Friday, December 21, 2007

M6N1f - Percents (2)

M6N1. Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and will use these concepts to solve problems.
f. Use fractions, decimals, and percents interchangeably.

To help us understand the nature of percents, let’s look at how percents are introduced in Japan. A very typical introduction to percents will involve problems like this one:
Here are the records of the basketball teams of three city schools. Which team has the best record?
East 15 7
West 10 7
North 15 10
Keep in mind that students do not know anything about percents at this point. Thus, students are initially encouraged to think about which two teams’ records are easier to compare. Students can easily see that the comparison of East and North is easy as they have played the same number of games – thus North has a better record with more wins. Some students will also notice that the comparison of East and West is also easy as they have won the same number of games – West needed fewer games to win the same number of games, so they have a better record than West. From this investigation, they generalize that if one of the two quantities (number of games played or the number of games won), the comparison is easy.

Students will then asked to think about how they can compare West and North. In this case, neither the number of games played nor the number of games won is the same. However, some students may realize that we can make the number of games same by pretending these two teams keep playing at the same pace. If West plays 30 games at this pace, they will win 21 games, while North will win 20 games out of 30 games. Thus, West has a slightly better record than North.

Thus, they can summarize what they have discovered so far:
* if one of the quantities are the same, the comparison is easy.
* if neither quantity is the same, then make one the same
Now suppose the team from South has won 12 games out of 17 games they played. We can compare West and South by making them play the same number of games (or make their numbers of wins the same). However, if we have to compare two teams at a time, the comparison may be a bit too tedious. So, what can we do?

One possibility is to think about what fractions of games played each team won. This can also be interpreted as pretending each team playing 1 game. However, it does not really make sense to think about a team winning a fraction of a game. But, the idea of making all teams play the same number is useful. An easy way to do this is to use a common multiple of games for the teams involved. However, that will mean we will use different number for different set of teams – or different points during the season. Instead, we can standardize the process by pretending each team played 100 games. The number of games each team won out of 100 games is a relative value – not an actual number of wins. However, because we standardized the total number of games, we can compare these relative values.

Note that in this situation, we are comparing the number of games won to the number of games played. When we are comparing two like quantities, we can calculate a relative value by considering the base of the comparison (in this case, the number of games played) as 100. That relative value is called percents. It is very important that we remember that percents are relative values.

Because we can also express relative values in terms of fractions or decimals, we can use fractions, decimals, and percents interchangeably WHEN WE ARE DEALING WITH RELATIVE VALUES. However, when we are looking at fractions or decimals as numbers, we cannot use percents. Thus, the GPS statement M6N1f should have included a phrase, “to express relative values.”

Tuesday, December 18, 2007

M6N1 f - Percents (1)

M6N1. Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and will use these concepts to solve problems.
f. Use fractions, decimals, and percents interchangeably.

Since I discussed ideas related to fractions and decimals, let me discuss another standard that deals with those ideas, M6N1f. As most of you are aware, the Georgia Performance Standards were heavily influenced by the 1989 Japanese Course of Study. There are many GPS statements that are identical to what you would find in the Japanese standards. However, this is one statement you will NOT find in the Japanese standards. Since a standard like this is probably very familiar to many of us, that might come as a surprise. Why doesn’t the Japanese standards expect students to be able to use “fractions, decimals and percents interchangeably”?

Before answering that question, let’s consider why fractions, decimals and percents might be used interchangeably. Some people claim that percent is just another representation of rational numbers. After all, “percents” means “out of 100.” Therefore, just like decimal numbers are based on powers of 10, percents are based on a particular power of 10, namely 100. Thus, 32% means 32 out of 100. M5N5a does state that students should be able to “model percent on 10 by 10 grids.” Thus, 32% may be represented by shading in 32 small squares on a 10 by 10 grids. Of course, the same model may be used to model a decimal number, 0.32, or a fraction 32/100. So, since percent is another way to represent rational numbers, decimals, fractions and percents are interchangeable.

However, if percents, fractions, and decimals are interchangeable, then we should be able to make statements like,
• I bought a 50% gallon of milk.
• The next exit is 75% mile away.
• My son is 350% year old.
These statements sound rather absurd, but why is it? If fractions, decimals, and percents are truly interchangeable, shouldn’t these statements make perfect sense? Or, consider these cases:

Jenny made 7 out of 10 (70%) free throws during the first half of a game. In the second half, she made 8 out of 10 (80%) free throws.
So, did Jenny make 70+80 = 150% of free throws in the game?
Can we express this situation as 7/10 + 8/10 = 15/20?

These situations suggest that there are some significant differences between percents and fractions/decimals, and they are not interchangeable all the time. So, when can we use these three interchangeably and why can we not? To answer this question, we must first understand what percents mean – they are not just another representation of rational numbers.

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Elaboration of Georgia Performance Standards by Tad Watanabe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.