MPM1D1 - Day 59 Knotted Ropes

The warm-up for today was this Which One Doesn't Belong:



It was a nice tie in to what we have been doing (slopes, intercepts, parallel lines) and I heard some great reasons for each one not belonging.

After the warm-up we consolidated the work that we did yesterday on parallel and perpendicular lines. 

The goal for today was to have students explore the relationship between the number of knots in a rope and its length (inspired by this post). Each groups received a length of rope and a length of string. 

They were to create a table, graph and equation to represent both their rope and string, then determine how many knots were needed to make both ropes the same length (same number of knots in each rope). We only had about 30 minutes to complete the activity. I knew that students wouldn't finish but was hopeful. Most groups worked very effectively, but a couple couldn't seem to measure accurately enough. I might need some larger diameter rope that doesn't stretch so much. A number of groups found that their rope was longer after adding a knot to it.

Nobody got past the graphing stage so we'll finish up tomorrow.



MPM1D1 - Day 58 Parallel and Perpendicular Lines

Today's problem was one that I was reminded of at a recent professional development session (thanks @chrisleechss ).

This question generated a lot of discussion. Is 1 a big slope? Does it matter if it's negative or positive?  There were lots of good conversations as students continued to solidify their understanding of slopes. For groups that finished early I asked how their answers would differ if they were allowed to use the numbers 0-9.

The consolidation of this problem led us to talk about horizontal and vertical lines. We talked (again) about the slopes of these lines but we also talked (unexpectedly) about the equations of horizontal and vertical lines.

The main lesson for today was investigating parallel and perpendicular lines. I gave this investigation. It was good for students to practice graphing lines but I think I need to rework it for them to get more out of the perpendicular lines portion.

We'll consolidate the investigation tomorrow.

MPM1D1 - Day 57 X- and Y-Intercepts

It's been a while since we did any work with percentages so today we looked at this Would You Rather problem:

Some groups were quick to choose a price and start working with it. Other found 40% of 15 and 50% of 25 and chose the smaller number :(. I asked what those numbers meant and eventually got them headed in the right direction. When the first group finished I asked if the price of the book made any difference? Was there better deal always a better deal regardless of book price.

Once groups finished I asked them to do the following:

I talked individually with groups about what they knew about the y-intercept. We arrived at the fact that they x-coordinate was always 0 and off they went. Having never done this before some groups struggled a bit. I brought the class together for a couple of minutes and we talked about what it means to be an x- or y-intercept. They went off and completed the questions above in their groups. When they were done they worked on question 2 from yesterday's handout. After everyone was done the questions above we consolidated and wrote a quick note. Students then finished up the practice questions.

MPM1D1 - Day 56 Another Kick at Success Criteria & Standard Form

Last week students looked at an incorrect (level 2) solution to the problem below.

They did a good job with it, but I wanted to take some time today to model how to solve a problem and to develop some success criteria for problem solving.

I gathered students at a whiteboard at the back of the room and we started working on the problem. As we worked through the problem I would periodically stop and ask what we had just done. The peer tutor wrote a list on a different board of all the things we talked about. Here are some of the things we talked about:

Once we were done most students really wanted to get a picture of the list. I guess something about our process spoke to them as being important. We'll come back to the criteria next time I give out a problem.

The plan for today was to look at the equation of a line in standard form. I had groups do the following:

Some of them struggled a fair bit with part b, but other groups blasted through it quickly. Once they were finished we talked briefly about what standard form is and consolidated how to convert from standard form to slope/y-intercept form (since many groups ended up doing some rearranging when making a table of values).

We did an example, then I gave out some practice questions (question 1).

MPM1D1 - Day 55 Error Finding and Angle Theorems

Today's warm-up was a little different. We started by looking at a problem with a fully worked solution. The solution contained errors (it was a level 2 exemplar) and students had to identify the errors then present a complete (level 4) solution.

I gave out a page with two problems along with the level 2 solutions.

The first problem served the purpose of practicing some of the algebra skills that we worked on earlier in the week, that need more work.

I had students work in groups of three up at the board to figure out what happened in the solution. This was difficult for some students. They just wanted to solve the problem their way, which was different from the solution. Once groups figured out the approach used in the problem they were able to determine where the errors were and correct them. There were some good discussions about how to fix those errors. I think students were able to solidify their understanding of the distributive property and collecting like terms.

As a result of this activity my class has now constructed an exemplar for solving these open response type questions. I'm hopeful that next time we solve a problem like this, we will be able to co-construct the success criteria for solving problems like this.

All groups completed the first problem, many were working on the second problem and one group finished both.

We then moved onto this activity (thanks @davidpetro314) to investigate parallel lines, transversals and angle theorems. This was review for most students and most of them seemed to remember doing it in grade 8. Once they were finished with the activity I had them create their own note for their notebooks to remind them of the theorems. Some of the notes were excellent, others were not, but who am I to say what type of note would be useful for all students. I then gave them some questions to practice.

MPM1D1 - Day 54 Finishing Up Barbie Bungee

Today we finished up Barbie Bungee. A couple of groups wanted to collect more data or double check some of their data from yesterday. Students began analyzing the data. They were creating graphs, both by hand on using Desmos. They were extrapolating using their graphs. They were coming up with the equation of their line of best fit. They were performing linear regressions in Desmos and comparing it to their findings. There was a lot more reasoning about the reasonableness of their answers than I expected. Once students had a number of rubber bands they were happy with they began writing their reports.

With about twenty minutes left we headed to the stairwell to see who could get Barbie the closest to the floor without hitting it. It was so much fun. We had one person at the bottom recording in slow motion. Some students were at the top of the stair watching, while others chose to observe from below. The closest group had Barbie touch the floor with her outstretched hand, but not touch her head. There was some debate about this should count or not. What do you think?

I haven't received a copy of the video yet but once I do I will post it.

What a great end to a great activity. I can't wait to read the write-ups.

MPM1D1 - Day 53 Barbie Bungee

We skipped the traditional warm-up today and got right into Barbie Bungee. Our warm-up was co-constructing the success criteria.

As students came into class they received a card (as they always do) that would assign them to a random group. After the bell went I had students gather at a white board at the back of the room. I explained that each group would receive a Barbie and they had to figure out how many rubber bands to tie onto Barbie so that she got as close to the floor without hitting it when she was dropped from the stairwell. I told them that this was an assignment and I wanted them to go to their boards and come up with a list of criteria that they thought should be included in their assignment. I felt like after the time we spent working on the success criteria  for the Pumpkin Time-Bomb assignment, students had a good sense of what should be included in an assignment.

I was not disappointed. The results were far better than they were the first time we co-created success criteria. Each group created a good sized list of useful criteria this time around. After about 10 minutes I stopped them and had them group their criteria into categories of their choosing. As it turned out most, if not all groups, created three different categories. The categories were roughly Data (needed/given/measured), Mathematics (graph, table, equation, line of best fit), Report (description of task and process used, showing your work, proper terminology, units etc.).

I brought the class together, we talked about the categories and some of the items in their categories. I told them that I would organize all of their ideas and send them a written copy via email.

I handed out the Barbies and seven rubber bands and let them go. There were lots of ideas floating around that led to some great thinking.

• We could measure one band and multiply by seven. 
• Should we measure them stretched or not?
• Let's measure from the floor up.
• Our measurements weren't very accurate (In one case adding a 10cm band only added 2cm to the distance Barbie fell).
• Let's do three trials at each level and take an average.
• We should each trial, then watch the video to see how far Barbie fell.

The data collection was time consuming and messy at times but in the end every group came away with a set of data they felt comfortable with. Although it was time consuming I think having students struggle through those difficulties and errors was very worthwhile.

Tomorrow they'll start to analyse their data and begin putting together their individual reports, which will be due sometime next week.

MPM1D1 - Day 52 More Optimization

We started with a quick Estimation 180 to order the glasses from smallest capacity to largest.

Then picked up right where we left off on Friday. Friday some groups did a cylinder question and some did not. It was great to have students working in different groups today. It allowed the expertise to flow through the room. They worked on the two problems below.

The groups that didn't have anyone who had seen a similar problem naturally struggled. I spent some time working with them, but I think they would benefit from some more practice. Once students were done they had some time to work on a couple of questions from each of the pages found on this document. Sadly, much of the individual work was seemed very unfocused. I'm thinking we'll have to revisit this topic at some point.

With about twenty minutes to go in the period I stopped them and we took up the algebra (collecting like terms and distributive property) mastery test that they wrote last week. After taking it up we wrote it again.

MPM1D1 - Day 51 Optimizing Volume and Surface Area

We started with these problems:

I gave the problems orally, one at a time and groups made their way through them. It was great to watch them work. For groups that made mistakes, it was often enough for me to say "Are you sure?" for them to think a bit about what they did and find their mistakes.

We then moved onto today's work.

This problem was closely related to yesterday's Dandy Candies only this time the side lengths didn't need to be integer values. Most groups easily made the connection to yesterday's work and quickly came up with a solution. One of the groups was really struggling so I spent some time walking them through it. 

Then we moved onto these problems:

Theses problems seemed to be at just the right level. Students seemed to be in flow for the entire period. When groups would get stuck I'd ask a question or provided a hint and off they'd go. 

The last problem I gave dealt with a cylinder rather than a rectangular prism.

A couple of groups didn't get this far today, but those that did made some great headway. A couple of groups struggled with what the cylindrical equivalent to a cube would be. I asked how they would approach the problem if the top and bottom were rectangles rather than squares. That was enough to get them going. One of the groups, on their own, actually drew a cylinder inside a cube. It was a thing of beauty. I wish I'd taken a picture of it.

We were out of time so I gave a couple of rectangular prism questions to practice for homework. We'll pick up with the cylinders again on Monday and consolidate all of the optimization.

The period flew by today. Students were right into the work. It was challenging but not so much so that they couldn't overcome the challenges. What a great period.

MPM1D1 - Day 50 Dandy Candies

We started by revisiting this type of problem:

I was curious to see how much of this work that we did a while ago they would remember. Some groups had it figured out right away. Others tried making a table which was great to see. They started with a pen that was 25 by 50 then increased (and decreased) the dimensions by 10. Which meant that they missed the optimal solution. We talked about the properties of the rectangle that seemed to give the largest area and what they noticed about it compared to the others. They quickly realized that their rectangle needed to be a square.

Before we moved onto the main event for the day we consolidated the work that we did on the distributive property yesterday. I also gave a couple of questions for them to try.

The main event for the day was Dandy Candies. I asked what they noticed and what they wondered. There were lots of good observations and a few good questions. The most common thing they wondered about was what question I was going to ask them.

We had some good discussion about volume and surface area and they had some practice calculating surface areas. Some went immediately for the formula at which point we had a discussion about what surface area actually means. No formulas were needed after that.

We finished up the class with a mastery test on collecting like terms, multiplying and dividing monomials and the distributive property.

MPM1D1 - Day 49 Area Models, Exemplars & Success Criteria

We started the class with a number talk. I asked students what 5 times 18 was. They thought quietly about it for a bit and when everyone had an answer we started sharing strategies. I told students that if they found an answer early they should try to come up with another way. I love that my students feel so comfortable with this. They work quietly and for the most part are willing to share their strategies. I also love how there are a huge number of ways to get the answer. On a previous number talk I mentioned the 'doubling and halving' strategy. It was neat to see some students using that strategy today. At the end of the talk I even had a student say "You could also double 18, halve 5 and multiply those together", which led so a good discussion about multiplying by half. Lots of great conversations.

After the number talk we revisited the distributive property, but this time using an area model. I pulled out the algebra tiles, we looked at an example as a group and then I let them try a few examples. Reactions were mixed. I heard "This is really easy" but also "I hate using these. Do we have to use them?". I think having multiple tools (the area model being one) to use can be very helpful.

Once we had practiced with the tiles we revisited our last assignment (Pumpkin Time-Bomb). The results of the assignment were not very good. My favourite was an email with one phrase in the subject and nothing else, followed by another email with another phrase in the subject and one final email with a link to a graph in the subject line. This was one student's assignment. I don't recall multiple emails using only the subject line being a success criterion.

I figured I had two options for this assignment: leave it and move on or spend some time getting it right. I opted for the latter and that's what we did today. I provided an exemplar to groups and asked them to identify the parts or characteristics that make it a good assignment. They came up with some ideas and I helped them notice a few others. They now have a good model. My only fear with providing this is that I'm going to get a class set of assignments that look essentially like the one I did. I'm willing to take a chance on this to see what happens. We will have more assignments later so I'm not too worried about a single assignment. They spent the rest of the period reworking their assignments.

I'm struggling a bit with wrapping my head around success criteria. I've had some great conversations both online and in-person with a ton of people who have more experience with this than I do. These conversations are helping me sort our some of the details but I think I'm just going to have to try a bunch of things, fail at some and repeat.

Some of the questions I had were:

• What happens when the success criteria is all 'fluff' (neatly written, includes units) and no math?
• Do I give the exemplar before we develop the criteria or after? If I give it before then am I just paying lip service to their contributions (since they have the standard in front of them)?

Some of the responses I have received are:

• Provide students with exemplars at different levels and have them assessed by students. -Melanie
• Try giving a Level 2 exemplar and asking what needs to be fixed. -@chrisleechss
• Model the creation of the exemplar with students. -@klaunderville

That last one is a big one! I think I'd like to try it but I'm worried that in doing so I will suck the thinking out of the task. I suppose I could model for a similar task and then give them the actual assignment, which would have the same or similar success criteria. However I decide to do it I'll think I'll capture some video and try to get some feedback from the video.

Thanks to all those helping me along this journey. If you have any other suggestions or comments please feel free to add them below.

MPM1D1 - Day 48 Distributive Property

We started with this Would You Rather question since we haven't done any proportional reasoning for a while.

It was interesting (yet frustrating) to watch how groups tackled this one. Some groups decided to figure out how many cups were in each package and use 'cups' as their unit rate. Some groups seemed obsessed with changing all the units to grams. Some groups wrote out some fractions but weren't really sure what they meant or what to do with them. Finally, a couple of groups figured out what they needed to do. It all took much longer than I expected it to. I sense more proportional reasoning in our future.

Things seemed to flow smoothly yesterday so we continued right along with the distributive property. Once again students worked in groups of three at the whiteboards. I gave out these problems one at a time. Some groups motored through the questions, while other groups seemed to be a little more dysfunctional. I think I need to do more get to know you type stuff every day. We did some at the start of the semester but I just kind of assumed everyone would be working well with everyone else by this point. I really like Laura Wheeler's idea of  having students introduce themselves every day and answering one icebreaker type question. It's probably a tough thing to be changing at this point but I'll keep it in mind for next time.

Once groups were finished we consolidated and I gave them some questions to practice individually.

The best part of the past couple of days was seeing students' confidence grow as they figured things out. That's what makes all of this so worthwhile.

MPM1D1 - Day 47 Multiplying and Dividing Polynomials

We started with Solve Me Mobile #64.

This is the most challenging mobile that we've done so far. Students were busy working away. I gave them a few minutes to think it through then I asked for solutions. I explained that I wasn't just looking for the answer I wanted to know how they did it. Some students started on the left side, others started on the right and still others started by subtracting two hearts from both sides. We talked about how we could use variables rather than symbols and about how it was important to follow and write down a process.

I was hoping to start Barbie Bungee today but some of my students have not handed in the most recent assignment yet and I didn't think it was fair to start another assignment without having returned the last one.

Instead we moved onto multiplying and dividing monomials. I tried to model the work today after the example I saw from Peter Liljedahl last week when he presented to our board about a thinking classroom. I had students work in groups of three at the board through these questions. I gave them one question at a time and I gave it to them orally. I had them create a picture or diagram for the first one so that they could see what was happening. Then they noticed an easier way. I talked to every group about their question to ensure that they were comfortable before moving on. Once they were ready they got the next question. A few groups went to help other groups who were struggling. Overall they moved through the questions quite nicely. We wrote a short summary note, where I levelled to the bottom (I didn't think we were quite ready for students to write their mindful notes). It was a great class.

We had some time left so we rewrote an earlier mastery test then I gave these questions for them to practice.

MPM1D1 - Day 46 Remebrance Day Assembly

We had our Remebrance Day assembly today so there was no math class.

MPM1D1 - Day 45 Exponents & Spreadsheet

Based on the results from yesterday's test my students need more practice with exponents. When they arrived I put them in groups and sent them up to the boards and I fired exponent questions at them. They worked while I walked around and questioned what they were doing. There was lots of great talk in some groups. A couple of groups had a hard time focusing.

After about 20 minutes of practice, we resumed work on Pumpkin Time-Bomb. When I gave this activity I purposefully gave them the entire spreadsheet of data. I wanted them to be able sift, sort and think about what data was useful. It was a bit of a frustrating experience but clearly they need more practice making sense of large data sets. They also need way more practice with spreadsheets. I told them that they should copy the relevant data to a spreadsheet so that they could clean it up. A few students asked what a spreadsheet was. Eeeek!

They worked their way through the activity. Many students finished their analysis as the bell went, a couple stayed to finish after the bell and the rest will have to get it done on their own time.

MPM1D1 - Day 44 Test Day

We had our second test today. Students seemed well prepared in general. I had a few say to me as they came in that they were going to fail, but this seems to be a normal approach for these students (which is kind of sad). It turns out they did alright.

I was very pleased with how well most did on the question about finding the equation of a line between two points. I often talk about this being the most difficult part of the course, but many of them said that they found it easy.

There are a few things that we need to work on, mostly number sense (exponents and solving equations) but we have lots of time to come back to these ideas. I'm also questioning how I assess. The test is a great place to assess the thinking, problem solving and communication skills but I'm wondering if smaller, more frequent (formative?) quizzes  might be a better option for assessing knowledge. What I mean by this is that most students seem to be able to solve an equation when given a context and as part of a bigger problem. They seem to be able to reason their way to the answer and they can check to see that it make sense. But, if I strip away the context and ask them to just solve a two step equation, a good number of them will struggle. We'll keep plugging away at it.

MPM1D1 - Day 43 Group Test

I've been away for the past couple of days and while I was away the class was working through some review. Ideally, had I not been away we would have reviewed on Friday, done a group test on Monday then done an individual test today.

I felt like a third day labelled as a review day might have been a waste of time so I decided today that we would do a group test. To be fair, my students were totally unaware of this until this morning. I decided that rather than assessing their work I would try to assess the conversations that were happening in the groups. I wanted them to help each other understand the concepts that were covered. I was actually surprised at how well they took to the group test without any prior warning. I didn't hear a single complaint.

The nice thing about this test is that I was able to put into practice some of the things that Peter Liljedahl showed HPEDSB secondary math learning teams yesterday. I gave the first question orally and they started working at the board right away. I circulated, listened, asked questions and tried to get groups to do a good job of documenting their solutions.

We had Maria Morris as an observer in the class today to see what a group test might look like. She also circulated, asked questions and helped out. It's always great being able to work with other teachers. I think it's something that we need to do more often.

Groups worked at their own pace and I gave them the next question as they were ready for it. Some groups worked very effectively, while others struggled a bit due to group dynamics.

Overall, I'd say it was a productive period. I did a terrible job of documenting the conversations that I heard. I wrote down a few things, but since it wasn't really a test I did a lot more talking than I might have. In any case, I did get some information down that may help me round out some test marks after tomorrow. I'm hoping to see some great results tomorrow.

I look forward to doing this again for our next test.

MPM1D1 - Day 41 & 42 Review For Cycle 2

I'm at the Ontario Teachers' Federation conference on Financial Literacy today and tomorrow. Hopefully my class was working on the review for their test next week. Most of this came from Denise Cowdry. Thanks Denise.

Monday I'll be away again. Peter Liljedahl is coming to our board. I'm really looking forward to hearing him talk about thinking classrooms. Hopefully my class will work on part 2 of the review for their test.

MPM1D1 - Day 40 Surface Area

We had a guest in our class today. I had a meeting with a teacher from another school this morning and he decided to stick around and check out my class. It's always nice to have visitors. Teaching is often done in isolation. It's nice when we can get together in a classroom and reflect afterwards. We need to do more of this. At some point I need to make time to get #observeMe going.

We started the day with this visual pattern:

I asked students to find an equation that represented the surface area for step n. I handed out the linking cubes and students went to the boards. Some began building the model, others quickly made a table (of values, not an actual table made of cubes). Many just made a table for the number of cubes. Others tried using a formula, without giving it much thought. Still others tried to reason their way through only to fall back on 'the formula'. When  groups got stuck with their chosen formula I asked what surface area was (we haven't really talked much about it in class). Everyone I asked was able to tell me it was the area of all the faces. I asked them to forget about the formula and find the surface area using the cubes they had in their hands and off they went. One group wanted to find the surface area of the rectangular prism then subtract the exposed surface of the cube that was missing. I asked what happens when they take out the middle cube and they realized that they would have to add in the surface area of centre. An neat approach.

Some groups finished the task quickly while others took a long time to get there, but did manage eventually. Once they were done I had them work on these problems at the board. There were lots of great discussions (especially about the Pythagorean Theorem). Today seemed to be one of those days that things just flowed smoothly. I guess having two adults in the room can do that.

With fifteen minutes to go we did a mastery test on finding the equation of a line and I handed out some practice questions for students to work on individually.

MPM1D1 - Day 39 Pumpkin Time Bomb

We started with this Which One Doesn't Belong:

I was very happy with the responses. Rather than just talking about which numbers were different in the equations, many students were talking the slope and y-intercept and how the graph of the equation would look. I think they're starting to make the connection between the different representations.

After the warm-up we moved on to Pumpkin Time-Bomb. I'm not sure what happened this year, but I was so unorganized that I didn't get any pumpkins. What kind of lame teacher does Pumpkin Time-Bomb without actually doing the activity, you ask? That would be me. We watched the video did some estimating, watched act 3 and calculated percent error. Then we fired up the Chromebooks and started looking at all the data. They were shocked about how much data there was and started asking some great questions. What school has the most? Is our school there? Why are some of the numbers weird looking? All of this without me prompting them about what they noticed. I love it.

We talked about how we should only consider rubber bands that were about the same size. I sorted the spreadsheet based on length, then on width. We talked about how some of the widths seemed to be measured in centimetres while others were in milimetres. I then told students to choose a variable (such as diameter, circumference, height, wall thickness) to compare to the number of bands required to explode the pumpkin.

Many students struggled with the messiness of the data. It took a good chunk of time for them to clean the data so that they could use it, but I think the process was a worthwhile one. That's about as far as we got today.