MFM2P - Days 8, 9 & 10

As I work through this way of teaching MFM2P I realize that some of my anxiety stems from my need for structure.  I like to know exactly what we will do in each class, each day.  Note that I am okay with controlled chaos too - teaching shouldn't be all neat and tidy.  Anyway, I have figured out that if I record what we are doing in class either by taking pictures of the whiteboard or by working on the SMARTboard, it helps me with that whole need-for-structure thing. And this blog is in large part for me to refer to when I teach this course again.  Frankly, I can't believe that anyone is reading it (really - why are you?!?)

Day 8:

We looked for patterns (see day 7) and someone noticed that the area of the big square was equal to the sum of the areas of the two smaller squares.  We drew the squares on the triangles to demonstrate this.

Somewhere along the way we had to go over the fact that, in a right triangle, the...

 And we worked through more examples in the same way - notice there is no a, b, or c here.  No x either.  We figured out the remainder of the table this way, which took a bit of time as some had a hard time figuring out when to add and when to subtract.

I then asked them to see if they could find a pattern among some of the 9 triangles. They needed more to work with so I gave them this:

It was interesting to see that some noticed that the first number went up by 3, the second by 4 and the third by 5, where I notice that the second row is a multiple of the first. They came up with the other three triangles in the same family and separated the remaining three triangles into 3 more families. (We also showed that we could keep generating triangles within a family.)

Day 9:

Students started day 9 by drawing the 9 triangles in their comp books, with correct side lengths. They then measured the acute angles - some needed to be shown how to use a protractor correctly - and recorded them on their diagrams.  They also had to calculate the area of each triangle.  

  This is what it looked like:

Note: I will get them to plot the areas of the triangles within a family based on scale factor and come up with the relationship. Not sure when we will do that, but it's part of the plan!

They quickly noticed that the angles within each family of triangles were approximately the same.

 We talked about the properties of similar triangles.  This was not very exciting for them or for me.  Less so for them, I think.  (if you know of a way of making it more interesting, please let me know)

Somewhere along the way I realized (duh!) that we should have MEASURED the angles, not just stated that they were equal (I am dumb sometimes) but I think we still got what we needed done.  And they had had enough after this!

Day 10:

Having received an email from my principal reminding us that progress reports are just around the corner (really? we have only had 2 weeks of classes!) I decided I better gather some "evidence" of what my students have learned so far. I spent part of my prep period this morning before class making a quiz to cover the types of the linear and quadratic scenarios we had looked at. I made sure they knew that they were allowed their comp books and any handouts for the quiz.  There was surprisingly little complaining and they got to work. They did a good job showing me a lot of what they know, with a little help from me along with way. There were a number of "I know this is wrong, but..." questions which I responded to by telling them to write down what they had just said to me - tell me that you know it's wrong then see if you can fix it. A couple of students needed encouragement to get going or keep going, but overall the results are fairly impressive - they worked for a solid 40 minutes!

I let them take a break for a few minutes after the quiz the we continued with similar triangles. We recapped yesterday's conclusions and started with the examples.

I suggested that adding symbols to the diagrams to help identify corresponding angles would be a good thing.

  

Happily most students remembered the sum of the angles in a triangle so they had no trouble with the next example.  They did need some help figuring out how to tell if triangles are similar if they are only given the side lengths. We will work on that some more next class.

So there is the end of last week.  Yesterday (Monday) was Family Day here so today is day 11 and I will try not to leave my blogging for so long this week, 'cause that makes it so long!

MFM2P - Day 7

Day 7

Today we pulled out the 26 squares and students were asked what they could make with them. There were houses, stars, stacks of presents, a horse and a triangle. They then all had to make triangles. Could they make a triangle with any three squares?  They said "Yes!". So I pulled out two small squares and one large square and could not connect the corners.  In groups, they had to figure out a rule for how to choose 3 squares that would form a triangle.  This was a little like pulling teeth, being a little after 8 am.  I circulated and choose examples like squares of side length 8, 13 and 21 and showed them that if I connected the outside corners I then needed to make the two shorter sides completely flat against the third side to connect those corners.  What was special about the squares I chose? Someone figured out that 8 + 13 = 21 (or whatever the values were for that group) and then they said that the two shorter sides had to be longer than the long side. "Write that down!"  We talked about what kind of triangles they would be able to make with the 26 squares which took us to right triangles.

Which squares could you use to make right triangles.  I offered them grid whiteboards to help form right angles and some groups did pretty well at finding combinations that worked.  Some were close so I pointed them in the right direction "It's not 16-20-25, but try replacing the 16."  Eventually this is what we came up with:

Tomorrow we will finish the tables and consolidate all of this.

MFM2P - Day 6

Day 6:
We started today with those "evaluate" questions.  If y = -5x² + 25x + 30, what is y when x is 6?  We substituted 6 for x in the equation and found that y = 0. I asked what that meant in the context of a ball being thrown so we then looked at the graph (they were working on their graphing calculators):

We found a number of important pieces of information. The y-intercept, or height intercept, is 30 which means that the person threw the ball from a height of 30 m. Actually, there were no units in the question and one of my students pointed this out which was really good.  We settled on time in seconds and height in metres. We then talked about that maximum point, the vertex (they had to look that up) and used trace to find its coordinates. We then interpreted those numbers - the ball reached a maximum height of 61.3 m after 2.5 seconds.

We also found the zero using trace, to confirm what we did algebraically.

We next worked through solving for y given x and solving for x given y, then I let them complete the other two examples. Reflecting about this I know some were not particularly interested in doing the algebra and I'm not sure it was the right time to be doing this. (Note to self: rethink this for next time.)

We then moved on to looking at quadratic scenarios.  We started with a fun shot from yesterday's Olympics - Spencer O'Brien's slope style run modelled with desmos:

We also talked about how parabolas are used in bridges and why. I love bridges. I have done workshops on the mathematics of bridges. There are posters of bridges in my classroom. Anyway, I think they got the point that quadratics are used in real life.

We looked at our first quadratic scenario and they had to create a sketch to represent it.

Then we made the connections between the key features of the graph and the context.

Then we used the graphing calculators to find the height of the ball at various times. And this is where it became all about button pushing and not about any of the math. I didn't give them any good reason to find any of these values. The students who follow instructions well did fine, the ones who have trouble following instructions likely didn't do any of it.  I did not do a good job of this.  Sigh...

and part (d) - well, we didn't even go there.  So I need to rethink how I would change this to make it more meaningful and interesting. I need to change the question so that they can demonstrate how symmetry is useful. I need to think about this some more...

Overall, today was "meh". There were some good parts but I sensed the drudgery in what I was asking them to do.  I think we will hit the 26 triangles again tomorrow, looking for those combinations that make right triangles.  And I'll throw another quadratic scenario their way - one that they can cut out and glue in their comp books. I think some of them need the time and change of pace that affords them.

MPM2P - Day 4 - 5

Day 4:
Yesterday's class was barely 24 hours ago and I am already having trouble remembering what we did (so tired!).  

We took up the questions relating to perimeter and area.  They started with simple questions and moved up to solving 2-step questions:

(I know there are no units - don't judge!)

We then looked at how to graph an equation and see the table of values on the graphing calculator (gc). They worked on this sheet, finding equations by doing regressions on the gc. I don't think anyone completed the first or last column, but they all got through some of the middle.  To be continued...

Day 5:
We started the class by having students cut out and glue the following linear and quadratic graphs in their comp books, leaving the page opposite each one blank.

We filled in the blanks and discussed slope. Some remembered that slope was rise/run - I tried to emphasize change in y over change in x, that slope is a rate of change.

Then we looked at some of the data from the sheet they were working on yesterday. 

Next we looked at some linear scenarios with the goal of finding an equation to represent each relationship. I know, I know, we didn't state what each variable represents. They were engaged (mostly) and I think I would have lost some of them had I insisted that they write "let" statements at that point. We did the first example together and they worked on the next two in their groups. There were many mistakes with the 3rd one... 

y = 1000x + 50

y = 1000x - 50 ...

which we talked about and they understood what the mistakes were.

We then filled in the blanks for the quadratic graph, talking a lot about symmetry and about the vertex. 

They spent the rest of the class completing the handout from yesterday. It was nice when they realized that they could complete the "context" column. I think we will start next class by looking at the "evaluate" column, as some were struggling with that.

Grade 10 Applied: Squares, Day 2 - 3

Day 2 of MFM2P:

Today students completed tables of values for the perimeter and area of a square, looking for patterns (constant first differences) and patterns within patterns (constant second differences).  They did this well, although many had the wrong perimeter for a square of side length 1, which I found interesting (and puzzling).  Having them draw a square and label the side length of 1 on each side quickly cleared that up.  It was also interesting to see that they could find any mistakes in perimeter or area by seeing that the pattern no longer held.  Next, they created scatter plots of their data by hand.  They were really good at this!  Some needed some help figuring out their scale, but overall, they did an amazing job.

They needed to find a formula relating side length and perimeter and another for side length and area.  This was followed with a number of exercises for them to work through.  Some students finished it all - one did so very quickly and I gave him some visual patterns to work on.  Others started the class by finishing getting their squares ready as they had not completed this yesterday.  I also got two new students so they had to start from scratch, but got some help cutting the squares from me.  They were all caught up by the end of the class.  Anyone who didn't finish today's work in class should be doing it as homework.  We'll see how that worked out tomorrow.

Day 3:
We started today by "formally" taking up some of yesterday's work (note to self: need to go over 2nd page tomorrow).  Here is what the board looked like once we had talked about perimeter and area:

Then I had them fill in this information on sheets that they glued in their comp books, followed by us coming up with some key characteristics of linear and quadratic relationships. I will have them fill in the second row of these sheets when we have seen some other linear and quadratic relations.

Next we put the data for side length and perimeter in the graphing calculators, created a scatter plot of the data and performed a linear regression.

And that was 75 minutes!  I think that taking the breaks when they cut and glue are important as it gives students who are behind time to catch up or time to get more individual help from me.

Going Around in Circles

Many years ago, while I was on the board of directors of the Ontario Association for Mathematics Education, we worked on the GUM project.  I was part of the grade 11 (I think) team and we were looking for a good introduction to sinusoidal graphs activity and some good evaluation questions to end the unit.  We did not reinvent the wheel (no pun intended - it's coming) but used the "Wagon Wheel" activity, also known as "Hula Hoop" and "Going Around in Circles". I have no idea to whom credit for this activity should go, but thanks nonetheless!

The idea is for students to work through this activity BEFORE they have seen the graphs of y = sin(x) or y = cos(x). Students should work in groups.  Each group needs some masking tape, a metre stick and a motion sensor (CBR2/GoMotion Sensor). 

The set up goes as follows: create a 1 m diameter circle on the floor using the metre stick (place on floor, use tape to mark centre and each end, rotate 90 degrees, mark each end then do 45 degree turns to get something that resembles a circle).  The alternative is to actually use hula hoops but I find that they don't travel well and are easy to trip over.  One student is the walker and one is in charge of the motion sensor but all have to predict the graph before starting (and they should switch jobs as they go through the activity).

Some advice - have the walker hold something like a large folder or whiteboard which will serve as the target for the motion sensor.  I have also found that the data is much better if the person holding the motion sensor rotates it to follow the motion of the walker.

Unfortunately I have not had the opportunity to use this activity for many years (I just haven't taught the course where this is introduced).  So here are some shots of the very old handout I have.

1. They get a feel for the shape of the graph - you will get all kinds of predictions before they do their first set of data collection. Then the starting position changes - this is a phase shift on the graph, but you will hear lots of different words to describe it as they have not learned the mathematical vocabulary yet.

2. Next, we go faster.  This is fun to watch!  And results in a period change.

3. The following questions have them thinking about changing the amplitude of the graph, again, without using the word amplitude.

4. Changing direction creates a reflection.

 5.  I am not sure why this is not up with #3...

6.  Changing the distance will creates a vertical shift.

The is an optional #7 which is to walk in the shape of an ellipse.  I recommend that the walker holds the motion sensor pointed at a wall to get good data for this.  You can do the whole activity this way, but I find it best for each member of the group to have a job.

By the end of the activity students understand how each of the parameters affects the graph without having seen a single equation.  This way, they have a foundation upon which to build when they see these parameters in an equation.  It's time well spent and it's fun!

Moving Forward through Collaboration

At a recent gathering of techie-minded math and science teachers I had the opportunity to work with Jill Gough (@jgough) which really helped push my thinking about PLNs forward.  Jill is very knowledgeable - part of what she does day-to-day involves teaching her entire school community about being part of a PLC.  She exemplifies what it means to share positive experiences and how to develop a great social presence in your school.

This working weekend also reminded me how much we can learn from each other. How you can take an activity that you have done *forever* and think about it in a whole new way.  Todd Morstein did just that and so I must remember to look at activities and lessons from multiple angles and see what I have not yet seen in them.

So my learning continues.  Today was the first day of 2nd semester. Today was the first day of me spiraling through the curriculum with my grade 10 applied class. I have taught this course many times and have been working more activities into it each time, but I knew that I could still do better for these kids.  Not to generalize, but many grade 10 applied students don't love math or school, for that matter.  There is a higher percentage of kids with IEPs in applied classes and there tend to be more behaviour issues.  These are great kids though, so when I saw what Alex Overwijk (@AlexOverwijk) and Bruce McLauren (@BDMcLaurin) have been doing over the past few years, I wanted to try it out too.  They spiral through the curriculum - there are no units, instead there are activities that each hit some of the 9 overall curriculum expectations for the course. The result is that students see something from each of the expectations within the first 6 weeks of the course and then see them again in more depth as they move through the course.  To me this is the perfect course for this approach.  A lot of group work.  A lot of talking. A lot of "doing" math.  Not a lot of sit down and take notes.


I am very fortunate to be working with Alex on this project.  We have met for a day to work through the first 4 weeks or so, and will continue to collaborate throughout the semester.  It is a little scary to not really know what I'm doing day-to-day, but I feel very positive about the experience and I am going to try to blog about it in a consistent way.

We are starting with 26 squares, which Alex blogged about here. Today, after dealing with paperwork and kids who hadn't returned books at the end of first semester, students started cutting out their squares. The instructions were to cut out each of the squares (1x1 to 26x26), write the side length, perimeter and area on each, then place them into the envelope provided.  We had a little chat about how to find perimeter: 
S: "Add up the four sides."
me: "Great! Is there another way?"
S: "Since all sides are the same length we can multiply the side length by 4."
They were quick to get the area for our example. 

They set to work cutting away and did a good job of their task.  Some did not finish so we will start there tomorrow.  Others got it all done and chose a folder and wrote their name on it and on the composition book (graphed) that I bought them.  I also had them cut out the printed 9 overall curriculum expectations and glue them to the second page of the comp book.  I am mindful that I will have to have extension activities for those who work faster.

Tomorrow they will fill in tables relating side length and perimeter and side length and area and look for patterns (1st differences) and patterns within patterns (2nd differences).  They will graph these relations and start working with graphing calculators (I spent a chunk of my day today updating the OS on my TI-Nspires and changing batteries).  And I will post more about that once I am on the other side of it!