I have not blogged in over a week. My sister-in-law passed away a week ago yesterday. It's been a hard week...

What happened at school this week was a bit of a blur combined with me being away. Last Friday my class worked on review. I was at the hospital that day. On Monday, they wrote their cycle 4 trig test, and cycle 4 analytic geometry test on Tuesday. I cancelled the quadratics part as I knew I would not be at school on Thursday or Friday to return their tests and didn't see the point if they could not get feedback. So, on Wednesday we worked on quadratics together and I circulated with a list of students whose work I needed to check for evidence of understanding. I liked this way of assessing and will have to consider how to incorporate it more in the future.

While I was away on Thursday and Friday they worked through a practice exam. I have no idea how that went.

Monday is our last day before exams. I imagine many students will choose to stay home and study, but I will provide help to those who are there. Their exam is on Tuesday morning.

Although I need to think over what changes I would make next time through, I really liked spiralling this course. I haven't heard any negative feedback from students or parents. I have heard positive things and think it really benefits kids who struggle with the pace at which the material is presented with traditional units. I may blog more about this at a later date - I will certainly reflect on the semester, but am not sure whether it will be a public reflection. At any rate, I'm not up for that now.

I would like to publicly say that my class has been fantastic. I feel very privileged to have spent the past 5 months learning with them - I could not have asked for a better group to work with.

## Saturday 23 January 2016

## Thursday 14 January 2016

### MPM2D - Day 79: Summative Day

Today was summative day. The tension level was high, but they seemed to jump right in and get going. I had told them that they should look at it as a conversation with me, as I will be the only one who sees their work, and they should explain what they are finding and why and reflect on their work as they go. From what I saw as my students were writing, I think they did a good job of it.

## Wednesday 13 January 2016

### MPM2D - Day 78

Today my students worked on what they needed to work on to prepare for tomorrow's summative. For some that meant creating more good questions and answering them, for others it meant going over homework sets with which they had struggled. I gave one-on-one help, as needed.

I don't know if it's just me, but I find it difficult at the end of the semester to give appropriate wait time when giving individual help. I will ask a question and want to jump in with the answer or will want to just tell them how to start a question rather than asking a question to get them to think about what they should be doing. I have to remember to keep myself in check and not rush through - even though the semester is almost over, my students are still learning.

I don't know if it's just me, but I find it difficult at the end of the semester to give appropriate wait time when giving individual help. I will ask a question and want to jump in with the answer or will want to just tell them how to start a question rather than asking a question to get them to think about what they should be doing. I have to remember to keep myself in check and not rush through - even though the semester is almost over, my students are still learning.

## Tuesday 12 January 2016

### MPM2D - Day 77: More Summative Prep

Not too much to report today. I started today's class by looking at some of the work they did last night. They shared their work in their groups and then we looked at a couple of student's work with the document camera. We commented on strategies students used to come up with values/equations in questions and that the work done to create a good question may demonstrate a whole lot of the math I hope to see. I told them that I would really like them to do their summative in pen so that I can see everything they do. I tried to show them how the process expectations fit in with their work and how these could be improved, especially how they can reflect upon their work. We looked at sample solutions before they worked on part (b) of yesterday's scenario. We also talked about how they can make this type of open-ended evaluation work for them. They can choose numbers that will either increase or decrease the level of difficulty of their question. They can practice certain skills (finding a triangle's circumcentre, finding the shortest distance between a point and a line, converting between forms of a quadratic, etc.) and see if they can make those work with the scenarios they are presented with on Thursday. I think that they left feeling a little less anxious and like they may have a little more control over what they will be doing on Thursday.

## Monday 11 January 2016

### MPM2D - Day 76: Summative Prep

I started today by going over what we will be doing for the remaining 11 classes before the exam. Then I had them continue the gallery walk from Friday, looking at each other's scenarios and questions. Next, we spent a little time recapping the curriculum expectations (although, because I spiralled the course, it was all pretty fresh) and talking about the process expectations.

I then showed them a scenario and some questions that I came up with a few years ago and tasked them with completing the questions and writing up solutions.

While they started working on that, I discussed homework progress with individual students. Their homework is to complete question (a), above.

I then showed them a scenario and some questions that I came up with a few years ago and tasked them with completing the questions and writing up solutions.

While they started working on that, I discussed homework progress with individual students. Their homework is to complete question (a), above.

## Friday 8 January 2016

### MPM2D - Day 75: Summative, Day 1

Unbeknownst to my students, today was day 1 of their summative. Our school board (district) requires us to have two final evaluations. For this class, that means a summative task and an exam. They are both board-wide evaluations so I can't really share them, but will try to give an idea of what we do. I didn't warm my class that this would be happening today as there was nothing they could do to prepare for today's work and I didn't want to raise anyone's anxiety level.

I organized my students into not-so-random groups of 3 and told them that their job today was to come up with scenarios and then good questions based on their scenario. They were not answering any of these questions. I gave an example of a scenario: Laura is flying a plane from Ottawa to Boston. We then came up some questions that would work for this scenario. We talked a little about trying to make connections between various curriculum expectations and that they should be working on creating rich questions.

They worked really well and had great conversations. I loved that they asked to use the whiteboards even though they knew that they would have to produce their final work on chart paper. Once they had worked through this process, they did a bit of a gallery walk to see what other groups had thought of. They were also encouraged to add feedback and their own questions to others' work.

I think that's all I can say about this. They were awesome - I love how easily they talk to each other about math.

I organized my students into not-so-random groups of 3 and told them that their job today was to come up with scenarios and then good questions based on their scenario. They were not answering any of these questions. I gave an example of a scenario: Laura is flying a plane from Ottawa to Boston. We then came up some questions that would work for this scenario. We talked a little about trying to make connections between various curriculum expectations and that they should be working on creating rich questions.

They worked really well and had great conversations. I loved that they asked to use the whiteboards even though they knew that they would have to produce their final work on chart paper. Once they had worked through this process, they did a bit of a gallery walk to see what other groups had thought of. They were also encouraged to add feedback and their own questions to others' work.

I think that's all I can say about this. They were awesome - I love how easily they talk to each other about math.

## Thursday 7 January 2016

### Breaking the Sine Law

It all started yesterday. I was checking homework from my grade 10 class and saw different answers to the same question. I had found the diagram on-line on

Once I realized that something was up, I triple-checked my work calculating the length of side c, then found angles A and B with both the sine law and cosine law. I could see that for angle B I was getting the first and second quadrant answers, but because I had all three side lengths of the triangle, I could not see the ambiguous case. Nope. No way. What was I missing, I asked? Literally.

As always, the MTBoS came through in spades! Look at all these amazing replies!

Although I could understand the explanations, it wasn't until I saw the diagrams from Sean Sweeney that my brain clicked that for this part of the question, it was the ambiguous case because I wasn't using the third side. I don't know why I couldn't see that before, but I felt like such an idiot. I mean, I teach trig in grade 10 and in grade 12. I know trig. I get trig. I love trig. Why couldn't I figure this out on my own? I haven't taught ambiguous case for years and years (it's in grade 11 in our curriculum, which I never seem to teach), but still, I have taught it and really do understand it. I think the purpose of this post, along with singing the praises and thanking the awesome folks on Twitter, is to remind myself (and others?) that it's okay to not know all the answers all the time. It's okay to ask questions. I likely don't look stupid for asking the questions, despite feeling that way. I certainly never think that of others when they ask questions, so why does that not apply to me? As it turns out, I now know more about the ambiguous case than I think I ever have! Mike Lawler even wrote a cool blog post (link

Thanks to everyone who helped me think this through. I greatly appreciate it.

**mathisfun.com**. Ironically, when I "steal" questions like this, I do check for the ambiguous case as it is not part of the grade 10 curriculum. Clearly I didn't do this very well! Here was the diagram - I had asked my students to solve the triangle.Once I realized that something was up, I triple-checked my work calculating the length of side c, then found angles A and B with both the sine law and cosine law. I could see that for angle B I was getting the first and second quadrant answers, but because I had all three side lengths of the triangle, I could not see the ambiguous case. Nope. No way. What was I missing, I asked? Literally.

As always, the MTBoS came through in spades! Look at all these amazing replies!

Although I could understand the explanations, it wasn't until I saw the diagrams from Sean Sweeney that my brain clicked that for this part of the question, it was the ambiguous case because I wasn't using the third side. I don't know why I couldn't see that before, but I felt like such an idiot. I mean, I teach trig in grade 10 and in grade 12. I know trig. I get trig. I love trig. Why couldn't I figure this out on my own? I haven't taught ambiguous case for years and years (it's in grade 11 in our curriculum, which I never seem to teach), but still, I have taught it and really do understand it. I think the purpose of this post, along with singing the praises and thanking the awesome folks on Twitter, is to remind myself (and others?) that it's okay to not know all the answers all the time. It's okay to ask questions. I likely don't look stupid for asking the questions, despite feeling that way. I certainly never think that of others when they ask questions, so why does that not apply to me? As it turns out, I now know more about the ambiguous case than I think I ever have! Mike Lawler even wrote a cool blog post (link

**here**) inspired (that seems a bit of a strong word but I can't come up with a better one) by my original tweet:Thanks to everyone who helped me think this through. I greatly appreciate it.

### Combined Functions Desmos Activity

Last night I decided that it would be useful for my students to have more practice with the graphs of combined functions. We have come to the end of the unit on this topic so they *should* be able to pull their skills and knowledge together and figure out what functions have been combined and how to create some interesting graphs.

My students did well with the first, second and fourth, but needed help with the third. I have since added a hint that it is a composition of two functions. I think that will help.

I asked them to pair up to do this activity and I really liked the conversations that came out of it. Here is some of the reasoning they provided for the first graph (top left):

And a few for the second (top right):

I hope others might find it useful. You now have the ability to edit within DAB so feel free to make it work better for your students. Also, if you want answers, email me :)

**Here**is the link to my Desmos Activity Builder file. There are four graphs of combined functions and their job is to figure what two functions were combined and how.My students did well with the first, second and fourth, but needed help with the third. I have since added a hint that it is a composition of two functions. I think that will help.

I asked them to pair up to do this activity and I really liked the conversations that came out of it. Here is some of the reasoning they provided for the first graph (top left):

And a few for the second (top right):

I hope others might find it useful. You now have the ability to edit within DAB so feel free to make it work better for your students. Also, if you want answers, email me :)

### MPM2D - Day 74: y = 2^x and Negative Exponents

We have what I think is a weird curriculum expectation lumped in with quadratics in grade 10:

I feel like it's a one-off lesson so I kept it until the end in the hopes that my students wouldn't forget it.

We started by looking at patterns.

They did well at seeing the pattern and answering the questions that follow. We jumped onto Desmos and looked at the two graphs as we zoomed further and further out. We also looked at what happens to the exponential as

Next, we watched James Tanton's video. I really like his explanation of 0 and negative exponents related to the piece of paper (around the 3 minute mark).

We looked at y = 3^x next to see that the same patterns hold true and then looked for a way of evaluating a power with a negative exponent without having to make a table and use the pattern.

We went through some examples together - I chose students to answer using my popsicle sticks. These were fine.

And the first row of the next set of examples went well, also. Then we got to (-2)^(-4) and my ability to take a quick question and turn it into a 10 minute class discussion hit. (Seriously, other teachers seem to finish a lesson with 20 minutes to spare and I don't get through the lesson.) Anyway, we had a lot of discussion about whether the answer should be positive or negative and whether the brackets were needed. I should say that the students were discussing it as I only asked what they thought of each others' answers without adding to the discussion. In the end, those who said yes to brackets convinced those who were saying that brackets did not matter.

I consolidated what they had figured out here:

And that was all we had time for.

I feel like it's a one-off lesson so I kept it until the end in the hopes that my students wouldn't forget it.

We started by looking at patterns.

They did well at seeing the pattern and answering the questions that follow. We jumped onto Desmos and looked at the two graphs as we zoomed further and further out. We also looked at what happens to the exponential as

*x*becomes a very big negative number, and tied that into the pattern above.Next, we watched James Tanton's video. I really like his explanation of 0 and negative exponents related to the piece of paper (around the 3 minute mark).

**Here**is the link. My students didn't want to stop watching as they wanted to know about exponent ½, which is not in the curriculum for this course :)We looked at y = 3^x next to see that the same patterns hold true and then looked for a way of evaluating a power with a negative exponent without having to make a table and use the pattern.

We went through some examples together - I chose students to answer using my popsicle sticks. These were fine.

And the first row of the next set of examples went well, also. Then we got to (-2)^(-4) and my ability to take a quick question and turn it into a 10 minute class discussion hit. (Seriously, other teachers seem to finish a lesson with 20 minutes to spare and I don't get through the lesson.) Anyway, we had a lot of discussion about whether the answer should be positive or negative and whether the brackets were needed. I should say that the students were discussing it as I only asked what they thought of each others' answers without adding to the discussion. In the end, those who said yes to brackets convinced those who were saying that brackets did not matter.

I consolidated what they had figured out here:

And that was all we had time for.

**Here**is today's homework.## Wednesday 6 January 2016

### MPM2D - Day 73: Cosine Law (day 2)

We jumped right back into cosine law today with this example:

They are really doing well with this. We went over what it means to solve a triangle before they worked on the next question.

As we had not practiced using the cosine law to solve for an angle very much, we did a couple of those types of questions.

They seemed to be getting the hang of seeing when they could use sine law and when to use cosine law. We talked a little about the upcoming summative where they will need to show multiple skills, not the same one over and over.

That example flowed nicely (pun intended) into answering "When do I use what?".

They are really doing well with this. We went over what it means to solve a triangle before they worked on the next question.

As we had not practiced using the cosine law to solve for an angle very much, we did a couple of those types of questions.

They seemed to be getting the hang of seeing when they could use sine law and when to use cosine law. We talked a little about the upcoming summative where they will need to show multiple skills, not the same one over and over.

That example flowed nicely (pun intended) into answering "When do I use what?".

**Here**is the link to the page with the following flowchart. I like its simplicity.**Here**is today's homework.## Tuesday 5 January 2016

### MPM2D - Day 72: Cosine Law (day 1)

As with yesterday's class, I started today by asking them to find the side length of a triangle.

They tried various strategies but they could not determine the value of

I was surprised where the hurdles appeared in this process. This was the first one:

I assumed that they would all write the ratio without any difficulty, but student after student (that I chose using popsicle sticks) could not come up with this. There were strange combinations of trig and the Pythagorean theorem, proportions involving trig somewhere along the way, and other convoluted (incorrect) answers. I gave them a few minutes to discuss what was being asked after which we got to the equation we needed.

The second hurdle came when I asked them to expand and simplify (

Seeing that you could substitute expressions from the left side into the right side proved to not be an issue which also surprised me.

They talked about the patterns they saw in the equations before we tackled our first example. I love that the student who wrote this solution made the mistake you see below. So many forget, or miss, the order of operations when they write out all the steps, which leads them to the wrong answer. I suggested that they should just put the entire expression in their calculator and let it do the work. I also reminded them that using the ANS key on their calculators can be really helpful.

I worked on one more example that was a little less straightforward and was their first attempt at solving for an angle using the cosine law.

We will do more work with the cosine law tomorrow and talk about when to use what in trig.

They tried various strategies but they could not determine the value of

*x*. When I asked why they said they didn't have enough information, which we then turned into not having the right information for the tools they had. We then developed the cosine law.I was surprised where the hurdles appeared in this process. This was the first one:

The second hurdle came when I asked them to expand and simplify (

*a*-*x*)². The errors didn't bother me so much as the lack of confidence working this out. We worked through it on the whiteboard and then moved on.Seeing that you could substitute expressions from the left side into the right side proved to not be an issue which also surprised me.

They talked about the patterns they saw in the equations before we tackled our first example. I love that the student who wrote this solution made the mistake you see below. So many forget, or miss, the order of operations when they write out all the steps, which leads them to the wrong answer. I suggested that they should just put the entire expression in their calculator and let it do the work. I also reminded them that using the ANS key on their calculators can be really helpful.

I worked on one more example that was a little less straightforward and was their first attempt at solving for an angle using the cosine law.

We will do more work with the cosine law tomorrow and talk about when to use what in trig.

**Here**is today's homework.## Monday 4 January 2016

### MPM2D - Day 71: Sine Law

I started with this picture and explained that the surveyor was trying to determine the width of the river (if someone can give me the source of this question I would greatly appreciate being able to give credit):

They then worked in groups to find the width of the river using this diagram (it was so nice to see and hear them talking math!). Most groups started by finding the missing angle in the triangle and then got stuck. They knew they needed to use trigonometry but either tried with this triangle as-is, or realized that they didn't have a right angle. I circulated and asked them what they needed to use trig and each group replied that they needed a right triangle. So I asked how they could create one and walked away.

Many groups added altitudes that were not helpful as they didn't know a side length of the right triangle they had created. I asked if they could create a different right triangle (and walked away).

Here is the (edited) solution of the group that didn't need any hints from me:

I told them that they had solved for a side in a non-right triangle! I asked if they would like to not have to always draw the triangle and add the altitude and so on. This led us to...

We talked about when each form would be useful and what information you needed to use the sine law before working on some examples. Many had a really hard time with the first example. I tried to help guide them by writing the sine law on the whiteboard and putting check marks next the information we knew so they could see that we only needed the proportion using A & B.

The next example required them to notice that they didn't have an angle-side pair so needed to find the third angle before using the sine law.

In the final example they found the measure of an angle.

They then worked in groups to find the width of the river using this diagram (it was so nice to see and hear them talking math!). Most groups started by finding the missing angle in the triangle and then got stuck. They knew they needed to use trigonometry but either tried with this triangle as-is, or realized that they didn't have a right angle. I circulated and asked them what they needed to use trig and each group replied that they needed a right triangle. So I asked how they could create one and walked away.

Many groups added altitudes that were not helpful as they didn't know a side length of the right triangle they had created. I asked if they could create a different right triangle (and walked away).

Here is the (edited) solution of the group that didn't need any hints from me:

I told them that they had solved for a side in a non-right triangle! I asked if they would like to not have to always draw the triangle and add the altitude and so on. This led us to...

We talked about when each form would be useful and what information you needed to use the sine law before working on some examples. Many had a really hard time with the first example. I tried to help guide them by writing the sine law on the whiteboard and putting check marks next the information we knew so they could see that we only needed the proportion using A & B.

The next example required them to notice that they didn't have an angle-side pair so needed to find the third angle before using the sine law.

In the final example they found the measure of an angle.

**Here**is today's homework.
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