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.
Today's plan: warm-up, one more cup stacking system, then move on to volume and surface area of pyramids.
The warm-up was this Daily Desmos, where they have to determine the equation of the line and we check it on Desmos:
I had them work in table groups on the big whiteboards. Here is some of their work:
Next, I had them go around the room looking at other groups' solutions and making comments on the work. Some fixed errors while others wrote comments like "where did this come from?" and "show your work". It was interesting. One group had a blank board (they were a little off-task) and they actually took the time when students were looking at each other's work to find the slope. We went over some of the errors and cleared up some misconceptions. I also had them go to Desmos on their phones to check their equations.
Next we went back to the cups. I put up the equations for the Styrofoam and red cups on the board. I assigned them random groups to answer this question: if the red cups started on the desk and the Styrofoam cups started on the floor, when would both stacks have the same number of cups and reach the same height? Many groups answered a different question - they made the height of each stack 76 cm (the height of a desk) and solved for the number of cups.
I let them struggle with this a little longer after explaining the difference between my question and the one they answered. I had my red cups travelling around the room with me as props to help me explain what I wanted them to find.
One group thought they had it solved so I pulled out the cups and we tried it out - and one stack came up short. Back to the drawing (white) board.
Eventually I had to stop the whole class and talk about what we knew and what we were trying to find. We went over what each of the numbers in the equation represented and whether it would change if you started the stack at a different height. We got to this point together and I let them continue in their groups:
Here is what they got:
The last picture, above, was for a group that calculated that they would need 65 Styrofoam cups to reach the desk so they were changing their Styrofoam cup equation by subtracting 65 from 7 - great strategy, but they did not finish : (
The consensus was about 133 cups and this is what it looked like:
Very close!
We still had a few minutes left and they clearly needed a specific task so I gave them this question to do as an exit card:
Based on the results, I think they understand how to solve these equations (yay!). So much for pyramids today. That will now happen on Monday.
Today was the first day of the new school year. Last semester I blogged daily about my grade 10 applied experience as I was spiralling through the curriculum with activities for the first time. I don't plan on blogging daily again, but really do feel that it helped me reflect about my practice, which I believe is really important and often neglected aspect of our profession. So my plan is to blog when I do something that I think is interesting, however often that is.
In Ontario, most schools are semestered so we teach the same students every day for half the year, then get a new crop of kids in February. We see each of 3 classes for 75 minutes and have a 75 minute prep period each day (when we are not supervising or covering someone else's class). This semester I have MHF4U (grade 12 advanced functions), MPM2D (grade 10 academic math) and MFM2P (grade 10 applied math) first semester.
At the end of last year, one of the cards I got from a graduating student said something to the effect of "I will never forget the first day of math class in September". At the time I read it, I had no idea what we had done back on September 3rd! I did figure it out though - we did the Marshmallow Challenge. And we did it again today in MHF4U! I stole this from someone (thanks and I'm sorry I don't remember who you are). I put students in random groups of (mostly) 4 and gave each group a large whiteboard.
I actually gave them 20 minutes to build as I remember time being really tight last year. Here are some of the final products. These are the sturdiest ones:
and here is the winner:
It was really interesting to watch them interact and I loved hearing the numerous "What if we ...". I was impressed with their efforts and collaboration. We also talked about mindset and learning from mistakes. It was a good first class.
Next up, grade 10 academic. I have done the "Don't Lose Your Marble" activity on the first day of MPM2D for ever (well, almost). Students have to find the relationship between the height of a ramp and the distance a marble will roll. I like this activity because I think it gives students a chance to remember something about linear relationships in a low pressure situation. (Note: I know that it is actually a quadratic relationship and I will tell them that tomorrow. For the small data set that they collect, a linear model works quite well.) They work in groups and help each other remember things like dependent & independent variables, slope, etc. And I get to observe them which is a great way to start to get to know them. I plan on doing many more activities with this class this year so we are off to a good start.
Last period of the (very hot) day was grade 10 applied. The girls all sat together and the boys did the same. I moved a few people around to even out the groups. We talked a bit about spiralling and the fact that we will be working through a lot of activities, which was met with a positive response. Then I gave them Fawn's Noah's Ark problem to work on in groups. I think my next comment is a reflection of why many of these kids are in the applied (vs. academic) math. It is like they have have the curiosity zapped out of them. It is sad that they gave up so easily, or tried to. That they did not want to solve the problem. That they thought they were not smart enough to solve the problem. I continued to encourage them and kept saying "Convince me!" when they came up with an answer. I tried to point out some of the good strategies they were using and nudged them in the right direction if they were really stuck. No one finished the problem in class so I said there would be a prize if anyone comes in with a correct, well written up solution tomorrow. We'll see. There is always a lot of work to build up the confidence in many of the students in MFM2P. I believe they can all succeed and will work toward having them believe the same.
I did accomplish my day 1 goal: learn the names of ALL my students. Now to finish planning for day 2!
