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.
Friday, 8 January 2016
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 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.
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.
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 :)
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 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.
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.
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.
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?". Here is the link to the page with the following flowchart. I like its simplicity.
Here is today's homework.
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 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:
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 (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.
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.
Here is today's homework.
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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