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!

Sunshine Blog Awards

For those of you who don't know David Petro, he is an amazing guy.  I have had known him for many years - I believe we met while serving on the board of directors of the Ontario Association for Mathematics Education.  He impressed me there and continues to do so with his weekly posts that keep Ontario math educators (and many outside our province) up-to-date with great math resources from the web.  He is one of those people that is creative and works hard and shares.  A great combination. I was shocked when he nominated me for a sunshine blog award - I mean, I don't blog that often and don't feel like I have contributed much...  However, I don't want to think that I am not honoured by the nomination so I will not ignore it, but I'm also not going to follow the rules.  David did a great job of pointing out that this is a pyramid scheme here so I will just answer his questions and maybe give some random facts about me.

From David...  My questions for my nominees :

1. How long have you been on Twitter?  June 24, 2012 was my first tweet.  I'm sure that I was lurking for a while before. I just learned that you can type something like from:MaryBourassa since:2012-01-01 until:2012-07-31 to search for tweets between a set of dates. Thanks, David.  I would not have known how to do that otherwise! I sent my first tweet to Dan Meyer - it was this link.  My second tweet was about desmos <3.

2. How many email addresses do you actively use? 3. One for school, one for personal use and one for "junk".

3. What's the name of the street you lived on when you were born? I have no idea.

4. Coke or Pepsi? Neither. I drink water. I live on the edge a little and sometimes have sparkling water. And the occasional venti-3-pump-extra-hot-no-whip-cinnamon-dolce-latte from Sbux.

5. When did you first get an email address? 1993? Maybe even before that. On Freenet. That was a long time ago.

6. First concert you went to? David Bowie? Again, that was a long time ago. 

7. Mac or PC? PC. I have an iPad, but I did not drink the Apple KoolAid.

8. Favourite Olympic Sport? I love equestrian events (I used to ride, before I got a mortgage). I also like watching alpine events (skiing) and curling (I am Canadian!).

9. Favourite Online News Portal? Who has time to read the news? I use Flipboard when I'm bored : )

10. Name of a teacher who inspired you? Madame Dezainde, back in Montreal where I grew up. Even though she made me cry.

11. Favourite Holiday Movie? Hmmm. I don't think it's actually a holiday movie but it always plays around Christmas so I'm going to go with Sound of Music. It's fun to watch with the kids.

There are a bunch of random facts about me contained in those answers. I'll give you one more: I don't really like to write about myself!

How Many Each?

After a great day at EdCampOttawa, I came home to a roast chicken dinner (yes, my husband is a great cook) along with roasted veggies.  While helping himself to seconds, my 8-year old, Noah, complained that he hadn't gotten enough potatoes.  My husband said that he had cut 4 potatoes into 4.  The kids quickly figured out that there had been 16 potato slices.  I then asked how many we would each get if they were equally distributed among us (us being 6 people).  They tried 2 and counted up to 12, then tried 3 and counted up to 18 (in 3s) so they knew it was more than 2 but fewer than 3 pieces each.  Noah then said that it wasn't 2 and a half.  I asked him how he knew.  He got mad at me, screamed that he wasn't doing this anymore and stomped away from the table.  So I turned to Jacob, who is 6, and asked him the same question as I had seen him making gestures with his hands that looked like he was figuring it out.  He told us that if they each got 2 and a half pieces, that would be 15 pieces.  At some point my 10-year old said she had figured it out, but actually managed to not yell out the answer (good job, Isabelle!).  So back to Jacob.  We went back to knowing it was more than 2 pieces each which he told me would be 12 pieces. I asked how many pieces were left to be divided among us.  He said 4.  He wanted to give each kid 1 and the adults none, but I said we each had to get the same amount.  Then he thought Noah shouldn't get any since he had left the table.  With a little encouragement he said we could "chop, chop" the pieces into 3.  So how many pieces would that make?  12, he counted. And then he figured that once around the table was 6 so he could around the table another time and we would each get 2 of the cut-up pieces.  So 2 and two-thirds each.  Just as we were saying this Noah yelled from the next room - 2 and a half and one sixth each!  Yes, Noah.  Yes :)  Then he sat back down and finished eating.

Completing the Square with Algebra Tiles

There seems to be some interest in how to use algebra tiles in the MTBoS so I thought I would attempt a blog post.  I apologize now if I confuse anyone - I don't claim to be an expert, but I do find they help kids make connections.

For years I avoided algebra tiles. It didn't help that one of my previous colleagues told the story of how one of her students managed to choke on a tile! Thankfully, none of my students have done that.  Our tiles are red and blue so they all feel the need to look through them as if they are 3D glasses (sigh). The kids get to choose which should be positive and which should be negative.  They invariably say red is hot so positive and blue is cold so negative so that is what I am using here.  

We introduce the tiles in grade 9 when we simplify polynomials.  They use them to multiply binomials (it makes a rectangle) and to factor trinomials (find the length and width of the rectangle).  So my students are very familiar with algebra tiles by the time they get to completing the square.

I start with a warm-up to make sure they remember what is special about perfect square trinomials. They work in groups of 4 - each student does one question then they add up their answers.  If the sum is correct, we can move on, if not, they have to find the error.  I stole this from someone at TMC13 (who stole it from someone)^n, who stole it from Kate Nowak.

Then we tackle our first example:

Students understand that instead of making a rectangle, they need to make a square with their tiles.  They each have their own set to work with.  We will place the 7 unit tiles off to the side and work with the rest.

They see that they need to add 1 unit tile to make a square.  In order to do that we have to add a zero pair so a -1 unit tile goes with the 7 off to the side.

We can then write the area of the square a its side length squared and simplify the unit tiles.  And just like that we have vertex form!

We work through a couple more examples in the same way, with students working with their tiles then consolidating with the whole class.  Each time they have to divide the x-tiles - half for the length and half for the width so that they make a square.  Each time they have to add unit tiles.  I get them to notice patterns in what they are doing.   

Next, we use a chart to connect the algebra tiles to the algebra.  The cool thing is that they actually understand why we are dividing 'b' by 2 and squaring it because they have done it with the tiles.

The next day we extend to quadratics where 'a' is not equal to 1.  Again we start with tiles and connect to algebra.  You need identical tile diagrams for each x^2 you have, but it's the same process.  It connects factoring out the 'a' value to the tiles.

I find algebra tiles really help explain why we are doing what we are doing. It helps that my students are asking how to change a quadratic from standard form to vertex form. Well, some of them are anyway!

My SMART Notebook file is here. I left off the extra practice for day 2 as I am not really happy with it, but am not sure how to change it.

If you are looking for on-line algebra tiles, The National Library of Virtual Manipulatives is a great resource.

Working with Integers

How many of us struggle with explaining integers?  I know that I do.  I find it most difficult to teach topics that are completely obvious to me, that I don't remember ever learning.  So integers are right up there.  Many years ago my friend, Tom Seidenberg, told me that he used the idea of a hot air balloon and sand bags.  I ran with that and have been using it ever since.  I am by no means saying that this is the best way to teach integers, but it does give students something concrete to hang on to.  I must also say that my students have already been taught about integers, but many did not understand them the first time.  This gives students a different way of thinking about integers.  It takes a lot more than this to get them comfortable working with integers, but this is how I start.