I start class by showing a series of really tough looking Exponential Examples. I give students five inutes to work on the equations on their own and then another 5 minutes in partnerships. I circulate and give hints and record any interesting strategies being used by the kids.
Later, as we solve each problem together, I highlight effective ideas and strategies. I list these out on the board, because the students' job today is to construct a problem that can be simplified with our strategies. This list is a reference for them during their investigation.
For Example 1 I want students to identify the importance of the commutative property. That allows us to rearrange the terms and cancel them out. In constructing the problem, I want students to use reciprocals. So a major strategy here would be to "plant reciprocals." The key is for them to understand why ones "cancel out." The example is also constructed to show the various ways in which a reciprocal will appear. I speak about each of these with the class.
For Example 2, I mix multiplication and addition, so students need to recognize that one can't be completely eliminated, since adding one will change value. The second series of terms is also used to highlight the single most important strategy that students can utilize with fractions. Here we have 4^4/3^3 being multiplied by 27/64. Instead of simplifying each fraction or instead of multiplying the two fractions, we can rearrange the numerators and cancel out. 27/3^3 = 1 and 4^4/64 = 4. The second fraction will be tougher for them, but it is a critical example.
So example 2 shows us that we can make a problem tougher by using multiplication and addition. It also shows us that we can make a problem a bit tougher by introducing cross canceling.
For Example 3, I make things look super complicated by adding nested brackets. The conversation here is, "how do we deal with that many series of parenthesis?" Simple! We start at the inner most bracket and work our way out. However, we don't want to blindly go through that process each time. We want to check and see how the exponents relate to each other (in this case working from the inside out will really help.)
Example 3 also introduces something that slows most students down, the division of fractions. However, the division process here is simple if you resist the urge to expand your terms. So instead of computing 9^2/3^-36, we can rewrite it as 3^4/3^-36 and then simplify to 3^40. This reduces to 1/3^80. So now I challenge them, "rewrite the problem. Keep the numerator exactly the same, but change the fraction in the denominator so that the answer is 1. I left space on the sheet for this work.
After we have shared how to simplify the three examples from the Start Up, we will move on to creating Challenge Problems. Before we do so, I want to show students one last technique for constructing a problem: finding zeros.
"What types of numbers can we add to always get a zero? What types of numbers can we subtract to always get a zero?"
I try and demonstrate how to place the addition of opposites in an equation, like having 2^2 + 3 + -4 and how to place the subtraction of equal values in an equation, like having 2^2 +3 - 4. Here the two expressions are equal, but the strategies in constructing them might be slightly different.
I ask students to use the different strategies from our discussion (which we listed on the board as we spoke) to create a problem that can stump the class. A benchmark rule is that it needs to look complex but break down to a simple result through the laws of exponents and the canceling out of terms to 1 and zero.
To close today's lesson, I pick one or two problems to share. We post the problem and give the class five minutes to solve. For the groups whose problems are presented, they circulate with me and help students who are stuck. We discuss the answers and talk about the strategy used to construct the problem.
We ask, "How we could have made it cancel out more nicely?"
I collect all the problems and post them to our blog as extra credit challenges for the class.