Category Archives: Middle School

Some of My Students Failed Today! Woo Hoo!

A new semester has just begin here at my high school, and one of my classes is a co-taught course we call Prob/Stat.  The Prob/Stat course is one we offer to our 9th graders, as a follow up for Algebra 1.  It includes concepts in probability and statistics, along with algebraic concepts like systems, polynomial operations, and matrices.  The students in this academic class will take the Pennsylvania Keystone Exam in May, a graduation requirement, so this course is quite important for them.

My math department colleague and I, along with both co-teachers, agreed that we did not want this to “feel” like math courses they had taken up until now.  We wanted our students to become more reflective in their approach, think about their strengths and weaknesses, and devlelop their own learning paths.  We have embraced Standards-Based Grading and a policy of re-dos and retakes to help meet our ideals for this course.

On the first day of the class, I wanted to set the tone that communication and discussion would be valued in my classroom.  I asked the students to arrange their desks in a circle, which brought many questioning looks and rolling eyes.  But once we established our circle (actually, it was more like an oblong), I passed around small slips to paper to every student.  I asked the group to list any factors which had caused them to not perform well in their past math classes.  Many students were willing to share their stories: “I don’t do homework”, “Teacher X didn’t like me.”, “I don’t like to ask for help.”…the list was rich.  Placing a trash can in the center of the floor, I instructed students to ball up their slips of paper, and toss them into the bucket…they are in the past! Chum BucketI stole this idea from my time at the Siemens STEM Academy, where we started the week by catapulting our educations hold-backs into the chum bucket (it was Shark Week at Discovery Ed).  You can read more about the chum bucket activity on the Siemens STEM Institute blog.

Next, I asked the students to write something they could do, moving forward, to improve their math outlook.  What an awesome conversation!  One student shared her fear of reading problems in math, but a desire to work through it and seek help.  Many students confessed their need to complete assignments.  Others communciated the need to start self-advocating, asking more questions.


For many students in my class, this is their first experience with Standards-Based Grading.  Before the course began, I took all course concepts and arranged them into 4 anchors, mimicking the anchor language of the PA Keystone Algebra 1 content.  Each anchor contains 5-7 standards, written as “I can” statements.  The document also contains room for multiple attempts on the same standard.  As students complete notes or assignments, I instruct them to write the standard we are working on clearly at the top of the page.

In this course, we start off with the probability sections, so we actually led off with 4.5 “I can find the probability of a simple event”.  Probability is a topic which haunts students of all ages, sizes, and ability levels.  And while many students did just fine on their first quiz, a number of students struggled.  Under normal circumstances, this would cause deep sighs from me, and steamrolling on. But, to be honest: I HAVE NEVER FELT MORE ENERGIZED ABOUT STUDENTS STRUGGLING IN MY CLASS!

All students in the class have their own binder, which houses the Standards Tracker, and all assessments. During the next few class meetings, my co-teacher and I will develop groups for small group instruction to discuss mis-conceptions, and work towards the re-do on their 4.5 quiz.  At the same time, we have moved forward into 4.6, multi-stage events.  We are striving to set-aside time each Friday to be reflection and redo time, in order to establish regularity with these new grading concepts.  I find myself looking forward to students dicussing their needs, and working with them to do better next time.  It’s early in the semester, but already things feel different.

Check out some of my earlier blog posts on Redos, Retakes, and Standards-Based Grading:

Rick Wormeli – Redos and Retakes

Standards-Based Grading, twitter chat recap

Quality Assignments, #sbgchat

The Puzzle Solutions You’ve Been Waiting For….

This post will present solutions to two puzzles I have presented here on the blog, and some ideas for extending the learning with your students.


This is a game I proposed a while back on the blog, and it took my friend Anthony from Twitter’s gentle nudging to ask for a solution.  Here’s a re-cap of the game:

  • 23 marks are placed on a board
  • On each turn, a player must remove 1, 2 or 3 marks
  • The player who clears the board wins

I often challenge classes with this game, letting kids play me and try to figure out the secrets.  The beauty here is that my students are often polite souls, and will let me go first…which leads them to doom.  It will often take a few days before a student can conquer me.

Here’s a video example:

And here is the message I sent back to Anthony, who requested the secrets to the game.

Here’s the secret to the take-away game – if I can make it so that the number of dots remaining is a multiple of 4, then I will win the game – guaranteed.  If they take away 3, I remove 1.  When they remove 2, I remove 2.  If they remove 1, I remove 3.  The game is all about groups of 4.  

With 23 dots to start the game, I will offer the student a choice; who plays first.
If they let me play first (which they often do because they want to be polite), then I erase 3 dots (leaving 20) and wait for their eventual demise.  If they play first, then I need them to slip up and make it so that I can control the multiples of 4.  If they erase 2 on their first move, then I remove 1, leaving 20.  
It often takes a few days of playings at the end of class for kids to develop the strategy.  They will first realize that being left with 4 is sure doom.  Then 8….then eventually they get it.
Have fun
And here are some ways to tweak the game, and see if students can develop strategies:
  • Change the number of starting marks
  • Change the number of marks allowed to be removed
  • Play with 3 players (actually, I’ve never tried this…)
I posted about the Hot Seat game recently.  Time to reveal the secrets to this interesting pattern game –

This table shows the winning chair in the game, based on the number of chairs which start the game.  Note that if the number of chairs is a power of 2, the winning chair will equal the number of chairs.  From any multiple of 2, the winning chair goes up by 2, until the next power of 2 is reached.

How would your students express this pattern?  What vocabulary would they need to use in order to communicate the pattern?  After a class develops the data table, challenge students to write a concise rule which will identify the winning chair.  Then, have students trade their explanations and critique them.

Could an algebra student develop a formula which outputs the winning chair, if the number of chairs is given?  

Take a look at the winning chairs, plotted vs the number of chairs.  Can we write a function which follows the pattern?  Share your function ideas in the comments, and enjoy the challenge.


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.


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.