In this lesson, I talk about the rule for dividing powers with the same base, which is motivated by using examples that invoke the term “cancelling”. I thought I’d talk about that term for a bit here. When talking about quotients such as 5∙2/5, we can divide the 5’s obtaining 1 in the numerator and denominator, and leaving 2 as the quotient. I have heard the advice that we shouldn’t use the term “cancel”, but talk instead about “dividing”. I agree that it is important that students know what is going on with cancelling. On the other hand, with all the talk about getting students to “think like” or “be like” mathematicians (without going any further about the impracticality of imposing such a goal on novices), I note that mathematicians use the term “cancel” all the time, so it seems silly to hide from it.
In teaching, I make the distinction between cancelling by addition; i.e., 5 + -5 is such a cancellation, which results in zero. Cancelling by division on the other hand results in one. It is important that students know the difference and to that end, the term “cancel” may cause confusion as I’ve seen when students say that a/a + x = x, because they mistakenly assume the canceling of a’s is zero and not one. This mistake comes from seeing expressions like ab/a being simplified to b. The a’s disappear, and when something disappears, students may assume it has become zero. Of course this couldn’t be true with ab/a, because if a/a were zero, we would have 0∙b, rendering the whole expression equal to zero.
To avoid all this, I make it clear that we are dividing, and may say “We cancel the a’s here, meaning we divide them to obtain one”. Eventually I dispense with the explanation and just say “cancel” like the rest of the mathematical world.
This lesson, like the last, presents the basics. In this lesson, I look only at simplifications that result in positive exponents. Negative exponents are taken up in a later unit that goes into more detail with more difficult problems concerning multiplication and division of powers.
Warm-Ups.
1. The sum of two numbers is 180. The larger number is five times the smaller number. Find the two numbers. Answer: Let x = smaller number; then 180-x = larger number. 180-x = 5x; 6x = 180, x = 30, 180-x = 150
2. If x = 5 how would you represent the number 8 in terms of ? Answer: x+3
3. Solve. 8a – 5 = 3a + 25 Answer: 5a=30; a = 6
4. Write in exponential form.
Answer: 6^3
5. Write in exponential form. a^5a^3b^2 Answer: a^8b^2
Problem 1 is a “hiding in plain sight” type of problem. Problem 2 might seem confusing to some students at first. One hint might be “What do you need to add to the right hand side of x = 5 to make it equal 8?” Problem 4 will feature in today’s lesson and requires students to see that one of the four 6’s is cancelled.
Dividing. I refer them to Problem 4 of the Warm-Ups. “This is actually a fraction multiplication; we’re multiplying 1/6 by 6^4 which we’ve written as 6^4/1. And as with many fraction multiplications, we divided one of the 6’s in the second fraction by the 6 in the denominator of the first. What is six divided by six?”
They know it is 1 but are wondering where I’m going with this.
“When we cancel, we’re dividing, and when we divide by the same number, we get 1/1; so it looks like the number, or the variable disappears. I’m going somewhere with this so bear with me. Suppose now we have this”:
“What can I cancel and what am I left with?” I have them write the answer on their mini-whiteboards. I should see “7”.
“Now since we’re multiplying three fractions here, we could if we wanted to, multiply numerators and multiply denominators to get something like this”:
“I left out all the ones to make this simpler. Now can we cancel the 6’s by dividing like we did before?”
The answer is yes. I labor over this because students have been used to cross cancelling when multiplying fractions, but for some it may not be obvious that in the fractional expression above, the 6’s can be cancelled.
Dividing Exponential Expressions. “Continuing in this fashion, what can I cancel by division here?”:
I have them write the expression on their mini-whiteboards and show what is cancelled. I should see: 5∙5∙5, although some students may have written it as 5^3.
“You cancelled by dividing two 5’s on top, two on the bottom, leaving three 5’s on top.” And to address those who went a step further: “I see some of you wrote it as 5^3, which is also correct, and what we’re talking about today.”
“Suppose I’m dividing a^6 by a^2. If we were to write this out we would have”:
“After cancelling by dividing, what do we have left?” I have them write this in their notebooks.
I will see both a∙ a∙ a∙ a and a^4.
“We have four a’s left or a^4. Yesterday, we said when we multiply two powers with the same base, we add exponents. Here we are dividing two powers with the same base. So what do you think we do?”
I usually have unanimous consent that we subtract exponents.
“So what would a^7/a^2 be using this rule?” Answer: a^5
“What about x^5/y^3”? Since this parallels what they did in the previous lesson, the students generally see that the expression cannot be subtracted; the powers have to have the same base.
On the board and in their notebooks:
1. Write in exponential form: 9^7/9^5 Answer: 9^2
2. 8x^2/2x Answer: 8/2∙x^2/x = 4x
Homework. Problems should include both multiplication and division, as well as mixed variables and numbers.