Two of my favorite questions to use in class are: what do you notice? and what patterns do you see? It is amazing to see the range of perception, from the most trivial matters – you used all even numbers in your examples – to very deep understanding. I have had a lot of success in adding these two questions to teaching students how exponents work when multiplying monomials, dividing monomials, and with finding the power of a power.
Since my students love to work on whiteboards, I have the students use the individual whiteboards to expand the following problems and then put the answer in exponential form. As they are doing this, I ask students to look for any patterns.
I then start with, what do you notice? followed by what patterns do you see? Most students can see that the base stays the same and the exponents are added together. At this point, students can write the generalized algorithm and do several problems without having to expand each number out.
I start the next lesson on dividing monomials the same way. Students use individual whiteboards to expand the problems out, divide out the common factors, and put the answer in exponential form.
Again, I ask students what do you notice? and what patterns do you see? before we write the general algorithm and do some examples.
By the time we get to finding a power of a power, students understand how exponents are working and this lesson is a bit easier for them to comprehend. I still ask them what do you notice? and what patterns do you see? because I want them to verbalize their thinking and understanding.
These two questions made me realize the need to allow students to start the process of understanding, rather than me giving them the needed information. They need to work out the problems on their own so they can each see what is happening. Otherwise, there is usually the same few students who see it faster than the rest and they say the answer before everyone else can process the information at their own speed. Additionally, It allows me to gauge where each student is at, rather than starting them too far ahead in the concept. Their answers can tell me if I need to back up a bit before proceeding with the lesson I have for the day. I might need to slow down and ask more (better) questions.