Q: You say " Give a new, improved estimate for square root of 10" and
you give 3 and 1/12 as the answer.  How did you arrive at that?  3 and
1/6 is a much better answer.

A: Our only information is 3 < x < 3+(1/6), where x is the square-root
of 10.

So we can only say, from this information,  that x and 3+(1/6) 
are no more than 1/6 apart.

But we can say that x and 3+(1/12) are no more than 1/12 apart since
3+(1/12) splits the interval of width 1/6 into two subintervals of
width 1/12th.

Q: Okay.  But 3 and 1/12 is now the lower bound.  How can you tell
which bound it is?  Your reasoning sounds good but 3 and 1/6 is a much
better guess than 3 1/12.

A: Yes, 3 1/6 is closer to sqrt(10) than 3 1/12, nevertheless, 3 1/12
is the better guess; if I tell you that I'm thinking of a number
between 0 and 10 and you want to minimize the error, you would guess
"5" and tell me that you are confident that your answer is within 5 of
the true answer.

If you guessed "10" then you could only tell me that "I can bound my
eror by 10" while if you guess 5 you could say that "I cna bound my
erorr by 5".

Now, it may happen to be the case that, the number I was thinking of,
was 9, so "10" was in fact closer to the right answer than 5;
nevertheless, 5 was the better guess.

If studentss persist, I would add: "5 with an error bound of err<5" is
a  Better Estimate  than "10 with an error bound of err<10". If there
is Additional Information later that allowes us to give "10 with an
error bound of err<1" then this third estimate is better than the
first, but the first was and is still better than the second.

I suspect most students will take the exercise at face value -- I do
ask them, after they write "3