Ms Pac-Man – Transformations!

Here’s how I implemented Robert’s great Ms. Pac-Man lesson.  Going into it I wanted to do the lesson as Robert outlined, but also make game of it, as well as have students do some work at whiteboards.  All in all – I feel pretty happy with how it went:

Here’s a Pac-Man game board to use with Roberts Ms. Pac-Man lesson.  It’s nice because it doesn’t use all the ink of the black background in the actual Ms. Pac Man.  I asked my TA to draw a Pac-Man board and he responded “I’m on it”.  I told him a bunch of other math teachers would use it – so don’t make me a liar!

66CW Ms Pac Man Game Board

I recommend cutting out a bunch of 1/2″ Ms. Pac Man’s to use with it. Here’s my desk of supplies!


The small Pac-Man’s were an excellent manipulative for the students – every student was using the one I gave them.  There’s a big difference between imagining how Ms. Pac Man would be oriented when she rotates, and actually rotating her and seeing how she is oriented.

The larger Ms. Pac Man’s are because I am going to have the students go to whiteboards and discuss that first turn where Ms. Pac-Man both reflects and rotates.  But I’m not showing my hand there yet – meaning I’m not going to bring up that first difficult corner.  I’m going to wait until they find it.

To learn about the lesson just go to Robert’s site.  But here is the extension that my colleague Alex Wilson offered up:

Each student gets 20 transformations to get the highest possible score.  Each dot is worth 1 and the big dots are worth 10.  Game on.

I thought about going right to playing the game – but I would have had to tell them the rules of Ms. Pac Man’s motion.  By first starting with Roberts lesson they were able to discover her motion themselves.

Here’s my first question to them after showing the video:

“What does she do first?”

After collecting some ideas from the class – they seemed to all agree that this was not just about translations.  I showed them the video where Ms. Pac Man only moves with translations and discussed some of those properties.

We collectively talked about her first two moves, and then I left them to go.  We had not directly talked about rotations – so this activity was their introduction to describing rotations by degree and CW  / CCW.  I just let them get started and fielded questions as they arose.  Then after a couple minutes  I talked to them about how to describe the first rotation.

Definitely the part where she first flips and rotates is a big thinking moment and I was not going to miss it.  One tell that students were engaged is that they did ask me about it when they first came to it .   Once the question had been asked multiple times, I grouped them quickly and had them work out that first difficult corner on a whiteboard:



A couple groups hadn’t made it to that corner yet – so I first asked them something to the effect of “Why do you think that I picked this corner to bring you all up to whiteboards for collaboration?”.  They manipulated the Pac Man and then told exactly why.  So all the students can develop the question regardless if they are actually at that corner on their board.

Once they were up together I also asked them if we could make any generalizations about Ms. Pac Man’s movement.  Does she always go up and down the same way?

The next day I brought Ms. Pac Man back to Whiteboards and this was the opener – in groups of 2 or 3 they had to make Ms. Pac-Man move around the obstacle and collect all the dots.  Engagement was very high – helped largely by the fact that they all felt they could be successful.






This opener allowed me to go around an clean up some of their notation.  There were definitely lingering misunderstandings about what line Ms. Pac Man should reflect over, and they were forgetting to say what point she rotated around.

I decided to assign the game as homework.  20 transformations, extra credit to the top three scores.  I’m torn because part of thinks the game should be done in class rather than homework.  I probably should have done the game right after the whiteboard opener.   In my head I was thinking that I needed to move onto a different topic.  Stupid pacing guide and it’s subconscious control over me!  Here are some examples of the quality of their homework:

(I never got around to making those copies – so just trust me it was good shit)

The activity served to solidify their use of proper notation for translations and reflections, and also served as a great introduction to rotations.   I think it would work perfectly fine as an introduction to – or review of – transformations.  All in all – Thanks Robert!

p.s.  It’s also super easy to turn this into a test question – just make a small course for Ms. Pac Man and it’s all good.

lastly – here’s a handy makeup assignment for absent students:


It’s What The Teachers Are Themselves

“No written word, no spoken plea

can teach our youth what they should be.

Nor all the books on all the shelves

it’s what the teachers are themselves”

I was going through some old files on my computer and found this letter that I had received during my 3rd year of teaching:


It reminded me of the importance of the poem above.  That poem was recited by  by John Wooden – legendary UCLA basketball coach (and also high school english teacher) during a Ted Talk he gave.  If you have 17 minutes somewhere in your life to spare – I would spare it by listening to this talk:

John Wooden – Ted Talk

I know not all my students were born into great upbringings.  One of my philosophies in teaching is that I believe students probably have adults in their lives who are emotionally unpredictable – happy one day, sad or stressed the next.  So I will not be that for them – They will only ever see me in a good mood.  I will not carry any negative energy into the classroom.  And it’s kind of funny because on the other side it ends up being a self-fulfilling prophecy.

The year prior to receiving the letter above I had a 2nd period class that was particularly disruptive.  And I had a 3rd period class was well behaved to a point that I felt like I probably didn’t deserve them.  The problem was that the events from my 2nd period class often left me upset and stressed out, and I would carry that into my 3rd period.  Near the end of the year I realized that too often 3rd period didn’t get the teacher they deserved, because he was always altered by the earlier interactions.  That was never going to happen again.  It was a learning moment for me.  Don’t carry baggage class to class –  Let it go and start each anew.


Looking For Our Classrooms

A retiring colleague of mine gave the graduation speech last year.  He is someone who I looked up to a lot and always appreciated my interactions with him.    One point he made during his speech that struck me was when he spoke of how thankful he was to have spent his life in such a “spiritually challenging profession”.  He talked about how it made him a better person, and then a better teacher.  I understood myself to know what he meant because much of my own classroom clarity came from a similar realization.  There is a beautiful quote from a book I love by Robert Pirzig called Zen And The Art Of Motorcycle Maintenance which describes this spiritual challenge perfectly, albeit not from the perspective of a teacher:

“The application of this knife, the division of the world into parts and the building of this structure, is something everybody does.  All the time we are aware of millions of things around us – these changing shapes, these burning hills, the sound of the engine, the feel of the throttle, each rock and weed and fence post and piece of debris beside the road – aware of these things but not really conscious of them unless there is something unusual or unless they reflect something we are predisposed to see.  We could not possibly be conscious of these things and remember all of them because our mind would be so full of useless details we would be unable to think.  From all this awareness we must select, and what we select and call consciousness is never the same as the awareness because the process of selection mutates it.  We take a handful of sand from the endless landscape of awareness around us and call that handful of sand the world.”

It is so hard to know if you are really seeing your classroom for what it is – rather than what you want it to be or what bugs you about it.  Or as simply a collection of small events that deviated from your predictions of how you thought it would go, or heard it would go, or read it would go.  We share stories of our classrooms and those stories are a collection of things that stood out.  Little pieces of awareness that we brought into consciousness.  But who knows what we missed or how our consciousness mutated the actual events in the room.  They are by definition incomplete stories and maybe at the end of the day the best teachers are the ones who can reconstruct the clearest and most accurate picture of their class as a whole.

How do we thrive in a spiritually challenging profession?  These bullet points are my two cents:

Along the way I’ve realized that every time I felt out of control, or stressed, it could be alleviated by improved lessons, or new structures.  No doubt about that.  But somewhere below all of those great teaching strategies that we know work – there are still personal insecurities that need to be understood.  Teaching will make you confront those.  That’s the spiritual part.  I thank my profession and my students for exposing me to my own.   It made me a better person…  who then became a better teacher.




Letter To Myself – 2nd Semester Goals

This letter really is for me – I probably won’t even spell check it.  I’m certainly not going to tweet it.  I’m writing it because I feel I have to.  Even if only discretely – I want to go on the record somewhere as a challenge to myself.

My main goal for second semester is to increase the amount of thinking students do.

I want to decrease the amount of time they spend reproducing and practicing any particular procedure.

I want more of their procedural fluency to come from interesting questions, engaging tasks, or a conceptual understanding.

I need to be more purposeful with my questions.  When assigning tasks I want to have two good questions predetermined for each.

I want to keep track of each activity – specifically whether or not they had these three pillars of a great activity present in them:

  • open middle
  • low entry, high ceiling
  • solution method not immediately clear but solution seems attainable with effort.

Homework to be more review oriented.  1 review, 1 conceptual, 1 new.  Something like that.  Write it out a week in advance.

Continue to rely heavily on vertical non-permenaent surfaces and visible random groupings.

Give students time alone for quite contemplation before having them begin group tasks.

Focus on MP.4 and MP.3.

MP.4 Modeling:  What does it mean to model with mathematics?  Unanswerable Questions type openers.

MP.3 Viable Arguments.  – How often do you get students to compare and contrast approaches?  I want to maximize the effectiveness of this both at whiteboards and at their desks.

  • Meaningful discourse versus call and response, or low level transmitting of process.  An example of the level of communication I want to avoid:  I ask students to share with their partner I get this sort of dialogue:
    • Student 1: “I did this, and then this”
    • Student 2: “ok, I did this and this”
    • Student 1:  “ok”
    • Me: “Did every one get a chance to share with their partners?”
    • Student 1 and 2:  “yes”

Keep having fun and knowing that it’s impossible to reach every student.

Keep learning more and more and more about your students as people.

Read the standards you are being asked to teach.

Can we represent this differently?

I want to say CONVINCE ME a lot.

Focus on process versus answer getting.

Be more honest with myself when I am giving problems that have only one solution method.

Use Michael’s student teaching as an opportunity to study the effectiveness of lessons without having to facilitate the lesson.

Managing “flow” and allowing for the existence of “productive struggle”.

Allow the last 5 minutes of class for debrief / exit tickets.

Visit more classrooms.

Finish my presentation on modeling for TMC16.

Determine what a conceptual understanding of each topic would look like, and how would we use that to build procedural fluency?

Share everything with collegues.







Capture / Recapture – Proportional Reasoning Lab

This is a popular activity – here is a link to a post from Dan’s blog back in 2008 about the same lesson.  I’m writing about it now because I love it and want to remind people of it if they had forgot.

The lesson is great because…

  1.  It’s “real enough world
  2.  It’s a true mathematical model so
    1.  Students get to question the assumptions its based on
    2.  Students get to see what effect those assumptions have on the model
    3. Students get to validate the model through experimentation.
  3. Allows students to estimate the solution.  (Based on these results would we perceive the forest to have a high or low population of deer?)
  4. Has a high ceiling as part B above is a fairly high level evaluation of the model.

The lesson has a few drawbacks

  1.  It doesn’t have an open middle.  The only variation in methods comes from how they set up the proportion.

Brief overview of Capture / Recapture:   It’s a technique used to estimate populations.  For example, let’s assume we wanted to estimate the population of fish in a lake.  We would first capture and tag a set of fish.  After releasing them and giving them time to disperse back into the lake we would go back out and capture another set of fish and see how many of these recaptured fish had a tag from the first capture.  We would then assume that the ratio of tagged fish in our second capture, to the size of our second capture, is exactly the same as the ratio of fish we tagged in our first capture to the total population of the lake.

Is that model valid?  Turns out to be surprisingly accurate based on some interesting assumptions that the worksheet at the end will take the students through.

I use regular and pink goldfish to take students through a simulation.  In groups of two, each student gets a lake with a lot of fish in it, a lake with not very many fish in it, and a bag of pink fish.

Big and Small Populations


Then I have them capture 10 fish from each lake

(first capture)

(first capture)

Now they take each captured fish (by replacing them with a pink fish)

(tagging all the fish in the first capture)

(tagging all the fish in the first capture)

Each fish goes back into the lake (bag) and are given sufficient time to reintegrate into fish society (shake the bag).  Then go back out to the lake and again capture 10 fish.  Record how many of the fish in this second capture have a tag from the first capture.

(take a look at how many of the fish from the first capture are in the second)

(take a look at how many of the fish from the first capture are in the second)

Now setup your proportion and solve.  Math says that the proportion of tagged fish to non-tagged fish in the second capture is exactly the same as the proportion of fish captured in the first capture, to the total population.  That those little piles of fish up there are a model of the entire fish population.

Of course we want to verify our results so we can test the accuracy of our mathematical model.  After all we are trying to be the “least wrong” here, and ultimately accepting that our model doesn’t give the exact answer…  but it is close enough to be acceptable to us?  Let’s see, now I have them count to see exactly how many fish were in each lake:

(highly populated lake)

(lake with lots of fish)

(lightly populated lake)

(lake with a few fish)

Then the next day give them these two slides (also from Dan)

“What if this were the results of the second capture.  Would we predict the total fish population was large or small?” (red fish are tagged, blue are not tagged)

Screen Shot 2015-12-18 at 12.08.49 PM

“A lot or a little?”

Screen Shot 2015-12-18 at 12.09.07 PM

Ok that’s it.  Other than that I just recommend doing some research on capture / recapture because there are a lot of interesting research articles and youtube videos on the topic.

The Goods

Here is the worksheet that I give the students after completing the lab.  I’m pretty sure this originally came from Dan Meyer’s blog, I can’t find it on there now so…  yeah:




– B

Exponential Growth IRA Application – Real Enough World

This application isn’t real world but it’s real enough world.  It’s not real world in the sense that nobody needed to figure out this exact question at their job.  But it’s real enough world because a financial advisor at a well known wealth management firm told me he does calculations like this a lot.  He simply invented the following scenario because he thought young kids could relate to it.  I’m not sure if they can – but I’m damn sure they can learn from it.  Since he’s the expert I’ll just leave it all in his words:

Additionally, here are some well- known abbreviations that I’ll reference in two scenarios:
PV = present value
FV = future value
N = years to goal
i = assumed annual growth rate
PMT = annual payment
Scenario 1 – When his son is age 18, a dad opens a Roth IRA for the boy with a $1000 investment (PV).  The dad tells the boy “I’m giving you this money under one condition…and that is, you must contribute  $600 per year (PMT) and leave it alone until you turn age 65, which is 47 years from now (N).  We’re going to invest the money in an aggressive growth stock mutual fund that over time, I expect, should grow 9% per year on average (i).  At age 65, I expect your account value will be in the neighborhood of $433,535 (FV).”   Pretty amazing what time and compounding will do, huh?
Scenario 2 – A 13 year-old girl wants to purchase a used car at age 18, 5 years out (N).  She expects the car to cost $8000 (FV).   So far, she has saved $3000 (PV) and wants to know how much she must save annually (PMT) if her money is invested at a 4% annual rate (i).  Solving for PMT:
PV = $3000
FV = $8000
i = 4%
N = 5 years
PMT = $803.13
 I only ended up using scenario 1 and my teacher move was to block out the $433,535 and have students go through the estimation process about that account value after 47 years.  What’s it going to be?  Give me a couple dollar amounts it definitely won’t be because they are too high or low.  Brave guesses only.  (I heard Dan use the term “brave” after prompting for an estimation and it works well).



Well Flow Rate – “Real World” Math

Keeping with the easy definition of “real world” being “math someone needed to do at their job” – I actually prefer the term “industry math”, but regardless here’s the question that needed to be answered:

My wife’s working on a project where they are building a house on a property without a well or access to city water.  So they dug a 425ft well that was 8 inch in diameter.  When they finished the dig at 3:00 pm it was completely dry.  The next day, at 7:00 am, the well had filled with water up to 45ft below the surface.

How fast is the well filling up in gallons per minute?

They wanted 1.25 gallons per minute.  Do they need to dig another well?  Wells are about $50 a foot – so yeah, they would rather not dig another one.

(digging the well)

(well finished! That’s the cap they left on it)

They used a falling rock to determine how much of the well had filled with water.  Well (not the noun), they also used a cell phone connected to a string but isn’t that just too obvious?

Here’s the video of the rock drop:

Well Rock Drop



“Real-Life” Annulus Problem!

I just can’t type “real-life” without quotes because I’m yet to resolve what “real-life” means in a math class.  But for this post it means – “Math someone needed to do at their job”.  And in this post that someone is my wife – who much to her distain and my joy – does a lot of geometry as a project manager at a construction firm.

So here’s what we need to know:  How much cement is needed to make that border?  We need the answer in cubic yards because you buy cement in cubic yards.


The image below provides some of the context for the problem – The cement border is being used to circle an existing tree.


I was actually surprised how open the middle was on this problem.  Yet students used two main strategies.  The first was the standard subtract the areas and multiply by the height, or they subtracted the volumes of both cylinders directly.  The second was the find the perimeter, think of the wall as a rectangle (dimensions of 2*pi*r X 1′), find the area of that rectangle and then multiply by the height.

One student using the second strategy used a radius of 14ft and got a solution of 16.2 cubic yards.  He told me he knew the answer would be a little too small because of only using the inner radius.  Another students used 14.5 and got the same solution as the area subtracting syndicate.

Converting from cubic feet to cubic yards is a great time to practice your perplexed I wonder why you divide by 27? face.

By the way the wall ended up costing around $35000.  I can’t believe how long I would have to work to put a wall around a tree 🙁


– B


Quick, Fun, Artistic Geometry Review

Here you go.  It’s a review of the fundamental vocabulary of geometry.  Basically students need to use all of them to draw a picture.  It is a simple idea and it works very well.  When I described it as “quick” in the title I was alluding to the fact that students are quick to understand what they are suppose to do, and they just go for it.  You don’t have to do much from a teacher standpoint – just get out of the way and see what their creative energies do.   It was made by my department chair – who along with everyone else in my department is an amazing, blogless (is that even a word?) teacher.  I suppose that’s where I come in.





The Goods:

Review Drawing Assignment

Adding Value – A Starbucks Lesson

This lesson could be better – but it was my first shot at it. I still definitely recommend it because of the great math conversations we had around convenience and adding value. Here’s the driving question I used:

What is Convenience Worth?

But it didn’t work as well as I wanted it to because we were really only looking at the convenience of the k-cup vs. ground coffee vs. going to a coffee shop.  It’s a cool driving question, but not necessarily the focus of this lesson.  This lesson is based more on “adding value” – where value is defined as what people are willing to pay for.  Starbucks added value to it’s coffee grounds by putting them into K-cups, and that is why they are able to charge more for it.  So how much value did they add?

So here’s the lesson:

Angle 1:  K-cups vs. Ground Coffee

Starbucks makes K-cups and ground coffee, and both sell for $10!  [excellent] So are they the same price?  The K-cups are a package of 10, 12.5 gram cups.  And the ground coffee is in a 340g bag.  Starbucks is able to elevate it’s price for K-cups because they are more convenient than traditional brewing methods.  So what’s the value of that convenience?






Angle 2:  Coffee Shop vs. Brewing At Home

It is more expensive to buy a cup of coffee at Starbucks than it is to brew it at home.  Yet people still do it.  What does that cost difference tell us about the value of convenience?  Here’s one of Starbucks menus:


There was definitely a period at the beginning where students did not know how to begin – or perhaps what they were suppose to do.  It took some time for them, and to extent me as well. to define the question.  I told them that it is natural in any problem solving situation to spend time defining the question, framing assumptions, setting up your analysis.

My teacher moves were as follows:

– Write about a time when you paid extra for convenience.  Talk it out.

– Ok so the value of convenience depends on how convenient the thing is versus the alternative.  [Future project could be to quantify convenience itself, rather than only focusing on Starbucks pricing].   That means we should standardize the convenience we are analyzing.  So let’s look at convenience through the lens of Starbucks pricing.

– What does the pricing of Starbucks coffee tell us about the value of convenience?  How could we use Starbucks to quantify its convenience.  What information would we need?

– Give various pricing data

– Ask students to come up with a value for the convenience of the k-cup vs. ground coffee, or the value of going to Starbucks versus brewing at home.

– Helping individually.   My most effective question was “For $10 you get 340g of ground coffee.  How much would it cost to get 340g of coffee in K-cups?”

It is also an interesting conversation that ground coffee and whole bean are the exact same price.  So Starbucks is saying that the convenience of having the beans ground for you adds no value to the product.

After the activity we read an article from Proffer Brainchild titled “ADDING VALUE:  A Lesson From Starbucks”

Here are some examples of student work.  The first example was the highest quality paper I received.






Lesson Learned:

It’s clear that not all students understood what I was asking – and so I concede that “What’s the value of Convenience?” was not a good driving question for this particular activity. I do see it that prompt as a great math modeling problem where students try to quantify an inherently qualitative thing. But that’s not what they were doing here.

Next year I am going to try and focus first on the k-cup vs. ground coffee prices and stick the economics concept of “adding value”. How much value is added when Starbucks puts their grounds in K-cups vs. ground coffee? And this is not just about the calculations, it’s about the argument.