Today is the last day of the Exeter Math Conference. We have shortened classes and everyone leaves by 1 pm. It has been a fantastic week - lots of interesting and fun math, great company and really good food (including a lot of ice cream!).
The participants in my class have been amazing - they have jumped right in and tried everything. We have had lots of productive discussions and really learned together and from each other.
We started by looking at a couple more examples of warm-ups: Would You Rather and Open Middle. My Thursday blog posts from this past semester have write ups about Would You Rather warm-ups. I am looking forward to adding some Open Middle questions to my warm-ups for September.
I had saved the Barbie Bungee activity for today and they loved it. The idea is that you have to determine how many rubber bands to use to make a bungee cord for Barbie so that she has the biggest thrill possible without getting killed.
We decided to drop from the 2nd floor (which is actually 2 flights of stairs up) of the science building. I measured the height, which was about 7 m, while my participants were working on their models. And then the challenge was on. I dropped each of their Barbies and am happy to report that they all survived the experience! This is a really fun activity. I have blogged about it here and here.
I had one last thing to show which is my day 1 of calculus activity. I have my students recreate a set of Desmos graphs. Some are to refresh the students on polynomials and some use animation, which is not something that my students have played with before.
The blog post, including a video, is here. If you want a link to the actual file, please email me.
And with that, my week at Exeter comes to an end. I really hope to have the opportunity to do this again next year. You can find information on this amazing conference here.
We started with a little 3-act fun from Andrew Stadel. Here is his blog post about Filing Cabinet with all the links to the videos. Here is my blog post from the latest time I have done it with my students.
I also showed them some of Nathan Kraft's craziness (that's crazy in a good way, of course). Toothpick insanity and Starry Night.
Then we played Polygraph: Parabola from Desmos. It was a lot of fun and everyone saw how the game encouraged the use of correct vocabulary and helped you create better questions by showing you what others had asked.
We also took a quick look at Central Park.
I probably sound like a broken record, but if you haven't tried Desmos Activities, you really need to check them out!
Time to get up and get moving! Tying Knots was next. The first part of this activity involves determining the relationship between the length of a rope and the number of knots in the rope. I really like this because, unlike most linear data collection activities, this has a negative rate of change.
We skipped the part involving putting everyone's data together to be able to find the relationship between the diameter of the rope and the rate of change (but it's on the handout) and moved on to figuring out how to get the ropes to be the same length with the same number of knots.
All but one of the groups got it to work which led to interesting discussions and to me adding what you see below for next time.
I have blogged about the Tying Knots activity here, here and here and the handouts are here and here.
We took a quick look at some Always-Sometimes-Never statements and discussed how these can be really good warm-up activities that help students think beyond just their initial reaction to the statement.
I then gave a choice of matching activities.Quadratic (credit to the teachers at Sir Wil for that one), rational, right-angle trigonometry or combinations of functions.
A couple of participants then had a quick but lively game of log war. I think I originally got these from Kate Nowak, so I'll give her due credit.
And, finally, I showed off some of my students' parabolic art - art work created entirely with quadratic equations. Here is that activity's blog post and this is one of my favourites:
We started the day by looking at the remaining visual patterns from yesterday. Participants shared their strategies and we talked about finding the simplified rule IN the pattern as explained by Hedge in her recent blog post here. There were really interesting strategies used for the last pattern - the one that I had really gotten stuck on this semester. I showed them my (Dave Lanovaz's) clever solution, which I blogged about here.
I wanted to get everyone up and moving so we did the "Don't Lose Your Marbles" activity next. I don't think I have blogged about this one, so I will try to describe it in more detail. I do this activity on day 1 of grade 10 academic. It allows me to observe students working in groups and really gives me a feel for the class without them even realizing it. They also talk about math and help each other recall some of what they learned in grade 9. Each group has to determine a relationship between the height of a ramp and the distance a marble will travel after going down the ramp. Here is the handout I give.
Although the relationship is actually quadratic, my students use a linear model which works well for the relatively small data that is collected. Also, my MPM2D students have only modelled linear data so they don't have anything else in their toolkit (you could throw this wide open in Algebra II). I tell them that there will be a competition at the end (with a prize!) where they will need to determine the height of a ramp that will allow the marble to travel a particular distance (I use masking tape to put a start line and finish line on the floor - they measure the distance).
It is really interesting to watch students collect data. Some will do repeated trials and average the results, others are not nearly as meticulous. The ironic part is that most school floors are nowhere near level so attending to precision while collecting data in this activity does not guarantee a win at the end. However, I really like this activity for the math it pulls out and for the opportunity to learn about my students.
The next activity we did involved looking at this picture:
Just as I do with my students, I asked what questions they had. The participants in my class came up with a good list which I wrote on the board. They included "How big is it?", "How old is it?", "How many creatures live in it?", "How many houses could be built from it?", and so on. Great! My students do this in groups on big whiteboards and then go around the classroom to see what questions other groups came up with and together, we choose the "best" question that we can answer. This question is usually something that requires finding the volume (how big is it? how many chairs/houses/toothpicks can you make from it?). I gave them some information to help them find an answer:
What happened next in today's class was awesome! Some chose to look at the shape as a cylinder, or more specifically, as three cylinders. Others looked at it as a cone. It is actually more of a truncated cone, or frustum. I have seen all this before. What I hadn't seen that was so cool was someone who did an exponential regression on the diameter vs. height data (or was it radius? - someone will have to help me out with that question). This is what it looked like:
Wow! A perfect fit! (I had no idea.) They then used calculus - volume of revolution (which is why I think we need radius, not diameter) to calculate the volume. How cool is that?!? Clearly, I don't teach teach volumes of revolutions of solids, but it was exciting for me to learn a new way of solving this. We did look at the "actual" answer on the website, but I think that as with my students, that was secondary to the work they had done. I have blogged about this, minus the calculus part, here and here.
We spent the last part of our class time on a Desmos activity:
All of the Desmos activities can be found at teacher.desmos.com and they are really well done. They allow students to go at their own pace and the teacher can see what each student is doing the whole way through. This lets you know who might need a little individual help as they work through the activity and when you might need to stop the whole class and work through a common error. If you have not checked out these activities yet, you need to do that!
I will be attending some CWiC sessions this afternoon and doing one of my own on Which One Doesn't Belong? I have blogged about WODB? a number of times... announcing the website and incomplete sets may be the most useful links to provide here. Oh, and the website itself is wodb.ca.
This is what I tweeted out this morning:
Speedy Squares starts off by having you build a 2x2 square, 3x3 square, 4x4 square, etc. to help determine how long it would take to build a 26x26 square (see here for why 26x26). The participants in my class used linking cubes and a timer (on their phone for the most part) to collect data which they then graphed and modeled. They did a fantastic job with their data collection and worked together really well. Many had never worked with linking cubes before. As an aside, I showed them that if you took each square and made it into a tower, the heights showed the quadratic function (area vs. side length). They used their model to answer the original question. Jon Orr made a video answer which you can see here. I didn't actually show the video as I was concerned about time (so much fun stuff to do!) but I really should have. That was part I. Part II uses the data from part I to determine the relationship between the number of blocks and the building time which is then used to figure out how long it would take to build a house of your own design using Lego. That's the general idea, but there are more specific blog posts about this activity here and here.
Next up: Visual Patterns. I started by showing them Fawn's site: visualpatterns.org. I love this site so much. The (optional) homework I gave yesterday was a set of visual patterns.
Each of these patterns is linear and we talked about how we saw each one growing. It was really useful to have the linking cubes when we talked about the surface area of the second pattern. The third pattern is my favourite because there are so many ways of seeing it and you can show that they are equivalent using algebra. If you want to know more, I wrote about it here.
Then I gave them the second set of visual patterns:
It turns out (not by accident) that each of these patterns is quadratic. My big message was that I wanted them to see the pattern in the pictures, not go straight to the numbers. The first pattern has a square in the middle that has side length equal to the step number, so the number of tiles in the middle can be represented by n^2. Each also has 4 extra tiles on the corners so the rule here is n^2 + 4. We found two ways of finding the rule for the number of football helmet in the second pattern. There is one way shown here. See if you can find another. Although some participants did not want to stop working on the patterns, I decided we should move on and look at the other three patterns tomorrow.
I introduced 3-act math tasks with basketball shots from Andrew Stadel found on this page (he is also the creator of Estimation 180). I mentioned that Dan Meyer has many 3-act tasks and that Kyle Pearce has also curated a great collection.
We were running out of time, but I quickly described speed dating. This is a fun way of having students practice a skill that may not be that exciting. I chose factoring trinomials. I have blogged about it here.
I then quickly referenced Michael Fenton's fun Desmos activity: Match My Parabola to end today's class.
I also went to a couple of CWiC sessions today which were great. Philip Mallinson's talk, Solving Quadratic Equations with Origami, was about how to find the roots of a polynomial geometrically. It was great. I'm sure I have seen him give this talk before, but it still made me think and was brilliant.
Julie Graves did a talk entitled Quadratic Models without Quadratic Regression. It was really, really good. We saw how, given a partial data set, we could determine where the vertex of the parabola was and come up with an equation to model the data using only linear regressions. This was repeated for an exponential decay function. I generally love all sessions given by the North Carolina School of Science and Math teachers - they always make me think and I walk away with new tools and ideas.
Today started with my class at 8 am. Just like school. We began by taking up homework. Unlike school! They shared what they took away from the blog posts they read last night. Several read posts by Dan Meyer, one by Bree Pickford-Murray, one by Ruth Hickman and one by Fawn Nguyen. All good things and a valuable use of their time. We briefly talked about subscribing to a blog vs. using something like Feedly.
We spent most of today's class on Stacking Cups. None had done this activity before so it was perfect. I have blogged about it starting here (2014) and here (2015). They started with the Styrofoam cups. They had to figure out the number of cups needed to reach my height (stacked one inside the other). I gave each group 10 cups and set out rulers and measuring tapes and let them go. Someone asked my height and I said about 5'7", but told them they were welcome to measure me. They did not feel the need to do this. They worked away measuring their cups, graphing, coming up with an equation and then determining the number of cups needed. Once each group had a number we started stacking the cups. That's when they asked how tall I was again, and felt the need to measure my height accurately : ) They were all within 1 cup of the correct number, which was awesome. We then repeated the process but made it wide open. They could use Styrofoam or red cups and could stack them any way they liked. They did lots of cool stuff.
They struggled somewhat with finding the equation to represent the quadratic pattern (for the picture immediately above) and I let them struggle knowing that we will be doing quadratic visual patterns tomorrow. I did, however, show them how to enter data in a table in Desmos and perform a quadratic regression.
I also introduced them (virtually) to Alex Overwijk whose students did a very impressive cup stacking job:
Next they had to figure out how to use the same number of the two kinds of cups to reach the same height, given that the red cups would start being stacked on the table while the Styrofoam ones started on the floor. I loved that one group had forgotten to account for the height of the table and when they changed their equation, their graphs intersected at a negative number of cups. Hmmm. They realized that they had added the height of the table to the wrong equation and when they fixed that, this is what they got:
We had just about enough time to do my favourite matching activity from Shell Center.
This is such a good activity. It combines linear and quadratic and they need to match up algebraic expressions, tables of values, descriptions in words and area models. Some cards have more than one match and some have to be filled in. It is really excellent as it ties together many concepts.
I left my class some linear visual patterns to work on for homework. This is fun homework!
I spent a good chunk of the rest of my day with the teachers in the Biology Institute as they were working on STEM activities. Jeff Lukens walked them through some data collection activities and I contributed some of the "mathy" side of things.
The evening talk was given by Frank Griffin from Cate School in California. It was funny and inspiring.
In between there was a lot of good food, including ice cream : )
I am at the Math, Science & Technology Conference at Phillips Exeter Academy in Exeter, NH this week. This is a pretty unique conference. It is a week-long, residential (we stay in the dorms) conference where each participant takes two 10-hour courses and also gets to go to a number (up to 4/day) CWiC (45 min) sessions as well as three featured speakers. I am leading one of the courses and will blog each day about that and possibly other cool stuff I see while I'm here. As the conference started yesterday, I am already a day behind with this blog post (!).
My course is entitled "Creating an Engaging, Inspiring and Collaborative Mathematics Classroom" and we will be doing lots of activities from the MTBoS. Here is the recap for day 1:
After introductions, we played Quadratic Headbanz. Each person put on a headband with a quadratic equation written on it. Their goal was to figure out the equation by asking questions with yes/no answers. I actually hand wrote the equations when I made that set, but here is the 2nd set I made with graphs. I love hearing people play and seeing them think of good questions. I love watching their faces as they process new information and figure out how that affects their equation. The take-aways, besides being a fun and engaging activity, is the need for correct vocabulary to both ask good questions and make sense of the answers your get. I really like that if you are struggling with what to ask, you can learn and "steal" good questions based on what others are asking you. You can also easily adapt this activity (which was for rational functions when I stole it from Sam Shah).
Next, we talked about Twitter and blogs and the MTBoS. Well, I think I did most of the talking. I tried to convey my love of Twitter and how much I learn from blogs. I assigned homework: read (at least) one blog post and share with the class on day 2.
Most participants had not brought a device to class, so instead of playing with SolveMe puzzles (which I blogged about a bit here) and with Desmos, I did some demonstrations and we worked through things together as a class. I love the SolveMe site and we talked about how to translate the pictures into algebra. I showed how it will let you transition to that by dragging each side of the balance.
You can also create your own, or have your students create their own which I love.
I then showed some Daily Desmos and talked a bit about how you can use it in class.
Somewhere along the way I also talked about warm ups and showed the warm-up book I made for this past semester. We also talked about whiteboarding (and used little whiteboards to play Headbanz) and spiralling. All this in a 50 minute session : )
I tweeted a bit about the evening speaker's talk. Helen Moore did an outstanding job of making the mathematics relating to the treatment of HIV+ patients accessible and interesting. She has written about it in this book.
Today is the last day of classes. My MFM2P class have their exam tomorrow morning. I really just wanted to say thank you to those of you who have taken the time to read my blog. It is nice to know that what I am sharing may be useful to others.
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!
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.
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.
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.
