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

The 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:

The 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 169: Pizza Doubler

Geometry started sectors and arc length today. I used Dan’s Pizza Doubler to introduce sectors.

Students stepped up to the extension question of “Would the best coupon for the slice above work for all slices or just some slices? Tell me under what circumstances I should use one coupon or the other”

Day 165: Dandy Candies II

Geometry finished up Dandy Candies from yesterday.

My district has a cross curriculum writing protocol that pretty much all students understand. I had my group write a paragraph analyzing if there was a better packaging option out there than the four we looked at yesterday.

They did a great job on their paragraphs, I was impressed with the level of understanding and reasoning they put into the problem.

Day 164: Dandy Candies

Geometry tackled Dandy Candies today. This 3-Act has a great flow and I love the animations in the video.

I made a couple mistakes along the way in asking students to solve for both surface area and ribbon used at the same time; too much was happening at once for some of ’em. However, they were able to answer both questions by the end of the period. I sent them home work the extension question of “Is there a packaging method out there that uses even less cardboard than option B?”.

Another great question just came to mind: Will the option that uses the least amount of cardboard always use the least amount of ribbon? These questions are scary close to maximum and minimum values of functions… Nice.

More to come on this tomorrow.