## Monday, October 13, 2014

### How Do You Know It's a Proportion?

I've written in the past of the mystery of when you'd want to use a proportion for an application problem, and what the benefits are for doing so (link). Once again, last week, one of my basic algebra students asked the question:
"How do you know it's a proportion?"
And once again I was unable to answer her. I've searched all through several textbooks, and scoured the Web, and I still can't find even an attempt at a direct explanation of how you know a problem is proportional. (Examples, sure, nothing but examples.) I've asked other professors and no one could even take a stab at it. Perhaps the student was looking at any problem such as the following:
A can of lemonade comes with a measuring scoop and directions for mixing are 6 scoops of mix for every 12 cups of water. How much water is needed to make the entire can of lemonade if there are 40 scoops of mix?

On an architect's blueprint, 1 inch corresponds to 4 feet. Find the area of an actual room if the blueprint dimensions are 6 inches by 5 inches.

The ratio of the weight of an object on Earth to the weight of the same object on Pluto is 100 to 3. If a buffalo weighs 3568 pounds on Earth, find the buffalo's weight on Pluto.

Three out of 10 adults in a certain city buy their drugs at large drug stores. If this city has 138 ,000 adults, how many of these adults would you expect to buy their drugs at large drug stores?

The gasoline/oil ratio for a certain snowmobile is 50 to 1. If 1 gallon equals 128 fluid ounces, how many fluid ounces of oil should be mixed with 20 gallons of gasoline?

Concisely stated, what is the commonality here? What is a well-defined explanation for how we know that these are all proportional problems?

1. Is the question "how does one recognize a proportion problem?"

The way I see it, they typically have two sentences. The first contains a ratio with stated quantities and units. The second also contains a ratio, using the same units but with an unknown quantity.

The blueprint problem is an anomaly. That one is much harder to identify as a proportion problem because it also contains an additional problem of computing the area.

Do you set up your proportions using units?

For example: (6 scoops mix / 12 cups water) = (x cups water / 40 scoops mix)

1. You know, that's a really good start: First teach how to dissect individual ratio statements, looking for the phrases "ratio", "corresponds to", "out of" and "for every". Then connect two such statements via matching units. (And also good point that the map question above is actually two proportions plus an area calculation.)

Personally when I write these I document the units in side notation (making sure that numerators & denominators match units). That way we're not committed to re-transcribing the units on every step of the solution, which would be unlike other problems in my basic algebra class. So I write:

6/12 = 40/x ← (scoops of mix)/(cups of water)
6x = 480 ← cross-multiply
x = 80 ← divide by 6
Interpret: We need 80 cups of water.

2. More complete list of English phrases indication a ratio relationship: "ratio of a to b", "for every", "per", "corresponds to", "a in b", "contains", "a out of b"...

3. But here are some other problems that seem harder to identify why we know they are ratios; consider especially when the units are all the same (Martin-Gay, "Prealgebra & Introductory Algebra", 3rd Ed., Section 6.1):

If a 170-pound person weighs approximately 65 pounds on Mars, how much does a 9000-pound satellite weigh on Mars?

A 16-oz grande Tazo Black Iced Tea at Starbucks has 80 calories. How many calories are there in a 24-oz venti Tazo Black Iced Tea?

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 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.

4. In a chemistry or physics class, students learn to do dimensional analysis in the stoichiometry unit. Dimensional analysis is just proportions on steroids. The repetition of units ensures that the ratios are set up correctly and that the answer you get hasn't been inverted. They're a great double- and triple-check.

(Oops, I have now just realized that in my previous examples above, the answers are inverted because the units don't match up.)

For the Mars problem where the units are both pounds, differentiate them by specifying which planet they apply to. I think it should be set up like this:

(170 Earth lbs / 65 Mars lbs) = (9000 Earth lbs / x Mars lbs)

And if your satellite is *on* Mars, it's spelled r-o-v-e-r. ;)

Here's the Liberty one:

(42 arm feet / x height feet) = (2 arm feet / 5 height feet)

If the juxtaposition of arms and feet in that problem isn't a condemnation of the US system of units, I don't know what is.

5. Forgot to mention: you are correct about identifying ratio statements by looking for the key words/phrases.

6. Good point about how to distinguish the units in the planet/statue problems. And excellent point about the "arm feet". :-)

7. Another question I've seen students crash and burn on from the testbank I use:

A punch recipe calls for mixing 4 parts of cranberry juice with 7 parts of lemon juice. Find how much cranberry juice should be mixed with 84 ounces of lemon juice.

The complication here is the use of "parts to parts" in the first ratio; that is, the units don't directly synch up between the 1st and 2nd ratios. You need to know that "parts" is a stand-in for a generic unit, that needs to get replaced by some more specific unit in practice. On the last test I gave, half the class answered something nonsensical like "48 parts" (instead of 48 ounces), even as I was walking around trying to explain this point during the test.

2. A proportion is just a function (f: x -> y) mapping one unit domain to another.

So for the water/scoops example (f: 6x -> 12y). For 40 scoops you let x = 40, and solve for y.

1. Forgot to mention that the function is a bijection.

2. That's a workable definition for a proportion. But the question is, for a basic math student, how do they know from reading a word problem that it's a proportion? The fact that multiple units are involved?

3. Consider also situations where the unit domains are the same; e.g., sides on a geometric figure, like aspect ratio for a TV or movie screen.

3. At the top of this post, you link to an entry from last year in which a student asks "what good are proportions?" I'll give you an anecdote.

In one of my programming courses, we spend a couple weeks using the Arduino. One of the projects is to use a light sensor to control a servo motor.

The light sensor can give us a reading from 0 to 1023, but in reality you'll get a more restricted range. Experimentally, we determine that the actual range is, say, 67 to 858.

The servo takes an output of 0-180 to control what angle the arm should point to. Zero is far left, 180 is far right.

So we need to convert the light sensor value to the corresponding servo value.

We work through the problem using proportions and come up with the formula.

When it all works beautifully, I always get at least one student who exclaims "Finally, a real-world use for that proportion stuff we learned in math class!"

1. Great example! I myself always found that math really got cemented in my head once I could write a computer application using it. Still do.

2. In fact... back in the mid-90's I made a real push to put BASIC program exercises in my college algebra classes that I was TA'ing at the time. (Not alone in that, of course; in fact the books of that era supported it.)

4. Proportions are relationships, so you're comparing things to each other and looking for the relationship between them.

5. Late comment, quote from Lunsford/Poplin, "From Research to Practice: Basic Mathematics Skills and Success in Introductory Statistics", 2011, p. 13:

"In general all except one of the questions (Question 7) that we found significant at the 5% level were questions involving ratios and percents. This suggests that basic facility with ratios have a positive association with our students‟ success in introductory statistics. We note that Johnson and Kuennen found questions 2, 4, 6, 10, and 12 to be significant at the 10% level. Even though they had a different student population, we find it interesting that we both found questions involving ratios (4 and 6) significant."

Is it possible that the unclarity of explaining ratios and proportions leads to this significance and trouble later on?

6. In light of the excellent observations from actual math people, I'm hesitant to offer anything, but might it be useful to compare proportions to analogies? "Light:dark as wet:(???)" would be a familiar question form to most students, and it seems proportions are something like that.

Also, and perhaps this is too elementary, but it seems that what is going on here is essentially just the identification of equivalent (is that the right term, technically speaking?) fractions. Prime the pump by reminding them of such elementary truths as 1/4 = 4/16, perhaps.

1. Nice timing on spotting this -- I've actually got a new post going up tomorrow with my current proposed solution (somewhat inspired from Common Core junior-high school work). The idea is to emphasize it as really being a pure multiply (or divide) by a fixed number; in graphical form, a straight line that runs through the origin. So the hint I'm going to pitch this week is: "Does zero give zero in this case?". In another week I'll know if that helped things or not.