Hilbert’s Infinite Hotel Paradox

TED-Ed made a very good video describing the Infinite Hotel Paradox, a thought experiment to describe how injective (one-to-one) functions can be used to examine countably infinite sets.

Mathematicians Explain Sports to Each Other

Just when I think I’ve heard every math pun under the sun, Math With Bad Drawings comes up with a new one. This week’s edition is called “Mathematicians Explain Sports to Each Other.” The one on basketball is my favorite.

To see the others (on tennis, soccer, golf, baseball, hockey, marathon, tee ball, cricket, and skiing), click here.

Another poorly written word problem (Part 6)

Textbooks have included the occasional awful problem ever since Pebbles Flintstone and Bamm-Bamm Rubble chiseled their homework on slate tablets while attending Bedrock Elementary. But even with the understanding that there have been children have been doing awful homework problems since the dawn of time (and long before the advent of the Common Core), this one is a doozy.

There’s no sense having a debate about standards for elementary mathematics if textbook publishers can’t construct sentences that can be understood by students (or their parents).

 

badproblem1

This one makes my blood boil. According to its advocates, the whole point of the Common Core standards was to increase the rigor in secondary mathematics. However, this one is SIMPLY WRONG.

The textbook does correctly note that the proper definition of a function is a set of ordered pairs. The “correct” answer, according to the textbook, is answer G — the plotted points do not match the ordered pairs.

However, answer H is also wrong. The textbook would have students believe that order is important when listing the elements of a set. However, order is not important — the domain of \{-3, 1, -1, 3\} is the same as \{-3, -1, 1, 3\} or latex \{3, -3, -1, 1\}$. This is standard mathematical notation — in an ordered pair (or ordered n-tuple), the order is important. For a set, the order is not important.

Specifying that the domain is \{-3,-1,1,3\} and the range is \{2,5,8,11\} does not uniquely determine the function. In fact, there are 24 different functions that have this domain and range (where we distinguish between the range of a function and its codomain).

In other words, in trying to be clever about properly defining a function and showing different representations of a function, the textbook promotes a misconception about sets… which makes me wonder if the textbook’s attempt at trying to be ultra-careful about the definition of a function is really worth it.

Venn diagrams

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xkcdgeeks_and_nerds

xkcduniversity_website

Sources: http://www.xkcd.com/668/

http://www.xkcd.com/747/

http://www.xkcd.com/773/