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\begin{center} {\Large \bf 105: Arithmetic} \end{center}

\vfill

\begin{enumerate}

\item Foundations
  \begin{itemize}
  \item Sets
    \begin{itemize}
    \item what is a set: notation, extension property
    \item subsets, unions, intersections, complements
    \item Venn diagrams (see also Counting: inclusion-exclusion, below)
    \end{itemize}
  \item Logic
    \begin{itemize}
    \item informal logic: parsing propositional statements 
    \item formal logic: and, or, not, if-then
    \item truth tables and deduction
    \end{itemize}
  \item Operations (see Whole Numbers: Properties of Operations, below)
  \end{itemize}

\item Whole Numbers and their Properties
  \begin{itemize}
  \item Counting
    \begin{itemize}
    \item one to one correspondence
    \item inclusion-exclusion
    \item cartesian products
    \end{itemize}
  \item Factors
    \begin{itemize}
    \item prime numbers
    \item unique prime factorization
    \item counting factors
    \item LCM: how to obtain it; properties of the LCM
    \item GCD: how to obtain it; properties of the GCD
    \end{itemize}
  \item Negative Numbers
    \begin{itemize}
    \item interpretation of negative numbers
    \item rules for manipulation of negative numbers
    \end{itemize}
  \item Properties of Operations
    \begin{itemize}
    \item abstract operations: commutativity, associativity, identities, inverses
    \item properties of addition, multiplication, subtraction, division
    \item consequences of these properties 
    \item distributivity
    \item mental math consequences
    \end{itemize}
  \item Place Value
    \begin{itemize}
    \item how our numeration system works
    \item other bases
    \item algebraic representation of numbers in terms of their digits
    \item negative place value and decimals (see Decimals: place value, below)
    \end{itemize}
  \item Common Algorithms
    \begin{itemize}
    \item justification of addition, subtraction and multiplication algorithms
    \item understanding alternative algorithms
    \end{itemize}
  \end{itemize}

\item Division and Fractions
  \begin{itemize}
  \item Division
    \begin{itemize}
    \item inverse of multiplication
    \item repeated subtraction
    \item division algorithms
    \item division of fractions (see: Fractions: multiplication and division, below)
    \end{itemize}
  \item Ratio and Proportion
    \begin{itemize}
    \item proportionality word problems
    \item recognizing when given data is in terms of proportion
    \item cross-multiplication and its justification
    \end{itemize}
  \item Fractions
    \begin{itemize}
    \item models of fractions: number line/measurement, ratio, quotient, operator
    \item equivalent fractions and cross-multiplication
    \item mixed versus improper fractions
    \item adding and subtracting fractions
    \item multiplication and division: models, procedures
    \end{itemize}
  \item Decimals
    \begin{itemize}
    \item place value
    \item terminating decimals to fractions
    \item classification of fractions as terminating or repeating decimals
    \end{itemize}
  \item Scientific Notation
    \begin{itemize}
    \item definition and conversion into and out of
    \item estimation of physical quantities
    \item approximate computation in scientific notation
    \item logarithmic scales: brief intro
    \end{itemize}
  \item Percentages
    \begin{itemize}
    \item representation as a fraction or decimal
    \item percentage increase and decrease
    \end{itemize}
  \end{itemize}

\item General Skills and Attitudes
  \begin{itemize}
  \item Problem Solving Strategies
    \begin{itemize}
    \item trial and error 
    \item reducing the problem
    \item numbers instead of variables
    \item harnessing physical intuition
    \end{itemize}
  \item Generalization and Abstraction
    \begin{itemize}
    \item observing patterns 
    \item expressing your observations
    \item using variables to express and justify observations
    \item broadening the scope 
    \end{itemize}
  \item Word Problems
    \begin{itemize}
    \item understanding what is asked
    \item unambiguous usage of variables
    \item units
    \item writing equations for a problem
    \end{itemize}
  \item Broader Context
    \begin{itemize}
    \item NCTM Standards for active learning, justification/proof, verbalizing
    \item articulate and grammatical mathematical expression
    \item argumentation
    \end{itemize}
  \end{itemize}

\end{enumerate}

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