Back to My "Roots"

Well... Somehow I made it through all of the blogging challenges (almost) on time, which means that in the last month I have blogged the same amount as I had over the past two years before that. So I guess that is a good thing, right? If nothing else, it has proven that even if my digital musings aren't stellar and I decide to post them anyway, no one is going to die! :)

This week, my poor chicken-scratch-covered remains of a pacing calendar had me covering simplifying radical expressions. Now, last Friday, I went to a fantastic PD by +Robert Kaplinsky where we talked a lot about HOW to bring problem-solving to life in the classroom and I was totally all about it! So I went home and hunted around for something that would relate and it was seriously like staring into a void. Now... I didn't search exhaustively because let's be real people...

At least, not in the middle of the school year.

But I am NOT KIDDING YOU - I was convinced that the #MTBoS would save me every time and I just didn't find something that I thought would build understanding of how and why we simplify imperfect square roots the way we do (then expand that to other radical expressions). 

So... I made something:

Here's a link to the entire document for those who are interested. I was having issues getting the word doc to show up correctly, so I had to convert to PDF, but if you want the word doc because you want to edit please just let me know and I can email it to you directly :)

Here's the rundown of what I was going for, how it went, and what I would do to make it better:

My Goals:

1. Build understanding that we can distribute roots over multiplication 
2. Extend understanding of equivalent expressions to include radical expressions
3. Use a specific progression of problems to demonstrate the pattern of "factoring out" perfect squares to simplify imperfect squares
4. Create "fertile ground" for students to develop the "rules" for simplifying roots without me explicitly teaching the procedure at the start. 

The Lesson Run Down:

1. Stood at the door and informed students that their warm up was the first two questions on the paper (we usually do a lot of our warm ups on whiteboards, so this was a change in procedure). I told them there was NO right or wrong answers, but they HAD to have their thinking written there. 
2. I asked what they noticed about the table of numbers. I was hoping they would see that they could multiply the numbers inside the radicals on the right to get the number inside the radical on the left. I was also hoping they would ask (or maybe even assume) that they were equal. *How it went: My first class got there and that was about it, but over the course of the day I had some really fantastic "noticings" - for example, I had students notice that the right side was always made up of a rational number times an irrational number but the left was all only irrational numbers (yay for some generalization!). I also had kids predicting what the next number(s) on the table would have been and passionately trying to convince peers who disagreed at first... it was glorious. And I didn't take a picture of our list. Because apparently I am forgetful and not quite as glorious as I wish. But you can imagine it, right?  Anyway...
3. I told them we were going to try to prove whether the two columns were equal. The students were set up in pairs today already, so I explained that each partner would use a calculator to find the value of the radial expression. We talked about order of operations and different calculators, and they set off. The DID complain when I told them that YES, in fact they did have to write that whole decimal number, but I also told them they only had to do their own column (they would compare with their partner's column after calculating). They were also to move on to the bottom two questions after comparing (so the whole front of the paper would be complete). *How it went: There were a couple of kids during the day who did weird things on their calculators and had to go back, but it was something like 3 out of 75, so not too bad there. Right away they saw that the two columns were equal, which was a blessing and a curse. That was kind of the point, but at the same time, many of them decided to stop there and turn off their thinking - like that was the pinnacle and they were done now. I had a hard time getting the early finishers to extend thinking at all and think about why I had chosen to break down the numbers on the right so that it was always a perfect square as a factor. I think perhaps I should add a question about this (i.e., "Why do you think Mrs. Aoki chose the numbers on the right the way she did?" or maybe, "Could you write another equivalent expression for the following?                            (imperfect square root) = (factored square root) = ____(fill-in-the-blank)___"   
4. I brought them back together again and took their noticing again as well as predictions for simplifying. I then had them flip over the paper and look at the example on the back. *How it went: At first, there was a lot of, "Wait, what the...?" but a lot of them started to see how it fit the pattern after a few seconds. Of course, there were also a lot who just looked confused. So we talked about how the example related to what we did on the front and it seemed like it wasn't too crazy of a leap. I do think I confused some of them because they wouldn't normally jump to 16 and 5 as their first choice of factors for 80. Most of them would have said 8 and 10. Perhaps I should have chosen a problem from the list that was a little more intuitive. Also, having the decimal approximation at the bottom was misleading. We covered approximating square roots before this lesson and I really wanted them to keep connecting that idea and practicing, but I decided it made more sense to actually do that at the top first (to the right of the original square root) and then go into simplifying. We discussed through a few and then I turned them loose again. Most got close to finishing before the end of class but didn't quite get there, so timing was pretty good. 

Overall: I was actually pretty happy with the result. I used to teach simplifying radical expressions in my Algebra I class four years ago, but having taught 7th grade Common Core for three years, it wasn't part of my duties any more. I feel like this little exploration really set them up for a stronger understanding so that when I went into more complex expressions (messier numbers and variables) students had the foundation and seemed to catch on a LOT faster than I remember in the past. 

Ok. If you are still with me - I seriously APPLAUD you because that was a long post. Thank you for your time and your eyes! If you want to really be awesome and go above and beyond, please share any ideas/improvements you might see so I can be better next year!