Grading Scale:

20 points for each problem (10 for each direction of the proof of 1-31)

Based on the logical construction of the proof, I gave each problem a 0,5,10,15,
or 20 (0, 5, or 10 for each part of 1-31). And I also took off a point or two
for very minor errors or if something didnt seem sufficiently explained.
Because grading these geometry problems takes so long, I paid less attention to
these details and focused primarily on the student's reasoning.



The low grade was a 0, and there was one perfect score. The average was a 72.

On 1-23 and 1-24, many students forgot to show that the statement applies to the
intersection of ALL angle bisectors/altitudes. All they need to say is that it
suffices to show it for two b/c the info in the problem is symmetric. This
isn't much, but it is very important that it is said. Some students just said
"WLOG" without any explanation;  it would be better to mention the symmetry
explicitly.  

There was also an issue on 1-31 and on the extra problem of students writing
explanations as opposed to proofs:  Rather than explaining why it seems like the 
two balls should intersect, PROVE that they do using mathematics.

Students have to also work on presenting their proofs more clearly. If a student
does the proof properly but fails to present it in a coherent manner, then they
will lose points. Students should try to avoid using their own personal
shorthand and drawing all kinds of arrows and whatnot all over the page.  If
it takes a long time to follow a proof, points will be taken off.