2. Number
Core Concept A
The real numbers include the rational numbers and are in one-to-one correspondence with the points on the number line.

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2. Number
Core Concept B
Quantities can be compared using division, yielding rates and ratios.

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2. Number
Core Concept C
A fraction can represent the result of dividing the numerator by the denominator; equivalent fractions have the same value.

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2. Number
Core Concept D
Place value and the rules of arithmetic form the foundation for efficient algorithms.

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3. Quantity
Core Concept A
The value of a quantity is not specified unless the units are named or understood from the context.

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3. Quantity
Core Concept B
Quantities can be added and subtracted only when they are of the same type (length, area, speed, etc.).

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3. Quantity
Core Concept C
Quantities can be multiplied or divided to create new types of quantities, called derived quantities.

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4. Expressions
Core Concept A
Expressions are constructions built up from numbers, variables, and operations, which have a numerical value when each variable is replaced with a number.

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4. Expressions
Core Concept B
Complex expressions are made up of simpler expressions.

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4. Expressions
Core Concept C
The rules of arithmetic can be applied to transform an expression without changing its value.

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4. Expressions
Core Concept D
Rewriting expressions in equivalent forms serves a purpose in solving problems.

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5. Equations
Core Concept A
An equation is a statement that two expressions are equal.

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5. Equations
Core Concept B
The solutions of an equation are the values of the variables that make the resulting numerical statement true.

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5. Equations
Core Concept C
The steps in solving an equation are guided by understanding and justified by logical reasoning.

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5. Equations
Core Concept D
Equations not solvable in one number system may have solutions in a larger number system.

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6. Functions
Core Concept A
A function is a rule, often defined by an expression, that assigns a unique output for every input.

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6. Functions
Core Concept B
The graph of a function f is a set of ordered pairs (x, f(x)) in the coordinate plane.

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6. Functions
Core Concept C
Functions model situations where one quantity determines another.

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6. Functions
Core Concept D
Common functions occur in families where each member describes a similar type of dependence.

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7. Modeling
Core Concept A
Mathematical models involve choices and assumptions that abstract key features from situations to help us solve problems.

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7. Modeling
Core Concept B
Even very simple models can be useful.

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8. Shape
Core Concept A
Shapes and their parts, attributes, and measurements can be analyzed deductively.
In this document, deductive analysis aligns with the notion of adaptive reasoning as defined in Adding it Up, and includes empirical exploration, informal justification, and formal proof.

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8. Shape
Core Concept B
Congruence, similarity, and symmetry can be analyzed using transformations.

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8. Shape
Core Concept C
Mathematical shapes model the physical world, resulting in practical applications of geometry.

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8. Shape
Core Concept D
Right triangles and the Pythagorean theorem are central to geometry and its applications, including trigonometry.

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9. Coordinates
Core Concept A
Locations in the plane or in space can be specified by pairs or triples of numbers called coordinates.

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9. Coordinates
Core Concept B
Coordinates link algebra with geometry and allow methods in one domain to solve problems in the other.

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9. Coordinates
Core Concept C
The set of solutions to an equation in two variables forms a curve in the coordinate plane—such as a line, parabola, circle—and the solutions to systems of equations correspond to intersections of these curves.

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10. Probability
Core Concept A
Probability models outcomes for situations in which there is inherent randomness, quantifying the degree of uncertainty in terms of relative frequency of occurrence.

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10. Probability
Core Concept B
The law of large numbers provides the basis for estimating certain probabilities by use of empirical relative frequencies.

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10. Probability
Core Concept C
The laws of probability govern the calculation of probabilities of combined events.

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10. Probability
Core Concept D
Interpreting probabilities contextually is essential to rational decision-making in situations involving randomness.

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11. Statistics
Core Concept A
Statistical methods take variability into account to support making informed decisions based on quantitative studies designed to answer specific questions.

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11. Statistics
Core Concept B
Visual displays and summary statistics condense the information in data sets into usable knowledge.

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11. Statistics
Core Concept C
Randomness is the foundation for using statistics to draw conclusions when testing a claim or estimating plausible values for a population characteristic.

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11. Statistics
Core Concept D
The design of an experiment or sample survey is of critical importance to analyzing the data and drawing conclusions.

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