New Cuyama

Task

On a road trip in California, Lisa saw a sign which made her laugh and take a picture:

What is mathematically suspect about this sign, eliciting Lisa's laughter?

IM Commentary

The purpose of this task is to provide a fun context to examine the pitfalls of disregarding units when reporting and manipulating quantities. Teachers might use this as a discussion-starter about appropriate and careful use of units. In this particular example, the units of the three quantities are so diverse that it is not surprising Lisa laughed when looking at the ''arithmetic'' on this sign.

Teachers are encouraged not to let students dismiss the example in a single sentence. For a famous example, lack of attention to units caused quite a stir with the loss of the expensive Mars orbiter described here http://www.cnn.com/TECH/space/9909/30/mars.metric.02/index.html?_s=PM:TECH. There are situations where objects of different kinds can be meaningfully added by viewing them from a different, more general perspective. For example, 10 hippos and 13 giraffes could be added to make 23 animals. In the case of this problem, however, it is hard to find a general category which encompasses the three types of units which are being added: people, feet above sea level, and years. We could, however, say that these are all ''things'' and that if we add them together we do indeed get 4663 ''things.'' While technically accurate, this is not a very interesting way of reasoning. Moreover, it can readily lead to further problems: we have 1 foot is the same as 12 inches but if feet and inches are both viewed as ''things'' then we find that 1 ''thing'' is equal to 12 ''things.''

A detailed discussion of the pitfalls of being cavalier with units touches on quite a few of the Standards for Mathematical Practice, e.g., MP3 (critiquing the reasoning of others), and MP6 (precision with language and units).

We note that the sign is intended to provoke laughter: more information about the town of New Cuyama can be found on the town's Wikipedia page.

Solution

The arithmetic is performed correctly on the sign (ignoring the units that should be attached to the numbers). So $$ 562 + 2150 + 1951 = 4663. $$

The problem with the sign is that all the numbers on the sign implicitly have units attached to them, and these are not taken into account when blindly adding together the quantities. The number 562 represents 562 people while 2150 measures feet above sea-level and the units for 1951 are years. It does not make any sense to add people, feet above sea level, and years. These are different units and we can only add the accompanying numbers when the units are the same. For example, if three separate towns had populations of 562, 2150, and 1951 respectively it would be accurate to say that there are a total of 4663 people in the three towns.