Monthly Archives: September 2014

Class Opener – Day 19 – A Blast From Geometry Past

During yesterday’s group work, which included a discussion of Pascal’s Triangle, I overheard some groups mention Sierpinski’s Triangle, which they had seen some in Geometry last year. That led to today’s opener, an applet from the awesomely mathy site Cut The Knot:


In the “Chaos Game” a point be-bops about a triangle under specific rules:

  1. The red point starts one of 3 randomly selected vertices of the triangle.
  2. Next, one of the 3 vertices is randomly selected, and the red point moves half-towards this new point.
  3. The process is repeated over and over, and all landing points are marked.

At first, I have the applet run slowly, and students don’t quite absorb what is happening. But as we speed up the animation, something interesting develops….


Our old friend, Sierpinski’s Triangle! Later in the period we saw this famous structure again when discussing Pascal’s Triangle and factors. Check out this cool coloring remainders applet and have fun!


Class Opener – Day 18 – We’re Going Bowling!

A unique sculpture greeted students as they entered class today:


There’s a lot of math goodness happening in this picture, but I don’t want to steer conversation in any particular direction right off. Time for some Noticing and Wondering! Students shared their thoughts on the back board:


Most of our class time today will be spent completing a jigsaw activity which guides students through many of the rich connections between Pascal’s Triangle, Combinations and the Binomial Theorem.  Knowing that I would eventually talk about Pascal’s Triangle (one of my favorite shares of the year!), I was hoping to see if we could generate ideas on triangular and tetrahedral numbers organically.  This visual opener did the trick. And while I ran out of time today for my Triangle chat, it’s in my pocket for tomorrow!

After sharing this experience on Twitter, Annie Fetter (the queen of noticing and wondering) chimed in with her ideas:

So many great ideas for packaging to be had here, but thinking I share it and leave it to my geometry colleagues to explore.

Class Opener – Day 17 – Time for a Card Trick

Today is the last day of school before a 4-day weekend. Since we have been talking a lot about cards and probability in class, it’s a great time for a card trick to maintain enthusiasm before a few days off. Enjoy a glimpse into the craziness which is my classroom…

A “how to” guide for this trick is given here by Mismag822 on YouTube:

In the instructional video, you find that the secret to this trick lies in the cards being face-up the first time around, which does transform the order of the cards during a “switch”. Later, when the cards are dealt face down, two single cards placed face down individually is equivalent two cards switched and placed faced down. For my geometry folks, there is a cool transformation introduction here.

Class Opener – Day 16 – A Revealing Discussion of Factorial

I opened today with what I had hoped would be a rich discussion concerning a past contest problem, but turned into something more substantial.  Here is the problem:


I wasn’t too surprised when many, many students reached for their graphing calculator…this is what freshmen do. I was however surprised when I asked the class to volunteer their ideas on how they could simplify the expression – placing ideas on the board.  Some appear below, and we discussed why or why not the procedures were valid.

photo1 photo2

Many misconceptions regarding what it means to “cancel” in a fraction were revealed; in fact, the very nature of what it means to reduce was the star of our discussion.

2014-09-23_0003Later in class, I provided a hint which I hoped would provide some clarity with our factorial challenge. Some students immediately saw the link to the original problem and simplified. But how much closer are wo to finding the largest prime divisor? After simplifying, it was back to the calculators….which doesn’t really help much here.  Listing the remaining factors of 16! after dividing common factors of 8! leaves us with the clear answer : 13.

The moral of the story – those who never touch a calculator discover the mechanics of this problem much more quickly than those who take a sledgehammer to it with a Nspire CX.

Class Opener – Day 15 – Paint it Black

Students arrived in class today to find tic-tac-toe boards on their desks, and a challenge on the board:

A 3×3 square is partitioned into 9 unit squares.  Each unit square is painted either white or black with each color being equally likely, chosen independently and at random.  The square is then rotated 90 degrees clockwise about its center, and every white square in a position formerly occupied by a black square is painted black.  The colors of all other squares are left unchanged.  What is the probability that the grid is now entirely black?

This is a scary, scary, looking problem, which I have shared before on the blog. I learn much about my classes by observing the reactions to these sorts of problems: who reads carefully, who dives right in, who turns to share thoughts with their neighbors, who gives up immediately, and so on…so much problem solving comfort revealed in one problem.

So how do we start? After a few reads, I asked students to experiment with their boards, and discover some patterns which meet the problem’s constraints.


As students made discoveries and found boards which met the problem’s requirements, I invited a few up to the board to explain their work. This led more students, many who were apprehensive at the start, to think about the problem and the rotations.

Students began to discuss their findings, and some agreements were reached:

  • The center square must be black
  • There must be at least 5 black squares

But do ALL grids with at least 5 square work? This led to one last challenge for the day – find a grid with 5 black squares which does not work? This was quickly tackled by a few groups:


We haven’t tackled the randomness and probability aspects of the problem yet – that will resume tomorrow. But hopefully less apprehension over complex-looking problems and some developing teamwork!

Class Opener – Day 14 – Many, Many Meals

Starting one of my favorite units of the year: permutations, combinations and the binomial theorem. I stumbled upon this article proclaiming the arrival of 140 Million Burger Combinations, heading to New York City – and when I see combinations in the news, it’s time to investigate.

burgersThe article comes from 2010 and heralds the opening of an exciting new direction in burger construction (which has since closed). The website for 4food is sill active for now, and has a cool applet where you can build your own burger. There are many decisions to be made, and some exotic choices (a scoop of roasted brussels?).  I offered students the opportunity to create their own burger, with stations set up on my laptop and on my ipad. Much pro-con debate over the appropriateness of burger pickles ensued!

The choices 4food offers (or, offered, as they are closed…) were summarized by my students:

  • 4 choices of bun
  • 4 choices of “add-on”
  • 10 condiment choices
  • 5 cheese choices
  • 3 “slice” options
  • 12 “scoops”
  • 6 patty options

But multiplying these numbers does not get us near 140 million…so what gives? My classes will explore this problem deeper in the coming days, but for now some seeds have been planted. Soon, we will consider the possibility that you could select multiple condiments, cheeses, and scoops, and work to derive the final count.

This scenario brings to mind a counting principle challenge I have provided classes in the past:

The Tastee Donut Shop charges eighty-nine cents for its Mix N Match selection, which allows you to select any three doughnuts from among the following varieties: plain, maple, frosted, strawberry, blueberry, vanilla, chocolate, glazed, and jelly.  How many different Mix N Match selections are possible?

Here is a printable version of this problem you can share with classes.

I enjoy this problem because students need to think beyond a one-step counting problem. This challenge is more sophisticated than many worksheet problems in that we need to consider a number of possibilities – could a customer buy 3 of the same donut? 3 different donuts? 2 and 1? In the end, the solution comes down to the sum of 3 distinct possibilities, each more challenging:

  • Buy 3 of the same donut (easy): 9 ways
  • Buy 3 different donuts (medium): compute 9 choose 3
  • Buy 2 of one type, a 1 of another (hard): we need to pick two flavors. But picking 2 glazed and 1 jelly is distinct from 2 jelly, 1 glazed. Order matters. Compute 9 pick (permuation) 2.


Class Opener – Day 13 – Serenity Now!

Is Mr. Lochel asleep?

Today is the first major test for my freshman classes, and for all of them it is their first big test as high schoolers. And after a Back-to-School Night the evening before where I discussed some study strategies for the 9th graders, it’s a pretty stressful day for the young ones.

As students entered I heard the usual cacophony of frenzied papers being shuffled and concept cramming. It’s just too much noise and too much distraction before a test. Time to change the culture some with a video:

5 minutes of restful waves and ocean breezes to clear the mind. But it took a few minutes to take hold. At the start, as I sat in the front, eyes closed, silently contemplating the day, most students continued their frenzied studying. But eventually a few joined in, resting their heads, shushing each other, and taking advantage of a few moments away from math.

I’m hoping that today’s test will mark improvement for a number of my struggling students. I find that students coming from middle school often suffer from a similar mindset when it comes to taking math assessments: every problem must be done rigidly, teachers grade with an eye for missign nuance (arrows at the ends of lines, that sort of thing), papers are returned and go into a folder, and we move on.

This cycle isn’t good enough if we want students to reflect on their progress and grow.  The usual test study formula, where students shuffle through notes and seek more practice problems, isn’t sufficient. And while it is difficult to cause students to completely change study habits, I provided some tips for students as they progressed through the unit:

  • Every time you encounter a sticky classroom or homework problem, place a star next to it. In the days leading up to a test, redo these problems. Have the concepts had time to marinate? Are you now able to complete these problems with less difficulty?
  • Reflecting upon past classroom quizzes is essential. This year, my students are required to re-do all missed quiz items (excluding minor errors) as homework and attach them to their original quiz. I’m happy that a handful of students visited to discuss their corrections, while many more re-visited their previous math sins.
  • Deep breaths and long pauses matter. Undo obsession over that one test question, the one you have been working on for 20 minutes, is probably not healthy. Think about the warm ocean breezes, move on to items you CAN do well, and remain upbeat.