Category Archives: High School

Compute Expected Value, Pass GO, Collect $200

Photo Oct 11, 7 08 49 AM.jpgExpected Value – such a great time to talk about games, probability, and decision making!  Today’s lesson started with a Monopoly board in the center of the room. I had populated the “high end” and brown properties with houses and hotels.  Here’s the challenge:

When I play Monopoly, my strategy is often to buy and build on the cheaper properties.  This leaves me somewhat scared when I head towards the “high rent” area if my opponents built there.  It is now my turn to roll the dice.  Taking a look at the board, and assuming that my opponents own all of the houses and hotels you see, what would be the WORST square for me to be on right now?  What would be the BEST square?

For this question, we assumed that my current location is between the B&O and the Short Line Railroads.  The conversation quickly went into overdrive – students debating their ideas, talking about strategy, and also helping explain the scenario to students not as familiar with the game (thankfully, it seems our tech-savvy kids still play Monopoly!).  Many students noted not only the awfulness of landing on Park Place or Boardwalk, but also how some common sums with two dice would make landing on undesirable squares more likely.

ANALYZING THE GAME

After our initial debates, I led students through an analysis, which eventually led to the introduction of Expected Value as a useful statistic to summarize the game.  Students could start on any square they wanted, and I challenged groups to each select a different square to analyze.  Here are the steps we followed.

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First, we listed all the possible sums with 2 dice, from 2 to 12.

Next, we listed the Monopoly Board space each die roll would causes us to land on (abbreviated to make it easier).

Next, we looked at the dollar “value” of each space.  For example, landing on Boardwalk with a hotel has a value of -$2,000.  For convenience, we made squares like Chance worth $0.  Luxury Tax is worth -$100.  We agreed to make Railroads worth -$100 as an average.  Landing on Go was our only profitable outcome, worth +$200. Finally, “Go to Jail” was deemed worth $0, mostly out of convenience.

Finally, we listed the probability of each roll from 2 to 12.

Now for the tricky computations.  I moved away from Monopoly for a moment to introduce a basic example to support the computation of expected value.

I roll a die – if it comes out “6” you get 10 Jolly Ranchers, otherwise, you get 1.  What’s the average number of candies I give out each roll?

This was sufficient to develop need for multiplying in our Monopoly table – multiply each value by its probability, find the sum of these and we’ll have something called Expected Value.  For each initial square, students verified their solutions and we shared them on a class Monopoly board.

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The meaning of these numbers then held importance in the context of the problem – “I may land on Park Place, I may roll and hit nothing, but on average I will lose $588 from this position”.

HOMEWORK CHALLENGE: since this went so well as a lesson today, I held to the theme in providing an additional assignment:

Imagine my opponent starts on Free Parking.  I own all 3 yellow properties, but can only afford to purchase 8 houses total.  How should I arrange the houses in order to inflict the highest potential damage to my opponent?

monopoly-back-row

I’m looking forward to interesting work when we get back to school!

Note: I discussed my ideas about this topic in a previous post.  Enjoy!

Pulling In To the Station

My school isn’t 1-1 with technology yet, though there are rumblings we will get there next year….or the year after….or 2031…anyway, it’s time to get techy!  My new classroom features 4 computer stations in the back – nice to have, but not super-helpful with classes of about 24 each. Station-model classroom structure has been super-helpful in my pre-calculus class in the first month. Besides the chance for all students to participate in rich technology-based activities, I’ve had the opportunity to carve out valuable small-group time with students.  Here’s an example:

In our first pre-calc unit, we review functions and their shirts, folding in new ideas like the step function, piecewise and even/odd functions.  My objective for the class was for students to consider functions in varied forms.  As students entered class, playing cards were drawn to establish their groupings, so there were 3 groups of 7 or 8.  With 15 minutes on the classroom clock, students started on their first station:

  1. Group 1 gathered in a small group with me in a circle of desks, where we worked through proving functions even or odd, and sketching their graphs.
  2. marbleslideGroup 2 worked at the computer stations on a Desmos Marbleslides featuring quadratic functions, with many students pairing up to work together. If you have never tried a Marbleslides, run and play now – we’ll wait for you to come back…
  3. Group 3 worked out in the courtyard (hey, my new classroom leads outside – which is nice) on a group task involving a piecewise function.

After groups had rotated through all 3 activities, we had time to recap / share and assess our learning over the hour.  Here’s why I need to do this more:

  • The small group station let me touch base with every student, assess strengths, find out what we need to work on, and provide feedback to everyone.
  • Marbleslides is sneaky awesome! When students begin to obsess over function shifts and how to restrict domains and don’t want to peel away from their computer, you know something is going right.
  • Class went fast! It felt like the mixed practice from Let It Stick was now becoming part of my classroom culture.
  • My pre-calc is mostly 11th and 12th graders, who have had a pretty traditional classroom experience in their math lives.  I can sense they appreciate that something difference is happening.
  • All students are responsible for their learning.  Even the least-active task, the piecewise function, was used the next class for sharing out and a jumping-off point.

 

Activity Builder – Classroom Design Considerations

This past summer, our forward-thinking math-teacher-centric friends at Desmos released Activity Builder into the wild, and the collective creativity of the math world has been evident as teachers work to find exciting classroom uses for the new interface. Many of these activities are now searchable at teacher.desmos.com – you’re welcome to leave now and check them out – but come back…please?

Its easy to get sucked in to a new, shiny tech tool and want to jump in headfirst with a class. I’ve now created a few lessons and tried them with classes which range from the “top” in achievement, to my freshmen Algebra 1 students. In both cases, I’ve settled upon a set of guiding principles which drive how I build a lesson.

  • What do I want students to know?
  • What path do I want them to take to get there?
  • How will my lesson encourage proper usage of math vocabulary?
  • What will I do with the data I collect?
  • How does this improve upon my usual delivery?

It’s the last question which I often come back to. If making a lesson using Activity Builder (or incorporating any technology, for that matter) doesn’t improve my existing lesson, then why am I doing it?


One recent lesson I built for my algebra 1 class asked students to make discoveries regarding slopes and equations of parallel and perpendicular lines. Before I used it with my class, a quick tweet 2 days before the lesson provided a valuable peer-review from my online PLC.  It’s easy to miss the small things, and some valuable advice regarding order of slides came through, along with some mis-types. The link is provided here in the tweet if you want to play along:

The class I tried this with is not always the most persistent when it comes to math tasks, but I was mostly pleased with their effort. Certainly, the active nature of the activity trumped my usual “here are bunch of lines to draw – I sure hope they find some parallel ones” lesson.

As the class finished, I called them into a small huddle to recap what we did. This is the second lesson using Activity Builder we have done together.  In the first, the students didn’t know I can see their responses, nor understand why it might be valuable.  In this second go-round, the conversation was much deeper, and with more participation than usual. In one slide, the overlay feature allowed us to view all of our equations for lines parallel to the red line:

parallels

We could clearly see not only our class successes, but examine deeper some misunderstandings.  What’s happening with some of those non-parallel lines?  Let’s take a closer look at Kim’s work:

Parallel 2

What’s going on here? A mis-type of the slope? The students were quite helpful towards each other, and if nothing else I’m thrilled the small group conversation yielded productive ideas in a non-threatening manner – it’s OK to make errors, we just strive to move on and be great next time.  The mantra “parallel lines have the same slope” quickly became embedded.

The second half of the lesson was a little bumpier, but that’s OK.  Before questions regarding slope presented themselves in the lesson, storm clouds were evident when the activity asked students to drag a slider to build a sequence of lines perpendicular to the blue line.  Observe the collective responses:

perp2

So, before we even talk about opposite reciprocal slopes, we seem to have a conceptual misunderstanding of perpendicular lines.  I’m glad this came up during the activity and not later after much disconnected practice had taken place.  In retrospect, I wish I had put this discussion away for the day and come up with a good activity for the next day to make sure were all on board with what perpendicular lines even look like, but I pressed ahead.  We did find one student who could successfully generate a pair of perpendicular lines, and I know Alexys enjoyed her moment in the sun.

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What guiding principles guide you as you build activities using technology? How do they shape what you do?  I’m eager to hear your ideas!