Multiplication of monomials using the rules of exponents, like other aspects of algebra, have rules. And, like many aspects of learning, as familiar as these rules are to those of us who have been working with them for many years, students new to them may be overwhelmed.
When I was last teaching this topic, I brought up the difficulty with a fellow math teacher at my school who had been teaching for twenty years. He told me that most algebra books present the rules of multiplication in two short lessons, and then follow it up with the rules of division, also in two short lessons. Even my beloved 1962 Dolciani algebra textbook falls prey to this. He said his solution was to spread out the information more slowly—teach the introductory amounts early and then keep giving them problems. Later, start to teach the rest and give them the more involved problems. In other words, scaffold and ramp up the problems and instruction over time.
I have therefore included two lessons at the end of this first chapter of algebra. The lessons provide the introductory basics of the rules for exponents, with beginning type problems.
Warm-Ups.
1. Write an equation and solve. The length of a rectangle is 6 feet more than three times the width, and the perimeter is 188 feet. Find the dimensions of the rectangle. Answer: Let x = width, then 2(6 + 3x) + 2x = 188; 12 + 6x + 2x=188; 12 + 8x = 188; 8x = 176; x = 22, 6+3x= 72
2. The sum of two numbers is 78. If three times the smaller is increased by the larger, the result is 124. Find the smaller number. Answer: Let x = smaller number; then 78-x = larger number. 3x + 78 –x = 124; 2x + 78 = 124; 2x = 46;
x = 23
3. Solve. 3/10 (t) = 6 Answer: t = 6(10/3) = 20
4. Simplify. -4(3x-12) + 9x -47 Answer: -12x + 48 + 9x – 47 = -3x + 1
5. Solve. 7(b+2) – 4b = 2(b + 10) Answer: 7b+14 – 4b = 2b +20; 3b+14=2b+20; b = 6
Problem 1 relies on students knowing the formula for the perimeter of a rectangle; some students will need to be reminded of this. Some students may need more reminding than others throughout the school year. Problem 2 is the “hiding in plain sight” type of problem talked about in the previous lesson. Even though it was talked about previously students still have difficulty recognizing when this structure is to be used.
Exponents. Students have learned what exponents are in the seventh grade. I like to remind them of this before I ask them what 8^3 represents. They usually remember, but will say it incorrectly. I often hear “It’s 8 multiplied by itself three times.”
Before I offer them the correct way to state it, I write on the board:
“The little number is called what?”
They generally know it is called an exponent. “And the number eight is called the base.”
“It tells us that eight is used as a factor three times. That is how you say it, by the way; not eight multiplied by itself three times. Let’s see how you do with this: 7^5; can you describe it in that way?”
I want to hear “seven used as a factor five times”, which I might hear mixed in with “7×7×7×7×7.
“This can also be called seven to the power of 5, or seven to the fifth power. Any number that has an exponent is called a power. How would I write a to the fourth power? That is, with an exponent.” I ask them to write it on mini white-boards
They should write a^4.
“What if I have aaaaa….. with n factors. How would I write this?” I want to see a^n.
“What is 3×a? Does anyone remember how we write this?” They should remember it is 3a.
“What about 3×a×a×a?” There will be various guesses. Prompts may be “How many times is a a factor. How many times is 3 a factor?”
More Examples. I continue in this fashion, using the examples as tools of instruction.
1. aa × bb Answer: a^2b^2.
2. 3aa × bb Answer: 3a^2b^2
3. aaa × bbbb ×cc Answer: a^3b^4c^2
4. What does 2^5 equal? Answer: 32
5. What does (½ )^3 equal? Answer: 1/8
6. 2a × 3a This may take some prompts. This can be rewritten as 2 ∙ a ∙ 3∙ a and we can rewrite this by putting numbers next to each other and letters next to each other. How would this look? 2 ∙ 3∙a ∙ a. Now multiply the numbers. We get what? And how do we write a ∙ a? Put it all together: 6a^2
7. 2∙a∙3∙a∙b∙6∙a∙b Answer: 36a^3b^2
Common mistakes: Seeing a∙3 and writing a^3.
Rule for Multiplying Powers. I write on the board:
“Let’s write out what this means.”
Starting with a^2 and then continuing, I end up with:
“How many times is a used as a factor?” It is used nine times.
“So how could I write this more compactly? That is, a used as a factor nine times is equivalent to a raised to what power?”
I’m looking for a^9 which students usually get.
“If I have b^3b^4 what’s a more compact way to write this, as a power?” Answer: b^9
“So I think you can figure out what this is leading up to. When we multiply two powers that have the same base, like a^8∙a^2, instead of writing out all the factors, what’s the quicker way of doing it?” There are usually more than a few students who will say “Add the exponents.”
On the board and in their notebooks:
“What about a^2b^3? Can we simplify that?”
There will be some who say yes, and some who say no. I call on those who say no to find out why. Generally, they will say that the bases have to be the same, though some may do it the long way by saying that a^2b^3 is aabbb and they can’t be combined. Which is a fairly effective way of saying why the bases must be the same. I have them write in their notebooks:
Exponents are added only when the bases of the powers are the same.
Examples. Again, the examples serve as methods of instruction.
1. a^2a^5 Answer: a^(2+5) = a^7
2. dd^2 Answer: d^3. A hint I give students is that d is used as a factor one time, so it is d^1—we just don’t write the 1, just like we don’t write the 1 for 1x. Some students will need to be reminded of this many times.
3. x^2x^3y^2 Answer: x^5y^2.
4. abab Answer: a^2b^2
5. 5xy∙2x^2 Answer: 5∙2∙xx^2y = 10x^3y
6. x^2x^2x^2 Answer: x^8
Homework. The homework should be fairly simple multiplication of powers like the examples. Some numeric problems should be included like 9^2∙9∙9^3 abd (1/3)(1/3)(1/3)