Teaching the Teacher

After last weekend’s meltdown, I assured my principal that I mentally compose 3 or 4 resignation letters a year. Don’t worry about them unless you see me making reservations to a country that doesn’t extradite to the US. (Really, just kidding about that!) Today I had a fun experience. One of my students taught me something.

I always tell my students that I expect to learn something from them each year. This year it came in an unexpected way. I have the kids solve various math problems at the end of each class. Today’s problem was to find out the next 5 numbers in this list: 1, 4, 9, 16, 25, __, __, __, __, __. Now, I hope you know this one, because if not, this is going to be a spoiler. (If you don’t know it – change your FB profile to a zebra and tell me if it’s black with white stripes or white with black stripes. Somebody already took the giraffes.) Now, my understanding was that you would get the next number by continuing to square the numbers in the sequence. 1^2 = 1; 2^2 = 4; 3^2 = 9, etc. That would make the next numbers 36, 49, 64, 81 and 100.

So I called on someone who said that they had the right solution in the one class that we had time to discuss this. They had it! When I asked them to explain, they said, “Well, you add 3 to 1 and get 4. You add 5 to 4 and get 9….and you just keep going” I replied “That’s not right what are your numbers?” And then I started figuring. It works. I told them that wasn’t the answer I was looking for, but it worked at least with these numbers and I had learned something.

So, when my planning time came along, I set it up in Excel…and it worked! As I told my wife about it, we figured out the formula, and then a visual for why it worked. The formula for the process is n^2 = (n-1)^2 + 2*n – 1. Does that make sense? Does that make sense? In words, if you want to find the square of the next number, take the square of the number you already know. Take the next number, multiply by 2, subtract 1 and then add it to the previous square. So why does this work?  Let’s see if I can make a text graphic. Let’s start with 7^2

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XXXXXXX   That’s 49. Now, according to the theory, add 2*8 and then subtract 1 for our answer. Here’s why. The green X’s are the 15 we add. We add 8 across and one less than 8 going down to make sure each row has 8 Xs.

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XXXXXXXYou now have 8 rows of 8 or 8^2 that gives you 64 Xs. If you count the green Xs, you will find 15. You are creating the new row and then adding one more X to the already present rows. My wife came up with this explanation. That being said, I was grateful to that student for the challenge! We’ll go over it in class tomorrow. I really do love it when my kids teach me something!. Makes me so proud!

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