Yea, that’s right. I just had 2 classes in a row teach themselves and others how to complete the square with circles and ellipses. How did I accomplish this miracle, because I really do consider it to be a miracle.
In my Advanced Algebra Class we have the following problem called, creatively enough, The Lost Hiker.
Suppose you need to find yet another lost hiker. Fortunately you have information from 3 different radio transmitters. From this information you know that he is:
25 miles from Transmitter A
5 miles from Transmitter B and
13 miles from Transmitter C
These radio transmitters are at the following coordinates:
Transmitter A – (25.5, 7.5)
Transmitter B – (-2.5, 3.5)
Transmitter C – (6.5, -11.5)
Use this information to find the coordinates of the lost hiker using the intersections of the theoretical circles. Show your work below.
There are 4 major steps here; 1. Write the equation of 3 circles, 2. Expand the circle equations, 3. “Collapse” the circles through polynomial subtraction to 2 lines, and 4. Solve the system of lines.
We worked on these for 3 days. These problems are huge and complex, and one single minus sign incorrect means the whole thing works out wrong. But they persevered for three periods in class, until every single learner could do 3 unique problems on their own, and get the correct answer.
I have an excel spreadsheet programmed to calculate an infinite number of nice problems with solutions. The solutions are posted on the board w/ a magnet, and I walk around with a bunch of problems in my hand. Get one right, here is another.
Well, at the end of 3 days, I wrote the following equation on the board and asked them to turn it back into a circle.
x2+ y2 – 6x + 8y = 12
It took them all of 1 minute to have the entire class finished. One learner who failed Alg2 taught the rest how to do it because it was, “Easy, just take the -6, divide by two, because when you square it you have two of them.”
Then I wrote 5 problems on the board.
x2+ y2 – 4x + 12y = 10
x2+ y2 + 2x – 10y = -8
x2+ y2 – 4x + 7y = -20
2x2+ 3y2 + 6x + 15y = 0
4x2+ 5y2 + 16x – 100y = 27
The first two were just some more practice. The question was asked, “Can we have a decimal?” When the answer was yes, they didn’t bat an eye.
The last two did make them think, but they had the factoring done correctly on the left side, they just needed a little hinting for the right side of the equals.
Why did it work so nicely? I think because they really were engaged with the lost hiker problem and honestly worked them. And they worked. They agonized over why they didn’t get the correct answer. They gave me dirty looks, and when we found the minus sign they missed or the extra number they wrote in, they were angry at themselves.
But they persisted! It was a thing of beauty and I loved it.
When they did the completing the square on their own, with only some gentle nudging from me, I told them how proud of them I am. They needed to hear it.
Heck, they EARNED it.