Lesson 22 of 23
Objective: SWBAT simplify expressions with the multiplication and division of same base powers by adding or subtracting the exponents.
Students work for 5 - 7 minutes on the warmup exponent rules while I circulate to check in homework and spot check a couple of problems. I am checking a couple of problems which will show me that they are using the distributive property correctly and not making mistakes in combining like terms. Mostly I expect to see terms out of order or careless errors.
Before we go over the warm up I point out two things that I have written on the board.
- One is the objective, which is actually a couple of questions that students generated after our "consecutive sums" problem (Number System Assessment, Garden Design, Power of Factors). I tell them we will do an exploration today to try to answer the questions "can zero be used as an exponent?" and "what does the number 1 have to do with powers of 2?"
- The other thing I have written on the board is a definition of an exponent as that which tells us the number of bases being multiplied repeatedly in the power. This definition helps us discover and make sense of the exponent rules or laws which apply to today's lesson.
As we work through the example in the warm up problems (3^3x3^2) I model use of the definition of an exponent by asking "how many bases does this exponent tell us to multiply?(3^3)", "how many more here?" (3^2), "how many total are being multiplied? (3x3x3x3x3)"
Then I ask if there is a simpler way to express the problem. I expect them to come up with the single power 3^5 and I have them explain how they got it and why it is simpler. My hope is that someone will ask if they can just add the exponents.
When we get to the bottom section I ask them for several ways to represent each expression. What I want is for someone to ask about subtracting the exponents. If no one does I may just ask if they notice a pattern similar to the one they noticed for the top section. The key here is to use the definition of exponent to focus on why the shortcut rules make sense. This is especially true for 3/3 which, when we use the rule results in a zero power. If they don't make sense of why the rule is true it will just become one more thing for them to commit to memory.
I start by asking students to do a 2 minute silent write explaining why it makes sense that we can add the exponents when we are multiplying powers with the same base. I tell them it may be useful to look at the work they did when they expanded the powers to help them find ways to explain. I am hoping they will explain that when they expand the power they use the exponent to tell them how many bases to multiply and, if the bases are the same, the exponent on the next power tells them how many more of the same base to multiply. Since it is just adding more of the same factor the sum of the exponents tells us the total number of factors being multiplied. This may be really hard for students to articulate and it may take input from many students. Having them go step by step explaining how to expand each one will help. Prompting questions like "what does this exponent tell us?", "what do we write when we see this exponent?", "Is this exponent telling us to multiply more of the same base?", "does this second power increase the number of bases we are multiplying?" will help them make sense of the shortcut.
I ask students to do a second silent write to make sense of subtracting exponents when dividing same base powers. Again I tell them to pay attention to the process of "eliminating factors of 1". This might be even harder for students. They may come up with the idea that when we multiply repeatedly we are increasing the number of factors and when we are dividing we are decreasing the number of factors. They may just say the numbers are getting smaller when we divide. In this case I would ask them to give me an example of what they mean or to show me where they see that happening. They may say that we can eliminate the factors of one, or that 3 over 3 is just one, so it doesn't count, or that they just cancel each other out. I may need to ask if simplifying or dividing removes or takes away factors. Then I would ask if this is the same or the opposite of what is happening when we multiply.
Lastly I ask them to write down what it looks like when we remove or cancel out all the factors. I remind them to review the work we did on the warm up and they should notice the last two. I ask them how many of the original bases are left (0), what the final power looks like (3^0 or 2^0). I ask them what the original problem might look like to result in 5^0. They will have various ways with different numbers of factors, but they should see that each possibility shows the same value in the numerator and denominator and that it equals 1. To wrap it up I ask them how they might write the number 1 as a power of 2. At this point they realize they have answered the two questions we started with. I remind them that the questions we explored in this lesson had come from them and that we might not have learned any of it if it hadn't been for their questions. I think it is really important to highlight the learning that happens as a result of their questions. I want them to view questions as a way to push learning forward not as something to make them feel stupid for having to ask, so they feel responsible for making everyone smarter.
white boards & homework
I have students complete one problem at a time with their math family groups and hold up their answers on the count of the 3. This way I can see that no one is opting out and I can give individual feedback.
Have students simplify:
4^3x4^5 3^6/3^2 2^3x2^4X2 n^4/n^3 8^20/8^20
I purposely alternate the multiplying a dividing, because I don't want them just following a pattern, I want them to have to look at the operation being done and figure out what to do. Some students will expand to simplify and others will try to use the "shortcuts". I encourage them to predict with the shortcut and double check by expanding. I tell them to look at their partners work and, if they think a mistake was made, to show them using the long way. If they mix up the rules I ask them if this problem is increasing the number of factors or bases being multiplied, or decreasing them. For the third practice problem they will need to remember the exponent on 2, which I remind them if I see an answer of 2^7, but hopefully one of their partners will catch it first. I will use the second to last one to help remind them that when the exponent is 1 the answer can just be written without it. The last one can be written as 1 or 8^0. If they put just 8 I tell them they are telling me that the answer is 8 to the power of 1, is that what they mean to tell me? I also ask how they know the final value equals one.
Students often get confused that when the exponent is 1 they can put nothing, but when the exponent is nothing, they have to put something (0). If this is confusing them I go back to the definition of an exponent and point out that if there is a lone base, then we count it as 1 base, which needs an exponent of 1. I also remind them that it is similar to the coefficient on "n". Then we would practice on a couple more like: 3x3^4 and x^5/x^4
If they are not having any confusion on that I might extend it and ask them to simplify:
I don't expect to get to this, but I want to be ready in case. Before they begin I might ask them how this one is different then the others to make sure they notice that all the bases are not the same. I think most students should write 2^5x4 or 2^5x4^1. Some might add in the exponent on the 4 (2^6^) in which case I expand it and ask how many bases of two are being multiplied to show that the exponent can only count one type of base. If anyone writes 2^7, I ask them to explain to the class, because they may have noticed that 4 is equivalent to 2^2. If they don't notice, I don't mention it.