Friday, 12 June 2015
MFM2P - Day 85: Exam Prep
Today we jumped right in with a practice exam. I really like having my students work through an actual old exam so that they can get a feel for the types of questions and the length (and how comprehensive it is). I let them work on question 1, then we took it up together. They worked on question 2, then we took it up together. And so on. It is a little odd to spend 75 minutes like this with this class where there has been so little "look at what I'm doing at the front" going on, but it allows me to ensure that they have all gone over the curriculum in some way. We got through the first section on measurement and trigonometry today and will continue on Monday.
Thursday, 11 June 2015
MFM2P - Days 81, 82, 83 & 84: Summative
Stomach flu hit our house on Sunday so I have not been much use so far this week. Hence no blog posts on Monday or Tuesday and yesterday I was trying to get caught up on all the marking that had piled up while I was home. Here's the 2P recap for the week, so far.
On Monday they worked on their end-of-course review with a substitute teacher. It was supposed to be day 1 of their summative, but that had to be postponed.
I dragged myself in to work on Tuesday, only for my 2P class, and we crammed two days worth of group work into one day. I arranged them into groups (of my choosing) and had each group start at one of six different stations. Everything was colour-coded - if they started at the blue station, they wrote on blue paper for every station, with a blue marker. The goal was for them to come up with as many (mathematically-related) questions as possible based on the pictures provided at each station. I gave them about 6 minutes to do this at each station after which time they put their questions in that station's envelope and then they moved on to the next station and started again. At the end of this process they were back at their "home" station and took out the six pieces of paper with questions from their envelope. Their next task was to organize the questions based on the curriculum:
Here is what one group's work looked like:
That was the end of day 1. Ideally (had I not been sick), they would have had the opportunity to go around to read and rank each others' questions (top 2 for each curriculum box). They were supposed to then choose, as a group, the best 2 questions for each curriculum box for their station and write down what information they had and what they needed to know in order to answer the question. They would also have looked for additional questions where needed. We skipped that, but they had at least invested themselves in the process of creating questions and understood where each question fit within our curriculum.
Yesterday and today were for individual work. They each received 8 questions chosen by me, along with the original picture with sufficient additional information provided. They were allowed to use their notes along with any manipulatives that would be helpful.
Those who finished early worked on creating posters as part of their exam review.
On Monday they worked on their end-of-course review with a substitute teacher. It was supposed to be day 1 of their summative, but that had to be postponed.
I dragged myself in to work on Tuesday, only for my 2P class, and we crammed two days worth of group work into one day. I arranged them into groups (of my choosing) and had each group start at one of six different stations. Everything was colour-coded - if they started at the blue station, they wrote on blue paper for every station, with a blue marker. The goal was for them to come up with as many (mathematically-related) questions as possible based on the pictures provided at each station. I gave them about 6 minutes to do this at each station after which time they put their questions in that station's envelope and then they moved on to the next station and started again. At the end of this process they were back at their "home" station and took out the six pieces of paper with questions from their envelope. Their next task was to organize the questions based on the curriculum:
Here is what one group's work looked like:
That was the end of day 1. Ideally (had I not been sick), they would have had the opportunity to go around to read and rank each others' questions (top 2 for each curriculum box). They were supposed to then choose, as a group, the best 2 questions for each curriculum box for their station and write down what information they had and what they needed to know in order to answer the question. They would also have looked for additional questions where needed. We skipped that, but they had at least invested themselves in the process of creating questions and understood where each question fit within our curriculum.
Yesterday and today were for individual work. They each received 8 questions chosen by me, along with the original picture with sufficient additional information provided. They were allowed to use their notes along with any manipulatives that would be helpful.
Those who finished early worked on creating posters as part of their exam review.
Friday, 5 June 2015
MFM2P - Day 80: Ropes (Systems)
Today was pretty terrible so you may want to stop reading now! I felt somewhat like I was trying to herd cats - and was about as successful as I would have been at that task.
We started with the following Balance Bender. I asked them to solve it and write an algebraic solution. Most of my students figured out the correct answer, but very few could translate it to algebra.
Having students come up with algebraic expressions for each of the balances was quite a challenge. I found myself asking "What do you have on this side? How could we write that? What do you have on the other side? How would we write that? What should we do to find... ?" But we did get there and I reminded them that if they had correctly solved it in their heads, clearly they could solve these equation and they needed to figure out how to write down their thinking. I also suggested that if they were having trouble solving an equation, they could turn it into a balance picture to help them visualize.
Then I took the ropes back out. Based on how the class was going so far, I opted to work through the rate of change vs. thickness modelling as a whole class. I pulled up the table from yesterday and asked what they noticed about the equations in the red rope row.
We then did the same for the white rope row and decided that one equation was in inches, while the other was likely in mm.
I pulled up the graph I had done in preparation for this activity, one that I did not intend to use, but given the lack of good data from my class, it was helpful. We found the point where the diameter was 2.54 cm (1") and determined that each knot in the 1" rope would cause it to shorten by 21.08 cm. We could then use this information to answer the original question.
We then moved on to the main focus for today:
Amid the balloons floating around (and later confiscated), ropes being flung and blue hair dye that appeared from someone's backpack, there was little work being accomplished. I even had an observation sheet to help me keep track of what had done what (solved by graphing, solved algebraically) and who could answer my questions (How did you choose your ropes? Why does it matter? If you have a different length of the same rope, how does that change your equation?). A couple of groups did some really good work. One tried to solve graphically but the intersection of the lines was beyond their grid so they solve using Desmos. They were also able to explain the conditions under which ropes would and wouldn't work and related these to their graphs. Sadly no groups were able to test their solution out. As a side note, the other teacher had much more success with her group.
This is what it should have looked like for the thick rope and a red rope:
Algebraically:
And proof:
What I will change for next time:
1. Identify each rope with a letter. So the red ropes would be A, B and C, the thick rope would be D, the white ropes would be E and F, and so on. Students can then easily identify the rope(s) with which they work and I can easily tell if their equation is on the right track.
2. With the ropes identified I could then have a table with the equation for each rope. This could be displayed when we do systems for any students who had not completed the first part of the activity.
3. I need to consider whether doing the rate of change vs. thickness of the rope part of the activity is worth doing.
4. I may get rid of one of the smaller ropes as the two smallest ones are very close in diameter.
I still really like this activity and will reflect more on what I can do to make it run more smoothly next time.
We started with the following Balance Bender. I asked them to solve it and write an algebraic solution. Most of my students figured out the correct answer, but very few could translate it to algebra.
Having students come up with algebraic expressions for each of the balances was quite a challenge. I found myself asking "What do you have on this side? How could we write that? What do you have on the other side? How would we write that? What should we do to find... ?" But we did get there and I reminded them that if they had correctly solved it in their heads, clearly they could solve these equation and they needed to figure out how to write down their thinking. I also suggested that if they were having trouble solving an equation, they could turn it into a balance picture to help them visualize.
Then I took the ropes back out. Based on how the class was going so far, I opted to work through the rate of change vs. thickness modelling as a whole class. I pulled up the table from yesterday and asked what they noticed about the equations in the red rope row.
We then did the same for the white rope row and decided that one equation was in inches, while the other was likely in mm.
I pulled up the graph I had done in preparation for this activity, one that I did not intend to use, but given the lack of good data from my class, it was helpful. We found the point where the diameter was 2.54 cm (1") and determined that each knot in the 1" rope would cause it to shorten by 21.08 cm. We could then use this information to answer the original question.
We then moved on to the main focus for today:
Amid the balloons floating around (and later confiscated), ropes being flung and blue hair dye that appeared from someone's backpack, there was little work being accomplished. I even had an observation sheet to help me keep track of what had done what (solved by graphing, solved algebraically) and who could answer my questions (How did you choose your ropes? Why does it matter? If you have a different length of the same rope, how does that change your equation?). A couple of groups did some really good work. One tried to solve graphically but the intersection of the lines was beyond their grid so they solve using Desmos. They were also able to explain the conditions under which ropes would and wouldn't work and related these to their graphs. Sadly no groups were able to test their solution out. As a side note, the other teacher had much more success with her group.
This is what it should have looked like for the thick rope and a red rope:
Algebraically:
And proof:
What I will change for next time:
1. Identify each rope with a letter. So the red ropes would be A, B and C, the thick rope would be D, the white ropes would be E and F, and so on. Students can then easily identify the rope(s) with which they work and I can easily tell if their equation is on the right track.
2. With the ropes identified I could then have a table with the equation for each rope. This could be displayed when we do systems for any students who had not completed the first part of the activity.
3. I need to consider whether doing the rate of change vs. thickness of the rope part of the activity is worth doing.
4. I may get rid of one of the smaller ropes as the two smallest ones are very close in diameter.
I still really like this activity and will reflect more on what I can do to make it run more smoothly next time.
Thursday, 4 June 2015
MFM2P - Day 79: more fun with ropes
We did today's warm-up together:
We had really good conversations about height being important and whether you would be willing to carry around that many quarters. I asked who thought the quarters would give them more money and who thought the $225 would be more money by a show of hands. The majority went with $225, although it was interesting to hear many say: "Well, we can't know which is best yet." I asked what information they needed and they said the thickness of a quarter. One student took out some quarters so that we could measure, which was great! We actually used the data from the mint to be as precise as possible.
Here is what we did:
We chose to work with a height that wasn't too tall, as the tallest student in the class had already calculated that he would get about an extra $85 if he took the quarters.They came up with two ways of converting the height to cm (or mm) - one (in orange) was a little more exact, but I really liked the reasoning both students demonstrated. They then worked out how many quarters they would need to reach that height and how much money those quarters represented.
I loved that one of my students worked out the break-even point all on his own:
Back to the ropes!
I tried to be clearer with my instructions today and gave them the new handout. I told them to choose a rope that was different from the one they used on Tuesday and that their goal was to determine the relationship between the length of the rope and the number of knots in the rope. They worked in the same pairs as last time. I had a table ready to go on the front whiteboard for them to enter their information. I also had the observation sheet I prepared, filled in with who had completed what the first day of this activity. I used this sheet to monitor progress as I circulated and make notes. I found it really helped me make sure I got back to groups that were progressing more slowly. Here is what it looked like at the end of class (the students' names are covered up along the left):
It's a little hard to see, but I checked off when they had completed their table (and made a note if they worked out their 1st differences as they made their table), their graph and made a note of the equation they got. The groups that did not have the correct equation for their rope got feedback from me and I made note of their revised equations.
Here is the table for the class, so far:
The group whose equation was y = -1x + 51 was working in inches, so this will provide opportunity for some good discussion.
Overall, they worked much better today and made good progress.
I think that the observation sheet, beyond being good practice, helped me feel more structured in my normally quite chaotic room (it is controlled chaos). It provides me with information that I would likely not remember and helps me better plan the next class. I think this was my first step in more formally recording observations and conversations as evidence of learning.
We had really good conversations about height being important and whether you would be willing to carry around that many quarters. I asked who thought the quarters would give them more money and who thought the $225 would be more money by a show of hands. The majority went with $225, although it was interesting to hear many say: "Well, we can't know which is best yet." I asked what information they needed and they said the thickness of a quarter. One student took out some quarters so that we could measure, which was great! We actually used the data from the mint to be as precise as possible.
Here is what we did:
We chose to work with a height that wasn't too tall, as the tallest student in the class had already calculated that he would get about an extra $85 if he took the quarters.They came up with two ways of converting the height to cm (or mm) - one (in orange) was a little more exact, but I really liked the reasoning both students demonstrated. They then worked out how many quarters they would need to reach that height and how much money those quarters represented.
I loved that one of my students worked out the break-even point all on his own:
Back to the ropes!
I tried to be clearer with my instructions today and gave them the new handout. I told them to choose a rope that was different from the one they used on Tuesday and that their goal was to determine the relationship between the length of the rope and the number of knots in the rope. They worked in the same pairs as last time. I had a table ready to go on the front whiteboard for them to enter their information. I also had the observation sheet I prepared, filled in with who had completed what the first day of this activity. I used this sheet to monitor progress as I circulated and make notes. I found it really helped me make sure I got back to groups that were progressing more slowly. Here is what it looked like at the end of class (the students' names are covered up along the left):
It's a little hard to see, but I checked off when they had completed their table (and made a note if they worked out their 1st differences as they made their table), their graph and made a note of the equation they got. The groups that did not have the correct equation for their rope got feedback from me and I made note of their revised equations.
Here is the table for the class, so far:
The group whose equation was y = -1x + 51 was working in inches, so this will provide opportunity for some good discussion.
Overall, they worked much better today and made good progress.
I think that the observation sheet, beyond being good practice, helped me feel more structured in my normally quite chaotic room (it is controlled chaos). It provides me with information that I would likely not remember and helps me better plan the next class. I think this was my first step in more formally recording observations and conversations as evidence of learning.
Wednesday, 3 June 2015
MFM2P - Day 78: Quadratic Visual Pattern & Linear Systems
I have been reading 5 Practices for Orchestrating Productive Mathematics Discussions and participating the in the book chat moderated by Jeff Lay of #OKMath (Tuesdays at 9 pm EDT). We are about half way through the book and there are already so many good strategies around giving a task. The huge amount of preparation suggested in the book pays dividends in terms of knowing what to say to students who are on the wrong track and guiding discussions. I planned on trying to incorporate some of the ideas in my MFM2P class today, but changed plans half way through class. I hope to share that with you tomorrow. Instead, rather ironically, I was under prepared when it came to today's visual pattern. I had found one way of expressing the pattern, but did not give myself time to look for more ways (which is precisely what I should be doing).
Here is today's visual pattern:
The version the students were working with was in grayscale so the patterns formed by the different coloured circles were not evident. They recognized that this was a quadratic pattern, but when I asked how they knew many said that it was because the number of circles was not increasing by the same amount. So I threw some numbers (something like (1,5), (2,15), (3,23), (4,63)) on the board and asked if they represented a quadratic pattern and again, got lots of "Yes, they don't go up by the same amount.". Yikes! I pushed further and asked them to tell me what they knew about this "pattern". They found the first differences and were even more convinced. Someone said something about second differences so we found those. Someone said that the second differences needed to not be constant and I decided to just correct that misconception. We could have looked at other known quadratic patterns and verified one way or the other, but I wanted to keep moving forward. We looked at the number of circles in each step, adding the 4th step to get more data, and showed that the 2nd differences were, in fact, constant.
Then we looked at different ways of seeing the pattern growing and found a rule.
This all took quite a bit of time which is why I decided to keep the ropes for tomorrow. I explained the goals for the ropes. Firstly, to find a relationship between thickness and diameter so that we could extrapolate to find the length of a thicker rope needed for a given number of knots. Secondly, given two different ropes, determine how to make them the same length with the same number of knots. The latter will require students to solve a system of linear equations which we have not done for a little while. So the rest of today's class was spent practicing solving systems with this handout that my colleague put together (thanks, Michelle!). Many students could not remember how to solve these, but with a few questions from me (What's the same in the two orders? What's different? What does that mean?) they did quite well.
Here is today's visual pattern:
The version the students were working with was in grayscale so the patterns formed by the different coloured circles were not evident. They recognized that this was a quadratic pattern, but when I asked how they knew many said that it was because the number of circles was not increasing by the same amount. So I threw some numbers (something like (1,5), (2,15), (3,23), (4,63)) on the board and asked if they represented a quadratic pattern and again, got lots of "Yes, they don't go up by the same amount.". Yikes! I pushed further and asked them to tell me what they knew about this "pattern". They found the first differences and were even more convinced. Someone said something about second differences so we found those. Someone said that the second differences needed to not be constant and I decided to just correct that misconception. We could have looked at other known quadratic patterns and verified one way or the other, but I wanted to keep moving forward. We looked at the number of circles in each step, adding the 4th step to get more data, and showed that the 2nd differences were, in fact, constant.
Then we looked at different ways of seeing the pattern growing and found a rule.
This all took quite a bit of time which is why I decided to keep the ropes for tomorrow. I explained the goals for the ropes. Firstly, to find a relationship between thickness and diameter so that we could extrapolate to find the length of a thicker rope needed for a given number of knots. Secondly, given two different ropes, determine how to make them the same length with the same number of knots. The latter will require students to solve a system of linear equations which we have not done for a little while. So the rest of today's class was spent practicing solving systems with this handout that my colleague put together (thanks, Michelle!). Many students could not remember how to solve these, but with a few questions from me (What's the same in the two orders? What's different? What does that mean?) they did quite well.
Tuesday, 2 June 2015
MFM2P - Day 77: Tying Knots
I decided to change things up a bit today and started with a Which One Doesn't Belong? warm-up. They worked in groups on the big whiteboards on this:
I asked each group to just tell me which one they thought didn't belong - those are the blue check marks below. Then I asked for reasoning, which is in red. Lots of students told me that the bottom left did not belong because it didn't have an x-intercept. I found this very interesting. We had a conversation about what the arrows at the ends of the line mean and whether that line would eventually cross the x-axis. I asked them all to continue to think about reasons why top left and bottom left did not belong. The 2nd round of reasons are in brown.
I then randomly paired them to work on the Tying Knots activity. I struggled with how much to scaffold this one. I have done it in a very scaffolded way in the past and wanted to get away from that while still providing a context for what they were doing. It was not a roaring success. One group did get right to work and were doing an awesome job. But so many other just sat there wanting to be told what to do and where to write it. I assume this is of my own doing, as I have given them structured handouts for much of the semester. To be fair, some were confused by the context and couldn't (didn't want to?) get past that. This is the handout they got today. I am going to work on it some more now... back in a few minutes. Here is the new version. I hope this will be clearer and help provide a lower entry point.
Here is some data collection going on:
Each pair got one of 5 different thicknesses of rope. They had trouble with things like understanding that it shouldn't matter how you place the knots along the rope and figuring out how to calculate the diameter of the rope. I feel like I need to get them to think about what they are doing more before they actually start, but I am unsure how to do that.
Here is some of their data:
One group started on their second rope today so we will need to continue this tomorrow. The goal is to then find a relationship between the diameter and the rate of change so that we can extrapolate to a thicker rope. And then we will solve systems with the ropes. Today's class was disappointing - I hope tomorrow's will be better.
I asked each group to just tell me which one they thought didn't belong - those are the blue check marks below. Then I asked for reasoning, which is in red. Lots of students told me that the bottom left did not belong because it didn't have an x-intercept. I found this very interesting. We had a conversation about what the arrows at the ends of the line mean and whether that line would eventually cross the x-axis. I asked them all to continue to think about reasons why top left and bottom left did not belong. The 2nd round of reasons are in brown.
I then randomly paired them to work on the Tying Knots activity. I struggled with how much to scaffold this one. I have done it in a very scaffolded way in the past and wanted to get away from that while still providing a context for what they were doing. It was not a roaring success. One group did get right to work and were doing an awesome job. But so many other just sat there wanting to be told what to do and where to write it. I assume this is of my own doing, as I have given them structured handouts for much of the semester. To be fair, some were confused by the context and couldn't (didn't want to?) get past that. This is the handout they got today. I am going to work on it some more now... back in a few minutes. Here is the new version. I hope this will be clearer and help provide a lower entry point.
Here is some data collection going on:
Each pair got one of 5 different thicknesses of rope. They had trouble with things like understanding that it shouldn't matter how you place the knots along the rope and figuring out how to calculate the diameter of the rope. I feel like I need to get them to think about what they are doing more before they actually start, but I am unsure how to do that.
Here is some of their data:
One group started on their second rope today so we will need to continue this tomorrow. The goal is to then find a relationship between the diameter and the rate of change so that we can extrapolate to a thicker rope. And then we will solve systems with the ropes. Today's class was disappointing - I hope tomorrow's will be better.
Monday, 1 June 2015
MFM2P - Day 76: Big Quiz
Today my students wrote a "big quiz" on quadratics and trig. They worked well, for the most part, and I was pleased that some took a set of algebra tiles to help them factor. There were 6 students away today. Once they have all written it, I plan to sit down with each student and go over their strengths and weaknesses and help them understand any of the material that is still not clear. This will also give each student an additional opportunity to show me evidence of what they know, based on questions that arise. By the way, I would be happy to send you any tests/tasks/quizzes, if that would be helpful. Just shoot me an email!
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