## Tuesday, February 19, 2008

### The temperature changed from -5 to 5 degrees. The percent increase was (A) 10% (B) 50% (C) 100% (D) 200% (E) 300%

Many readers of this post objected to the scientific fallacy in asking for per cent change in temperature even if the arithmetic is correct. I completely concur with these criticisms and have a posted a definitive explanation below quoted from a knowledgeable meteorologist...

[ALSO: IF YOU ENJOYED THIS POST, TAKE A LOOK AT A NEWER POST ON DEVELOPING RATIO SENSE FOR MIDDLE SCHOOLERS.]

Just something for you or your students to consider as we are still in a deep freeze in some parts of the country.

Some thoughts...

• Why multiple choice? Would the question be better asked in an open-ended way? Could this appear on a standardized test in this form?
• Does the negative sign affect the outcome?
• Is % change an important topic?
• When should students be expected to know how to do this?
• What are some effective instructional strategies and/or methods for this?
• (F) None of the above?

"Your arithmetic is correct but your result is arbitrary. When expressing a temperature change as a percentage, one must use a temperature scale whose zero point is the temperature of absolute zero, and then use that selected scale consistently. The Kelvin scale is such a temperature scale. All other temperature systems, like Fahrenheit and Celsius, have arbitrary "zero points," and calculations of percent temperature change using those scales will give arbitrary results.

In the example you provided, on the Fahrenheit scale the temperature rise is 100 percent (from 50(degrees) to 100(degrees) F.), but that same change on the Celsius scale is 280 percent (from 10 to 38 C.). Using the Kelvin scale, whose zero point is absolute zero (-460(degrees) F., -273(degrees) C.), the rise is actually 10 percent, from 283(degrees) to 311(degrees) K."

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Tom Skilling is chief meteorologist at WGN-TV.

Anonymous said...

(F) None of the above.

You can't look at temperature this way, unless you are using the Kelvin scale. For example, you can't say 60 degrees F is twice as warm as 30 degrees F.

Isabel said...

None of the above, of course. (Unless you're using Kelvins -- but then negative temperature is very rarely meaningful.)

I actually heard a TV weatherman yesterday say that it got up to 60 (Fahrenheit) on Monday, and by the end of the week it would only be getting up to 30, so temperatures would be "cut in half". For that error I'm tempted to exile him to a place where the temperature is -200 Fahrenheit -- that's actually half of 60 Fahrenheit. But that would be a bit mean.

mathmom said...

I think percent change is a very important topic, but I agree with others that it doesn't make sense here because of "meaninglessness" of where zero is placed.

I also think the negative number is a problem here. 10 is what percent of -5? -200%? It doesn't make sense.

I was trying to think of another problem where the negative might be used but the problem would make sense, and I'm kind of stuck.

What if we think of above and below sea level. It is reasonable to say that 10 feet below sea level is twice as "deep" as 5 feet below, and it is reasonable to say that 10 feet above sea level is twice as high as 5 feet above, but I still don't think there's anything meaningful you can say about how much higher 5 feet above sea level is than 5 feet below. It's 10 feet higher, but... as a percent, it doesn't make sense to me.

Totally_clueless said...

Interesting!

It always tickles me when I see news reports that say that a company's profits went up x%. Now, if the company had a zero profit the previous reporting period, and made a profit now, I always wonder what they would say (Of course, they avoid the issue of mathematical correctness, and say the company achieved profitability; the wimps!).

In some cases, it looks possible that logical interpretations can lead to mathematically correct results. For example, if a loss of -5 decreased by 200%, the resultant is -5 -(-5*2) = 5, which means a profit of 5 units. Of course, it would be incredibly confusing to many. I doubt you can say the same for temperature though since there are no two different words for positive and negative temperatures (heat & cold ???)

On a similar vein, you never see a contestant on Jeopardy! with a negative total says "Let's make it a true daily double" :-)

TC

Boian said...

Just to remind people, -5K makes no sense, at all. Temperatures, measured in Kelvin, cannot be negative.

Dave Marain said...

Questions like this always seem to provoke controversy and make some readers' temperatures increase!

First of all, I've seen questions like this on standardized tests. Issues of imprecise wording, inaccurate mathematics or inaccurate physics are generally ignored. Test constructors are instructed to formulate math questions using real-world contexts, so-called 'applied' problems.

The following is an actual test question I have seen:

The temperature decreased from 100 to 99 degrees. What was the % decrease? The anticipated answer was 1%, without regard for scale issues.

Here are some of the issues I see when considering temperature changes:

(1) Is it legitimate to ask about relative change in temperature if comparing temperatures on the same scale? Thus, if the question had specified the temperatures as taken on the Celsius scale and had asked for the % increase in the Celsius temperature, would that have made the question more legitimate or does it still not address the underlying issues?

(2) is it legitimate to consider relative or % error in temperature measurements? Thus, would it be appropriate to say that a temperature was measured as "25 degrees Celsius with 1% error", meaning the actual temperature is between 24.75 and 25.25 degrees Celsius? I believe it is, but I know there are complications here.

(3) If the relationships among the different temperature scales did not involve an additive constant (or 'offset'), like 273.15 or 32, would there be as much of an issue here? [See my comment below]. Note that an increase in height from 12 inches to 24 inches can be considered a 100% increase, just as the percent increase from a measurement of 1 foot to 2 feet is 100%].

Suppose the relationship between a unit on the Fahrenheit scale and a unit on the Celsius scale ere related by F = (9/5)C, without the additive constant. Then ΔF/F = ((9/5)ΔC)/((9/5)C) = ΔC/C. Thus, if the scales varied directly (or, equivalently, 0 degrees F., corresponded to the same temperature as 0 degrees C.), the relative or % change would be the same on both scales!

(4) All of the above doesn't even address TC's issue of % change when one begins with an initial value of ZERO OR dealing with a change from negative values to positive values.

What an innocent little question! Hey, it's all relative!

mjswart said...

Richard Feynman relates very well the problem with these sorts of problems. As people have already pointed out. The answer to this question is almost never meaningful.

Here's what Richard Feynman had to say on his experience evaluating textbooks:

Finally I come to a book that says, "Mathematics is used in science in many ways. We will give you an example from astronomy, which is the science of stars." I turn the page, and it says, "Red stars have a temperature of four thousand degrees, yellow stars have a temperature of five thousand degrees ..." -- so far, so good. It continues: "Green stars have a temperature of seven thousand degrees, blue stars have a temperature of ten thousand degrees, and violet stars have a temperature of ... (some big number)." There are no green or violet stars, but the figures for the others are roughly correct. It's vaguely right -- but already, trouble!
...

Anyway, I'm happy with this book, because it's the first example of applying arithmetic to science. I'm a bit unhappy when I read about the stars' temperatures, but I'm not very unhappy because it's more or less right -- it's just an example of error. Then comes the list of problems. It says, "John and his father go out to look at the stars. John sees two blue stars and a red star. His father sees a green star, a violet star, and two yellow stars. What is the total temperature of the stars seen by John and his father?" -- and I would explode in horror.
...

It was perpetually like that. Perpetual absurdity! There's no purpose whatsoever in adding the temperature of two stars. Nobody ever does that except, maybe, to then take the average temperature of the stars, but not to find out the total temperature of all the stars! It was awful!

Dave Marain said...

Dear mjswart--
Thanks for that insightful comment! Math textbooks often attempt to include 'applied' problems that don't really make sense in the physical world.

Note that there was no attempt in the original question to imply that a temperature like 60°F is twice as warm as 30°F.

Do you believe that one can say that a temperature of 5°F is numerically 10 Fahrenheit degrees more than a temperature of -5°F? There's no suggestion here of quantifying 'how much warmer' one temperature is than the other. Of course, the question I'm asking doesn't have any real physical significance, but I'm asking it from a purely arithmetic perspective.