Monthly Archives: April 2013

Monopoly Math

The big Monopoly battle is coming near its end, and the iron and racecar are battling for Monopoly supremacy.

Monopoly Board

Both players own properties on the next block, and have some spaces they’d like to avoid.  For the car, here are the spaces he’d like to avoid.

Car Spots

And for the iron, there are a few spaces to avoid.

Monopoly - Iron 1

Since there some houses and hotels on some of the spaces, they are worth different amounts.  Below, here is how much each player will have to pay if they land on the “bad” spaces.

Car Board B

Monopoly - Iron 2

So, here’s the question:  which player is in “worse” shape?  Which player should be more worried about their upcoming turn?

Let this stew with your classes, and would enjoy hearing some class reflections.  The big reveal will come in a few days.

Linear Programming with Friends

An early morning post from Nik_D from the UK led to sharing class activities for linear programming, and provided a great example for me to share with colleagues on the value of twitter:

The activity on Fawn’s blog invites students to build Lego furniture and find the combination which will maximize profit.  I love the idea of handing students baggies with the “supplies” and having students build the chairs and tables.  But, without having Legos around, both Nik and I sought a way to approach these linear programming problems with a different hands-on approach.  In previous years, I had used sticky dots to help students visualize constraints and a fesible region.  Nik posted about his experiences last week, and now I am happy to share the U.S. point of view

Here, I worked with a freshman-year teacher who was eager to try something different to open linear programming.  As students wandered into class, they were given the initial problem.  The Powerpoint slides are available for you to use.


The class worked in teams to consider the problem.  Many start off by making data tables of the possibilities.

Student work 1

As the teacher and I circulated the room, we found that eventually, students consider algebraic models.

Student work 2

There was agreement on solution: 2 chairs and 2 tables are ideal.  The teacher asked students to share their ideas, which were written on the board, and led to new vocabulary: constraints, profit function.  We’re now ready to tackle our next challenge:


Many groans were heard, as students understood that “guess and check” would no longer be a great idea.  Note that the chair design also changes, which a student cleverly noted was from the “Game of Thrones” collection.

First, the class agreed on the constraints:

Then, for this part of the activity, the class is split into two groups, one for each constraint, which both the lead teacher and I worked with to explain the guidelines.  A spreadsheet with 50 identical “strategically selected” points were given to both groups, along with a pack of stickers.  The group task: test each of the points for their given constraint, and place a sticker on the wall if it satisfies the constraint.  The “small block constraint” group was given blue dots, while the “large block constraint” group had red dots.  After a few moments of organizational chaos, leaders emerged, and points were distributed nicely to the team.  Soon, dots made their way to the board.


After both groups were satisfied with their work, the teacher (Joe, below) discussed the dot patterns.  Where do the dots share space?  Where are there only reds, blues?  What parts of their graph are most important for this problem?  Then, I grabbed Nik_D’s idea by turning on the Desmos calculator and super-imposing  the inequalities onto the graph.  There was some prep work needed here, as Joe and I made sure the grid paper was placed nicely on his SMART board.  Also, please note that I seem to suck at taking clear pictures….it’s a probem.

Joe G.

One thing we would do differently here is letting students see the inequalities.  We hid them, so as to maximize screen space.  This would allow the teacher to turn the inequalities on or off, and emphasize where the colored dots reside.

The class discussion continued with an argument of how to identify the “maximizing” point, and the corner-point principle.

One last thought here.  The power of Desmos is evident for linear programming problems.  The teachers I work with agree that having students graph these sorts of problems by hand is not only time-consuming, it is silly.  By letting students experience the Desmos calculator, not only can we have real discussions of problems, we can tackle problems which may not be so graph-friendly.

Thanks to Nik D. and Fawn for the sharing!

Those Funky, Funky Exponential Functions!

A neat math discussion came from an unexpected place today when a teacher in my department sought me out with the TI Nspire of one of her students in hand. The student was in a pre-calc class where exponential functions were being examined, and attempted the graph the following:

What should we expect to see?  How does this graph behave?  Here is what my new Nspire app gave me, which matches what the student calculator showed.


There seems to be a little funkiness around the origin which confused the Nspire, but the bigger issue is that the meat and potatoes of the graph is just wrong.  This about these values of this function and you’ll see why:

This function be-bops around in a quite interesting manner, and the TI-84 shows the graph nicely, as individual dots.  After going through some usual diagnostics in my head, and the list of dumb things kids sometimes do to calculators which cause them to act funny, the problem seems to be with Nspire. But this got me thinking about this strange function, and it’s behavior.  What happens if x = 1/2?  2/3?  3/2?  What’s the domain of this function?  And how do some of my other online math tool friends handle this one?

Wolfram|Alpha is our first contestant.  Show me your stuff:


How cool!  And what a neat discussion of complex numbers, and an interesting overlap between real and complex parts.  Wondering if anyone has insight into the domain and range though.  Is it that this function has no domain?  Or is it that  the domain is simply too difficult to express nicely?

Next up is my old friend Desmos.  I know you won’t let me down.  First, entering the function, Desmos does nothing (trust me, no screencap…nothing happens).  But, activate the table and you can plot some points.  I also added a few of my own at the end of the table:


A good effort, but wish there was some indication of the graph’s behavior without the table.

Overall, this is a tricky little function with a lot to talk about.  Put it on the board for your classes and let them think about:

  • What rational values cause the function to be undefined over the reals?
  • What rational values cause the function to have negative value?  Positive value?

Then, the plot THICKENS!  Later in the day, I was showing of my Nspire app and the goofy function to some math friends at a meeting, when an English teacher collegue joined the fray.  After giving us the obligatory “what a bunch of geeks” staredown, she grabbed my iPad and gave a few finger swipes at the graphs, changing the window values and……


Holy crud!  How cool, yet….pretty much not useful at all!  During the day, I also sent a note out to TI about what I had found, and a response was given later in the day.  Thanks for getting back to me so quickly TI folk!

This Week’s Required Reading for Algebra Teachers!

Mid-April, that time of year where teachers and students start to see the finish-line of the school year.  Everyone feels the burdens…state testing, class distractions, covering all the “material”….teachers have a lot on their plate.  But it’s also a great time to reflect upon the past year, work in teams to consider best-practice, and plan changes for next year.  Two intriguing blog posts by Grant Wiggins this week should be required reading for all secondary math teachers.

First, Grant Wiggins rants against courses we call algebra 1.  What could be wrong with Algebra 1?  We all took it, we all agree kids “need” it, and isn’t a proven gate-keeper to college success?

Algebra, as we teach it, is a death march through endless disconnected technical tools and tips, out of context. It would be like signing up for carpentry and spending an entire year being taught all the tools that have ever existed in a toolbox, and being quizzed on their names – but without ever experiencing what you can craft with such tools or how to decide which tools to use when in the face of a design problem.

Amen, brother.  In algebra, we move from the unit of linear functions, to the unit on systems of equations, to the unit on exponents, then the unit on polynomials. At the end of each unit, we duitfully give the unit test, get some number score back, then move on to the next unit.  We have trained students to think this way:  that algebra means mastering one skill, then the next.  How often do we provide rich tasks which allow students to reflect upon their cumulative skills set?  I appreciate the work of many math folk out there to change the nature of Algebra 1 from a rigid sequence of skills to a course which encourages application and reflection, driven by interesting, authentic problems.  Some examples of outstanding math educators working to promote inquiry in math class are listed at the end of this post.


For many special education students, chunking is a device used to “help” students in algebra.  By continued pounding of square pegs into round holes, using worksheets of similar problems (i.e. solving a one-variable equation, with variables on both sides), students can achieve temporary, recordable “success”.  The students most in need of seeing auhentic problems are often those least likely to move past the chunking, and into authenticity.  Fortunately, to help sort out the madness, Grant Wiggins provided a second great article of required reading for math teachers this week:

Grant Wiggins on turning math classes into bits of disconnected microstandards.

What’s so harmful about taking a broad subject like Algebra and breaking it into pieces?  What is the consequence?

Take a complex whole, divide into the simplest and most reductionist bits, string them together and call it a curriculum. Though well-intentioned, it leads to fractured, boring, and useless learning of superficial bits.

Hallelujah!  Make sure you check out Grant’s driver-ed analogy for the full effect.  More ammunition for us to develop math courses rich with interesting, relevant tasks, where algebra is the tool, not the star of the show.

Fortunately, there are many educators out there working to develop tasks which develop algebraic thinking, and encourage the use of algebra as the tool, rather than the exercise.  Keep them in your toolbox for future planning.

Dan Meyer: the king of perplexity.  If you aren’t visiting Dan’s blog at least semi-regularly, then start now.  And check out his spreadsheet of tasks for the math classroom.  In the same theme, visit Timon Piccini, and his many on-point 3-act tasks.

Sam Shah:  Sam leans more towards the pre-calc, calc end of the math spectrum, but I apprecaite Sam’s constant self-reflection and great ideas for engaging kids in math discussions.

Kate Nowack:  sometimes task-oriented, sometimes ranting on policy, but always interesting.

NCTM’s reasoning and sense-making task library has a number of problems around which algebraic ideas can be wrapped.

Expected Value and Analyzing Decisions, part 1

As an A.P. Statstics reader, I am excited to see the increased emphasis on statistics and probability in the Common Core standards.  Ever better, the standards specifically ask our students to be able to reach conclusions based on data:

CCSS.Math.Content.HSS-MD.B.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

CCSS.Math.Content.HSS-MD.B.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

This is a great start, and requires that we move beyond just flipping coins and drawing beads from bags.  We need to get our students writing about what they see, and provide strategies for developing clear, statistical thinking.

In this first post, we’ll look at a famous game show, and examine possible decisions.  Next, the game of Monopoly will provide a more complex argument in expected value.  Finally,  in a third post, we’ll look at past Advanced Placement Statistics items and adapt them for use in non-AP clasrooms.

The game show “Let’s Make a Deal” provides a simple example in decision-making based on values and probability.  At the end of the show, contestants who won prizes during the show are asked if they would like to give up their prizes for a chance at a bigger prize by choosing one of three doors:

Should a player give up their prizes?  How much does the “big deal” need to be worth in order for a player to be tempted to give up what they won?  How about the other two doors…do they matter?  Show the first video to your class, and let them debate whether they would go for it, and take a class vote.

In the next stage, we can show what all 3 doors have behind them, and begin to consider the game as a whole:

Does this new information change any decisions?  How willing are you to risk your prizes if you know there are some other nice prizes to be won, which may or may not have the same value of what you already have?  The slides below let you walk through a discussion with your class.  Use dry erase boards or even Google Forms to have students share their ideas.

There are a number of approaches which can be taken here.  Hopefully your students develop one of these on their own, or have creative, new ideas.

  • In this example, 1/3 of the doors hold the Big Deal, but another door has a prize of essentially equal value, the “medium” prize.  It may be difficult for us to predict the value of the middle prize, but it seems plusible that 2/3 of the doors will have value at least equal to what we are giving up.
  • In the worst-case scnenario, the non “Big Deal” doors have minimal value to us, essentially zero.  So, we have 1/3 chance of selecting the Big Deal, and 2/3 chance of winning nothing.  We can compute the expected value:

  • In the video above, we have some insight into the three doors on the show, and there always seems to be a “middle” prize and a “small” prize.  We can compute the expected value based on this information:

In any case, it seems reasonable for us to consider giving up the $7,000, but next we can think about the limits of our decisions.  What is the most you would give up to go for the Big Deal?  Can students create a general rule for making the decision?

And finally, a few parting shots for discssion:

  • It’s easy to think of the prizes as a cash value only, but does the prize you are giving up matter?  What if you always wanted to go to Paris?  Does that add to the value?  Or are there prizes you would never give up?
  • Expected value gives a nice summary of what we should expect to see in the long run, after many, many trials.  But on Let’s Make a Deal, you only have one shot…the chance of a lifetime.  Does this change your approach to the decision-making?

Would enjoy hearing your class experiences in sharing the Let’s Make a Deal examples.  Stay tuned for the next post on Monopoly.

Quality Assignments at #sbgchat

Another great topic last week at #sbgchat (9PM on Wednesday nights), where quality assignments were the theme.  There are so many people providing excellent ideas and thoughts each week, the action fast and furious, I have now found myself “favoriting” many tweets,  taking time on the weekend to read through the good suff, and reflecting upon what it all means to me as a math teacher.  You can review last week’s action on Sortify, and thanks to Tom Murray for hosting the recap.  Each week, there are a number of questions you can respond to, but I am going to focus on just one of them this week.


Over my years of teaching, I have seen my approach to assignments change some.  As a beginning teacher, I did what I suspect many math teachers do: find a “good” worksheet which has practice problems tied to the lesson, or give the odd problems in the textbook.  In recent years, I added more reflective pieces to assignments, eventually using Google Forms to have students contribute ideas, in a move away from static assignments.  And my philospohies towards grading assignments has also changed to the point where I rarely grade nightly asignmnents.  Some of my favorite responses to this week’s #sbgchat question are helping to refine my attitudes further:

If you ask students and teachers separately what the goal of an assignment is, how would the answers from each group be different?  Undoubetly, many teachers would point to the need for practice and their students to learn responsibility.  But what would students say?  Do students see the need to practice skills as a prmiary outcome of assignment completion?  Are your students asked to reflect upon why an assignment is valuable?  Have the learning goals been communicated and understood?  And finally, how do the math assignments math teachers give today look and feel different from those given 25 years ago?

What I really apprecaite about #sbgchat is that I am challenged to think about my classroom practices.  Sometimes, these are not comfortable reflections.  Often, the hard work required to shift to effective pracices seems monumental, and I wonder who is up to the task – me, my colleagues, my students.


  1. Students should understand how assignment completion will help (or not help) them develop skills, and this should be the primary motivation for assignment completion.
  2. Students should have the opportunity to personalize assignments, selecting problems and/or experiences which move them towards their goals.
  3. Students should reflect upon their choices, and communicate how their choices helped them (or did not help them) reach their skill goals.
  4. Teachers have the responsibility to provide appropriate options for skill mastery, and discuss those options with students.
  5. Students should be allowed to mess up.  It’s natural, and all young people will make a bad choice.  Learning to move on and adjust from bad choices is a lesson unto itself.  

In a post from a while back, I provided some ideas for differentiating assignments, and some of the ideas seemed to be quite popular.  I would add now that perhaps students should also reflect upon their assignment choices and be asked to justify them.   Are students choosing to path of least resistance?  Or are they choosing assignments based on their perceived areas of need?

To incorporate many of these ideas will require a change in culture from both teachers and students.  Why do we provide assignments?  And why do students complete them?


  1. We have trained our students to play the school game.  Many assignments with point values cause students to play the point-gathering game, rather than reflecting upon their progress.  

Hadley Ferguson, a teacher near Philadelphia, has summarized her experiences with a non-graded 7th-grade class.  It’s inspirational reading.  The dedication to reflective practice, and creating a culture of saefty and authentic learning, have clearly changed the 7th grade.  It’s certainly not easy chaning a culture.

Here are more resources to help you assess and develop your own assignment philosophies:

Joe Bower – Real assessment for learning: Joe provides an outline of routines used in his classroom to provide feedback and information to students.

Creating Quality Classroom Assignments: Susan Brookhart provides a simple planning tool for evaluating classroom assignments.

Skills Mastery as the Beginning, Not the End: Justin Lanier provides his classroom experiences with a first attempt at standard-based assignments.  A sample checklist is given, and ideas of how to manage the grading.

ThinkThankThunk:  A wealth of resources and classroom experiences in SBG by Shawn Cornally.  The link here is for math, with ideas for fracturing your gradebook, but click around to find more resources.

Screencasting Follow-Up

About a month ago, I posted on screencasting tips and basics, using resources from an after-school PD session I facilitated for teachers in my district.  The fantastic TV crew in the district filmed my presentation, and edited it down to a nice summary of the session.  Click the link below to get to our district’s video PD page, and look for the “creating your own classroom screencast” video.

HHTV Professional Development videos

Also, feel free to enjoy a video from back in December on Writing Strategies for Math Class, and other topics from my district colleagues.

I usually hate watching myself in these videos, but appreciate the fine work of Andrew Morse, Bob Anderson, and the HHTV crew!