Author Archives: themathletes

Testing out my lessons from Dan

One of the downfalls of my job is that I don’t always get to spend as much time as I would like in classrooms – I don’t have my own students to test things out with. Luckily, my husband does so when I need some students I can sometimes borrow his for a while. I was feeling teaching withdrawal after a few days of meetings and after having the opportunity to finally hear Dan Meyer speak at OAME, I wanted to try out some new things he had me thinking about.
I knew the class was working on re-activating their prior knowledge of volume and surface area. They had looked at rectangular prisms the day before and the plan was to do some work with cylinders.
I had this image from 101qs on the screen when the students were coming into class.


As they came in, they were already asking the questions I would hope for:

– What is that?  Is that a sinkhole?  Where is that?  How big is that?

Once everyone was in the room I had the students write down one or two questions they had on their mini-whiteboards.  I recorded their questions on the board:

  • Where is that?
  • What is a sinkhole? How do they happen?
  • Why do people build houses in places where sinkholes happen?
  • How big is it?
  • What is the radius?
  • What is the volume?

When Dan did this with us he quickly moved to the questions that focused on his goal for the “lesson” and let the answers to the other questions come out or addressed them at the end.   We decided in the moment to have my husband answer the first few questions quickly (he was their science teacher as well) because we knew some of the students would actually have a hard time working on the other questions without the answers to the first few.

I asked them what they meant by “how big is it” and the students that asked that said that they felt that the next two questions would answer that.

I then had them focus on the volume of the sinkhole and asked them what information they would need to figure this out.    How wide and how deep came up.

I then took a move from Dan and had them try to predict the diameter of the hole.   They first estimates ranged from 10m – 300m but they narrowed it to 10 – 50m using other objects in the image to help (the street looks about as wide as 3 cars plus the two sidewalks).   I then gave them the dimensions from the National Geographic article of 18m wide and 100m deep.  I asked if they needed any more information and there was some discussion about if they needed a length too but they decided that just those dimensions would be enough.  I was surprised that no one asked for a formula but they did have a formula sheet available to them from another day.

Students worked for a while and sorted out a variety of issues by working with a partner.  Some accidently found surface area but were corrected by a peer.   We discussed the answer but the number was so large it was hard for them to imagine how much dirt it would to take to fill that hole.  One student said “I wonder how many truckloads of dirt that would be”

We decided to help them make this volume more understandable by having them think about how many classrooms full of dirt that would be.   We estimated the dimensions of the room and then had the calculate the number of classrooms of dirt it would take to fill the hole.

This was an in-the -moment decision that we were happy with because it helped them connect even further with the problem and gave them some additional practice with volume from their previous days lesson.

For the students that had calculated surface area we tried to brainstorm reasons why you might want to know the surface area – we didn’t come up with many reasons other than one suggestion “if you wanted to turn the sinkhole into a very deep pool and tile the entire inside walls…..”   At least they understand what surface area is!

They then had time to do some practice with calculating volumes and surface areas of other objects.   They worked well the rest of the period, I think largely because they saw the connection to the rest of the world.


Would you rather….

In addition to her already fantastic blog, Fawn Nguyen has been blogging about Math Talks that she is doing with her classes.   A Math Talk is simply an opportunity for students to think, and share their thinking about mathematics, generally without paper and pencil.

We’ve been doing a lot of work around proportional reasoning in our board and in the province.  My latest favourite question is one generated from a team of teachers that were working on some problems to use early in the grade 9 course.

Would you rather have $500 or the value of a stack of quarters as tall as you?

The question seems simple enough but as I’m using it with various groups, both teachers and students, I’m amazed at the wide variety of strategies used to when thinking about this problem.   It is a nice opportunity to get at ideas of proportional reasoning, multiplicative thinking and unitizing.  

When I’ve posed this problems with groups I usually make them guess first based on gut instinct without any time to think.  

After some time to think and then share with a partner I have them share their ways of thinking about this:

  • a roll of quarters is worth $10 and about 2.5 inches high.  So that’s $20 for 5 inches…..$40 for 10 inches, $240 for 60 inches
  • 6 quarters is about 1cm high, so that is $1.50 for 1cm, $3.00 for 2cm, If I’m 180cm tall, that is about $270 
  • A roll of quarters $10, so you would need to stack up 50 rolls to make $500, and I’m definitely shorter than 50 rolls of quarters…
  • my foot is about 3 rolls of quarter long and I am about 7 of my feet tall, so $30 x 7 = $210

What I love about this discussion is the wide variety of “units” that people use – based on their schema.   Some work in inches, some in centimetres, some (mainly those with retail experience) talk about rolls of quarters and some use personal referents.

I also love the amount of gesturing that occurs when talking about this problem.  I think we underestimate the power of gesturing when thinking and talking about mathematics – in particular when it comes to proportional reasoning.     

After having time to think about the problem, I have had some ask whether $500 is too large of a value to use since most people seem to estimate their “quarter worth” to be around $250.    I point out that they didn’t realize this until they had time to think about the problem so I’m not sure it matters.  In fact I don’t want it to be too close because the problem is not about exactness but rather about some general thinking about proportions.

Inevitably some start thinking about how tall you would have to be to make the quarters a better decision and the discussion continues.

If you are looking for more ideas for Math Talks check out Fawn’s blog!















EQAO Review

I have always found it difficult to review for EQAO.  Other than giving students lots of opportunities to see EQAO-type questions throughout the year, doing lots of EQAO questions before the test, offering lunch time help sessions, etc., I haven’t really known what to do.  I’ve tried a few different things over the year, but never really felt that what I was doing was very effective or useful.  I’m excited to share that I have some new strategies!  Some strategies that not only allow students more exposure to multiple choice and long answer questions, but that also hilight their misconceptions and encourage a deeper understanding of the grade 9 math curriculum.  These aren’t my own ideas.  They are from many conversations with colleagues around the board about best practices in teaching grade 9 math.  That’s the best part about this job… I get to learn so much because I get to talk to so many teachers!

Yesterday we worked with a grade 9 Academic class and used a strategy that Liisa had told me about that she has used in her classes.  We wanted to help students develop strategies for answering multiple choice problems as well as some overall review of the course.  The first question we asked students was “How do you think teachers make up multiple choice questions?”

One student replied that he thought we probably mostly copied and pasted from textbooks (yup, definitely a strategy I’m sure we all use from time to time!).  Another thought that we picked common mistakes students might make when solving a problem and worked out the answer that would lead you to.  My colleague, Elizabeth (who has experience writing for EQAO), was able to expand on this for the students.  We then told them that we were going to now let them do some multiple-choice question writing.

Students were seated in homogeneous groupings and we gave each group one of six different questions.  The questions were the stem of a real EQAO question with the four answers blank.  They looked something like this:

eqao practice 1

Students had about 15 minutes with their partner to come up with the right answer, but also three good distractors.  As we circulated around the class, we heard some fantastic conversations.  It is one thing to be able to find the right answer to a question, but it takes some deep thinking sometimes to think about the mistakes that others might make.

After they had their four answers, we had students meet with the other group that had the same question as theirs to compare their options and come up with 4 final answers that they thought were the best.  We then collected the “final answers” and had students go back to their seats with one clicker per pair.  It was now time to see how good their distractors were!

For question 1, we wrote the four options on the SMARTboard.  Students were given a few minutes to talk with their partner about what the right answer was and “click in” their answer. For example;

eqao practice 2


About 86% of the class chose (a) and the remaining 14% chose (d).  I asked for someone to explain their choice, whether it was (a) or (d).  One boy raised his hand to say that he had picked (a) because he had distributed out the numbers to get 12x-15-63x+14.  He then simplified to get answer (a).  I asked if anyone else had thought about it differently.  A girl in the class raised her hand and said she had done the exact same thing as the other student, but she picked (d) because it was just another way of writing it.  This girl saw answer (a) and (d) as being equivalent expressions!  I asked the class what they thought.  There were some puzzled looks so we gave them a moment to talk about it with their desk partners.  Walking around and listening to their conversations, we realized that this was a misconception that several of them held.  They didn’t know that brackets beside brackets was a convention we use for multiplication!

Elizabeth orchestrated a discussion about this with the class to clear things up.  It felt good to be able to find out about a misconception in the class and to clear that up.  The thing that struck me though was that if it hadn’t been for the nature of the activity – giving students an opportunity to speak and listening carefully to what they were saying – we never would have come across that!

There were many other great moments during the class, but there’s not enough time to mention them all here.  This is a simple activity, but definitely one that I’ll be using again in the future!


Things to Focus On

A couple of weeks ago we ran a workshop with all of our secondary schools and talked about some things that can help us help kids learn math.  Here’s a summary of some of the points we discussed along with some relevant articles/videos/etc.

1)  Closing the Gender Gap

In general, our boys are doing better than our girls, particularly in the applied stream.  Jo Boaler’s book, “What’s Math Got to Do With It?” raises some good points on why girls sometimes fall behind boys in mathematics.  Her research has shown that while boys tend to be content following a set of rules, girls need to know the “why” behind the mathematics.  She advocates for “classes in which students discuss concepts, giving them access to a deep and connected understanding of math [which] are good for girls and for boys.”

Jo Boaler also has a video (from the course she offered this summer) talking about stereotypes and growth mindset. 

The research she quotes was done by Carol Dweck and can be found in this article: Is Math a Gift? Beliefs That Put Females at Risk

2) Creating Opportunities for Discussion

Discussion of math concepts is important in helping students develop a deeper understanding of the mathematics.  Malcolm Swan’s Standards Unit offers some great suggestions for incorporating and encouraging discussion in the classroom.

Cathy Bruce has a video about Math Talk on the Ontario Ministry of Education’s “Webcasts For Educators” site. There is also a talk by Dan Meyer about inquiry, the power of predicting and the importance of having a clear goal in mind.

One way to encourage discussion in math class is by giving students problems that encourage discussion.  The Grade 9 Writing Team that got together in the summer came up with some great problems that are available in First Class.  There are also some great blogs that offer some other suggestions for “perplexing problems.” A few to check out are Dan Meyer, Fawn Nguyen, Robert Kaplinsky, Sam Shah, and of course our very own What Did You Do in Math Today blog.

See OAME 2013 Presentation – Talk is Cheap so Let’s Make the Most of it

3)  The Importance of Mistakes

Mistakes are imperative for learning.  This video, again from the course Jo Boaler ran this summer, explains the science behind mistakes.

There is actually a blog about making mistakes! It is appropriately named Math Mistakes and contains pictures of errors students make in their work.

4) From Patterns to Algebra

This resource is invaluable for teaching grade 9 math in either the academic or applied stream.  It not only connects what students already know about linear relations to allow them to go deeper in grade 9, but also allows them to develop really flexible thinking around the multiple representations.  See You gotta try this!

5) Growth vs Fixed Mindset

The other piece we talked about is student (and teacher) mindset.  This is a new idea that we were introduced to during Jo Boaler’s summer course that has a lot of implications for teaching and learning.  This is the video we showed explaining the science behind brain plasticity and what that means for the teaching and learning of mathematics.

The tasks that we give students can be designed to encourage a growth mindset in our students.  Tasks of this nature include: (taken from Jo Boaler’s summer course)
1. Openness
2. Different ways of seeing
3. Multiple entry points
4. Multiple paths strategies
5. Clear learning goals and opportunities for feedback

6) EQAO Strategies Across Schools

We also spent some time talking about strategies that schools across the board are having success with when it comes to EQAO. These included:
1. Logistics – writing during one day, time between booklets, etc.
2. Support from across the school – peer tutors, LRT, greater awareness to school and parents
3. Using exemplars – give students exemplars (without the codes) and have them try to determine the coes, have students complete an open problem and compare their solution to the scoring guides to determine their score, etc.
4. Familiarity with EQAO-type questions – use previous EQAO questions on tests, reviews, diagnostics, as problems to investigate during class; have conversations about multiple choice distractors
5.  Extra support for students – EQAO midterm, MSIP, after-hours numeracy


The Border Problem

I have a new favourite problem!!  It looks like this:

Without counting one by one, determine how many shaded squares are in this 10×10 grid.


This is not a new problem.  I think it’s appeared in many books and possibly even textbooks throughout the years.  My first exposure to it was actually in Jo Boaler and Cathy Humphreys’ book “Connecting Mathematical Ideas”.  I loved watching the video of middle school students explaining the way they saw the problem and then eventually generalizing the methods for an nxn square, but I always wondered what it would look like in a high school class.  Luckily one of my good friends allowed me to come in and try this problem with her class.  It was awesome!  Like exactly what I was hoping it would be!  Maybe even a bit better!  Here’s how it went:

1.  We put the graphic above up for a moment and asked kids to give us a thumbs up when they knew the number of shaded squares.  I asked someone to explain how they were thinking and then wrote that method on the board.  “So is this how you thought about it?” I asked.  They agreed.  “Did anyone think about it differently?”  It kept going until we had a list like this:10x10

2.  I asked students to imagine they had a 6×6 square now.  “How would Jessica figure out how many shaded square there are?” I asked them.  I gave them a moment to talk about it with their partners.  When I did this again with Angelo’s method, the conversation got interesting.  The kids started doing funky things with the numbers “6 x 10 – 6” was one of the ones I heard.  I could tell that they weren’t thinking about the visual, and definitely not thinking about the expression as a function.  I asked them how we could check and they told me that they could add up the numbers and see if it equaled what Jessica’s equaled.  Not a bad method – definitely was a step towards getting us to talk about equivalent expressions which was one of my learning goals. I kept referring them back to the “imaginary 6×6 grid” and we eventually ended up with a list like this:


3.  I asked students if we could do this for any sized grid.  They all told me we could for any number… even for a 100241 x 100241.  “So instead of making a list for every single possible grid, could we just say that we have like a say, nxn, square and n could be any one of those numbers?”  I asked them to talk about that in their partners.  These students all have had lots of exposure to variables in the past, but I find they often have started to forget what they really mean by the beginning of grade 9.  They start to think they are some fixed thing and then start doing all sorts of weird things with them.  The students all agreed that that would be alright.  They also said that we could pick any letter as the variable.  I asked them what Jessica’s method would look like for an nxn grid and gave them some time to talk about it and write it on their whiteboards. As I circulated to see what students were doing I saw a bit of this ” n+n+(n-2)+(n-2)” but a lot of this “n+n+p+p” or something to that effect.  I was expecting this, but I still didn’t know what the best way to deal with it was.  I didn’t say anything to individual groups, but once I saw every group had something written down I asked them to put up their whiteboards and look around the room at what other people had written.  They did, but there weren’t really any lightbulbs flashing, so I started the discussion “I see two different types of answers (wrote them on the board). Are these both the same?” Most students were nodding their head.  “Okay, so in the n+n+p+p, what does n represent?” A student raised her hand and said it was the dimension of the grid.  “Okay, so what does the p represent?” No hands.  And then one, “the p is like the inside part of the other two sides that the n didn’t count”.  I asked students to talk about this – is this true?  how do we know that this is what p represents? Some good discussion started happening.  Some of the ones that had the (n-2)’s already written became more convinced, others who had written the n+n+p+p’s started looking over to the (n-2) people.  I brought them back, “what did you talk about?” One student told me that she had converted, “we need to write the n-2 so that we know that it is a row minus the 2 squares we already counted.” Most students seemed convinced about this, others seemed a bit shaky.  I erased the n+n+p+p and left the other expression on the board.  Then we moved on to Angelo’s method.  As you can imagine, similar conversations ensued.  After some good discussion and some synapses firing we ended up with a list like this:nxn

4.  We had already touched on the idea of equivalence with the 6×6 list (the fact that all the expressions should equal the same number), so I brought it up again with the nxn list.  “How do we know that these expressions all are equal?” No volunteers, but I was out of time.  I left them with that challenge for homework – to pick two of those expressions and show to me how you know they are equal.  My purpose was to hopefully get kids to start thinking about equivalent expressions and also like terms, the big idea being that even though expressions look different they actually mean the same thing, and we can manipulate them to make them look differently. 

Because I wasn’t able to be there the next day, I’m not sure what kids did to show that the expressions were equal.  The classroom teacher told me that the next day they pulled the algebra tiles out to start modelling these expressions with them and proving equality.  I think that is a great way to prove this, and also a great lead in to combining like terms, distributive property, etc. 

Asking questions you don’t know the answer to.

This has clearly been an insanely hectic month as neither Jessica nor I have blogged. So today I have made it my priority.

I attended a session at OAME2013 Free-Falling – Letting Go Of Textbooks, Worksheets And Units by Bruce McLaurin (@BDMcLaurin)  which left me with much to think about. I intended to blog about it immediately after returning from the conference but I am now glad that I didn’t get a chance to because I think I’ve benefitted from thinking about it a bit more.

Early in the session Bruce asked us if we had ever asked a question in class that we didn’t know the answer to.  He gave us time to share with those around us and then we discussed as a group.   It took me a bit to think of an example of when I did this.  I think I thought I did it more than I actually do.   Interestingly, those in the room that have taught computer programming said they do it often in that class but not in math class.

The example I shared is one that I did quite a few years ago, when I had my own class still.   With all the learning I have done since then I would do it quite differently now, but I will share my “beginner version” with you because I think its important to be honest with where I started.

The problem came from a visit to the grocery store back in 2007 when I saw a sign that looked something like this:

billion bags

It was in the early days of moving people to using re-usable grocery bags instead of plastic bags.   I read the sign and then went on my way but on my way home I started thinking…

“Is eliminating 1 billion plastic bags really that impressive?  Plastic bags can be squished up pretty small.   Would eliminating 1 billion of them from a landfill make that much of a difference?  Is this company just using the number billion to make it sound good?  How much space would 1 billion bags actually take up?”

So that September I shared that poster with my students and I told them about all of the thoughts going through my head.   (I’m already cringing at typing this because it is so obvious to me know how much better I could have done this… but I said I would be honest about where I started).

I asked my students how much space did they think 1 billion bags would take up.  They were having a hard time explaining their ideas so I gave them something to relate it to – a classroom.   I simply said, “Do you think it would take up a whole classroom, part of a classroom, many classrooms?”.  I had them all make a prediction and I made one myself as well.  I told them I really had not tried the problem yet – we were going to explore this together.

I think provided students with the collection of plastic bags that I had accumulated over the years (which I still have, in case I do this problem again).  I also gave them some plastic storage containers of various sizes and rulers and tape measures.  Students then worked in groups to determine how much space 1 billion plastic bags would take up.

They persisted quite well, although working with large numbers gets challenging at times.  I found I often had to suggest to groups that they label their calculations along the way because they were losing track of what all the numbers on their pages represented.

We were all quite surprised with the result (I won’t spoil it for you) and it was nice to be genuinely surprised with them.

I have since run into other plastic bag videos, promotional info, etc. that could fuel a similar exploration in classes such as this one:

As I was thinking about this blog post I was having a bit of an internal struggle. I like the idea of asking questions that I don’t know the answer to because I think it helps students see me acting like a more true mathematician who is working with them.   However I know that I always encourage teachers to work through a problem themselves before giving it to students so that they can think about: the mathematics that will come out of it, what questions might arise, how they will facilitate the conversation, what strategies they might expect to see, etc.   Although I didn’t know the answer to the problem, because I hadn’t worked through all the calculations myself ahead of time, I did spend time thinking about all of those questions above as well as where it fit into the curriculum I was responsible for.

You gotta try this!

If you haven’t heard of Patterns to Algebra by Dr. Ruth Beatty and Dr. Catherine Bruce you have to check it out! It is a resource that transformed my teaching and filled in the blanks that I had always missed when teaching linear relations to my grade 9’s.
Linear relations is really one big concept: the fact that some things in life follow a linear pattern (or an almost linear pattern) and we can represent this type of relationship using a graph, table, equation, or story. The problem is, the way that I taught it (which was mostly following the general order of a textbook since I didn’t know what else to do) was teaching kids that these four representations were very different things. I did try to get them to connect the four, but the deep understanding of the inter-connectedness of it all was lacking. I can clearly remember being frustrated when my students could not understand why the first differences (when x is increasing by 1) is the same as the slope. I showed them with a graph, but I knew they couldn’t see the connections.
Fortunately I discovered Patterns to Algebra! Well, it actually discovered me. A fellow teacher had been asked to pilot the materials with her Grade 9 Applied class and I decided to try it alongside her with my Grade 9 Academics. I’m sure part of our success was the benefit we had of working together. We had the same prep and so we spent every day going through the materials we’d been given and planning what we were going to do with our students. We even did some cross-class stuff (since both our courses were running at the same time) and had our classes go see each others patterns and try to guess the “rules”.
I won’t attempt to explain the whole scope of the resource here. But basically it is a thoroughly researched, well thought-out plan for teaching linear relations through patterning. It starts off with students doing robot charts (or input/output charts, whatever you prefer) and patterns with a multiplier only. Then students do the same with a multiplier and constant. Students are writing “rules” for the tables and the patterns and with not much exposure are able to explain why Number of Tiles = Position # x 5 +2 grows faster than Number of Tiles = Position Number x 2 + 5. Eventually you lead into graphing (using the patterns) and story telling (using graphs and patterns) and it all connects together so nicely! By the end students have a clear understanding of what is really happening in a linear pattern and are able to connect that to any of the 5 representations (graph, equation, story, table, pattern). It’s so EXCITING!!!!
This resource is a grades 6-10 resource and often students remember patterning and “robot charts” from elementary school. I think that is a great thing… another connection! Kids have had so much exposure with patterning we need to be using it in high school because they understand it! That’s part of the power of this resource… it reaches ALL students. I found that my students with the most gaps were the ones that grabbed onto this and loved it. They were finally completely understanding something and doing well at it. The great part is too that as you continue to delve deeper into linear relations and analytic geometry (for grade 9 academic) you always have something concrete to refer back to. Kids never forget the patterns!
Try it… you’ll love it!!
(oh, by the way… now I barely teach first differences: it comes out when students notice that the multiplier (i.e. slope) can be looked at from comparing the position number to the number of tiles OR looking at how each pattern changes from one position number to the next. They also think it’s a “trick” for finding out the rule for an input/output chart. All I have to tell them is the vocabulary!)