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\begin{document}

\begin{titlepage}

\begin{center}
{\Large Meta-Rubric}
\end{center}

\begin{itemize}
\item Answering the right problem
\begin{itemize}
   \item Could we tell you understood what we were asking you to do?
   \item Did you show procedural understanding of any terminology?
[Procedural understanding means correct use, not recall of the definition.]
   \item Did you understand what information was given?
   \item Did you indicate any short answers that were asked for?
\end{itemize}

\item Explanation
\begin{itemize}
   \item Did you explain the important aspects of what you did?  
[interpretations, choices of units, assignment of variables, as so forth, 
are included here]
   \item Were your explanations coherent on a paragraph and sentence level?
[use of an illustration, when appropriate, goes here]
   \item Did you have a reasonable approach?
\end{itemize}

\item Justification

\begin{itemize}
   \item Did you justify each step you could?
   \item Did you acknowledge steps that required justification when
you couldn't justify them?
   \item Did you address all cases?  [some extra credit for finding or
classifying difficult cases goes here]
   \item Did your justifications use correct logic?
\end{itemize}

\item Right answer(s)
\begin{itemize}
   \item Did you state the right answer, or an answer as good as could
be expected?
   \item Extra credit, for answers beyond what was expected, goes here
\end{itemize}
\end{itemize}

\end{titlepage}

%%%\section{Poison}

\section{Hippos}
\begin{itemize}
\item Answering the Right Problem [2]
\begin{itemize}
\item Problem described in own words.  Especially state what is asked for [2]
\end{itemize}

\item Sufficiency of Explanation [11]
\begin{itemize}
\item State what variables represent (e.g., x = weight of lightest, 
   etc.)  [3]
\begin{quote}
Note that variables should represent quantities, not hippos.  
Also should not be randomly assigned
\end{quote}
\item Model problem with system(s) of equations [5] (Only one system 
   = 2 points)
\item Correct solution process (e.g., algebraic, guess 
and check, etc.) to find values of variables [3]
\end{itemize}

\item Sufficiency of Justification [8]
\begin{itemize}
\item Describe why system(s) of equations with the stated 
variables model the problem [6]:
\begin{itemize}
\item Describe why, for example, lightest 2 hippos must weigh 312 or why 
   2nd/4th hippos must weigh more than 2nd and 3rd hippos, etc.)  [2]
\item Not sure of weight order of 1st/4th hippos and 
   2nd/3rd hippos [2]
\item No other system models problem (can assign the given values to the 
   pairs in 720 ways.  But only two of these assignments can model the 
   given problem and are essentially different) [2]
\end{itemize}
\item Describe why process to solve set-up system(s) works. [2]
\end{itemize}

\item Achieving the Right Answer [2]
\begin{itemize}
\item Final set(s) of answers to the problem [2]
\end{itemize}
\end{itemize}

Out of: 23

\pagebreak

\section{Dots}
\begin{itemize}
\item Answering the right problem [2]
\begin{itemize}
\item Note the need to describe the triangular numbers through 
finding a pattern/formula for generating the numbers [1]
\item Note the need to find the number of 2-element subsets of a 
given-sized set [1]
\end{itemize}

\item Sufficiency of explanation [9]
\begin{itemize}
\item Develop and describe summation pattern in words or numerically 
(i.e., $1+2+ \cdots +n$) [1]
\item Note the need for simpler formula for finding nth triangular number [1]
\item Find the $100^{th}$ triangular number [2]
\item State simpler formula for $n{th}$ triangular number [2]
\item Find and describe pattern for the number of 2-element subsets 
of an $n$-element set; describe how it is similar to the triangular 
numbers previously found [3]
\end{itemize}

\item Sufficiency of justification [9]
\begin{itemize}
\item Describe why simpler formula for nth triangular number 
works (several ways to do this) [5]
\item Describe why the number of 2-element subsets is related to the 
triangular numbers. [4]
\end{itemize}
\end{itemize}

Out of: 20

\pagebreak

\section{Banned Books}
\begin{itemize}
\item Answering the right question [3]
\begin{itemize}
  \item Correct interpretation of the percentages that were given [2]
  \item Understanding what it means for some of the data to be needed 
or not [1]
\end{itemize}

\item Sufficiency of explanation [8]
\begin{itemize}
  \item Clearly stating what computations you did [2]
  \item Explaining your picture, if you drew one, or other representation
of the data [3]
  \item Explanation of what your computations represent, e.g., what does
a percentage over 100 represent, what are doublecounted people and why do
you subtract them? [3]
  \end{itemize}

\item Sufficiency of justification [6]
\begin{itemize}
  \item Why your additions and subtractions result in what you claim 
they do (algebra, or complete accounting for the additions and subtractions
of each person) [4]
  \item How you know whether the percentage of non-readers matters [2]
\end{itemize}

\item Final answers [4]
\begin{itemize}
  \item Correct percentage who read all three [2]
  \item Correct answer as to whether non-reader percentage matters [2]
\end{itemize}

\end{itemize}

Out of: 21

\pagebreak

\section{All in the Timing}

Problem 1:

\begin{itemize}
\item Sufficiency of explanation [4]
\begin{itemize}
  \item 7 is a prime factor in the prime factorization of a number 
if and only if the number is divisible by 7 [2]
  \item $2^3 \times 3$ is in the prime factorization of a number 
(i.e., at least three 2's multiplied together and 3 is a prime factor) 
if and only if the number is divisible by 24 [2] 
\end{itemize}

\item Sufficiency of justification: reason the above works in terms 
of the definition of ``factor'' or ``divisible''  [1]

\item Final answers: 98, 168, and 84 divisible by 7 (the others are not);
384, 168, and 144 divisible by 24 (the others are not) [1]
\end{itemize}

Problem 2:
\begin{itemize}
\item Sufficiency of justification [3]
\begin{itemize} 
  \item Different primes implies they are not both divisible by 7 
(or not both divisible by 13) [1] 
  \item The above implies the numbers are not equal because of famous 
result [2]
\end{itemize}

\item Final answer: no, not equal [1]
\end{itemize}

Problem 3:

\begin{itemize}
\item Sufficiency of Explanation [2]
\begin{itemize}
  \item List factors and count [1]
  \item The numbers each have six factors [1]
\end{itemize}

\item Sufficiency of Justification [4]
\begin{itemize}
  \item Use of common prime factorization form (or equivalent argument) [2]
  \item Why that prime factorization form yields six factors [2]
\end{itemize}
\end{itemize}

Problem 5:

\begin{itemize}
\item Sufficiency of Explanation [3]
\begin{itemize}
  \item	Reasonable use of guess and check [1]
  \item Use of prime factorization form with selective primes (or other 
reasonable strategy) [2]
\end{itemize}

\item Sufficiency of Justification: Explanation of why strategy yields 
correct answer [2]

\item Final answer [1]
\end{itemize}

Out of: 22

\pagebreak

\section{Counting Factors}
\begin{enumerate}
\item ~~ [3]
\begin{enumerate}
  \item Writing down two numbers with two factors each [1: justifying = .5]
  \item Describe rule for numbers with exactly 2 factors [2]
\end{enumerate}
 
\item ~~ [4]
\begin{enumerate}
  \item Write down a number with exactly 16 factors [1: justification = .5]
  \item Write down a number with exactly 32 factors [1: justification = .5]
  \item Write down the number of factors of  anumber whose prime factorization
is the product of $n$ primes [2: justification = 1]
\end{enumerate}
 
\item ~~ [3]
\begin{enumerate} 
  \item Write down two numbers with 3 factors each [1: justification = .5]
  \item Describe the rule for numbers with exactly 3 factors 
[2: justification = 1]
\end{enumerate}
 
\item ~~ [3]
\begin{enumerate}
  \item Write down the factors of 12 [1]
  \item Write the factors of $p^2 q$, where $p$ and $q$ are prime [2]
\end{enumerate}
 
\item ~~ [2]
\begin{enumerate}
  \item Write one number with exactly 9 factors [1: justification = .5]
  \item Write one number with exactly 27 factors [1: justification = .5]
\end{enumerate}
 
\item 
Write a general fule for the number of factors of a number given the 
number's prime factorization [3: justification = 2]
 
\item 
Describe the flaw in the argument for multiplying the number of 
factors of 6 by the number of factors of 4 to obtain the number of 
factors of 24 [2]

\end{enumerate}

Out of: 20

\pagebreak

\section{Stupid Number Tricks}
\begin{itemize}
\item Correct understanding of what the trick said to do to each number 
and what we need to show [1]
\item Algebraic (or equivalent) understanding, up to getting $10x+2$ [1]
\item Verbal explanation of arrival at $10x+2$ [1]
\item Knowing to state that $10x+2$ is $x$ followed by a ``2'' [1]
\item Why $10x+2$ always has a 2 at the end [1]
\item Why $10x+2$ always leaves $x$ when you eliminate the last digit [1]
\item General verbal organization and math grammar [1]
\item Generalizations, or limits to the validity [1*]
\end{itemize}

Out of: 7

\pagebreak

\section{Fractions!}
\begin{enumerate}
\item 3 points: 2 points for a reasonable picture(s) and 1 point for 
a description of how picture(s) were drawn and/or adjusted (e.g., 
splitting unit up into equal-sized pieces, what happened from 
1st picture to 2nd picture, what do shadings stand for).

\item 4 points: 2 points for description (mostly involving finding 
common denominator to make equivalent fractions and comparing 
resultant numerators).  2 points for a description of why this works 
(e.g., WHY we need a common denominator and why, once that is done, 
that the largest fraction is the one with the largest numerator.)  
NOTE:  Use of decimals not allowed unless could argue how we 
compare decimals and why THAT method works (use of comparison of 
fractions technique could be helpful in describing comparing decimals).

\item 2 points: 1 point for recognizing that we don't need to consider 
insignificant digits.  1 point for subsequent technique and answer.

\item 4 points: 2 points for process(es) (some dealt with mixed 
numbers separately).  2 points for why processes represent addition 
(e.g., WHY we need a common denominator and  WHY we only need to 
add numerators and keep the common denominator constant) and why steps work.  
SIDE NOTE:  Thought question:  Are there situations where "addition 
of fractions" could involve both adding the numerators AND adding the 
denominators?  Hint:  We've dealt with this possibility it on at least 
two problems recently (PT Cruiser and Condo).

\item 3 points: 2 points for process (Using the distributive property 
twice or "FOIL").  1 point for answer.

\item 4 points: 1 point for recognizing it's a division problem.  
1 point on WHY it is a division problem; 1 point for justifying process 
to answer (e.g., "flip and multiply" or repeated subtraction); 1 point 
for interpreting answer (e.g., what the 1/3 represents).

\item 2 points: 1 point for reasonable story problem. 
1 point for describing how story problem suggests a technique for 
dividing by a fraction ("flip and multiply" or repeated subtraction)

\item 2 points: 1 point for rewriting mixed number.  1 point for final answer.
NOTE:  The question only asked for the result and not for justification 
of the process.  However, you may want to consider why IT MAKES SENSE 
to define multiplication of fractions as "multiplying the numerators 
and multiplying the denominators."
\end{enumerate}

Out of: 24

\pagebreak

\section{Off base}

\begin{enumerate}
\item 3 points: answer = 1; work = 2.

\item 
\begin{enumerate}
  \item 2 points: answer = 1; work = 1.
  \item 2 points: answer = 1; work = 1.
  \item 3 points: answer = 1; work = 2.
\end{enumerate}

\item 
\begin{enumerate}
  \item 1 point.  
  \item 2 point: answer = 1; justification = 1.  
  \item 3 point: answer = 1; justification = 2.  
  \item 2 point: answer = 1; justification = 1.  
  \item 2 point: answer = 1; justification = 1.  
\end{enumerate}

\item 2 points.
\end{enumerate}

Out of: 22

\pagebreak

\section{Word problem review} 

For each problem, 

\begin{enumerate}
\item 2 points were given for a precise description of each variable you used.

\item 2 points were given for:  
\begin{enumerate}
  \item Equation(s) that accurately modeled the situation (1 point)
  \item Descriptions for why each equation models (part of) 
	the situation (1 point).
\end{enumerate}

\item 1 point was given for obtaining the correct answer in an 
algebraic manner consistent with the problem (i.e., no points 
for coincidentally obtaining the correct answer).
\end{enumerate}

Out of: 20

\end{document}

