Friday 28 February 2014

MFM2P - Day 18

Day 18
Today was about consolidating their knowledge of solving linear equations. We started by gluing some examples in the comp books and filing them in. (There was space between the examples.)

Then they spent most of the remainder of the class working through a handout while I circulated and helped. I am really impressed at how well they are doing with the algebra. I know, I know I was saying how poor their algebra skills were just a week ago, but I am surprised at how well they are picking it up.

The plan was to finish with one or more Estimation 180s, but my computer decided it needed to reboot so that plan was foiled. Instead we went over how to convert Mr. Stadel's height from feet and inches to metres. Not the fun ending to the week I had planned!

Thursday 27 February 2014

MFM2P - Day 17

Day 17
Instead of counting triangles or squares, etc. today, we counted vertices for the same shapes as yesterday. The handout looked very similar - they had to come up with three representations for the number of triangles (square/pentagons/hexagons) and number of vertices. They clearly did learn something yesterday as they had no trouble starting and many got the equations without any help. They were even using algebra to solve some of the questions! 

Next, we took up the two quizzes they have written. Well, I tried to but my SMARTboard decided it needed a break so we will go over the last two questions tomorrow.

We finished with a little Estimation 180 action. We started with day 1 and each student got their own small whiteboard to write their estimate. 

The twist I threw at them was that I wanted their estimate of Andrew's height in feet AND in metres. I asked them what information they needed from me. One student said he needed Andrew's height. Nice try! They asked for the number of feet in a metre and then for the number of inches in a foot. This was very informal - I did not check their calculations. It was meant to get them thinking about converting between units which is an expectation of the course. They held their boards up and for some, just doing that, was an accomplishment. I made them all hold their boards up whether they had a guess with both units or not - they had to make a guess. No one got his height correct though. We'll do another tomorrow.

Wednesday 26 February 2014

MFM2P - Day 16

Day 16
We changed gears today and moved on to looking at patterns using toothpicks.

They readily came up with the number of toothpicks for each step and had little trouble creating the scatter plots, but had great difficulty finding the pattern. We used used step 1, step 2, etc. to try to help them relate the pattern to what there would be at step 0. It was a tough sell - they really didn't like that there was 1 toothpick at step 0. 

After we had gone over the pattern for triangles and squares they came up with equations for pentagons and hexagons and moved on to the questions.  The questions were all of two types - how many toothpicks would you have if you had <insert big number> triangles/squares/pentagons/hexagons and how many triangles, etc. would have you if you had <insert big number> toothpicks. As expected, they could solve for the number of toothpicks, but had a hard time working out the number of triangles given the number of toothpicks. In fact, several students got decimal values for the number of triangles. I know their algebra skills are weak so I grabbed this opportunity to work through solving equations algebraically (since doing it "their way" wasn't working).

I will continue to work on algebra skills with them throughout the course and hopefully eventually it will stick.

Tuesday 25 February 2014

Finding my Groove

At my school those of us teaching the same course teach the same thing on the same day. That's the way it has been done so that's how it continues to be done. I am stepping out of the box with my grade 10 applied class this semester, spiralling through the curriculum while the teacher of the other MFM2P class teaches it in a more traditional way. Anyway, not to rock the boat too much, I am teaching calculus by units and in the usual "this is what we are covering today" way. I do many investigations with my classes (I have 2 sections of Calculus & Vectors). I try to make sure that my students understand the why, always. But I still felt like I could be doing better, even within the framework that we are using. 

Last year had added a "teamwork" portion to one of the lessons from last week - it was a more challenging problem that they worked on in groups on the big whiteboards. And, although this was not the first time I had done this, it struck me that this is what I am missing from the rest of my lessons.

These "teamwork" questions are harder questions, ones that make students think a little more, stretch them. They are the questions that when assigned for homework either get skipped or students copy a solution without understanding it. But they are the kinds of questions that students should be doing. The ones they must do. And now they are - every day! After we have gone over whatever concept we need to, the whiteboards and markers come out and the conversation gets lively. It is fantastic! Whiteboards are magical things in a math classroom. They give students permission to make mistakes and to try different ideas (and to draw pictures). They let students stand up, crowd around and point at what is good, at where there is a flaw in logic at something they don't understand and talk about the math! 

The added benefit to doing these "teamwork" questions is that it is (hopefully) helping prepare students for the unit tasks they write, usually the day after the unit test. I have been asked for years how students can better prepare for tasks and I think I have finally found the answer. Clearly I am a slow learner (!) but I'm very happy that I have finally figured this out and have found my groove in calculus.

MFM2P - Day 15

Day 15
Not an exciting day... I asked if they would rather do some more problems first or take the quiz first and they chose the quiz (that surprised me).  They worked away at it (open book) and I think did mostly good work. I will mark them later today. They worked on more word problems after the quiz and we took up a variety of them. That was it. Moving on tomorrow!

And since this is so short, here are my dogs. Not the greatest picture but, like taking pictures of children, it's difficult to get them both in the shot.

Monday 24 February 2014

MFM2P - Day 14

Day 14
School buses were cancelled on Friday due to freezing rain. As 100% of students at my school are bused in, there were no classes - it was a wonderful day to get ahead, with a little Olympic curling and hockey action on the side.

Today we started by cutting and gluing three trig examples into comp books. Again, I want to ensure that they have an example of each type of question we do in their books as they will be allowed to use them for some tasks and for the summative. The good news is that they have a solid grasp of which side is opposite, adjacent or hypotenuse given a marked angle. Many correctly remembered how to solve for an angle.  A few remembered how to solve for a side in the numerator. And we went over how to solve for a side that is in the denominator together. I know this is still causing issues, but we will move forward and come back to it when we have done some more algebra work.
They spent the rest of the class working on problems relating to similar triangles, trig and sum of squares. Some of these are word problems that we have not done previously so I circulated helping them with their diagrams and figuring out what type of problem they were dealing with. It is good to see them "getting it" with a little nudge from me.

Thursday 20 February 2014

MFM2P - Day 13

Day 13
We spiralled back today and looked at similar triangles again.  I really wanted them to have a full example for similar triangles in their comp books so I quickly made one up for them to cut and glue.

They did pretty well matching up the angles and I heard some conversations about "scale factor" (yay!).  

Then we continued with trig. They started to have trouble when they realized that they couldn't just do the same thing for every question. They were okay with the difference between finding the length of a side and finding the measure of an angle, but solving for different sides caused problems. Their algebra skills, for the most part, are so weak that they don't have a foundation on which to develop understanding for these questions. They follow when solving an equation like:
and seem to understand why we are multiplying both sides by the denominator. However, when the variable is in the denominator like:
they have no tools to work with. I went over a case with just numbers with them... 14 divided by what gives 7? How could we write this another way? But they had a hard time transferring that to the example shown above. And they just want to memorize what to do if the variable is in the numerator and what to do if the variable is in the denominator. Sigh. I will see if I can figure out a way to better help them tomorrow...

Wednesday 19 February 2014

MFM2P - Day 12

Day 12 started with a review of "opposite, adjacent, hypotenuse". They are improving but I think having a practice sheet just for naming sides might have been a good idea.  We picked up where we left off yesterday with the table which we completed with a row of student data:

I think I will reorganize this table for next time... I like that the hypotenuse is in the middle but some students had a hard time understanding how to correctly complete it.

We then moved on to calculating ratios:

I asked them what they noticed and they said that some of the numbers repeated. They noticed a pattern - that they "switch".  I'm not confident they understood the "why" here. To be reviewed. We also discussed why the first two columns of ratios are all numbers between 0 and 1, but that one of the ratios in the 3rd column is greater than 1.

On the back of their sheet are trig tables with the headings that match the ratios - no mention of "sine", "cosine" or "tangent". They noticed that their results matched the numbers in the table fairly closely.  I talked about how we can use these tables to help us find unknown side lengths or angles in right triangles.

We worked through two examples together, starting with "compared to the marked angle, label the sides opposite, adjacent or hypotenuse" and moving forward from there.  

I am fairly confident that they can at least label the sides correctly, but know that we will have to do quite a bit more work together to get solid on how to use trig. Then we will put it together with sum of squares and similar triangles.

We spent the last 25 minutes of today's class preparing for the OSSLT (Ontario Literacy Test) which I think was a welcome break from math for some!

Tuesday 18 February 2014

MFM2P - Day 11

Day 11
We started by discussing how you can tell if two triangles are similar. Despite saying that they didn't want to do any work today, my students managed to come up with equal angles as one possibility quite readily. When prompted about side lengths someone piped up about using a scale factor to find missing side lengths. Okay, we were on track.  So I asked them to work on a few practice questions. And they had (for the most part) no idea what to do. So we worked on the first one together, marking up the vertices to know which ones corresponded and finding the scale factor and ...

Then I got them working on the remaining 3 practice questions and some did a great job.  Others hadn't been listening and still had no clue. I worked with some of them and got them going and took up the last example for anyone who was still unsure where to start. (In case this is coming across - I'm tired and a little grumpy as I write this having just finished marking 2 sets of calculus tests and having fought with blogger all day to post my last post. I probably shouldn't be blogging right now but I don't want to get behind like I did last week - procrastination is a bad thing!)

Then we started with a little vocabulary... hypotenuse, opposite, adjacent.  "What does adjacent mean?" "Next to, beside." "Great!"  And then we applied it to a triangle or two:

Students then had to apply this to two right triangles of their own. They drew them on grid paper and had to measure and label the sides and angles then fill in a table. 

They did not understand what to do.  I went around helping each group, but clearly I need to make this more "user-friendly" for next time. Anyway, they have at least one row filled in (hopefully two) so we will start looking at ratios of sides tomorrow as we actually venture into trigonometry!

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!

Tuesday 11 February 2014

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.

Monday 10 February 2014

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.

Friday 7 February 2014

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.