Day 176: Perfect Squares

In geometry I put this up on the board as they walked in.

1, 4, 9, 16, 25…

I asked ’em what patterns they saw.

  • Perfect Squares
  • First difference is always odd
  • Constant second difference of 2
  • y = x²

I was wondering if they would come up with this:


I eventually put the pattern up and asked if any square multiplied by 4 is always a square. They said yes and talked about the shape a dot representation of the number would make when multiplied by 4.

Then a student asked “Is a perfect square multiplied by a perfect square also always a perfect square?” 

Well (a²)(b²) = (ab)²

We had a great 5 minute discussion about this, I had never thought of the product of two perfect squares, it is always fun when I learn along with them. I don’t talk enough about number theory, this was a great exercise in developing both number theory and sense.

Day 174: Guess My Rule

I put this up as the opener for algebra today:


A large chunk of students quickly jumped to “the previous number times two”.

That wasn’t my rule.

Then they shifted to +2, +4 +6…

That wasn’t my rule either.

Silence for about 2 minutes… Then a student asked “Is the next number 10?”

“That fits my rule” I replied.

This threw ’em all off… They dabbled around with different numbers for a bit, I had fun with it, especially when they pitched crazy numbers like 54, 81, and 1092, which fit my rule.

Eventually, they drifted into trying to find numbers that didn’t fit my rule… which is awesome problem solving.

This really is the heart of what I want to get at through all this factoring, solving, graphing and so on. To me math isn’t so much about memorizing or reproducing a certain skill, but more about taking what you know and tweaking it to solve some crazy problem you have never seen before.

That moment where you have abandoned all hope and try some crazy technique in a problem that you learned months or years ago, which ends up working is what I love about math. The struggle leading up to that moment; following hundreds of self-imposed rules and just sheer grit isn’t easy. It is even more difficult to learn and teaching it takes someone really special.

Everyday I try and very delicately move students towards that direction. With the hopes that maybe, at some point in their lives, in a situation that isn’t even close to math related, they will be able to use their critical thinking abilities, which took years to develop, to solve a difficult problem and experience that felling of having everything fall perfectly into place.

For me, that hope makes everyday worth it.

By the way… my rule was each number had to be larger than the previous. They went crazy over this, some nasty reverse psychology on my end: They automatically see something math related and dive into testing different equations and rules, when really the rule doesn’t require anything fancy. Credit. 

Day 173: Staggered Starts

In high school, I ran the 1600 and 3200 meter races. In one particular 3200m there were over 50 runners on the track. I was placed in an alley made up of lanes 9 and 10 and was out in the middle of nowhere, already halfway around the curve. We cut in at half a lap, it seemed like I was way ahead but ended up landing in the middle of the pack.

Here is what geometry looked at today:

*I mashed together a few problems/resources to put together this problem: 1        2        3

TrackThe first question I asked students was “If all the runners started at the same place, how much more distance would the runner in the 8th lane cover compared to the 1st lane runner?”

We put a few guesses on the board then they asked for some info, I ended up giving ’em this:

Track2The surprising piece is the straights of the track aren’t 100 meters… I think there is the possibility of an awesome problem within that piece of information that would really get students thinking (maybe designing a track or something). Not quire sure how to approach it though.

They went on their way, after a few minutes I introduced some structure.


They finished the table then I asked ’em to make it fair and on a picture of a track, find where the staggered starts should be so every runner ran exactly 400 meters. I lost a few of the students at this point, they were pretty stubborn in moving forward. so the problem just hung loose at this point for the rest of the period. When I try it again next year, I might stretch the thing into a period and a half or so.

Day 172: Substitution Puzzles

For the last few weeks I have been using half the period on Fridays to review. A couple of weeks ago I came across some great puzzles at MathArugments180 (the actual credit seems to go to Mimi). These helped students ease back into solving systems by substitution today. I want to also make a note here for next year to use these as a follow up to Noah’s Ark.


A colleague showed me these puzzles about 4 months ago also… I missed out not using ’em earlier.

Day 171: Graphs

The last week of algebra consists of an introduction to statistics. They start the whole process by reviewing (for most) how to interpret graphs. Normally, this isn’t a very exciting process but I found some fun graphs to spice things up.

First I put up a blank slate and them ’em guess what it could be:


Turned out to be something like grams of fat for ingredients of a sandwich.

After we went through all the different types of graphs, here are my favorite examples of each:





Gotta keep content interesting with the end of the year looming….

Day 170: Programming

Algebra is wrapping up work with the quadratic formula. Today we went over how to write a program for the quadratic formula on Ti calculators.

I started ’em off with a step-by-step explanation of writing a program to find the slope of a line between any two given points. After testing it a few times, I passed around this sheet of code;


Students either went at it themselves or looked up to the smartboard where I was working through writing the program with a document camera. I know there are programs out there with less lines of code and if then commands for the discriminant, but for only one day of programming I felt this was more than enough.

They then used the program to correct an assignment from yesterday. For the last 5 minutes of class, we worked through writing code for this mystery program: