Year: 2013

Categories
Algebra Middle School

My “Fake World” Task

Dan Meyer’s recent post on “fake world” math tasks has me thinking about many of the openers and games I have used in my classroom.  I have written about The Take-Away Game before, and I still use it often…until the kids learn how to beat me and the strategy is revealed.  This next one is not so much a game, but more of a task, similar in some ways to the Locker Problem.

THE HOT SEAT

In this task, chairs are placed in a circle.  Chairs will be removed from the circle using the following rules:

  • Chair #1 is removed first.
  • The next remaining chair is skipped, and the next chair removed.
  • This continues, with chairs skipped and removed until only one chair remains.
  • Once a chair is removed, it is “out” of the circle
  • Whoever is sitting in the last remaining chair “wins”

Here’s a brief Doceri video which shows some game playings:

Like the “Take-Away Game”, I can’t recall where I first encountered this problem.  They have both been sitting in my files for over a dozen years.  If anyone can name a source, I’d be happy to award some credit.

Why I enjoy this problem:

  1. It’s not intimidating.  We have a chance to draw, get out blocks, magnets….whatever we want to use to model the problem.  Great for working in teams.
  2. I can let the problem marinate.  On one day, I may ask the class “Where should I sit if there are 8 chairs?”, and come back the next day with “How about 24 chairs?”  If it seems like discussion is flowing, I can put my foot on the gas.
  3. I can use this problem with all levels of students.  If we need to create a data table and look for a pattern as a class, that can happen.  If my honors kids want to fly with it, that can also happen.
  4. The answer is not obvious, but a clear pattern eventually emerges if you model enough circles.  And there will be some nice vocabulary opportunities as the payoff.

There are a number of ways to express the solution.  Later this week, I will post the “answer”.  Until then, have fun moving around the furniture.

Categories
Algebra

Student-Created Polynomial Digital Notebooks

Over half-way done my first time teaching Algebra 2 in over 8 years (under block scheduling), and it’s amazing how much technology has changed many of my former approaches.  Nearing a chapter on polynomial functions, I was somewhat dreading the experience.  This is a pretty dry chapter…synthetic division, rational root theorem, complex conjugate roots…there’s a lot of rules and regulations here, and not much room for engagement.  Here’s what I settled upon to avoid dry lectures, promote student ownership, and encourage a serious review of resources:

DIGITAL NOTEBOOKS FOR POLYNOMIALS

On day 1 of the unit, I gave students their assignment for the chapter, which you are welcome to download: Polynomial Notebooks.  The link describes the assignment:

This chapter contains many landmark theorems and ideas for analyzing polynomial functions.  Your job is to create a digital notebook of the ideas below, using a Google Document to house your information, and eventually share with the class.  You may use online resources, or examples you create, to serve as examples for each idea.  The goal of this document is to serve as a resource which demonstrates an understanding of the ideas, and which helps you study their meaning and usage.

To be honest, this did not start off as I expected, as my students were simply not accustomed to not being fed material.  The idea of seeking outside resources, being able to weigh their merits, and summarize them for use was foreign.  But our second day in the computer lab was fruitful, as students had great questions about the Intermediate Value Theorem and Descartes’ Rule of Signs; what was really fasicnating were the student attempts to think about the ideas in language which made sense to them: “Does this mean that…?”, “Does this example work here…”.

Not all class time was spent doing research.  We started class with sythetic division problems done on boards around the room.  Also, I developed 3 “food for thought” questions each day during the unit, which helped drive discussions.

  • Name a polynomial with roots  2i and -3 (multiplicity 2).
  • How many possible negative real zeroes does the following polynomial have?
  • Show that polynomial given must have a real zero between 2 and 3.

Enjoy all of these Food for Thought questions: food_for_thought_daily_qs2

It’s also interesting how different groups have chosen to format their notebooks.  While some are going through my list in a linear fashion, others have developed connections and examples to help make sense of the rules.  One ambitious young man has developed an outline of the topics, rearranging the ideas, providing examples, and organizing links.

Tomorrow we will have a group quiz on the material, and I feel like this experience has helped students personalize their needs, and think about their gaps.  I also feel more confident in the aiblity of the class to think about these  polynomial rules as a whole, rathern than as a set of disconnected ideas.  Hoping for a good day!

 

 

 

Categories
High School Statistics

Matched Pairs with Hallway Bowling

The experimental design unit in AP Statistics is a fun one, with lots of opportunities to design activities, discuss possibilities and collect data.  For a few years, a “Hallway Bowling” activity I created has been one of my favorites for discussing matched-pairs experiments.  This year, I added a new wrinkle to this activity day, in order to economize class time.  As students entered the class, they drew a playing card, each having one of three suits which determined their group assignment for the day.  Each group had 7 or 8 students.  Groups then rotated through 3 stations, with 15 minutes on the clock, and with each activity designed to review a different aspect of the chapter.

In Station 1, students met with me in a small group, where we discussed experimental design, writing ideas and experiment trees on desks.  This was a departure from whole-class discussions, and more students had the chance to share their ideas on experiments dealing with clothes washing temperatures and drug trials.  Experimental design vocabulary like blocking and matched-pairs were clarified, and the small-group discussions were rich.  At the end of the day, the students shared how much they liked being able to share in a more intimate setting.

In Station 2, the group completed an actual AP item dealing with experimental design.  Papers were collected as a group, and I will randomly choose 2 paper from the group to grade.  Students knew this going into the activity, and this procedure holds all students accountable for the group grade/

In Station 3, the group went out of the room to play and collect data with “Hallway Bowling”.  15 minutes was enough time for students to practice, play, and collect data.

You can down loading the rules here:  Hallway Bowling

Here’s how Bowling works.

  • 2 markers are placed 5 meters apart (I had pre-taped blue X’s on the floor)
  • players stand behind one marker, and roll a golf ball as close to the other marker as possible.
  • During the data recording, players will roll 4 times; alternating hands and measuring the disatance to the marker.

BowlingAfter the activity, a whole-class discussion is held to talk about Hallway Bowling as an experiment.  What are we trying to prove?  How does our activity provide data for the experiment?  Where is the randomization?  What could be done to improve the design?  Here, we are looking to encourage “matched-pairs thinking”; where all subjects are exposed to both treatments (rolling with dominant and non-dominant hands), and we are interested in those differences.  We can also consider blocking here if we feel that males and females may be effected diffferently by the treatments.  We can also revisit the data later when we look at hypothesis testing procedures.

And about that data we collected?  My kids entered their data into a Google form.  There are some great comparisons to consider: right hand vs left hand, boys vs girls.  But how did the distances come out for dominant hands vs non-dominant hands?

Graph 1

Note the difference in medians here.  But can we directly compare individual player performances?  To do this, we can subtract dominant and non-dominant hand scores, and observe the differences:

Graph 2

If players are truly better with their cominant hands, we should see many negative differences here.  We see over 50% negative, but is there enough evidence to prove a mean difference for ALL players?  Time to start linking to inference.

So have fun with hallway bowling, and try some classroom stations!