Tuesday, April 21, 2015

The Family of Nobodies

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Here is a question from last January's Algebra I (Common Core) regents Exam in New York state. Read it carefully.

This "predicting function" is based on household size as input and number of devices as output. Hence the household size falls into the domain.

Here I show a quote from www.cliffnotes.com (see it here)
When you first learned to count, you started with 1, 2, 3 and kept going until you couldn't remember what came next or grew tired of counting. These positive counting numbers (1, 2, 3, 4, ...) are called natural numbers. The ... means the number list continues on infinitely.
If you add the number 0 to the natural numbers, you get the whole numbers (0, 1, 2, 3, ...). You also get an example of how a number can be classified as more than one type. For example, the number 2 is both a natural number and a whole number. In fact, all natural numbers are whole numbers, but not all whole numbers are natural numbers. Why? The number 0 is a whole number but not a natural number.
I may be missing something here, but it seems to me that question 6 above is missing the correct answer. Unless you find it common to talk about the number of families with zero members, the answer to question 6 must be the natural numbers ("counting numbers" is an accepted synonym). The question asked for the most appropriate. As long as you are willing to include elements in the domain that are nonsensical, any of the answer choices are okay.

As for a benefit of including households with 0 members: Uninhabited homes can be considered as householder-occupied, vagrant iPhones can be given a place to live. Wolfram (seen here) even acknowledges lack of a standard, and recommends the following:

I do grant Wolfram a certain amount of respect in the world of mathematics.

While we are at it, a a standard convention in mathematics is, in the absence of a specified domain, to use the largest domain for which the function makes sense. In this case, that is the positive integers, or natural numbers, or counting numbers. However here, as in many elections, the correct answer is just not one of the choices.

Now we have question 13 from the same test.

Give me a break: 79 cents?  Not 75, not 80. Seventy-nine cents?

This question writer(s) seem to have concluded that 75 or 80 would have made the question too easy. For someone who can do arithmetic mentally, it would be easier. Or would it? This problem requires absolutely no calculations of any kind! That 79 just sends up a red flag saying "don't take this test seriously".

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