This topic was discussed in Post 11 for seventh grade and, as many topics in this first chapter in an algebra course, it is a review for students. The difference is going into more detail with equations that have fractions. Given how the class moves through the material I may use this opportunity to show how multiplying by the reciprocal of a fraction can serve as a proof of the invert and multiply algorithm for fractional division. If they are having difficulty with the material, I may present it at another time, as I describe in the discussion below.
Students have no trouble in solving problems by division, but have a harder time recognizing that equations of the form x/n = k are solved by multiplication. I continue to repeat that a fraction x/n is the same as 1/n (x).
Warm-Ups.
1. Five equals 2 times a number decreased by 1. Write an equation and solve for the number. Answer: 2x-1 = 5; 2x = 6; x = 3
2. Simplify. 2(t+4) – 3 Answer: 2t + 8 -3 = 2t + 5
3. 4/5 × 5/4 = ? Answer: 1
4. Simplify and write in lowest terms. n/4 + 9/4 (n) Answer: ¼(n) + 9/4(n) = 10/4(n) = 5/2(n)
5. If b = 3 ½ and c = 4, then 16(c-b)^3 equals what number? Answer: 16(1/2)^3 = 16(1/8) = 16/8 = 2.
Students will know how to do Problem 1 which is also this lesson’s topic. It can serve as an entre to the lesson. Problem 3 is discussion in this lesson when showing how to solve equations of the form ax = b where a is a fraction. Problem 4 serves as a reminder to students that n/4 is the same as ¼(n) Problem 5 requires computation and an understanding of raising a fraction to a power.
Division Property of Equality. After going through the Warm-Ups, I point out that Problem 1 is solved using the division property of equality, which they’ve been using for some time. On the board and in their notebooks:
The division property of equality: For each a, b, and c and c ≠ 0 if a = b then a/c = b/c
“So just as a review, because you’ve all had this before, what we’re doing when we divide both sides by the same number, is undoing the multiplication. If I want to solve 6x = 84, I know I can divide both sides by 6 because the Division Property of Equality says I can do that. Now why would I want to do that?”
The usual reply is that it gets rid of the 6, or it cancels the 6. “Yes, we are cancelling the 6. We’re saying 6x/6 = 84/6. We saw in Problem 4 of the Warm-Up today that n/4 is the same thing as ¼(n). That means we could also write 6x/6 as 1/6(6x). And I could also write this as 1/6(6/1)x. I’m not asking you to do this, I’m just showing you something: what is 1/6 times 6/1?”
I hope and pray that someone will say “one”. Some will say 6/6 which is correct, and I’ll ask what that equals, and they know it is “one”.
“That’s why you can cancel the 6’s. We end up with 6/6 which is one. And 1x, we simply write as x. We’ve isolated the variable.
I give a few more examples like 5x = 25, 7x = -49, and even a problem that has a fraction for an answer, like 4x = 25 so they get used to an answer being a fraction. I include problems like 0.2x = 5 and 0.03x = 2.4 in which they divide both sides by the decimal coefficient.
Multiplication Property of Equality. “You’ve had this before as well, so write this down and don’t throw it away this time!”
Multiplication Property of Equality. For each a, b, and c, if a = b, then a ∙ c = b ∙ c
“We just solved problems like 6x = 84, where in order to isolate x, you had to undo the multiplication. That is, 6 is being multiplied by x. So how did we undo the multiplication?”
Response should be and usually is “divided”, or “division”.
“And in Problem 4 of the Warm-ups we said that n/4 is the same thing as what? How did we rewrite it?”
I want to hear 1/4(n) and I take note of who answers in my “formative” type observation in my continual assessment of where students are mathematically—their interest and ability to absorb information.
“Let’s say we now have this problem: a/5 = 6. Knowing how we rewrote n/4 How can we rewrite a/5?”
I should have more students volunteering an answer, so I may venture a cold call at this point. Hearing 1/5(a), I continue.
“In this problem a/5 = 6, we want to undo the division, just like we undid the multiplication when we had a problem like 6x = 84. We can rewrite our problem as 1/5(a) = 6. What can I multiply 1/5 by, to get 5/5 or 1?”
“Five” is the answer I want and I usually hear it.
“We multiply both sides by 5. This cancels the 5 in the denominator.”
I write on the board: 5/1∙ a/5 = 5 ∙ 6.
“The 5’s cancel, and we’ve isolated a. What do we have on the right hand side?”
The solution is a = 30.
“I purposely rewrote a/5 as 1/5(a) to make it clearer why we are multiplying both sides by 5. And the Multiplication Property of Equality says if we multiply both sides by the same thing, we still have equality.”
Examples: These are more of the same, so they are comfortable with either rewriting it as we did, or they can just multiply by the number in the denominator.
x/7 = 5 Answer: x = 35
t/6 = -5 Answer: x = -30
y/14 = 3 Answer: x = 42
Fractional Coefficients. “Can anyone tell me what the reciprocal of 3/5 is?” I will announce. Silence usually follows. In case no one answers: “I’ll give you a hint. We just were working with some. What did we multiply 1/5 by to get 5/5? OK, we multiplied by five. 5, which I can write as 5/1 is the reciprocal of 1/5. It’s a fraction turned upside-down. So what is the reciprocal of 3/5?”
Hearing 5/3, I will then ask “And what do we get when we multiply a number by its reciprocal. If you’re not sure, try it with 3/5 and its reciprocal.” The answer is usually loud and clear as “one”. They’ve had this in sixth and seventh grades, but without using these concepts on a frequent basis, they will not stick, so repetition is key and these concepts come home to roost in algebra.
On the board and in notebooks.
A number times its reciprocal equals 1.
“If we have (1/4)n = 5, we multiply both sides by 4, which is the reciprocal of ¼, which gives us 1 times n. That’s how we isolate the variable. So how would you solve (2/5)x = 16? What would you multiply both sides by?”
They are starting to see the pattern by now, and more students volunteer the answer: 5/2.
“Let’s do it. We have (5/2)(2/5)x = (16/1)(5/2)(16). Do it in your notebooks.”
They should get x = 40.
“What about (3/8)x = 4?” They will get a fraction: 32/3, which I tell them to keep in fraction form, rather than a mixed number.
Examples:
(3/7)x = 21 Answer: x = 49
(4/5)x = -2 Answer: x = -10/4 = -5/2
(3/2)x = 5/6 There are likely to be questions, like “How do we do this?” My response is always the same. “What did you do last time? You multiplied both sides by what?” The reciprocal of the coefficient.
“So let’s write that. Tell me what it looks like.” I write on the board as I call on students to tell me what to write. “Left hand side, what do we do?”
(2/3)(3/2) x
“Now I do the same to the right hand side.” 5/6 ∙ 2/3 =10/18 = 5/9
When all is said and done x = 5/9.
4/5(x) = 5/7; Answer: x = 5/7 ∙ 5/4; x = 25/28
Homework. Problems should be as above, including those whose solutions are negative numbers, as well as fractions.
Supplemental: Proof of Fractional Division Algorithm. The discussion above may or may not lend itself to a discussion of a proof of the fractional division algorithm of invert and multiply. It depends how long we have spent going over the above. If it moves quickly and they don’t seem to be having difficulty with the concepts, I move ahead. If not, I reserve it for a Friday and after the presentation have the students recreate the proof as an extra credit quiz. The proof, as I teach it, is given below.
“In the problem we just did, (4/5)x = 5/7, I’ve sometimes had students ask me why we don’t just divide both sides by 4/5 using the Division Property of Equality, rather than multiplying by the reciprocal.”
And if in fact a student does ask this, refer to the question and use it as an entre.
“This is a good question. As it turns out, we are doing that. And if you’ve ever wondered why when we divide fractions we ‘flip and multiply’, we now have enough experience and mathematical tools to find out why.”
“I want to prove that 5/7 ÷ 4/5 = 5/7 × 5/4.
“First let’s write the problem: (4/5)x = 5/7
“If I divide both sides by 4/5, I’ll get x = 5/7 ÷ 4/5. Let’s pretend for the moment that we don’t know how to do this.
“Now tell me how to solve it the way we’ve been solving it. What do I need to do?”
By now they have it down well enough to tell me to multiply both sides by the reciprocal of the coefficient.
“OK, good. The coefficient of 4/5 is what?” Answer: 5/4
I write down: (5/4)∙(4/5)x = 5/7∙5/4 or x =5/7∙5/4.
“Now we said before that x = 5/7 ÷ 4/5. I’m going to write it as 5/7 ÷ 4/5=x. This gives me the following information:
“Anyone remember the transitivity property?” Someone might, but in the interest of time, I say it—partially: “If a = b, and b = c, then what?” They usually remember at this point: a = c.
QED