Adding To Ms Pac Man

This post is about two new additions to way I taught Robert Kaplinsky’s Ms Pac Man lesson and I recommend them because they made the experience in my class even better (and it was already one of my favorites).  I wrote about the lesson previously here.  And Deb Boden wrote about it here.

New Addition #1 – Conditions for the double move

I thought of this the day before I did the lesson and I was excited about it – but I didn’t even realize the two improvements it would lead to (namely providing strategy to the game, and providing a way for students to check their solutions).

Here’s what I did:  After a few students start to run into that first corner where Ms. Pac Man needs to do more that just rotate – I still waited for a critical mass and then grouped students and had them navigate that turn together at whiteboards.

But when they sat back down I hit them with this challenge (which is the new element):

“Find other conditions for a double move”

We had established that if she is moving up and wants to go left that she will need to do more than just rotate.  When else will this be true?  I gave them about 10 minutes of alone time and after that had a class discussion.

You will have students who don’t feel comfortable enough with the transformations to handle a question like “Find other conditions for a double move”.  If I saw students not progressing, I would walk by them and something like “If she is moving left and wants to go down, will this need to be a double move?”.  Then come back to them and ask “What other times will she need to do a double move?”

Turns out that every left turn and every turn when she is moving left will require two moves.  There are two reasons why that is a great discovery from the lesson perspective:

#1:  Now when they play the game to score the most possible points in 20 moves they can have a strategy.  Minimize these moves that take two transformations.  And they did!  This year over half the class was even mapping out their entire path before actually recording the individual moves.  They were counting out the moves per turn and trying to minimize the number of left turns.  The students recording each move also seem to be doing so with increased strategy.

#2:  If you are doing the activity Robert suggests where they figure out all the transformations to complete the path on the screen, you’ll always get some form of this question:  “Mr Miller I’m done but I don’t know if it’s right”.  Now you can challenge them to count up how many moves it should take based on knowledge of which turns are a double and which are a single.  If they used a different number of moves then they probably made a mistake somewhere.  This type of self check on the answer helps extend the activity because the early finishers have something to do.

A Subtle Distinction

If you notice in the image above I had written “triple” move instead of double move.  That’s because students disagreed on whether they should count the translation as a separate move or not.  That happened because one of my students offered the beginning move as a double because it’s a reflection and then a translation.  But if that’s a double, then our “double” moves (reflection, rotation) were actually triple (reflection, rotation, translation).  This was probably one of the most subtle distinctions that I have had an active group conversation about with my students.  Loved. Every. Minute.


New Addition #2 – Draw the resulting orientation of Ms Pac Man after each move.

I got this idea from Alex Wilson and it was very effective.  I recommend it highly.  Last year I just had them write all the transformations without the additional image of Ms. Pac Man.  Have them keep track of Ms Pac Man’s orientation after each move.

I also – mostly in the moment – came up with a way to address a very common mistake without telling them the answer, so here it is:

Anticipation – Here is a way to navigate a common mistake:

Lastly – I definitely recommend having them say exactly what line they are reflecting over, rather that just vertical or horizontal reflection.  And when you do that you are sure to have some student always write “reflect over y-axis” for every vertical reflection, and “reflect over x-axis” for every horizontal reflection (when really they are reflecting over y=h or x=k).  So what I did this year after I noticed a critical mass doing that in one of my classes, I asked everyone to sketch an x-y axis and put Ms. Pac Man in the first quadrant.  Like so:

Then I told them to reflect her over the y-axis.

That was pretty much all they needed to see for them to realize that Ms. Pac Man is not reflecting over the y-axis in the game.  Then I had them draw a line through Ms Pac Man and reflect her over that line, as well as the x-axis.  The next day, and I’m not kidding on this – not a single group of 2 students had her reflecting over the wrong line (although one group had her reflecting over points).  Turned out to also be a great spiraling of reflecting shapes.

This post is really about having more in our back pocket as we try provide this learning experience to our students as best we can.

By the way – I did the lesson over two days.  I did the opener that I blogged about previously on day 2, where they send Ms Pac Man around the plus sign.  Here are 9 of the 12 whiteboards groups I had (that’s the most I can fit on an image)




Sideboard Design – Scaling & Similarity on VNPS

I love this lesson for a variety of reasons – one being that they initially scale down a rectangle, and then they scale up a design and watch it fit perfectly into their original rectangle.  Scaling down and up, woohoo!!!

The abstract would call this a four step process:

1.  Draw a primary rectangle on the board that will represent the desired size of the sideboard.  It should be actual size – they are usually 30″ to 36″ tall.

2.  Scale down that rectangle to fit on a piece of paper using a scale factor they create.

3.  Design a sideboard on paper and measure all relevant dimensions.

4.  Scale the design back up to it’s actual dimensions and fit it into the original primary rectangle.

The Workflow:

I had them in groups of 2 and 3.  Everyone in the group starts with the same primary rectangle.  They choose a scale factor and scale it down.  Now each member of the groups designs their own sideboard.  They measure the necessary dimensions and then scale them up to what they should be in real life.

Once each group member had a design and knew what the scaled up dimensions were going to be, the group choose which design they were going to actually draw full size.

Once students were finished with the design I asked them to calculate the area of the primary rectangle in ft^2 and in^2.

In Retrospect:

Some students designed in only integer dimensions.  I would require one of the dimensions in the design be a non-integer value.

Only two groups ended up actually using a primary rectangle that didn’t have integer values.  Next year I think it would be cool to do this activity again but require them to use one of the dynamic rectangles as the primary rectangle.

Also they would make design decisions based on aesthetic reasons, rather than making decisions based on the simplicity of the measurements.

Students could also look up sideboards online and to get design ideas.

Two Sneaky Reasons Why It’s Great:

1:  Beginning with the primary rectangle offers an aha moment when they scale their design up and it fits perfectly inside the original rectangle.  If they just scaled their design up based on a scale factor they would say – “yeah, well this looks right”.  But since their rectangle is already on the board, the fact that the design fits right into is feedback that they scaled it correctly.

2:  Kind of similar to above but they get to see the actual area.  So when I ask them to find the area of the primary rectangle and they calculate 15ft^2, they can look at it and say “yeah, that’s actually what 15ft^2 looks like”.  I mean how often do they calculate an area and never experience the actual size of the thing they are calculating [answer:  too often].

Teaching Moments:

One of my favorite was when students began drawing their rectangles on paper and I got to ask them “What’s your scale factor?” and listen to them justify their selection.  Not each group went with 1″:12″, because it made their drawing too small.  A lot of groups went with 1″:6″, and some of those groups communicated that as 2″: 1′.


Meter sticks, rulers, and tape measures are great.  Also if you have whiteboards loose they can also be used as a straight edge.


I did the following two assignments the day before.  I had also done a vase design that began with a primary rectangle, so they had experience.

65CW Activity door design

65CW Activity scale drawing

More Student Work:

For The Absent Kids:

I got you covered!  Here’s a version of the activity that doesn’t need whiteboards.  This worksheet was the inspiration to this lesson and it was given to me by Dave Casey.  When I received it I thought to myself, “how can I make this great for a vnps classroom?”  I love editing existing lessons rather that creating them from scratch.

66CW MU Activity sideboard design




Origami Heart – A Geometry Lesson

I made this video / blog post because I remember wanting to do origami in class but being unsure how to connect it to content.  Until recent years – teaching math through art was pretty foreign to me.  But mostly with the help of Dave Casey I have been able to open up that portion of my curriculum and my students are better off because of it.

Origami lessons can be almost whatever you want them to be.  With each fold you can step back and ask questions, and take observations of the resulting geometry.  There are lots of bisectors, scaling of shapes, isosceles right triangles, trapezoids, kites, squares, and on and on and on.  Take this video as just one approach out of many.

As a rule I tend to ask the most questions during the first few folds and then pick up the speed as the folds get to the finish – but that’s not a rule or anything.  I also go back and forth on whether to have them unfold and observe the folding pattern that results.

This year I spent 35 minutes on this lesson.

Here’s the folding diagram for the heart – drawn by Dave:

The Extension:

Have students make a rectangle that is similar to 8.5″ by 11″ and then fold a heart with that rectangle.  Have them record the ratio and comment on the area differences.

The Goods:

CW Origami Heart




“Alone” Time In A Collaborative Classroom

I have three different “times” during my class this semester:

  1. VNPS.  Group work at whiteboards.  Groups of 2 or 3.
  2. Individual work.  Students are at their desks working on their own piece of paper, but they still collaborate.
  3. Alone time.  During alone time everyone puts on headphones or ear plugs and works alone.  No questions are allowed.  If they have a question they write it down on paper.  `

So what’s this “alone” time all about?  This semester I have been focused on a question:  What is the role of working alone in a collaborative environment?  If I focus that question a little more it would be:  Can we use music or silence to improve student perseverance, creativity, and mathematical understanding?  If I was to toss one more question in there:  How can we improve the transfer from what they can do at whiteboards in groups, to what they can do alone on paper?

During “alone” time every student either puts on headphones and listens to music or they put on ear plugs.  The students who listen to music are encouraged to find music that helps them concentrate.  The majority of my students are using Focus At Will for their music.   Focus at Will is a website that is designed to provide music that helps people concentrate for longer, and improve their creativity and perseverance during that time.  Although Focus At Will is a pay site, Spotify has free “focus” channels that work good too.  Focus At Will does offer advantages over Spotify but I’ll compare music options in a different post.

Here are the rules for “alone time”

  1. No talking, no questions.  If you have a question you need to write it down.  I generally break this rule if they raise their hand, but definitely no questions of peers.
  2. All students put on headphones or earplugs.  Amongst other things, these serve as physical barriers to communication.  There are no social pressures to communicate during these times.
  3. They do not have to finish whatever worksheet they are doing (if I’m going for procedural fluency).  They should relax and focus on understanding.  If they are expected to finish, they will rush.  And the need to finish will cause stress if they are unable to ask questions.
  4. I recommend Focus At Will if they use ear buds at this time, and remind them that it’s not about finding music you like – it’s about finding music (or ambient sounds) that help you concentrate and make you relaxed and creative.

I generally run “alone” time for about 5 to 15 minutes.  It depends on whether we are doing a procedural fluency type lesson, or a problem-based lesson.  Here’s my basic outline:

Procedural Fluency:

I show them how to do it.  They go to whiteboards and try it in groups.  Then they have “alone” time, and lastly individual time.


Sometimes I start with “alone” time so they can develop their own thoughts. Then they go to whiteboards, and end with alone time during the reflection.  Other times I do the opposite – begin and whiteboards and then have them go to alone time.

The Data:

After the first 3 weeks of implementing “alone” time – at about a frequency of once a class – I gave a quick survey.  The survey was a rating of 1 through 5 where I asked them to rank how helpful the “alone” time has been for them.  Here’s the scale:

1 – Negative effect.  You would be better off without “alone” time.

2 – Not helpful.  This is the “meh” answer.  You can take it or leave it.  It doesn’t help or hurt.

3 –  A little helpful.

4 – Helpful

5 – Very helpful.

Here are the results:

I have 3 preps this year and here are the breakdowns for each subject:

The above data was referencing “alone” time, rather than the idea of listening to music specifically.   I also asked the students if they had used Focus At Will outside of class.  Here were those results:

Over half of my students are using Focus At Will outside of the classroom now.

I’ll categorize some of their reasonings in a different post.  Students have found listening to music helpful, but also the other elements of “alone” time.  Multiple students commented in the survey that they loved the fact that they did not have to finish the worksheets during alone time.  They enjoyed be able to go at the pace of their learning.




I didn’t mention that although Focus At Will is a pay site, I was able to give all my students a free one year subscription.

I forgot another important rule to “alone” time:  When I do need to cut in and talk to the class they are not required to take their headphones or ear plugs off.  Since the music is suppose to help them concentrate, then presumably it helps them listen to me as well.


Japanese Temple Geometry – Quick History

I am attaching the article that I use in class.  My students read parts of it before we start these problems.  I was just going to link to it but for some reason it is surprisingly hard to find online.  I’m glad I have a copy – here it is:


The author of the article – Tony Rothman – also has a book called “Sacred Mathematics: Japanese Temple Geometry” that covers Sangaku and can be found here:

I will write about the pedagogy a different day – but if you are interested in these problems I wanted to share the article above.




My Spiraling of Vroom Vroom

I titled this “My Spiraling of Vroom Vroom” because I highlight how I used this lesson to teach (introduce) line of best fit and statistics.  The original lesson uses line of best fit, but I also wanted to teach stats as well.

This would be the first time my students ever did a line of best fit and used it to make a prediction, and so I needed to add in some teaching around line of best fit.   I also wanted to use the pull back cars for a lesson on mode, median, range, and mean – so I used a different day 1 activity.  I’ll mention all the different “thinking” moments I cut out and where I implemented procedural fluency.  I got the activity from Fawn’s blog which you can see:  Vroom Vroom.  The difference between this post and Fawn’s is initial activity, as well as the need to bring in a Desmos activity builder since they had never used a trend line to make a prediction yet.

Pro tip:  This will go over several days, so if you have cars that look alike do not forget to number them!


Vroom Vroom is a lesson using pull back cars from Fawn.  Check the link for her implementation.  The premise of her lesson is that kids get a pull back car and placed in a competition for who can get their car closet to the finish line.  At first they just get the car but do not know how far away the finish line is going to be and so collect data about how far the car travels from various pull back distances.  Then armed with just their data and regression, the are told how far away the finish line is going to be and they must decide how far they are going to pull back the car.  Closest wins.  I did do that same thing, but not until day 3.

My implementation is different because I also wanted to use it as a vehicle for an introduction to statistics.

Day 1: Collecting Data From A Single Distance

I told the kids that they (groups of 3) were going to get a pull back and I wanted them to take 15 measurements of how far the car travels when pulled back 5 inches.  I gave them time to discuss and decide if they needed any clarification.  These are the types of questions I got:

  • Do we measure from the starting line of from the release point?  (my answer was starting line)
  • What do we do if it doesn’t go straight?  (still measure how far it went.  Try your best to point it straight and if it doesn’t go perfectly straight then that’s just an attribute of your car)
  • When it’s done moving, do we measure it from the front of the car to the finish line? (yes)
  • Do we put the front wheels on the starting line or the front of the car?  (front of car)

Teacher Move I Needed:

  • After the first couple measurements I got the class’s attention and reminded them not always have the same person measure.  Learning how to measure the distances to the nearest quarter inch was one of learning goals and everyone should take turns doing it.  It seemed like each group had given themselves roles – pull back person, measure person, and recorder person.  I didn’t want that.



The reason I didn’t start with the competition was that I didn’t think they would have a natural sense of the variability of the cars performance.  By beginning this way I believe they will do a better job with the competition data collection.  We’ll see  tomorrow.

At this point I decided to give a mini lesson of how to find range, mean, median, and mode.  I showed them how to do it and gave them a worksheet for practice. Initially the students worked “alone” (told to put on headphones or ear plugs and no talking or questions)  When they were sufficiently far I had them go back to their whiteboards and find the mean, range, median, and mode of their data set and graph the numbers on the number line.

Then told them how to make a box plot of the data




Thinking Questions: Here are some great questions to get the students thinking.  I always recommend teachers come up with these types of questions and turn them into “talking points” – it’s the best way I know to get everyone one record and discussing.

  • How confident are you that if you try it again the distance the car travels will be in the box?
  • If the prize for having the car end up in the box was $100, how much would you be willing to spend to play the game?
  • Is the mean or median the most appropriate measure to base our predictions on.
  • Let’s say you pulled the car back one more time and by some act of God it went 25000 inches.  What would your new mean and median be?

The final part of this day the students pull their car back one last time and see if it falls within their box.  I tell them to imagine they actually made the best, that money was on the line.  Once back to their seats we reflected on the activity and possible sources of error.

I should add that during the reflection I asked them how confident they would have been if I said the car had to travel anywhere between 0ft and 300ft.  They were 100% confident.  I used that to tell them informally about the 95% confidence interval.

Day 2 The Competition Begins

Today I tell them that they are in a competition.  Which ever group gets closest to the finish line wins.  They get 20 minutes with their car.  The catch is that they don’t know where the finish line is going to be until after the 20 minutes.  At which point they need to decide how far they are going to pull back their car to achieve the proper distance.  One chance.

I gave them 2 minutes at the beginning to write up some thoughts on what they are going to do once they have the car.  I was able to learn that some students were planning to pull it back as far as it would go.  They had not connected to the activity and thought they were suppose to make the car go back as far as possible.  Here’s some of the data that was collected:



As teams began to finish I had them begin to enter their data into Desmos.  Then I collected the cars and told them the distance was 5′ and they had to come up with a pull back distance.

Day 2 Line Of Best Fit Procedural Fluency

This lesson marked the first time my students used a line of best fit, so I didn’t feel they would be able to use it to help them make their predictions yet.  I decided to scaffold that by created an activity in Desmos activity builder.  I walked them through the first 3 screens, and they finished it on their own.  Then I had them go back and use their new skill to help them make another prediction for the pull back distance.  They would be able to test both distances.

Here’s the link to my activity:

I used real-world climate data off the EPA’s website.  I wanted to have this learning segment to be a conversation that mattered and top of being fun and playing with pull-back cars.

After the students finished the line of best fit Desmos activity I had them use their new skill on the data they collected from the previous day.  They each did a line of best fit and got their second estimation for the pull back length.

Day 3 

With their estimations in hand – we setup the track and did the competition.  The champ got pretty close.

Day 4

I didn’t plan on this day, but the Desmos activity highlighted that the students where unable to describe what the slope of the line of best fit meant in the context of the question.  So I handed out two of the graphs from the Desmos activity and taped them to whiteboards and asked the following questions:

The high exit question – which I delivered orally – was to ask them what does the line of best fit predict for year 0?  And if they thought it was reasonable to use our line of best fit for a prediction about year 0.  We discussed domain and how it’s probably not best to use this post industrial revolution data as a predictor for before the industrial revolution.

I ended with them doing some worksheets “alone” (wearing headphones or ear plugs, no talking, must write questions on a paper) and then “individually” (able to talk to neighbors).

In Retrospect:

Here’s what I didn’t do well:

  • Having students keep a written or digital record of the activity.  Since it went over several days and was done a lot at whiteboards and on Desmos, there wasn’t much official to turn in.
  • Similar to above – but having a more official paper for them to write they group members, car number, daily reflections, and so forth.
  • Although they dealt with the repeatability issues in the first activity, I didn’t really go back to that idea and discuss what is the optimum number of measurements for the challenge.
  • I wanted to cut out more alone time.
  • My students mostly were unable to think consider the nature of the relationship.  They tended to assume linear and go with it.  I clearly did not do a good enough job last semester having them contrast the different relationships – linear, exponential, quadratic, other.


I’m happy.  I think the goal was achieved of having them understand what a line of best fit is and how to use it to make decisions.  Next year it will be better thought.  And there will be a next year with this one.




Toasters And The Purpose Of Math Education

When I was in college I studied Electrical Engineering.  I got an MS.  I spent 5 years in the field before getting laid off.  Then I switched to teaching for reasons I usually make up during interviews to sound good.

I only introduce my past like that because when I was studying electrical engineering I was told a story that I would only later come to realize how it not only defined engineering – but also defined math education.  And if I was really being bold – education.

It came from Professor Pedrotti at UC Santa Cruz, who was telling us about a time in his life when he was asked to fix a toaster.  His grandmothers toaster had stopped working, and Prof Pedrotti was home for the holidays while getting his Ph.d.  So his grandmother asked him – “Ken, maybe you could fix my toaster?  Have you taken a class on toasters?”

There is no class on toasters.

When learning electrical engineering, you learn basic fundamentals about how electricity works, and how different circuit components behave.  You learn about when assumptions about behavior are valid, and when they are not valid.  You learn how things work individually, and how they should work when you put them together.  And you are expected to apply these fundamentals to a wide variety of applications.  For example – here is how such and such behaves.  Here is how it is used to blah blah.  Now use such and such to design a circuit that’s going to muck muck.  Or something like that.  You get the point.

That’s conceptual understanding.  And that’s mathematics.  And that’s education.  What mathematical reasoning allows this process to work?  What mathematics does this new application remind me of?  And if there is some component missing – what might that component need to do?

The story’s conclusion fits in nicely with the idea of a thinking classroom.  But on a more micro level probably – in our profession it’s more like “there is no lesson on toasters”.  We can’t teach you how to do every specific thing.  We can teach you to apply fundamentals so you have the ability to do whatever it is you have been asked to do.

“I must have been absent when we covered this?”  Nope.  You’re here now.

– B


Licensed to Ill – Conversations That Matter

Here’s a post regarding the My Favorite I did at TMC16 on how I used the Mathalicious lesson “Licensed To Ill” to have a conversation that mattered.   And since Karim inspired me to go this way with his blog post On Purpose – I’ll defer to him for the reminder about why it’s important:

The video of the presentation is here: (Thanks Glen!)

I’m going to highlight what changes I made from the suggested lesson delivery.  Go to the Mathalicious website for the specifics of the lesson.  I took the pulse of the room and a lot of students had opinions about health insurance and Obama care but did not understand what it was or know any of the details.   Mathalicious provides a great video introduction to health care that allowed me to make sure every student had a basic idea of it before going into the scenarios outlined in the lesson.

The most effective thing I added to Mathalicious lesson was the physical simulation. There was a story telling benefit, a pedagogical benefit, and a mathematical benefit:

  • Allowed me to increase the drama around denying Daniel.
  • Anytime you put something in the students hands it is a win.
  • We got to use all the simulation data to better discuss it’s relationship to the expected value.

I think there could be a cool teacher move where the students come up with how to simulate these probabilities, but I choose to have that all setup from the start.  I made sure the room was setup a bit different than normal, and each table already had a decks of cards, a coin, and dice.

TMC16 My Fav.008

Alonzo’s probability on the card draw isn’t 1%, so I talked to the students about whether or not we thought it was sufficiently improbable for our purposes.  My student teacher said we could have them draw from a 50 card deck, and then if they draw correctly then they flip a coin.  I thought that was cool – but the idea came after we had already started so I didn’t go with it.

I did have the students keep a record of all their simulations, which allowed us at the end to compare our expected values with the results of all our simulations.   This is particularly helpful in the “set your price scenario” where they expect a profit of $0, but the only possible results with a single Daniel is +/-$20,000.

One edit I did which was subtle and only done for storytelling purposes was to make the students the insurance company.  You can make the story more interesting to students by injecting them into it.  Now instead of asking the students to calculate the expected profit of the insurance company, I would instead ask “calculate your expected profit”.

Although I did most the work at VNPS – I did do the reflection portion on the videos with the exact same handout Mathalicious gives.

Here’s the final grid.  I started with just two columns and had the students add a column with every new situation.  That way they didn’t know how many we were going to do and it kept the direction of the lesson more surprising.

TMC16 My Fav.010

Out of all the scenarios, the only one I actually added was the generalization at the end.  I also changed the instruction on the second scenario – preferring to have them “set their own price” to try and fix the problem from the first scenario.  Mathalicious is more explicit with their instruction.

I am asking students to figure out the profit for the best and worst case scenarios.  I’m undecided if this actually provides them insight into the situation and I’m mostly doing it because I want to ask them to calculate the probability of the best and worst case scenario which will allow me to see if they are adding or multiplying probabilities.

In the moment of giving the lesson I would periodically put the mandate in effect.  For instance, when you can deny Daniel I told the kids to have Alonzo buy.  Sometimes the mandate was still in effect, other times it was more a better safe than sorry thing.  Once the mandate is first introduced, the students wanted to know if it was in effect for each of the subsequent scenarios.

The questions I keep repeating over and over again are “what are the advantages and disadvantages?”, “who wins, who losses?”, “Is this fair to _______ “, “Do you agree with… “.

The Conversation That Matters:

I frame each scenario to the students pretty much like I did in the video.  I had to skip a few elements of each scenario in my TMC presentation in order to make the 10 minute window.

The mathematics has to better inform the students of how heath care works, and how changing specific parameters effects the situation as a whole.  I am not trying to tell students that the Mandate is fair or unfair for Alonso, or that it’s fair or unfair to deny Daniel coverage just because he is likely to have a heart attack.  I am trying to give them a sense of how insurance works, and to look at how changing certain parameters effects other parameters – sometimes in positive ways, other times in negative ways.  Then I want that understanding to then form the basis for an informed opinion of “fairness” and ultimately whether or not something is a good idea.

Here’s some more of the slides from my presentation.

TMC16 My Fav.011


TMC16 My Fav.013

TMC16 My Fav.014

And here’s the comic that a girl drew up on her whiteboard after we went through the scenario of denying Daniel health care for his pre-existing condition:

TMC16 My Fav.015





Real World Math – The Retaining Wall

Keeping to my simple TMC15 definition of “Real World” as being math that someone needed to do at their job – I love this one.  I believe these types of math problems deserve to be a spice in our curriculum.  Especially in a geometry class!   I used this problem at the end of Math 2 because it does a nice job of connecting angle measure and slope.  I used it on an exam, so I will also show the two scaffolds I used.

Basically my wife is building a giant barn / man cave and they need to construct a retaining wall.  Here is how my wife explained the problem, and how I gave it to students.  I gave them the below diagram as well.  Too see how she delivered the question to me take a look at the text message at the bottom of this post.

Here’s how my wife wrote it:

A General Contractor is building a retaining wall in Sonoma County. The General Contractor is required to have the portion of the retaining wall which is over 3’ tall designed by a licensed engineer. The base of the wall follows a driveway at a 6% slope. The top of the wall is level. How much of the wall needs to be engineered?

Here’s how I gave it to my students on their exam last year:

A General Contractor is building a retaining wall in Sonoma County.  The General Contractor is required to have the portion of the retaining wall which is over 3’ tall designed by a licensed engineer.   The base of the wall follows a driveway at a 6% slope.  The top of the wall is level.  The short side of the wall is 0.75 feet tall, and the tall side is 6.5 feet tall.  How much of the wall needs to be designed by the licensed engineer?  (Hint: When is the wall 3 feet tall?) (diagram not to scale)

PT Trig Building

I ended up loving this problem because it connected slope to angle measure, algebra to geometry.  I got to tell them that in Math 1 we always talked about slope, but the angle measure was always there.

I used this problem as an exam question.  In order to scaffold it I did the following two activities as openers.  Both of them combined did a nice job of familiarizing the students with the language of the question.  I also had gone over what a general contractor is before the exam.

Opener Scaffold #1

What would this road sign look like if we used slope or angle measures for gradients instead of grade?  Draw all three.  I didn’t run straight to this question – I developed it a bit first by asking questions like “what do you think this sign means?”.

Here’s an example of student work:

95SW_Grade Sign

The point of this first activity was for students to become familiar with grades – the fact they are just the rise divided by the run in percent.  Also I wanted to make the connection between slope and angle measurements, algebra and geometry.

For the question on the final I want them to have the ability to convert percent grade to an angle measure.

Opener Scaffold #2

San Francisco Streets

Pick two of these streets and draw a diagram of them using the angle measure instead of the percent grade.   If you finish early build an accurate physical model of the street using string and tape.

This opener went well – I didn’t take any pictures.


Here are some of the results of their work on the exam.  I was surprised how many different ways the students went about solving it.









The Real Problem:

Here’s the actual text from my wife.  Notice the dimensions she gave (0.8 and 6.56 ft) were different than the ones I used (0.75 and 6.5 ft).  I changed the measurements because I didn’t feel I had done enough work with measurements for students to be able to deal with the actual dimensions.  I still the question qualifies as real-world.  Real enough world.


A Note On Construction Convention:

In construction – they would say “one over seven” to describe a slope of 1/7, and it would likely be written 1:7.  A slope of 4/9 would be called “four over nine”.  I talked about slope using that language during these activities.

The Goods:

PT Trig Building



Car Jack Problem – Math Modeling

My colleague Dave Casey designed the initial lesson.

I enjoy this modeling activity.  It’s based on developing a relationship between the width and height of a car jack – which just happens to be a perfect shape to be modeled by a rhombus.  I bring a car jack, but also like to show these videos because the jack opens a little slow for the students to get a great feel of the movement.

Or check it out in double time

Check out the cool graph a scissor jack makes (height as a function of width)


Here is one of the guiding questions I use – it’s also the big thinking moment that I make sure to stay very intentional about:

Is it more difficult to raise the car in the beginning or the end?

I phrase it that way as to make students have to define what it means to be difficult.  Some may think the end is more difficult because it takes longer to lift the car, and others think the beginning is harder because it takes more force (since you are lifting the car more).  Either way, they need to resolve what it means for it to be difficult.  Here is another way I phrase the question:

Let’s say 16 of us each signed up to move the jack screw 1″.  Would you rather be the 1st person, the last person, or does it not matter?

Here is an activity handout you can use.  I didn’t use it – but it was nice to have for students who were absent that day.

CW car jackI start by talking about the shape.  I have the students draw the shape and we make conjectures about the key characteristics of the shape.  The students have graph paper, so I recommend a uniform figure – I tell them to make it eight gridlines long, and +/- two gridlines up and down.

A critical point here is let the students know that the car jack is actually not a rhombus because the sides don’t actually touch.  So we are modeling the real car jack as a rhombus even though that is not actually the shape.  The rhombus is a mathematical model of an actual car jack.   Even the word “modeling” itself implies that we are not working with the real thing – rather a model of the real thing.

To improve the visual I usually draw a representation on the board of the jack close to starting position and ending position.  I add the length of the sides as 8″, and then I ask the these two questions:

  1.  What is the largest width we need to consider?
  2. What is the shortest width we need to consider?

I then put them at whiteboards and tell them the whiteboard number they are at is the width of the car jack they need to calculate the height for.  I also tell them to pick one other width.  I do not tell them how to do it – just what to do.  I’ve done this with classes that knew the pythagorean theorem and used it, and I’ve done it with classes that didn’t and we used this a vehicle to reteach the pythagorean theorem.

Common Mistake Alert:  A few students will anticipate a linear relationship and just quickly make a table that goes 16-0, 15-1, 14-2 etc.

Once they have the table, I have them graph it.  I show the Desmos graph, talk about how to write the function.

Once the graph is up, and the table is complete, I give them a couple minutes to consider when is it the most difficult to lift the car.  I survey the entire room.  Once each student is on record with an opinion – moving them into a talking point on the following prompt:

“It doesn’t matter.  It will be the exact same difficulty to turn the crank at all times because the car’s weight doesn’t change.”

Once the talking circle is over I am generally satisfied that they have all had sufficient opportunity to think and discuss.  I go around the room again and tally the results of the talking point and have a group discussion, calling on various students to give their reasoning.

I also like to bring up the slope of the lines, and ask them “What does slope represent in the context of this problem?”.   Between the first and second person is the largest slope – what does that represent in this context?” (By the end most of us are thinking of the scenario where 16 of each are each going to turn the screw one inch)

My goal by the end is to get the students to see that the car moves more in the beginning than in the end, and understand how that validates arguments for the beginning or end being more difficult based on how the word “difficult” has been defined.

Here is a students work – but a lot of the great student thinking that happened here today was not caught on paper.  I didn’t take any pictures of the whiteboards.

CarJackSW_Page_13 CarJackSW_Page_14

Regrets:  I never had them write their arguments for when they thought it was the most difficult to use the Car Jack.  At least not formally.  I need to do a better job with the written communication piece of modeling.





And even my textbook thought this was a good problem: