MFM2P Update

I started spiraling my grade 10 applied math class a number of years ago and have written daily blog posts for the course twice. The second time, which starts here, also contains links to all the resources I have used.

Every time I teach this course, I tweak and adjust based on the group I have. This semester I have decided to give homework that supports the spiraled approach I take with this course. I am creating one double-sided page of homework per week. Each homework set contains a little of what we are doing in class that week, along with previously learned topics so that students stay on top of the material. So far I am pleased with how it is going.

Here is the link to my Google sheet. There is a column with links which is where you can get the homework sets if you are interested. They are PDFs as I know some people have trouble seeing equations I create using MathType. If you would like the Word files, feel free to send me an email. There are also tabs for the last couple of years so that you can see an overview of what each cycle looks like.

Skyscrapers

I spent a lot of time thinking about what activity I should do with my grade 10 applied students on the first day of semester 2. I wanted them to be engaged in mathematical thinking, preferably with something hands-on (but nothing that would cause complete chaos!) and I wanted them to work with someone else in the class. What I ended up choosing was Skyscrapers from BrainBashers - here is the link to their site. These are logic puzzles with only a few rules:

A completed board would look like this, where the numbers in the grid represent the height of each skyscraper:

I first learned about these puzzles from Alex Overwijk last year and I'm fairly certain that he heard about them from Peter Liljedahl (I spelled that correctly this first try!). Alex let us try them at our math PD day last year using linking cubes as the skyscrapers.

I set up the skyscrapers ahead of time for my class, using a different colour for each height so that it would be easy to see if they had more than one skyscraper of a particular height in the same row or column. It turned out that I found this feature more useful than them as I circulated and checked their work.

Each pair got their first puzzle and these:

Here is an example of one column. We reasoned through the fact that there is only one way to place the skyscrapers if you can see all 4.

Here are a couple of views of the completed puzzle:

I had printed out 6 different puzzles for them to work through, each on a different colour of paper so that I knew which one they were working. I also had the solutions printed on the same colour of paper to make checking their work faster. I checked the first two and then let them go. I should have printed some harder ones as some groups flew through these. I did have blank ones for them to make their own, but I really think this could have been a much richer experience if they had tried some of the harder ones, like these:

This is what I tweeted out:

What you may not realize is that very few grade 10 applied students ask for more "work", so it was awesome that they wanted more! Day 1 was definitely a success. I got to interact with all my students and see a little of how they think, whether they are able to follow directions easily, and how they work with others. It was a good day and a great start to the semester.

Grade 10 Applied Math, February - June 2016

At some point during this semester I said I would blog about what I did with my grade 10 applied class. I have spiralled this course for a number of years now and felt that it was time to change things up a bit. Here is my attempt to remember what I did and why. Firstly though, here is the Google sheet that I used to plan and record what I would be doing during each cycle. There are pictures of the cycles included below.

The first big change I made this time around was to start with trig. I have found that starting with 26-squares did not pose enough of a challenge to some students who decided right away that they didn't need to do any work in this course and behaviour issues ensued. Trig, being new to all students, was a great way to see how they learned and dealt with new content. I also introduced sine, cosine and tangent right away. I used to wait until cycle 3, but found that students tried to pick it up and then forgot how to use the tables. I taught with both trig tables and SOH-CAH-TOA the entire way through the course, and students generally chose one way and stuck with it.

Another change I made had to do with warm-ups. I still did them daily, but I created them as we worked through the course so that they would connect more with the content, either as practice for what we were doing or as lagged practice for a topic that we hadn't looked at for a bit. Here are this semester's warmups. There are no warmups for weeks 6 and 12 because I was testing those weeks.

Here is a look at cycle 2:

Here is a look at cycle 3:

And end-of-year stuff:

I didn't make any other major changes, but did try to spend a bit longer on each topic in cycle 2. I am sure I added a few new things but when you wait several days or weeks to blog about them, you forget. Well, at least I do. If there is anything that you want to know more about, please get in touch!

Quadratic Visual Patterns

You all know how much I love Fawn Nguyen's Visual Patterns site. I use them a LOT. They have been part of my warm-ups for years now and have been some of the best moments of my class each week. I have been recreating my warm-ups for my grade 10 applied class this semester (no, I can't leave things alone). I decided to do this so they align more with the curriculum expectations we are working on or provide lagged practice for other expectations. The warm-ups have included quadratic visual patterns for a few weeks now and I decided to step it up a little this past week with with a couple of patterns from Michael Fenton. If you haven't tried these ones before, I encourage you to do so before you scroll down.

We didn't actually work with the colour-coding, instead looked at the squares that overlapped by 1 each time. We worked with the number of circles first, established that this is a quadratic relationship and then found the rule by comparing the "side length" of each square to the step number.

I really also wanted to look at this pattern using the colours as a guide so we started over and found that we ended up with the same simplified rule.

I am totally impressed that some of my students can do these as they are not easy, especially for students who have struggled a lot with math and have trouble making connections. They have shown incredible progress and I love how willing they are to try.

Here is the next one we did:

 There is a lot going on with this pattern, but the colours really help show the squares emerging.

I should note that these "warm-ups" took about 45 minutes to work through. It was definitely time well spent. 

Linking Cube Towers

I am not doing a daily blog about my grade 10 applied class this semester. This is not because everything is the same as last year or eve last semester. In fact, I have changed almost everything so far in this first cycle. I have different warm-ups, am doing topics in a different order and have made some new resources, too. If I'm not happy with things, I cannot leave them alone. I have thought about sharing what I have done at the end of each cycle - if that's of interest to you, please let me know.

Last year during a lesson study process we looked at an activity that involved students creating towers out of linking cubes and competing to see who could get the tallest tower. This is what I mean by linking cubes (also known as cube-a-links and unifix cubes):

I believe the students all had cards that told them how many cubes to begin with and how many to add each time. It went fairly well, but once a student was "out" (because their tower fell), they were no longer really engaged. Anyway, that is what inspired today's activity which will be my students first look at solving systems of linear equations. I am trying to have them really understand how the starting value and rate of change will affect their towers (and corresponding graphs). We have done a few visual patterns and solved some equations with a variable on both sides, so I think they will be ready for this.

Here is the file. I would love feedback. I will add a postscript if it's a disaster ;)

MPM2D - Day 41: Similar Triangles

We changed tacks today and returned to similar triangles. We had not proven that triangles were similar before solving for missing sides in cycle 1, so that was the focus today. We began with this handout. We looked at the first case (SSS~) in detail together and then discussed AA~ and SAS~. The highlighters came out again to help them identify corresponding sides.

Next, we put this into practice with a couple of examples which allowed students to get a refresher on how to solve for a missing side length.

Part way through those examples I lost half of my class to Inside Ride. The timing wasn't too bad as there an investigation up next. I "stole" this from myself - I made it for my grade 10 applied class. I think it is a nice combination of looking at relationships (linear & quadratic) and similar triangles. They got a lot of the first page done (perimeter) so will continue with area tomorrow.

Here is today's homework.

MPM2D - Day 40: Quiz & More Quadrilaterals

Today started with a quiz. Once they had finished, we looked at one of the quadrilateral questions from yesterday:

We then brainstormed ideas on how to verify that PR bisects QS and came up with a couple of strategies. I asked them to follow through with this one, as I want them to practice these skills.

I didn't give any new homework tonight as I think many needed more time to finish the quadrilateral questions. We will move on to something new on Monday.

MFM2P - Day 86: Reflections

Today and tomorrow are our last two days of classes for this school year. We continued taking up questions from the 2013 exam today, and will finish up posters tomorrow. I thought I would take some time to reflect on this semester's MFM2P class and think about what I want to change for the fall. It is very much looking like I will have a section of MFM2P each semester next year, so it is important to think of ways to improve the course.

Overall, I think I chose good activities and will tweak/replace the ones that didn't meet my expectations. I will also think about where new activities might be beneficial and see if I can create those. Again, I feel like cycle 3 could have been better. Somehow the momentum drops during cycle 3 and I feel like I lose some of my stronger students. Perhaps it is my enthusiasm that drops? This was certainly a rather politically charged and stressful semester... I think that the fact that there was not that much "new" material in cycle 3 played a role, especially since we did all the new stuff right a the beginning of the cycle as a "just in case". Some more specific thoughts:

- I want to do more with the exercise books next time around - they barely made an appearance for the last couple of months. 

- I think that making class posters at the end of cycle 2 would be more beneficial than waiting until the end of the course.

- I will continue doing the warm-ups. I liked the duo-tangs I made and will adjust some of the content for the fall. The mathematics that came out of the warm-ups was really good and helped make connections between the mathematics expectations in the course and between the math and the real world.

- I love the beginning of my recording of observations and conversations journey. I will definitely expand on this in the fall and hope to either create a binder of obs & convs checklists/comments or use the TOTS app that students at my school have created. The entire process of creating these checklists will force me to take the time to really think about what I expect to see from my students and determine specific questions or prompts to help students in need.

- Going along with observations and conversations, I would like to start to use video as a means of evaluating students. Someone at EdCamp last year suggested making a roster and each person would video the person after them. If carefully planned this could work well. Students are generally much better at explaining their work orally than they are in written form. Likely more effort for me, but if I could better gather evidence of their understanding (and misconceptions), I think it would be worth it.

I will continue to think through the changes I would like to implement - it's great that we get to try again and see if we can do better!

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.

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.

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.

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.