Apparently sometimes $1/2 < 1/4$? My son brought this home today from his 3rd-grade class. It is from an official Montgomery County, Maryland mathematics assessment test:

True or false? $1/2$ is always greater than $1/4$.

Official answer: false

Where has he gone wrong?





Addendum, at the risk of making the post no longer appropriate for this forum:

Questions about context are fair. This seems to have been a one-page (front and back) assessment. Here is the front, notice the date and title:



Based on the title, it seems to me that this is an assessment about the number line in which case my son's picture and written proof are inappropriate and better would have been to locate $1/2$ and $1/4$ on the line and state something like "No matter how many times you check, 1/2 is always to the right of 1/4." However, based on the teacher's response it seems the class has entered into a quagmire and is mixing up numbers with portions.
Simply put this is a travesty. The question asks, what appears to be a simple question about two real numbers, $\frac{1}{2},\frac{1}{4} \in \mathbb{R}.$ In particular, it appears to ask if $\frac{1}{2}&gt;\frac{1}{4}.$ The answer to this question is clearly, under normal construction and ordering of the reals, a resounding YES.

What the question meant to ask is a question about fractions of potentially different quantities. In particular, from the teachers drawing, the question meant to ask if $$\frac{1}{2}\cdot x&gt;\frac{1}{4}\cdot y, \qquad \forall x,y \in \mathbb{R}.$$ The answer to this is again, very obviously no, but this is not what the question asked...
Possibly another way to refute the counterexample given, is to reduce it to a statement that's more clearly untrue. Suppose $\frac{1}{2} &lt; \frac{1}{4}$. Then, multiply both sides by 4 to get $2 &lt; 1$, which is more clearly false. 

So at this point point, the teacher either has to say that the multiplication doesn't preserve the inequality, or (less likely) that the resulting statement is false.

(Though I wouldn't be surprised if they try to refute modus ponens, or the law of the excluded middle, or so on)