In Algebra 2, I start with my 3 rules. They really are not “my” rules they are just restatements of the multiplicative identity, additive identity and balancing equation. I believe how I use them to set the tone and stage for the entire year is different, however.
On the second day of class (the first day I usually do a problem solving activity, cell phones, or other type of activity) I introduce the “rules”.
You can zoom in, or you can grab the files from here. I had a long discussion with several teachers about the wording, and Meg Craig made up the files once we settled on the phrasing. Can they be tweaked more? Absolutely. I would love to improve them. Leave the suggestions in the comments.
Back to how I use them, and why I think I use them differently. I will say with 100% assurance that I use these rules differently than I have seen others, and I absolutely use them differently than I used to.
This grew out of the frustration of having a nice, simple equation to solve like, y=5x+2, and needing to solve it for x. I really wondered why learners would mess up the math on such a simple equation so frequently. And don’t even get me started if the equation looked like v=ba-d, because that was impossible.
And I realized that although I was teaching the idea of inverses and identities, I was not connecting learners with or building the idea that these things are used.
So, I turned to the “SADMEP” idea. (this link is a google search for SADMEP. That is sad, huh.) But as I worked with this a year or two, I realized that the SA always made a zero, and the DM always made a 1, so I added that to my SADMEP poster (sadly, there are no pictures of this, I threw them all away several years ago).
Which lead me to the idea that it is NOT subtraction or addition that is important, it is the ZERO! Same thing with the ONE, those are the important ideas. Those are the identities. Why do we subtract 2 from both sides of the equation above? Because 2 – 2 = 0. No magic. We can actually subtract ANY number, but we chose to subtract 2 because that is the most convenient way to reach zero in one step.
And then I stumbled upon another magic word that goes hand in hand here; convenient. Why do we chose the values we chose to add, subtract, multiply, or divide? Because those values are convenient ways to make a zero or one the fastest.
Back to the ugly equation above: v=ba-d. Solve it for ‘a’. Add ‘d’. Why? because -d+d=0. Do we care what d is, or represents? No, we know how zero works. Same goes with multiplying by 1/b.
Then, show this video: http://www.youtube.com/watch?v=seUU2bZtfgM up to the point where he goes into transcendental numbers (approximately minute 6).
I reach for my physics and chemistry books right about now, and find some ugly equations. These in fact. This is the file I start with to get the learners thinking about solving.
And then I go nuts. Put the formula sheet for AP Physics under the elmo. Do the same with AP Chem. Pick one. Pick another. Solve for any variable. Then solve the same equation for a different variable. For every single function / formula, the only thing they can write to justify their steps are “blah because it makes a zero” and “bleep because it makes a one.”
And we discuss that every single problem I can possible give them is solved simply by using these three simple rules. I make a huge deal of this in the log unit because they learn a NEW way to make a 1 in that unit. That is exciting.
All year long, my learners are shouting out, “because it makes a one” when we are working with exponent procedures (note they are not exponent ‘rules’) because that is how math works. Why do square roots and squares “cancel?” Well, they don’t. 1/2 exponents raised to the power of 2 means 1/2 times 2 which equals 1.
That’s it. I use it all year long. I rarely write a radical symbol, just fractional exponents. It just makes sense.
This is a couple of days of work, and I really think pulling from physics and chem texts helps. I have never had such success with solving and literal equations (in fact, they stop thinking literals are any different) as I have had the last two years.
There is a reason they are framed so nicely at the front of my room. They matter, and they solve every problem we encounter.
Some of the CCSS standards this idea hits: