Analyzing Mistakes

My 9th graders have been learning right triangle trigonometry. We decided to include this topic in the 9th grade math curriculum because they are also taking physics. An understanding of trig will help with analyzing two dimensional motion and also with analyzing forces.

So, we did a bunch of problems and they mostly got it. But not all of them and not all the time. I used Kelly O’Shea’s Whiteboarding with Mistakes idea and had them produce solutions with common mistakes that students might make when solving these kinds of problems. Then the other groups had to identify the mistakes in a given solution. It led to some interesting discussion.

“Why would you want us to deliberately make mistakes?”

What a great question, I responded. Why do you think? Here’s a sampling of their responses:

  • To make us aware of mistakes that we can make.
  • To make us pay closer attention to our work.
  • To have fun.
  • To challenge each other.
  • To teach us how to analyze work.
  • Because without mistakes there can be no learning.

A little side note.

The 9th graders at my school also take an engineering class where they practice and practice the engineering design cycle. They identify a problem, design a solution, test it out, see where it fails, make improvements, and begin again. The teacher is very clear about learning from mistakes. Apparently, that message is being heard as evidenced by the last comment.

“Because without mistakes there can be no learning.”

I’m not sure that I agree with that exactly; I don’t think that mistakes always have to present for learning to happen). I do know that I tend to learn more from situations that give me unexpected results. But the better thing here seems to be that we are helping our students to understand that their work doesn’t have to be perfect the first time. They are kind and curious and smart – and not afraid of making mistakes.

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Function Carnival

I haven’t been doing a good job posting this year. Something awesome happens in class and I think, “I have to write that up.” Then I get home, and start planning the next few lessons, and I forget all about the awesomeness. It’s been a busy year.

This morning, on CBS Sunday Morning, I learned about a truly extraordinary man, Jim O’Connor, a high school math teacher who volunteers his time at the local Children’s Hospital. What made me sad, though, was his comment, “It drives me crazy when people say that school should be fun. I mean it’s nice if it could be, but you can’t make school fun.” Watch the video. Mr O’Connor really is an amazing man. I just think that it might be time for him to retire from teaching.

I mean, if learning math can’t be fun, then why should anyone consider doing it? Kids and their parents already think that learning math is a drag, so shouldn’t we math teachers be working hard to change that thinking, not perpetuate it?

I’d like to think that my students have had fun learning this year. From dissecting chocolate chip cookies to writing graphing stories to rolling balls down ramps, they’ve collected and analyzed data and created function models. They’ve studied some statistics and some functions (linear and non-linear) and now we’re working on right triangle trigonometry. With 9th graders. I’ve worked hard to make learning fun and challenging.

Thankfully, others are also working hard to make school mathematics not only interesting and fun, but helpful for us teachers to diagnose student difficulties. Take the Function Carnival currently under development by Christopher Danielson, Dan Meyer, and Desmos. Honestly, I don’t know how they do it over there at Desmos, but these little animations will tell me more about what my students understand about functions than anything I could have come up with. And the beautiful thing is that they’re engaging for physics, too. That’s awesome for me and my students because at Baxter Academy, my 9th graders are also learning physics. Imagine my glee at learning about this interesting new tool. I will definitely have them exploring (in a few weeks) and sharing the results with my physics teacher colleagues.

In response to suggestions from the many commenters to Dan’s post, the Desmos team got busy creating more scenarios, including graphing velocity vs. time along with height vs. time. I’m looking forward to these new situations being included in the current Function Carnival site. Maybe they’ll be ready when I need them in a few weeks. It will also be fun to have my students attempt these graphs before we go off to Physics Fun Day in May.

Here are a few more challenges in development:

Try them out. Give feedback. Encourage your students to have fun while they learn.


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What Time Will the Sun Rise?

This week I begin Exploring the MathTwitterBlogosphere. I’m looking forward to these missions and challenges because I need someone pushing me to find the time to write in this blog. It’s good for me. Like spinach.

This week’s mission: What is one of your favorite open-ended/rich problems? How do you use it in your classroom?

One of my favorite open-ended/rich problems comes at the end of a unit on trigonometric functions. After exploring, transforming, and applying trig functions to Ferris wheels, tides, pendulums, sound waves, … I assess my students’ understanding by giving them some almanac data of  sunrise and sunset times for a specific location on Earth. Their job is to analyze the data and create a trig function to model either sunrise times, sunset times, or hours of daylight – their choice.

The data looks like this

and that makes it somewhat challenging for students to even begin. They are reminded that they should have “enough” data to know if the model they develop fits well. I point out that the times are given to them in hours and minutes, but that they probably want a single unit (hours or minutes after midnight). From there, they are on their own to solve the problem. Usually, they work with a partner.

In the classes that I’ve used this task with, we’ve modified the amplitude, period, and midline of the sine and cosine functions. We haven’t introduced phase shift, yet. So, there is also a reminder about selecting a convenient “Day 0″ for the function they choose to model with.

What I love about this task:

  • Students are talking math, asking each other about the number of data points they should use: “Should we just pick the same day every month? Are 12 data points enough?” or “Do we just go every 20th day?” or “What should we use for the first day?”
  • Students are problem solving. They have to convert the times into a single unit. They have to make decisions about which variable to model, when to start, which type of model to use. Then, they can collect the relevant information to modify their chosen function.
  • Students are using technology. Although they don’t have to, it’s really easiest to have the kids making scatterplots on calculators or computers and then graphing their model on top of that. Then they have a built in way to check their work – they don’t have to ask me (the teacher) if they are correct. It shows up in the picture that they create.
  • Students think that working with trig models is really hard, so they feel very proud when they are able to complete this task without any help from the teacher.
  • It’s really easy to grade. Either the model fits or it doesn’t. Kids turn in their data tables and work showing how they calculated the necessary values for their model. This precludes anyone from using the old SinReg command.
  • Even though I’ve used this task for about ten years, it’s a perfect fit with the Common Core math standards (trigonometric functions) and practices. And since I live in a SBG world, this is a very good thing.

My favorite kind of assessment is one where students have to apply what they’ve learned to a different situation. Even though we create lots of different trig models in class, sunrise, sunset, and daylight hours represent a new application. And a new challenge.


Filed under MTBoS Challenge, problem solving, technology

Week Two @ Baxter Academy

Wow. What a week. We had our first week of classes and our first Flex Friday.

Our schedule is an interesting mix of current tradition and pushing the envelope. The students have eight classes and there are no study halls. There are four classes every day, so the days alternate. Classes meet either Monday & Wednesday or Tuesday & Thursday. So, during this past week, I met all of my students twice. I am teaching seven classes this year. Yes, seven of eight blocks. It’s a lot. But it’s what’s necessary during this first year. So we are all doing what’s necessary. I am teaching all of the 9th graders and 2/3 of the 10th graders. That gives me about 105 students, which averages to about 15 per class. Not bad, really. I just wish I had a little more time to chat with colleagues. I’m hopeful that will come.

So what did we do this week?

9th grade

We had a bit of fun with Dan Meyer’s Pyramid of Pennies problem. After watching the video a couple of times, I had them write down the first question that came to mind (and then a couple more). They shared their questions with each other and with me. Many questions were common from class to class, but some were quite different. Here’s a listing:

  • How tall is it?
  • How much does it weigh?
  • How much money is it?
  • How did they keep the pennies from falling?
  • How many pennies is that?
  • How long did it take?
  • Where did they get the pennies?
  • How many pies can you buy with all those pennies?
  • How many pennies are in each row?
  • Why a pyramid?
  • How many people did it take to build?
  • How old were the creators?
  • What’s the oldest penny?
  • Why time lapse instead of sped up video?
  • Were there any patterns to the penny placement?
  • Are they real pennies?
  • Who has that much time?
  • Is the change between each layer a constant?
  • What did they do with the pennies afterwards?

As they set out to figure out how many pennies were in the completed pyramid, I heard lots of great discussion about how the layers were formed. If the bottom layer was 40 stacks by 40 stacks, then was then next layer up 38 by 38 or 39 by 39? If there were 1600 stack in the bottom layer, then did the next layer up have 1596 because it lost four stacks from the corners? Each team made some assumptions and then proceeded with their computations from there. Several teams were able to see their plans through, making some adjustments if the numbers didn’t seem to be making sense. Other teams ran out of time. But that was okay. They had done enough to share strategies. I especially liked that when one group said, “We multiplied 40 x 40 x 13 to get the number of pennies in the bottom layer, then we did 39 x 39 x 13 to get the next layer, then 38 x 38 x 13, and so on. Then we added up all the layers to get the total,” another group said, “We did the same thing. We just multiplied by 13 at the end.” Huh? How can it be the same if you multiplied by 13 at a different time, I asked. The response: “They found the number of pennies in each layer. We found the number of stacks in the pyramid and then multiplied by 13 to get the number of pennies.” Isn’t that beautiful?

On the second day of class (Wednesday/Thursday) we played around a bit more with the patterns of pennies. Since just about every group had focused on squaring numbers to figure out how many pennies were in each layer, I asked them to take a look at these numbers: 1, 4, 9, 16, 25, … and describe any patterns they discovered. Looking for and describing patterns is key to thinking critically about mathematics (as well as lots of other things). Most groups found the difference pattern. At least one group in each class found a pattern by looking at the final digit of each square number. Interestingly, no groups attempted to represent these numbers visually. I guess we’ll have to go back to that. They were also able to tell me that the number of stack on layer n would be  n^2. (Maybe someone can teach me how to show this properly using latex. I tried and tried and couldn’t get it to work.)

Then I gave them this pattern: 1, 5, 14, 30, 55, 91, 140, … and asked them to find the next few numbers in the sequence. Again, this involves looking for patterns. I also challenged them to come up with a formula, suspecting that they would not be able to. They’re 9th graders, after all. I’m happy to say that nobody gave up. They really tried to come up with a formula. They were thinking recursively, of course, but don’t yet have any language for that. Again, that’s okay. I found out a lot about these students during those two classes.

10th grade

Again, I turned to Dan Meyer. But this time we tried out the Penny Circle. It was really the first time I’d ever asked my students to do math through an online guided activity. I’d been through it myself, first, and it seemed pretty straight-forward and reasonable. And while the conversations during class were all good, the data that I received on the back end (teacher dashboard) was not so helpful. It’s not any fault of what Dan & Desmos put together. I love what they put together. I just forgot that I would be using it with 10th grade boys. (Yes, most of our 10th graders are boys – there are only a handful of girls in the 10th grade.) So, I got some very silly, anonymous results. Thankfully, nothing was school inappropriate!

However, it’s now difficult for me to use their data to figure out who needs some help understanding the relationship between diameter and area of a circle. Although the easy answer is: most of them. I’m not sure that this new presentation of the original problem prompted creative problem-solving and curiosity in the way that the penny pyramid problem did for the 9th graders. Maybe that wasn’t the point. But part of me wishes that I had given them some real pennies and real circles and had them collect the data that way. These kids didn’t have the opportunity to think about their own questions after watching the video. I’m just not sure their curiosity was sparked.

I’ll keep working at it. I know that I can spark some curiosity around math for this group. But they will be a bit tougher than the 9th graders. I have my work cut out for me.

Flex Friday

One of the founding principles of Baxter Academy is that students work on projects. Big projects. Long term projects. Meaningful projects. To give kids time to work on these big, long term, meaningful projects we have Flex Friday. We have no regular classes on Friday. Instead, the time is devoted to project work (mostly). Since it was our first Friday and students do not yet have projects to work on, we teachers gave some presentations of possible projects. The ideas ranged from building a noise & dust containment system for our CNC router, to figuring out the best possible lunch program for our school, to building a greenhouse, to designing a video game, to researching the ethnomusicology of Maine. Some students have their own ideas that they are hoping to pursue this year, but the rest now have lots of good ideas to choose from.

Since we have this gift of Flex Friday time, we also thought that it would be good to get out and about into Portland. We are only a couple of blocks from the Old Port, after all. So, on Friday afternoon groups of kids with their advisors went to different locations around town. I was with the group the went to the Portland Public Library. They’ve had a recent renovation and the new building is awesome. Bright and inviting, this is a place where I would not mind spending an afternoon. We signed all the kids up for library cards – I got one, too. The little city of Portland has a great library full of wonderful resources. I can’t wait to begin exploring them all. Now, to find the time …


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Week One, Two First Days

It was an absolutely beautiful day.

It was an absolutely beautiful day.

We (the faculty at Baxter Academy) decided that we actually have three 1st days of school this year. Nathaniel called them 1D1, 2D1, and 3D1. The names stuck.


On 1D1, we all gathered at Fort Williams Park in Cape Elizabeth. This beautiful seaside setting was the backdrop of our work for the day. The school contracted with Rippleffect whose fabulous staff designed a series of five challenges for the kids and faculty.

Clumping and acting seemed to be the key.

Clumping and acting seemed to be the key.

One challenge was called, “Where’s my water bottle?” Similar to the old red light, green light game, this time the kids had to figure out how to grab said water bottle, return to the starting line, and let everyone have a chance to hold it, all while making sure that Toby, the Rippleffect guide, did not see them moving or guess who was holding the water bottle. It was interesting to see them devise a plan that completely fooled Toby. It turns out that Toby was a student in the first year of the Francis W Parker Charter Essential School, the charter school in Massachusetts founded by Ted and Nancy Sizer. It was an interesting side note for me – during my first year of teaching, I met Ted Sizer during a Coalition of Essential Schools function and read his book, Horace’s Compromise. He made a tremendous impact on the direction my teaching ultimately took.

It quickly became a salvage operation.

It quickly became a salvage operation.

In a different challenge, the group was divided into four engineering/design teams. Their task was to design, build, and market a raft from three pieces of float foam, three 6 foot long 2×4 pieces of wood, and three lengths of rope. The ultimate test, of course, was to take them on the water to see how far they could go (and how quickly they would fall apart). There were some solid designs in my group. Unfortunately, the construction process didn’t go as well as planned. All four rafts fell apart in the water. Kudos to the kids who volunteered to test the rafts. Sure, it’s September and the weather was beautiful. But this is still Maine – that water isn’t very warm.

It was a fabulous first day. Lots of laughs, smiles, and new friends.


Our second first day was at the school. As students entered the front doors to wild applause and noise making, a reporter from Maine Public Broadcasting caught the racket from outside. He was later allowed to enter the building and speak with a couple of teachers. Here’s the report. Quite nice, actually, considering some of the turmoil that has surrounded the opening of this school. Once the opening ceremonies were over, we got down to business: building furniture and arranging our rooms. We had about 7000 pounds of tables, chairs, cabinets, and bookshelves to assemble. Computers needed to be set up in the lab. Off we went in our advisory groups and got to work. My classroom started out looking like this:

IMG_0301 IMG_0302 IMG_0303

A few hours later it looked like this:


And the view out my window?



Sure, there’s a hotel under construction and a lumber yard next door (they own the building, after all). But there’s also water. I’ve missed the water. It’s good to be home again.

Stay tuned for 3D1 – that happens on Monday – and I get to teach some classes!

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First (Teacher) Week Down

Whew! This starting a new school thing is hard work. I know that’s not news to anyone, but I had to just say it.

What’s on the list of things to work out right before school starts?

  • Special Education – the difference between IEP, 504, intervention
  • Standards - what are they; knowledge (content) vs skill
  • Scheduling - fitting in all those classes; what does the day look like; placing kids into classes
  • Courses - what are the rough outlines of our classes; what are the natural integration points
  • Grades - what kind of grading and reporting system will we use; will it be rubric-based
  • Flex Friday Projects – a major cornerstone of the school; how do we design and manage them effectively
  • Advisory - what does it look like; what’s a better name than “advisory”
  • Getting to know your Chromebook – what are some apps that kids can use
  • Administrative tasks – parking (in downtown Portland without a school parking lot); insurance; harassment training
  • Planning the first few days of school

Many of the items on this list would look familiar to any teacher new to a school. When I first began teaching at Poland Regional High School, I spent two days in new teacher workshops (even though I was not new to teaching) and four more days in PRHS teacher workshops. A public “thank you” to my PRHS colleagues, mentors past and present, for preparing me for this moment. The twelve years I spent at PRHS was the best teacher education I could ask for. You pushed me to think in ways that were completely unknown to me.

Learning and sustaining new models of teaching is hard work and deserves time for proper deliberation. We will be continuing many of these conversations as we move forward. There is great excitement and energy among this group, but there are some things that we just can’t finalize until we meet our students. That happens on Wednesday.

Bring on the kids!


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T-minus one day and counting

baxter logoEven though I have been doing work for the opening of Baxter Academy all summer, my new year “officially” begins tomorrow. We will have five full teacher workshop days to prepare for the day the kids arrive. There’s still much to do, but I am confident that this group of teachers will get there. It is truly an amazing crowd and I am honored to be one of them.


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