Let me again ask the question, "How Do You Know Something is a Proportion?". For the math practitioner this is one of those things that requires no explanation ("you just know", "it's just common sense"). But for the remedial college algebra student, running into these common questions on tests, it's a real stumbling block to recognize when the proportional exercise is being asked. Actually, I've found that it's a stunningly hard problem to explain when we know something is proportional -- I've asked the question online twice here (first, second), I've asked around to friends, and rarely do I get any really coherent answer back at all. (One professor colleague investigated the issue and then said, "It's the question on the final that doesn't have a clear direction to it, that's how you know."). Any college basic-math textbook I look at has a stunningly short attempt at a non-explanation, as though they're keenly embarrassed that they really can't explain it at all (usually in the form of, "Proportions are useful, for example in the following cases..."; see first link above for specifics). Let's see if we can do better this week.
Proportions in the Common CoreFirst, let's look at what the up-and-coming Common Core curriculum does with this issue. In CC ratios and proportions are a 7th-grade topic, and there is a presentation specifically on the subject of "identifying proportions". This is uniformly given by two methods: (1) a table of paired numbers, and (2) a graph of a relationship. The key in the first case is to see that the pairs of numbers are always related by the same multiplication factor (usually a simple integer), i.e., y = kx. In the second case you're looking for a straight line that goes through the origin (0, 0), which I think is not a bad tactic. These materials very consciously avoid relying on the cross-multiplying ratios trick, and seek instead to develop a more concrete intuition for the relationship. I think that's a pretty solid methodology actually, and I've come to agree that the common cross-product way of writing these obscures the actual relationship (see also work like Lesh, 1988 that argues similarly).
Some references to Common Core materials where you see this strategy in play:
In College Remedial CoursesSecond, let's observe that the exercises and test questions in our remedial algebra classes at the college level are not given in this format of numerous-data-points; rather, a much more cursory word problem is stated. In fact, let us admit that most of our exercises are at least somewhat malformed -- they are ambiguous, they require some background assumption or field-specific knowledge that things are proportional; they fail to be well-defined in an almost unique way. Here are a few examples from Elayn Martin-Gay's otherwise excellent Prealgebra & Introductory Algebra, 3rd Edition (2011), Section 6.1:
49. A student would like to estimate the height of the Statue of Liberty in New York City's harbor. The length of the Statue of Liberty's right arm is 42 feet. The student's right arm is 2 feet long and her height is 5 1/3 feet. Use this information to estimate the height of the Statue of Liberty.Notice that this doesn't assert that the person and the statue are proportional. In fact I can think of a lot of artistic, structural, or biological reasons why it wouldn't be. In this way the problem is not really well-defined.
51. There are 72 milligrams of cholesterol in a 3.5-ounce serving of lobster. How much cholesterol is in 5-ounces of lobster? Round to the nearest tenth of a milligram.As someone who's eaten a lot of cheap lobster growing up in Maine, again I can think of a lot of reasons of why the nutritional meat content of lobster might not be proportional across small and large lobsters (the basic serving being one creature of whatever size). For example, the shell hardness is very different across different sizes.
57. One out of three American adults has worked in the restaurant industry at some point during his or her life. In an office of 84 workers, how many of these people would you expect to have worked in the restaurant industry at some point?My immediate guess would be: definitely less than one-third of the office. It seems that a sample of current office workers are somewhat more likely to have worked in an office all their career and therefore (thinking from the standpoint of statistical inference) fewer of them would have worked in a restaurant than the broad population. For example, simply change the word "office" here to "restaurant" and the answer is clearly not one-third (specifically it would be 100%), cluing us into the fact that former restaurant workers are not spread around homogeneously.
I don't really mean to pick on Martin-Gay here, because many of her other exercises in the same section avoid these pitfalls. The very next exercise on the Statue of Liberty says, "Suppose your measurements are proportionally the same...", the one on skyscraper height says, "If the Empire State Building has the same number of feet per floor...", others give the mixture "ratio" or medication dose "for every 20 pounds", all of which nicely serve to solve the problem and make them well-defined.
But I've seen much worse perpetrated by inattentive professors in their classrooms, tests, and custom books. So even if Martin-Gay fixes all of her exercises precisely, plenty of instructors will surely continue to overlook the details, and continue to assume their own background contextual knowledge in these problems that their remedial students simply don't have. In short, for some reason, we as a professorship continually keep writing malformed and poorly-defined problems that secretly rely on our application background knowledge of things as proportional.
Proposed SolutionHere's the solution that I've decided to try this semester in my remedial algebra courses. To start with, I'll try to draw a direct connection to the Common Core exercises, such that if someone ever has or does encounter that, hopefully some neurons will recognize the topic as familiar. Instead of saying that a proportion is an equality of ratios (as most of these college books do), I will instead say that it's a "relation involving only a multiply/divide", emphasizing the essential simplicity of the relation (really just one kind of operation; no add/subtracts or exponents/radicals), and covering the expression of it as either y = kx or a/b = c/d.
The hope here is that we can then inuit in different cases, asking, "does this seem like a simple multiply operation?", with the supplemental hint (stolen from Common Core), "would zero result in zero?". For example: (a) paint relating to surface area (yes; zero area would require zero paint), (b) weight and age (no; a zero-age child does not weigh zero pounds). In addition -- granted that students will be certain to encounter these frankly malformed problems -- I will list some specific contextual examples, that will show up in exercises and tests, to try to full in the gap of application-area knowledge that many of us take for granted. Here's what my brief lecture notes look like now:
Proportions: Relations involving only multiply/divide (also: “direct variation”). Examples: Ingredients in recipes, gas consumption, scale comparisons. (Hint: Does 0 give 0?). Formula: a/b = c/d, where a & c share units, and so do b & d. Ex.: If 2 boxes of cereal cost $10, then how much do 6 boxes cost? 10/2 = x/6 → 60 = 2x → 30 = x. Interpret: 6 boxes cost $30. Ex.: (YSB4, p. 68) 22; 24. [4/7 = 12/x. Interpret: 12 marbles weigh 21 grams.]
So that's what I'll be trying out in my courses this week. We'll see how it goes; at least I think I have a legitimate answer now when a student asks for these kinds of problems, "but how do we know it's proportional?".