Math instruction these days aims to help students understand what is happening in a math problem, so that they can solve it in a number of different ways. That's great. Unfortunately, the rapid-fire pace at which these solutions may be presented, coupled with the natural limits of the younger students' cognitive development, can sometimes just cause the students to not understand any of them.
Take this 6th grade problem, for example: Sixty percent of Mr. Hall's math students have pets. The actual number of students in his math classes who have pets totals 45. How many math students does he have in all?
One sixth grade class was recently presented with two methods for solving this type of problem: the "tape method" and the "proportion method." Because of the short time frame in which they were expected to master both methods, some students were left completely befuddled.
Given that many 6th graders are only beginning to develop their abstract thinking skills, it would stand to reason that the best method to solve the problem would begin with something more concrete. As I've mentioned in a previous blog post, the Singapore Math curriculum is very successful at getting students even younger than sixth grade to solve problems which would traditionally require algebra; they do so by visually representing the problem with diagrams. The tape method mentioned above is classic Singapore. But in the long run, does it really matter what method is taught? Let's compare these two in particular.
Here's the proportion method:
1. Create two equivalent fractions based on the numbers and percentages given. In this case, the number of students with pets is the numerator (45), and the total number of students (x) is the denominator. Likewise, the percentage of students with pets (60%) is the numerator, and the total percentage of students (100%) is the denominator:
We know that these two fractions are equivalent because they are referring to the same groups in the same ratio-- i.e., students with pets: all students.
2. Connect opposite numerators and denominators diagonally:
3. Now, multiply the diagonal numbers (45 x 100) and divide the result by the lone number (60). The answer is the x, or in this case, the total number of students.
Of course, this is the same as the algebra algorithm which sets 60x equal to 45(100), and then solves for x by dividing 4500 by 60. Will a typical 6th grader understand this? No. Even a demonstration that multiplying the opposite parts of two equivalent fractions will produce equal numbers-- if 2/3 = 4/6, then 2 x 6 = 3 x 4 -- may not transfer in their minds to (2 x 6) ÷ 3 = 4. In fact, I bet I even lost YOUR attention in that explanation! When it comes to math concepts, the fewer words that are necessary to explain them, the easier young children will understand them.
Without the background in algebra and a cognitive level that allows them to analyze abstract concepts, they will likely not understand what's happening in the proportion method at all.
In an attempt to simplify the process for sixth grade consumption, one teacher offered this formula:
My math sense is somewhat disturbed to see that "part" and "whole" are identified in only one of the fractions here-- when the whole reason the formula works is that the "percent" is part of the whole 100%-- the two fractions are equivalent.
Also, many students may not recognize that the word "what" represents a number. They may not understand which number the "is" or "of" goes with. For example, the first and last questions here both contain the words "what number is." In the first question, "is" goes with "what number." In the last, "is" goes with the 3. So the student has to analyze the meaning of the sentence just to use the "shortcut;" for some, this is beyond their skill level. (Ask me how I know.)
And then there are problems that don't fit the mold at all. Where are the "is" and "of" in this problem:
Sixty percent of Mr. Hall's math students have pets. The actual number of students in his math classes who have pets totals 45. How many math students does he have in all?
A mature brain can analyze the problem and restate the problem as "45 is 60% of what number?" But again, this requires cognitive maturity beyond many 6th graders. The problem with this method is that it is completely word-based-- the numbers represent only words, so the concrete-thinking student easily becomes lost, not knowing what the numbers actually mean.
Worse, the words are actually assigned a false meaning-- looking at the formula above, a student will conclude that "is" and "of" both represent numbers. If you really want to translate "is" and "of" into math, "is" becomes equals and "of" most generally indicates multiplication -- in this case, multiplication by the decimal equivalent of the percent. So:
What number is 75% of 4? translates into (pre-algebraic) math: ____ = .75 x 4
3 is what percent of 4? translates into math: 3 = ____ x 4
75% of what number is 3? translates into math: .75 x ____ = 3
And, from the original example, 60% of what number is 45? .60 x ___ = 45
This method of finding percentages of whole numbers could easily follow the mastery of multiplying and dividing fractions, and converting between fractions, decimals, and percents. It would make more sense, because a typical sixth grader has actual experience with the concept of "half of 100" and can understand that, if we translate "of" as multiplication, 1/2 x 100 is the same as .5 x 100 or 50% of 100. There is no formula to get confused about. On the other hand, the proportion method (as presented) offers a formula that is easily misapplied by preteens because the concept is not fully explained, and it is not fully explained because the students are not cognitively ready to understand the abstraction.
Now, let's take a look at the more visual "tape method:"
Sixty percent of Mr. Hall's math students have pets. The actual number of students in his math classes who have pets totals 45. How many math students does he have in all?
1. Draw a long rectangle (like a tape) which represents 100% of the whole quantity-- in this case, all of Mr. Hall's students. Divide it into 10 segments. Each segment represents 10% of the whole.
---------All (100%) of Mr. Hall's math students---------
2. Shade in the percentage indicated in the problem. In this case, the yellow indicates students who have pets, which is 60%, or six of the 10% segments:
---------All (100%) of Mr. Hall's math students---------
------the ones with pets------
3. Label the part. The yellow also indicates the actual number of students who have pets, which is 45:
4. Now divide the part by the number of segments shaded to find the value of each segment:
45 ÷ 6 = 7.5
5. Finally, multiply the value of each segment by 10: 7.5 x 10 = 75.
This method is very visual, so it is more likely that a student who is still developing abstract conceptual skills will understand it. They will SEE that the 45 and the 60% actually represent the same value-- an idea that can easily be lost in the previous method. If a student prefers to work with manipulatives, the diagram is easily converted to a paper strip, math blocks, modeling clay or anything else you might have.
It also is easier to explain how this same representation can work for the other percentage questions. The key each time is to figure out the value of the10%:
Q: What is 60% of 75?
---------------------------75----------------------------
2. Shade in the percentage indicated in the problem. In this case, 60%:
3. Now divide the part by the number of segments shaded to find the value of each segment:
75 ÷ 10 = 7.5
4. Finally, multiply the value of each segment by the number of shaded segments, or add up the shaded segments: 7.5 x 6 = 45.
1. Draw a long rectangle (like a tape) and divide it into 10 segments. Each segment represents 10% of the whole.
= 752. Divide the whole (75) by 10 to get the value of each segment: 75 ÷ 10 = 7.5
3. Determine how many segments it takes to get 45 (either by adding, or dividing 45 by 7.5)
4. Count the number of segments. Each segment is 10%, so 6 segments = 60%.
While this method is not completely foolproof-- there will still be students who are initially confused as to how 100% can represent any "whole" quantity-- the visual component allows a student to see what is happening to the numbers as he works with them. He is then much more likely to be able to transfer that understanding to algebra as he moves forward with math. By introducing the former method too early, however, the student may simply learn that math is a series of incomprehensible tricks and formulas that must be memorized.
