After the brief foray into more complex equations, I now settle in to the unit on percents. Equations will not be as complex as those just covered; in fact they are rather straightforward, and students feel confident solving them.
One of the goals in teaching this unit is to present an algebraic approach to solving percent problems. By that I mean translating a percent problem into an equation. I do this in two ways. One way is an interpretation of what’s going on mathematically, and translating that into an equation. For example, a problem may ask for the original price of a lamp selling for $200 that was discounted by 25%. An equation-oriented approach would be to let x represent the original price prior to discount. Then the problem would be modeled as:
This is confusing to students, but I want them to see it.
The second method uses proportions, which ultimately yield the same equation. Students prefer the proportional approach. The final equation is equivalent to that obtained in the first example. (I will get into this more on the chapter on discounts and mark-ups.)
The entire unit is a balance between the equation-oriented algebraic method and proportional method. I have found that this works well, and although most students prefer the proportional method, some will use the algebraic method. Ultimately, both methods produce the same equation.
Warm-Ups. These should touch on some aspects of percent that will feed into the basic review at the beginning of the lesson.
1. Solve. 3/2 (2x -3) = 9 Answer: 2x-3=9∙(2/3); 2x -3 = 6; 2x=9, x = 9/2
2. Write 64% as a fraction in lowest terms. Answer: 64/100 = 16/25
3. Solve. 65/100 = 130/x Answer: 65x = 13000; x = 200
4. 16/25 of 50 = ? Answer: 32
5. Which fraction is greater? 5/7 or 4/5? Answer: 5/7 = 25/35, 4/5 = 28/35. 4/5 is greater.
Students should remember enough about percents that they can do Problem 2. I still get questions about such problems in the beginning, however. Problem 3 is a proportion which they have learned how to solve. Problem 4 is simple fraction multiplication which they also have learned but may need a prompt such as expressing 50 as 50/1.
What is a Percent. Students have learned about percents in fifth and sixth grades. Nevertheless, a review of the basics is essential. For purposes of review, a basic definition of percent is: A ratio of a number to 100. Fractions can be expressed in terms of 100; 4/5 equals 80/100 or 80%.
My initial explanation of what percents are used for is as a common currency which makes it easier to compare numbers. One way is to find a common denominator as was done in one of the Warm-Ups which asked for the larger of 4/5 to 5/7. It can be solved by making equivalent fractions: 28/35 and 20/35. Alternatively, percents can be used: 80% and 71% represent 4/5 and 5/7.
This may raise the question among some students of why we can’t just compare using common denominators other than 100? My explanation is that using 100 as the denominator makes some comparisons easier, as well as computations.
“Suppose we had 30 different fractions that we had to order from greatest to smallest. It would be far easier to use percents, than to find a common denominator for 30 different fractions.”
I then continue with a short review of essentials:
Converting fractions to percents:
We can easily convert 2/5 to a percent because we can multiply the denominator by 20 to get 100; therefore 2/5 is equivalent to 40/100 or 40%. But we can always divide: 2÷5 = 0.4. Divide and multiply by 100 to obtain 40%.
Taking the percent of a number:
What is 90% of 30? I’ll ask if anyone knows how to do this. Usually at least one person does, but sometimes I get blank stares. “What is 90% as a decimal?” It is 0.9 which is then multiplied by 30.
Finding the percent of one number to another:
17 questions out of 25 is what percent? As we did earlier, we can divide: 17 ÷ 25 = 0.68, which is 68%.
I tell students that though they are familiar with this, today’s lesson presents a more formal way of thinking about this. This way of thinking will help solve other problems. For example, “19 is 95% of what number?” We will use algebra to solve percent problems.
Putting Sentences Into Equation Form. I start off by explaining that there is a vocabulary that we use when translating sentences into algebra, as we’ve done in the past.
“of” translates to times: Example: 30% of 50 becomes 0.3 × 50
“is” translates to “equals”: Example: 19 is 95% of 20 becomes 19 = 0.95 × 20
“what” as in “what number” or “what percent” translates to a variable, like x
“There are three main types of percent problems that we’ll be doing: Finding a part of a quantity, finding a percent of a quantity, and finding the whole amount.” I lead them through the following worked examples:
Finding a part:
What number is 8% of 75? “In this equation, we’re finding the ‘part’ of the whole. The whole is 75. The part is what we want to find. We can do that by making an equation. First, we need to change 8% to decimal form which is what?”
Response: 0.08
“What do we use for ‘what’?”
Response: I generally hear “x”.
“How about ‘is’?”
Response: Some hesitance but they come around to “=”.
“And 8 percent OF 75: what does “of” mean?”
Response: 0.08 times 75.
We get to the following equation:
Finding a percent:
What percent of 40 is 6? Again, I provide prompts to guide them.
“How do I translate ‘what percent”?
Response: Answer: x (other letters will do, of course!)
“Now we have ‘of 40’. What does ‘of 40’ translate to?”
Response: Times 40.
“So we have “What percent” as x, and that’s times 40. How do I write that?”
Response: Inevitably I hear “x times 40”.
“And we write that as ‘40x’. Next: How do we write ‘is 6’?”
Response: Equals 6.
The equation then becomes:
Part of the difficulty with this type of problem is the way various textbooks word them. After the initial translation, I reword the problem so it now says: “6 is what percent of 40?” Students find this easier to translate.
Finding the whole:
The problem is: 19 is 95% of what number? “How do I translate ’19 is’?
Response: 19 =
“Now we have 95%. Do I keep it in percent form?”
Response: No, change it to a decimal.
“So how do we write ‘0.95 of what number?
Response: 0.95 × x.
Which I again correct to 0.95x.
The equation becomes:
I have them solve it; x = 20.
Again, the wording of the original problem may cause difficulty for some. I will ask students if there is a better way to phrase the problem. “95% of what number is 19?” may be easier
Examples:
I have them work in their notebooks and walk around to give guidance and answer questions. They are to write an equation for each problem even if they can do some of them as they have been doing. I will help them reword problems as necessary to help them with the translation as I discussed above.
1. What number is 10% of 920? Answer: x = 0.1∙920; x = 92
2. 120 is what percent of 150? Answer: 120 = 150x x = 12/15 = 0.6, 60%
3. 15 is 75% of what number? Answer: 15 = 0.75x; x = 15/0.75 = 20
4. 12 is 40% of what number? Answer: 12 = 0.4x; x =12/0.4; x = 30
Homework. The homework should be a mix of these type of percentage problems, with students required to write and solve the equations. Mixed in to the problems should be some word problems that involve more than direct translation, but require some interpretation to identify what type of percentage problem it is. Students will likely find these difficult, and they should be gone over the next day during discussion of the homework problems. There should be about three word problems. Any more than that may invite protests.
Part: Tom scores 80% on a test with 25 questions how many did he get right?
Rephrased: 80% of 25 is what number?
Percent: If you got 35 out of 40 questions right, what percent did you get right?
Rephrased: 35 is what percent of 40?
Whole: If Joan had 23 questions right and scored 92%, how many questions were there on the test?
Rephrased: 23 is 92% of what number?