My last post was about the three rules I use in my classroom. I developed the how and why in that post. In this post, I will explore some detailed “how I use them” in the classroom. I am careful to never say the word “rule” except for these three. We have exponent shortcuts, log shortcuts & properties, but never rules.
To do this right, I am going to use my Surface and do a lot of handwriting and posting of screenshots. If you are wondering, this is all done in OneNote with a Surface Pro 3. The bad handwriting is my own.
What really drove this point home for me, and made me codify it as something that needed to be talked about every day in class is the fact that if you do the “why?” for every step, you write down a “1” or a “0” for almost every step every time. Sure, you write down “I distributed,” “I found the value to complete the square,” or “I factored” but why you do the next step is almost always a 1 or a 0.
See what I mean below.
To see how I connect the rules, let’s start with an expression. It is not a very complex expression, but it sits solidly in the Alg 2 curriculum and throws learners for a loop often.
You can see that the expression is changed through the use of multiplying by one, and the convenient value we selected to use is a value that gets us a 1 when added. Why add? Because of the properties of exponents when we are multiplying bases.
Compare this to the rational expression of adding:
Here, we take the adding expression, and multiply both terms by 1, but a different one each time. Why? because we select the ones to use based upon the convenient terms to accomplish a common denominator.
Let’s move into some solving. Here is a straightforward quadratic that is in vertex form.
How about turning a quadratic into vertex form?
Here I explicitly used the first rule (I used it implicitly above as well, and I have the extra “I chose the 9 because it completes the square” step. Of course, these are bracketed by a zero and a one.
Finally, a log solving equation. I have just one, although I can do more. I chose an moderately ugly equation to solve, so it could not be solved any other way.
[OOPS! I just realized I switched the 3 and 5 in my bcs statement. Damn dyscalculia. Sorry about that.] Here we have a double whammy. I conveniently chose to use Log base 5 to do to both sides. Why? Because Log base 5 of 5 equals 1! From experience, I know that taking the log of the more complex side reduces the number of steps. I don’t tell my learners that. They play and figure it out by doing the one problem both ways. We also used Rule 1, do unto both sides.
In looking at the commonality between all of these problems, you can see the connection of “1” and “0” throughout. I stress this all year long, and have the learners write it all year long. This is the minimum requirement of writing I ask of my learners as they progress. We start writing much more, but I demand they write it. It reinforces the identities of addition and multiplication over and over again all year long. As the year goes on, they write less, but still write it.
Also, I almost never write a radical symbol until the final answers. All radicals are transferred to fractional exponents immediately all year. This helps explain why cubes are inverses of cube roots, and we don’t need to worry about notation. This is a big deal when dealing with some money problems and the exponent is 377 or some such nonsense and we are solving for “r”. The “you just raise both sides to the power of 1/377 because the exponent will be 1” is automatic at that point.
I hope this gives a better understanding of what I mean by “zero’s” and “ones”. Please leave me questions here or on Twitter; @gwaddellnvhs.