The course I’m teaching (11th grade PSSA prep) is basically a “review” that helps kids pass the state test. How I’ve decided to teach the course is a “problem solving/critical thinking” course and I’m allowed to do this because most people associate any problem with words with critical thinking *not true*. The cool thing is since I’ve deemed this as a critical thinking course and the students are required to actually take Algebra 2 concurrently I’ve requested to simply teach concepts and reenforce understanding rather than memorization/procedures. So far I’ve gotten the green light from my co-op.
So, right – square roots. Most students should know this by 11th grade but the truth is these students didn’t. They had heard of a square root, they’d seen the symbol and some even could list the perfect squares (not from understanding but by adding consecutive odd numbers) but they didn’t know what it was, why, or where it came from. Enter Student Teacher Molly.
I taught this lesson over 2 days and introduced square roots by relating finding area of a square with given side lengths and looking for the inverse/opposite, what if I give you the area of a square - how would you find the side lengths?
I started with the side lengths of 3, which gave us an area of 9. Then after doing side lengths of 3-5 I put up a square and said the area was 9, so what must my side lengths be? Since we had just done this they knew it must be 3. We did this a few times and I introduced the new notation saying that the side length for the square with area 9 is the sqrt(9) = 3, and I wrote that along the side of the square. We did area of 25 together and wrote it as sqrt(25)=5 on the side length. Then they did the 3 squares on their sheets alone. I easily saw the misconceptions of side length being sqrt(5) and so I was able to walk around and address that discussing area etc.
(**My only issue here was the introduction of the square root symbol which felt a little forced like, hey this is a quick little shortcut symbol we use to denote this idea and this is what it means, I would’ve liked to do some history or something else on the symbol in hindsight**)
Then we filled in the proper way to say square root of A and what it was, “the length of a square whose area is A” (most of the students actually did that on their own). Then we talked about perfect squares and how they get their names and we made our lists. We began to talk about estimation and what to do when it’s not a perfect square; the first day we just left the side lengths in square root notation (before we got ’2 answers sqrt(9)=3, but now we just get sqrt(10)) but the second day we started to estimate what they would be between.
The next day we talked about estimation which was actually a breeze. I started by actually drawing the squares next to each other so they could see porportionally what perfect square was bigger and smaller. After we did it with the phsyical squares, I just used our new notation and placed the question on the board, filling in the small square root above the one in question and the bigger one below (like a smush sandwich) and voila, you can find the whole numbers it lies between.
And the students really got it. When we went back to the PSSA book that asked questions about square side lengths and area the students were able to answer them easily (or maybe easier is being more honest) than if I were to have to introduced the square root more abstractly with all symbols and no squares. Thanks @dgreenedcp!
Here is the worksheet taking almost exactly from Dan Greene with some changes based on stretching the lesson and a few other things.