1. Foundations

• Sets

- what is a set: notation, extension property

- subsets, unions, intersections, complements

- Venn diagrams (see also Counting: inclusion-exclusion, below)

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Logic

- informal logic: parsing propositional statements

- formal logic: and, or, not, if-then

- truth tables and deduction

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Operations (see Whole Numbers: Properties of Operations , below)

2. Whole Numbers and their Properties

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Counting

- one to one correspondence

- inclusion-exclusion

- cartesian products

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Factors

- prime numbers

- unique prime factorization

- counting factors

- LCM: how to obtain it; properties of the LCM

- GCD: how to obtain it; properties of the GCD

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Negative Numbers

- interpretation of negative numbers

- rules for manipulation of negative numbers

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Properties of Operations

- abstract operations: commutativity, associativity, identities, inverses

- properties of addition, multiplication, subtraction, division

- consequences of these properties

- distributivity

- mental math consequences

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Place Value

- how our numeration system works

- other bases

- algebraic representation of numbers in terms of their digits

- negative place value and decimals (see Decimals: place value, below)

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Common Algorithms

- justification of addition, subtraction and multiplication algorithms

- understanding alternative algorithms

3. Division and Fractions

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Division

- inverse of multiplication

- repeated subtraction

- division algorithms

- division of fractions (see: Fractions: multiplication and division, below)

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Ratio and Proportion

- proportionality word problems

- recognizing when given data is in terms of proportion

- cross-multiplication and its justification

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Fractions

- models of fractions: number line /measurement, ratio, quotient, operator

- equivalent fractions and cross-multiplication

- mixed versus improper fractions

- adding and subtracting fractions

- multiplication and division: models, procedures

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Decimals

- place value

- terminating decimals to fractions

- classification of fractions as terminating or repeating decimals

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Scientific Notation

- definition and conversion into and out of

- estimatio of physical quantities

- approximate computation in scientific notation

- logarithmic scales : brief intro

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Percentages

- representation as a fraction or decimal

- percentage increase and decrease

4. General Skills and Attitudes

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Problem Solving Strategies

- trial and error

- reducing the problem

- numbers instead of variables

- harnessing physical intuition

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Generalization and Abstraction

- observing patterns

- expressing your observations

- using variables to express and justify observations

- broadening the scope

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Word Problems

- understanding what is asked

- unambiguous usage of variables

- units

- writing equations for a problem

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Broader Context

- NCTM Standards for active learning, justification/proof, verbalizing

- articulate and grammatical mathematical expression

- argumentation