Tell students to imagine that they have 80 friends on Facebook (or 80 other objects easily divided into two categories such as male/female).
“If I say 50% of your 80 Facebook friends are males, how many of your friends are males?” (40) Ask students how they arrived at their answer. Students may know that 50% means half; if they don’t, tell them that when we say 50%, we mean half.
“If 100% of your 80 Facebook friends are female, how many of your Facebook friends are females?” (80) Students should recognize that 100% means all, so all 80 are female.
“What if 35% of your 80 Facebook friends are female?” Students will most likely struggle with this. “This is the kind of problem we’re going to look at; we’re going to talk about percentages and how to solve problems using percentages.”
Activity 1
Write the word percent on the board. Underline the two syllables separately.
per cent
“What does the word per mean?” Students may struggle, so suggest some contexts such as miles per hour or a store charging, say, $1 per soda. Guide students toward the realization that per really means for each. If a car gets 55 miles per gallon, it travels 55 miles for each gallon of gasoline. A store charging $1 per soda is charging $1 for each soda.
“Now let’s consider examples of words where cent is the root. Raise your hand if you can think of an example.” Examples such as cents (pennies), century, centipede, centimeter, and centigram are likely to come up. Explain how these are all related to the numeric value 100. “The root word cent means 100.”
After defining what each syllable means put them together. “Per means ‘for each’ and cent means ‘hundred.’ So, the word percent means for each hundred.”
“Suppose you download 200 songs and 48% of them are rock songs. Remember that 48% means you have 48 rock songs for each group of 100 songs downloaded. How many groups of 100 songs did you download?” (2) “And for each of these 2 groups, you downloaded 48 rock songs. So you have 48 rock songs for the first group of 100 and 48 rock songs for the second group of 100. So how many rock songs do you have altogether?” (96)
“Notice that percentages are really rates; they’re rates per 100 of whatever you’re talking about.”
“Suppose you download 300 songs and 12% of them are jazz songs. How many jazz songs did you download?” Guide students through the steps of identifying how many 100s they have (3) and counting 12 for each hundred (12 + 12 + 12 = 36).
Work through some more examples with the same context, keeping the number of songs as a multiple of 100 (100, 200, 300, etc.).
“Now, suppose you download 150 songs and 8% of them are country music songs. How many country music songs do you have?”
Here, tell students that we know for the first 100 songs, we have 8 country songs. “How about those other 50 songs? How many of them are country songs?” If students struggle, point out that for every 100 songs, we have 8 country songs and that 50 is half of 100. Students should recognize that since 50 is half of 100, we have half as many (4) country songs. “So our total number of songs is 8 + 4 = 12.”
If necessary, do some more examples before having students work on the Percents with Hundreds worksheet (M-6-7-3_Percents with Hundreds and KEY.docx). Students should complete the worksheet without a calculator and without using any decimal or fraction multiplication, but using the method described above. Have students complete the work individually before comparing answers with a partner.
Activity 2
“When we talk about percentages, we’re always talking about finding some quantity out of 100. So, if we’re finding, say, 24% of 98, we’re really multiplying 24% by 98. Remember that in math, the word ‘of’ often denotes multiplication. So, in addition to the techniques you used on the worksheet, which work great with round numbers like finding percentages of 200 or 650, we can also rely on multiplication.”
Write “24% of 98” on the board.
“To figure this out, let’s rewrite 24% as a fraction.” Pause here to write, underneath “24% of 98,” the fraction 24/100 and then a multiplication symbol (×). “We’ll also change the word of to a multiplication symbol, and then just copy our 98, the amount we’re finding the percentage of.” Write the 98 next to the multiplication symbol and then an equal sign.
“To multiply, we’ll just multiply 24 by 98 and then divide by 100. So, using this multiplication method, what is 24% of 98?” (23.52)
“That’s an answer we couldn’t get with our previous method. However, we can estimate our solution to 24% of 98 by rounding the 98 to 100. If the problem was 24% of 100, keeping in mind that percent means for each hundred, we’d know the answer was 24. So, our answer of 23.52 makes sense. It’s really close to 24, but a little bit less, since 98 is really close to 100, but a little bit less.”
“Another example: what is 48% of 90? Does anyone have a method we could use to estimate this percentage?” Answers may include rounding 90 to 100 (to give an answer of 48) or rounding 48% to 50% (to give an answer of 45). “Based on these estimates, we know our answer should be right around 45. Again, to find the actual answer, write 48% as 48 over 100 and then multiply that fraction by 90. Multiply 48 by 90 and then divide by 100. What is 48% of 90?” (43.2)
“So, even though this is another problem we can’t solve using mental math, at least not very easily, we can still use rounding and our mental math techniques to come up with a good estimate of the answer. This helps if we make a mistake when performing the multiplication or division. We’ll at least know if our answer seems reasonable.”
Have students work in pairs on the Estimating and Finding Percentages sheet (M-6-7-3_Estimating and Finding Percentages and KEY.docx). Provide the following instructions:
“The worksheet has 20 problems. Each of you is going to estimate the answer to 10 of the problems without a calculator, and you are also going to answer the other 10 problems exactly (with a calculator, if permitted). With your partner, decide which problems each of you is going to estimate the answer to. One of you will estimate the answers to problems 1 through 10 and solve 11 through 20 exactly, and the other one of you will do the opposite, solving 1 through 10 and estimating 11 through 20. Don’t compare answers until each of you has finished all 20 problems, half by estimating and half by calculating the actual answer. Then, compare your answers and make sure the estimate and the actual answer are reasonably close.”
Make sure students get sufficient time and that each pair is checking their estimates against their actual answers.
Activity 3
“So far, all the problems we’ve done have been ones in which you’ve been given the whole. For instance, knowing how many Facebook friends someone has in total, we’ve been asked to find a part, say, 25%. But it’s possible that you could be given some other information. For example, maybe you’re given the percentage, like 25%, and the part; maybe you’re told that Jake has 10 family members who are Facebook friends and that makes up 25% of his friend total. The question is, how many friends does Jake have in total on Facebook? To write it out another way:”
Write 25% of what number is equal to 10? on the board. Spend some time making sure students understand the difference between that problem and the similar problem 25% of 10 is equal to what?
“For these types of problems, where we’re given the percentage and the part and need to find the whole, we’re just working backward compared to what we’ve been doing. So, before, we’d multiply our two numbers together (remembering that one is a fraction over 100 so we’d divide by 100 at the end). Now, we’re going to divide our two numbers and remember to multiply by 100 at the end. To solve this problem, we’ll just divide 10 by 25—making sure to divide by the percentage—and then multiply by 100. When we do 10 ÷ 25 × 100, we get 40. So Jake has 40 Facebook friends.”
Make two columns on the board. Label one Given the percentage and the whole and the other Given the percentage and the part.
Label the first row What are we asked to find?
“So, in a problem where we’re given the percentage and the whole, what do we need to find?” (the part) “And given the percentage and the part, what do we need to find?” (the whole)
Label the second row First Step.
“When given the percentage and the whole, what do we do first?” (Multiply the whole by the percentage.) “And when given the percentage and the part?” (Divide the part by the percentage.) Pause here to circle the differences (multiply and divide) and underline the similarities (in both cases, we’re doing something to the known quantity by the percentage).
Label the third row Second Step.
“After multiplying the percentage by the whole, what do we do?” (Divide by 100.) “And after dividing the percentage by the part?” (Multiply by 100.)
Point out the differences and similarities again and point out how, throughout the chart, the two types of problems essentially represent opposites of one another.
Work through the following three examples:
- 45% of what is 36? (80)
- 65% of what is 46.8? (72)
- 12% of what is 29.4? (245)
Make sure students are dividing the part by the percentage and remembering to multiply by 100.
Have students work individually on Finding the Whole sheet (M-6-7-3_Finding the Whole and KEY.docx).
The lesson is founded on a language-based approach and all of Activity 1 deals with using this language-based approach to solve percentage problems. Students who enjoy calculation and quick problem-solving will enjoy the remainder of the lesson. Students who are conceptual learners will appreciate the relationships drawn between the two main problem types.
Extension:
Use these suggestions to tailor the lesson to meet the needs of your students during the unit and throughout the year.
- Routine: Because percentages are so prevalent in the news, students can continually bring in news articles that use percentages to describe populations. These articles can also be used to introduce the concept of percent change (increase and decrease), a common topic in financial articles (the change in the Dow Jones Industrial Average is reported as a percentage in virtually every newspaper and news-related Web site.)
- Small Group: Using this Web site, http://www.webmath.com/wppercents.html, have students create problems for one another as practice. Students should mix in all the problem types they’ve learned (this lesson only covers the first and fourth types on the Web site). The Web site allows students to generate an infinite number of problems and be assured when checking their group members’ answers that they have the correct answer at hand.
- Expansion: The lesson can be expanded in several directions: Students can find the percentage given the part and the whole, which requires an exploration of decimals and converting percentages. Given fractions or decimals, students can convert them to percentages; with fractions, this requires understanding equivalent fractions as well as division with decimals for fractions that cannot be converted to have a denominator that is a power of 10.
