In a previous lesson, the multiplication of powers was introduced to front-end load the topic and give students opportunity to practice. This was done to familiarize students with the procedure of multiplying powers to avoid overwhelming and confusing them with the full set of rules, which we are now about to present.
In this lesson, the problems ramp up from the initial presentation. Students having already worked these problems (in Warm-Ups and in Friday review quizzes if those are given) are ready to take on more complex multiplications.
Warm-Ups.
4. In a race, Batman is 50 feet in front of Robin after 10 seconds. How fast can Batman run if Robin can run 20 feet per second? Answer: After 10 seconds, Robin has run 20 ft/sec × 10 sec or 200 ft. Batman is 50 feet in front of Robin after 10 seconds, so has run 250 ft in 10 seconds. Batman’s rate = 250 ft/10 sec = 25 ft/sec.
5. The length of a rectangular playground is 25 feet greater than twice its width and 650 feet of fencing are needed to enclose it. Find its dimensions. Answer: Let x = width of the playground; then 2x + 25 = the length. 2x + 2(2x+25) = 650; 6x +50 = 650; 6x = 600; x = 100 ft., 2x + 25 = 225 ft.
Students should be able to solve Problems 1 and 2 fairly easily since they have seen these types before. Some students may need a prompt to remind them to group the numbers together and multiply and then multiply the variables as shown. Problem 4 requires students to find how far Robin has run as shown. Possible prompt: “How fast does Robin run? What distance has he run in 10 seconds?” The problem also requires students to find Batman’s speed by dividing distance by time. Problem 5 is another in a series of rectangle problems that will continue through the year. Students may need to be reminded of the formula for perimeter of a rectangle, though there should be fewer students needing reminding at this point.
Multiplying Powers. Having gone over the Warm-Up problems, I state that Problems 1 and 2 are what this lesson is about. “Yes, it’s true. We’re having a lesson on something you learned before.” I pause for about five seconds; students are guarded.
“You’re waiting for the catch, I can tell,” I say and students agree and ask what it is. “The catch is that the problems will be a bit more complicated. And I know you’re just hiding your excitement.” There’s either laughter or anger or both or neither, so I quickly fill in to take advantage of the momentum of curiosity before it disappears or boredom before it develops.
I write on the board:
“I want you to multiply. The parentheses mean I’m multiplying each of the terms inside the parentheses by each other—we’ll be using that notation in this lesson.”
The answer is 2a³ which most students should have correct. “Now something similar, but a little different.”
“I’m going to rewrite the problem. This is actually a string of numbers and variable all multiplied by each other. And if I write it like that it will look like this:”
The answer is –2a³ which most students will get correct. “Notice I grouped the powers with the same base together.”
I put a few more problems on the board and have them write in their notebooks. Typically what I do is work with students as needed and have the weaker students who obtain the correct answer to put the answer on the board, grouping numbers together and power with the same base together.
With respect to the second problem, I remind students that –z is -1z.
Common Mistakes. A frequent mistake that I see is that the negative sign in parentheses is mistakenly taken as a signal to treat the multiplication as for a distribution. For example, in the last problem above, I see students trying to distribute 6x in various mistaken ways, like 6x - 3x²y. This is the reason why I have them write out the problem as a string of multiplications. In the above example, writing it out enables them to see that 6 is multiplied by -3.
Another mistake is not seeing that a term like –z carries a negative 1 that isn’t written, as I pointed out above. Thus, the product of –6x(–z) would be 6xz. Yet, I see it written as –6xz.
Students will ask “Do I have to write out the multiplication like we did today?” My answer is that for this initial homework assignment, they do, after which they can do as they please. I want them to see what it is that is being multiplied and avoid the mistakes discussed here. Nevertheless, the mistakes will still occur. For students who continue to make such mistakes, I will have them write out the multiplication.
Homework. I assign about 20 problems, which seems like a lot, but which I find is necessary to get students used to working with such procedures. As usual, we start on the homework in class. Upon hearing the initial complaint that it is a lot of problems, I remind them that the more they get done in class, the fewer they will have to do at home. This generally works as motivation for most students.