Tuesday, September 25, 2012

Student: Will That Always Work? OR I Flubbed It Up

I will be the first to admit it. I flubbed up a teachable moment. A student asked me, "Will that always work?" And rather than throwing the question back at the class, I got excited and used algebra to prove that "Yes, it will always work." So, here is what actually happened.

It is the beginning of the year in Algebra 1. Some students have completed Algebra 1, but still have some gaps in their understanding. Others have never had Algebra 1. The homework problems last night were very "traditional" (think Dolciani) and asked students to write equations for various consecutive integer problems. Somehow we came up with the following equation as we reviewed the homework.

(n+1)2 - n2 = n + (n + 1)

In other words, the difference of the squares of two consecutive integers is equal to the sum of the two integers. I had a student suggest a number for n and we saw that the equation was true. We then picked a different number for n and saw the equation was still true. This led to a student asking, "Will that always work?" At which point I got math-geek goosebumps (a proof in an algebra class!) and quickly expanded the left side and did some simplification and answered the question with a "Yes, it will always work." As I looked up, I could hear the crickets chirping in the silence and the deer-in-the-headlight stares of my students.

In my excitement, I had forgotten that my students didn't know what it meant to square a binomial, combine like terms or get a solution to an equation that was an identity. So, now I am going to rewind this lesson and start again.

Student: Will it always work?

Me: I don't know. How could we figure out if it always works?

Another Student: We could try different numbers.

Me: That sounds like a good idea. Turn to the person beside you and pick a pair of consecutive numbers. Then do the calculation of the difference of the squares and compare it to the sum. (Demo with two numbers suggested by a third student.)

Students work together and start to think that it will always work.

Me: Do a few examples show that something is true? We have only looked at a few cases. How can we know it is always true?

Student: You could look at lots of examples.

Me: But how many examples would be enough? Let's look at something simpler. Is 2(x+1) = 2x + 2? Always? How do we know?

And so on...You get the idea. The point of this blog is that I flubbed it up. But I recognized it and have thought about how I would do it differently. I can already tell by the questions students are asking this year that they have more background knowledge and more of a mathematical disposition than I originally gave them credit. This could make for an interesting year!

Friday, September 7, 2012

Technology is Great (When It Works)

My school has moved to a 1-1 laptop/ipad program. The 7th and 8th graders get ipads and the 9th graders get MacBook Airs this year. Being a PC person, I have found that items that used to be certain places aren't always the same place. Things that take 2 seconds to do now take 10 minutes to figure out. Luckily there haven't been too many things like this for me and my colleagues have been great at helping me out.

However, my biggest frustration to date is regarding this one website that I was planning on using at least once a week in Algebra 1 this year. When asked how our department was going to use the laptops, this website was one of the two main items I listed. (The second one was using TI-Nspire Publish View with a free 30 day student trial.) I logged onto the site today and nothing shows up. Yes, the site is there, but none of the interactive features work. I am hoping there is just a setting I need to adjust. (Keeping my fingers crossed.) But if it doesn't work, I will need to spend extra time looking for a replacement website. I really don't need to add more to my weekend "To Do" list. Hopefully my math twitter friends can offer some advice. I'll post a tweet after this.

Although it has been a long (and at times, frustrating) week, I did finish my 3rd TI-Nspire Quick Tip. I hope some of you give them a try. Less than 60 seconds for a quick helpful hint.