In the previous lesson, students were instructed on how to translate percent problems from English into algebraic equations. Some students find this easy and natural while others have difficulty with it. In this lesson an alternative is presented that uses proportions. Since both methods end up with exactly the same equation, it is a mistake to call one method algebraic and the other not. I teach both methods so that students can use the method they find easier.
In later lessons in this unit, many students tend to use proportional methods exclusively. Again, I have no problem with this because ultimately they will learn how to write equations from word problems using either method. I have taught students in seventh grade math who I taught again in eighth grade algebra. When taking algebra these former seventh graders remarked that the percent problems that they found difficult in seventh grade were now very easy. (I note also that these students were in accelerated Math 7; difficulty with equations was not confined to regular Math 7.) Part of the difficulty they experienced may have been because of their transition to and discomfort in working with equations. Another may be that with more experience with equations in algebra they had a better understanding of what is going on mathematically when solving such problems.
Recall that we discussed three types of computation with percents:
1. Finding a part:
What is 8% of 75? (Alternate wording: 8% of 75 is what number?)
2. Finding a percent:
What percent of 40 is 35? (Alternate wording: 35 is what percent of 40?)
3. Finding the whole. 24 is 95% of what number? (Alternate wording: 95% of what number is 24?
In translating these statements directly into equations, students did not have to keep track of what is the part and what is the whole. Using the proportional method it is necessary to know what is what. This is done through a mnemonic method that students gravitate toward. For some students, when they take algebra, identifying the part and the whole becomes easier.
Warm-Ups. Some of the Warm-Ups provide a review of yesterday’s lesson as well as a segue to the present one.
1. In the statement “3 out of 4” what is the part and what is the whole? Answer: 3 represents a part and 4 represents the whole.
2. Write as an equation and solve. 15 is what percent of 20? Answer: 15 = 20x; x = 15/20 = 0.75 = 75%
3. In Problem 2 what number is the part and what number is the whole? Answer: 15 is the part; 20 is the whole.
. Write as an equation and solve. 16% of what number is 4? Answer: 0.16x=4; x = 4/0.16= 25.
5. Write as an equation and solve. Mary gets 24 question correct out of 30. What percent of questions did she answer correctly? Answer: 24 = 30x; x=24/30 = 0.8 = 80%.
Many of these problems will elicit questions because students are still unfamiliar with recognizing parts and wholes. For Problem 1, I would hint that if there are 3 slices of pie out of 4 slices can they now identify what numbers are part and whole? Problem 3 may be approached the same way.
For Problem 5, a hint is “What are we trying to find? Part, percent, or whole?” It asks for percent so that’s fairly easy. Now it can be restated as “24 is what percent of 30?” or “What percent of 30 is 24?”
Going Over Homework. The previous homework assignment had some word problems which required students to analyze what it was asking—that is simplifying it to a part, percent, or whole question. Usually problems requiring finding the whole prove most difficult for students. In particular:
“If Joan had 23 questions right and scored 92%, how many questions were there on the test?”
I will ask “What are we trying to find: part, percent, or whole?”
They can eliminate “percent”, since the problem states she has 92% correct. That now leaves ‘part’ and “whole’. For those who say ‘whole’, I ask why. Generally the answer is along the lines that she had 23 correct out of some total number of questions. Anything resembling that as an explanation is acceptable.
“Let’s rephrase it using the template questions we’ve been using.” I’ll ask if anyone can do that. I’m looking for “23 is 92% of what number?” or “92% of what number is 23?”
Part/Whole Proportions. We get into the main part of the lesson with the introduction of the percent proportion:
I have them write the equation in their notebooks; it is also on the board. “We can use this proportion to find the solutions to the types of percent problems we’ve been doing.”
Finding a Percent:
I use one of the problems introduced earlier and write it on the board: 8 is what percent of 20?
“In the problem, what are we trying to find? Part, whole or percent?”
It’s easy to see it is percent, and upon hearing that answer I say “The percent value is what p represents. So if p were 25, then p/100 is 25%. Look at the problem andell me what the part is and what the whole is.”
There will be some back and forth and I might provide a prompt: “Are we looking at 8 things out of 20 things?” This helps students identify that 8 represents the part and 20 represents the whole. We use this information to plug in to the proportion equation:
They have been solving these using cross multiplication. I have them solve it; 20p = 800; p = 40. “So the answer is 40%.”
Some students might find this confusing since if p=40, why is 40 not the answer? I make clear that the problem asks what the percent is, which is 40/100 or 40%.
“In doing this problem, since we don’t know what the percent is—we’re trying to find that out—the first thing we write is p/100= ; and now we have to fill in the rest, by plugging in part over whole.”
I introduce a mnemonic device that helps identify part and whole. “If you have trouble identifying which number is the part, and which is the whole, look at the problem and focus on the words ‘is’ and ‘of’.”
Looking at “8 is what percent of 20?”, the word “is” is next to the number 8, and the word “of” is next to the word 20. “Think of the phrase ‘is part of’. The word ‘is’ connects to the number that is the part. The word ‘of’ connects to the number that is the whole. So we can think of the proportion as:
“Now let’s try another one. What percent of 30 is 24?” This is worded slightly differently.
“How do I write ‘what percent’?” Having just completed a problem in the same form, students quickly answer p/100.”
“This leaves the part and the whole, or the ‘is’ and the ‘of’ if that’s easier for you. How do I write that?”
The consensus is 24/30, though there may be a few who say is is 30/24. I will ask a student who had it correct explain why they wrote it as 24/30. The answer may be that it’s like the last problem, and we’re finding “24 out of 30” or some variation thereof.
I move on to the other types of problems which help solidify both the concept and the mnemonic. It is good to have both, since sometimes identifying what “is” or “of” is connected to can be confusing.
Finding a Part:
Problem on the board: What number is 8% of 75?
“Do we know the percent? If we do, that’s the first thing we write. How do I write 8% as a fraction?” I hear 8/100 and write it down and have them do so in their notebooks.
“Now we have to find the part and the whole. That leaves ‘What number’ and ‘75’. So which is which?”
I’ll go around to see what people write in their notebooks and have someone who had it correct tell the class what it is and how the student arrived at it.
The student might say “The word ‘is” is next to ‘what number’ which is x and ‘of” is next to ‘75’.” Occasionally they might say that we’re finding a number that is a part of the whole which is 75. But not often in my experience.
I will then point out that the word ‘is’ is also next to 8% but we’ve already taken care of that by writing it down. “That’s why I like to write the percent part of the proportion first, whether it’s something like 8/100 or x/100. That way, we know that what’s left is part and whole, or ‘is’ and ‘of’.’ ”
The equation is 8/100 =x/75; x = 6.
Finding a Whole:
Problem on the board: 39 is 78% of what number?
“What do we do first?” They know by now to write the percent: 78/100.
“Now we have to find the part and whole. What are they?”
This one is fairly obvious because of the placement of “is” and “of”. The equation becomes 39/x = 78/100, which they then solve; x = 50
Examples. I give various examples as I did in the previous lesson, which they should work out in their notebooks or mini-whiteboards.
Homework. The homework should be similar to the previous lesson’s, but they are to use the proportion equation. I make it clear that they are to write the proportion for each problem, and that is what I consider “showing work.”