the board and many decided that the one on the left was easier as the variable was in the numerator on that first line. This led to a good discussion about whether they could always make that happen. Others didn't care that the variable was in the denominator which also made me happy.

We then did the actual examples I had planned, which turned out to be a little repetitive as many use the tangent ratio.

I purposely removed the diagrams for as many questions as I could as drawing them and correctly labelling them is a big hurdle for some students. They worked through this one and then we consolidated together.

Here is the next one, again they had to draw the diagram:

Then they worked through three practice problems. Along the way we defined "stringer", "guy wire" and "clinometer". I randomly chose names of students to put up solutions to the practice problems and we looked over them together.

We had time to look at one more example together.

At this point, their eyes glaze over because they were overwhelmed by the sheer length of the question. I asked them to read it as many times as they needed to in order to draw a diagram. They tentatively did so. I showed them that they could break it up into two triangles, each of which they could solve. They seemed to understand that they needed to take the sum of the answers to get the height of the taller building.

Here is today's homework set. This is the end of cycle 1 so we will be doing review tomorrow and Thursday, and the test will be on Friday and Monday.