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12.4.1
Number and Numerical Operations
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12.4.1.12 A
Number Sense
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12.4.1.12 A.1
Extend understanding of the number system to all real numbers.
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12.4.1.12 A.2
Compare and order rational and irrational numbers.
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12.4.1.12 A.3
Develop conjectures and informal proofs of properties of number systems and sets of numbers.
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12.4.1.12 B
Numerical Operations
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12.4.1.12 B.1
Extend understanding and use of operations to real numbers and algebraic procedures.
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12.4.1.12 B.2
Develop, apply, and explain methods for solving problems involving rational and negative exponents.
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12.4.1.12 B.3
Perform operations on matrices.
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12.4.1.12 B.3.1
Addition and subtraction
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12.4.1.12 B.3.2
Scalar multiplication
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12.4.1.12 B.4
Understand and apply the laws of exponents to simplify expressions involving numbers raised to powers.
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12.4.1.12 C
Estimation
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12.4.1.12 C.1
Recognize the limitations of estimation, assess the amount of error resulting from estimation, and determine whether the error is within acceptable tolerance limits.
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12.4.2
Geometry and Measurement
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12.4.2.12 A
Geometric Properties
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12.4.2.12 A.1
Use geometric models to represent real-world situations and objects and to solve problems using those models (e.g., use Pythagorean Theorem to decide whether an object can fit through a doorway).
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12.4.2.12 A.2
Draw perspective views of 3D objects on isometric dot paper, given 2D representations (e.g., nets or projective views).
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12.4.2.12 A.3
Apply the properties of geometric shapes.
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12.4.2.12 A.3.1
Parallel lines - transversal, alternate interior angles, corresponding angles
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12.4.2.12 A.3.2
Triangles
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12.4.2.12 A.3.2.a
Conditions for congruence
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12.4.2.12 A.3.2.b
Segment joining midpoints of two sides is parallel to and half the length of the third side
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12.4.2.12 A.3.2.c
Triangle Inequality
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12.4.2.12 A.3.2.d
Special right triangles
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12.4.2.12 A.3.3
Minimal conditions for a shape to be a special quadrilateral
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12.4.2.12 A.3.4
Circles - arcs, central and inscribed angles, chords, tangents
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12.4.2.12 A.3.5
Self-similarity
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12.4.2.12 A.4
Use reasoning and some form of proof to verify or refute conjectures and theorems.
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12.4.2.12 A.4.1
Verification or refutation of proposed proofs
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12.4.2.12 A.4.2
Simple proofs involving congruent triangles
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12.4.2.12 A.4.3
Counterexamples to incorrect conjectures
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12.4.2.12 A.5
Perform basic geometric constructions using a variety of methods (e.g., straightedge and compass, patty/tracing paper, or technology).
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12.4.2.12 A.5.1
Perpendicular bisector of a line segment
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12.4.2.12 A.5.2
Bisector of an angle
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12.4.2.12 A.5.3
Perpendicular or parallel lines
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12.4.2.12 B
Transforming Shapes
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12.4.2.12 B.1
Determine, describe, and draw the effect of a transformation, or a sequence of transformations, on a geometric or algebraic representation, and, conversely, determine whether and how one representation can be transformed to another by a transformation or a sequence of transformations.
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12.4.2.12 B.2
Recognize three-dimensional figures obtained through transformations of two-dimensional figures (e.g., cone as rotating an isosceles triangle about an altitude), using software as an aid to visualization.
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12.4.2.12 B.3
Determine whether two or more given shapes can be used to generate a tessellation.
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12.4.2.12 B.4
Generate and analyze iterative geometric patterns.
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12.4.2.12 B.4.1
Fractals (e.g., Sierpinski's Triangle)
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12.4.2.12 B.4.2
Patterns in areas and perimeters of self-similar figures
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12.4.2.12 B.4.3
Outcome of extending iterative process indefinitely
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12.4.2.12 C
Coordinate Geometry
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12.4.2.12 C.1
Use coordinate geometry to represent and verify properties of lines and line segments.
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12.4.2.12 C.1.1
Distance between two points
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12.4.2.12 C.1.2
Midpoint and slope of a line segment
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12.4.2.12 C.1.3
Finding the intersection of two lines
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12.4.2.12 C.1.4
Lines with the same slope are parallel
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12.4.2.12 C.1.5
Lines that are perpendicular have slopes whose product is -1
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12.4.2.12 C.2
Show position and represent motion in the coordinate plane using vectors.
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12.4.2.12 C.2.1
Addition and subtraction of vectors
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12.4.2.12 C.3
Find an equation of a circle given its center and radius and, given an equation of a circle in standard form, find its center and radius.
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12.4.2.12 D
Units of Measurement
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12.4.2.12 D.1
Understand and use the concept of significant digits.
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12.4.2.12 D.2
Choose appropriate tools and techniques to achieve the specified degree of precision and error needed in a situation.
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12.4.2.12 D.2.1
Degree of accuracy of a given measurement tool
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12.4.2.12 D.2.2
Finding the interval in which a computed measure (e.g., area or volume) lies, given the degree of precision of linear measurements
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12.4.2.12 E
Measuring Geometric Objects
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12.4.2.12 E.1
Use techniques of indirect measurement to represent and solve problems.
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12.4.2.12 E.1.1
Similar triangles
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12.4.2.12 E.1.2
Pythagorean theorem
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12.4.2.12 E.1.3
Right triangle trigonometry (sine, cosine, tangent)
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12.4.2.12 E.2
Use a variety of strategies to determine perimeter and area of plane figures and surface area and volume of 3D figures.
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12.4.2.12 E.2.1
Approximation of area using grids of different sizes
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12.4.2.12 E.2.2
Finding which shape has minimal (or maximal) area, perimeter, volume, or surface area under given conditions using graphing calculators, dynamic geometric software, and/or spreadsheets
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12.4.2.12 E.2.3
Estimation of area, perimeter, volume, and surface area
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12.4.3
Patterns and Algebra
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12.4.3.12 A
Patterns
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12.4.3.12 A.1
Use models and algebraic formulas to represent and analyze sequences and series.
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12.4.3.12 A.1.1
Explicit formulas for nth terms
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12.4.3.12 A.1.2
Sums of finite arithmetic series
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12.4.3.12 A.1.3
Sums of finite and infinite geometric series
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12.4.3.12 A.2
Develop an informal notion of limit.
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12.4.3.12 A.3
Use inductive reasoning to form generalizations.
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12.4.3.12 B
Functions and Relationships
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12.4.3.12 B.1
Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs.
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12.4.3.12 B.2
Analyze and explain the general properties and behavior of functions or relations, using algebraic and graphing techniques.
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12.4.3.12 B.2.1
Slope of a line
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12.4.3.12 B.2.2
Domain and range
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12.4.3.12 B.2.3
Intercepts
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12.4.3.12 B.2.4
Continuity
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12.4.3.12 B.2.5
Maximum/minimum
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12.4.3.12 B.2.6
Estimating roots of equations
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12.4.3.12 B.2.7
Solutions of systems of equations
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12.4.3.12 B.2.8
Solutions of systems of linear inequalities using graphing techniques
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12.4.3.12 B.2.9
Rates of change
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12.4.3.12 B.3
Understand and perform transformations on commonly-used functions.
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12.4.3.12 B.3.1
Translations, reflections, dilations
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12.4.3.12 B.3.2
Effects on linear and quadratic graphs of parameter changes in equations
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12.4.3.12 B.3.3
Using graphing calculators or computers for more complex functions
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12.4.3.12 B.4
Understand and compare the properties of classes of functions, including exponential, polynomial, rational, and trigonometric functions.
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12.4.3.12 B.4.1
Linear vs. non-linear
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12.4.3.12 B.4.2
Symmetry
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12.4.3.12 B.4.3
Increasing/decreasing on an interval
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12.4.3.12 C
Modeling
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12.4.3.12 C.1
Use functions to model real-world phenomena and solve problems that involve varying quantities.
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12.4.3.12 C.1.1
Linear, quadratic, exponential, periodic (sine and cosine), and step functions (e.g., price of mailing a first-class letter over the past 200 years)
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12.4.3.12 C.1.2
Direct and inverse variation
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12.4.3.12 C.1.3
Absolute value
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12.4.3.12 C.1.4
Expressions, equations and inequalities
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12.4.3.12 C.1.5
Same function can model variety of phenomena
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12.4.3.12 C.1.6
Growth/decay and change in the natural world
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12.4.3.12 C.1.7
Applications in mathematics, biology, and economics (including compound interest)
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12.4.3.12 C.2
Analyze and describe how a change in an independent variable leads to change in a dependent one.
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12.4.3.12 C.3
Convert recursive formulas to linear or exponential functions (e.g., Tower of Hanoi and doubling).
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12.4.3.12 D
Procedures
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12.4.3.12 D.1
Evaluate and simplify expressions.
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12.4.3.12 D.1.1
Add and subtract polynomials
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12.4.3.12 D.1.2
Multiply a polynomial by a monomial or binomial
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12.4.3.12 D.1.3
Divide a polynomial by a monomial
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12.4.3.12 D.1.4
Perform simple operations with rational expressions
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12.4.3.12 D.1.5
Evaluate polynomial and rational expressions
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12.4.3.12 D.2
Select and use appropriate methods to solve equations and inequalities.
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12.4.3.12 D.2.1
Linear equations and inequalities - algebraically
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12.4.3.12 D.2.2
Quadratic equations - factoring (including trinomials when the coefficient of x2 is 1) and using the quadratic formula
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12.4.3.12 D.2.3
Literal equations
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12.4.3.12 D.2.4
All types of equations and inequalities using graphing, computer, and graphing calculator techniques
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12.4.3.12 D.3
Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.
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12.4.4
Data Analysis, Probability, and Discrete Mathematics
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12.4.4.12 A
Data Analysis
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12.4.4.12 A.1
Use surveys and sampling techniques to generate data and draw conclusions about large groups.
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12.4.4.12 A.1.1
Advantages/disadvantages of sample selection methods (e.g., convenience sampling, responses to survey, random sampling)
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12.4.4.12 A.2
Evaluate the use of data in real-world contexts.
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12.4.4.12 A.2.1
Accuracy and reasonableness of conclusions drawn
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12.4.4.12 A.2.2
Correlation vs. causation
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12.4.4.12 A.2.3
Bias in conclusions drawn (e.g., influence of how data is displayed)
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12.4.4.12 A.2.4
Statistical claims based on sampling
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12.4.4.12 A.3
Design a statistical experiment, conduct the experiment, and interpret and communicate the outcome.
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12.4.4.12 A.4
Estimate or determine lines of best fit (or curves of best fit if appropriate) with technology, and use them to interpolate within the range of the data.
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12.4.4.12 A.5
Analyze data using technology, and use statistical terminology to describe conclusions.
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12.4.4.12 A.5.1
Measures of dispersion: variance, standard deviation, outliers
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12.4.4.12 A.5.2
Correlation coefficient
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12.4.4.12 A.5.3
Normal distribution (e.g., approximately 95% of the sample lies between two standard deviations on either side of the mean)
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12.4.4.12 A.6
Distinguish between randomized experiments and observational studies.
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12.4.4.12 B
Probability
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12.4.4.12 B.1
Calculate the expected value of a probability-based game, given the probabilities and payoffs of the various outcomes, and determine whether the game is fair.
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12.4.4.12 B.2
Use concepts and formulas of area to calculate geometric probabilities.
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12.4.4.12 B.3
Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models, and solve problems using these models.
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12.4.4.12 B.4
Determine probabilities in complex situations.
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12.4.4.12 B.4.1
Conditional events
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12.4.4.12 B.4.2
Complementary events
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12.4.4.12 B.4.3
Dependent and independent events
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12.4.4.12 B.5
Estimate probabilities and make predictions based on experimental and theoretical probabilities.
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12.4.4.12 B.6
Understand and use the "law of large numbers" (that experimental results tend to approach theoretical probabilities after a large number of trials).
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12.4.4.12 C
Discrete Mathematics-Systematic Listing and Counting
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12.4.4.12 C.1
Calculate combinations with replacement (e.g., the number of possible ways of tossing a coin 5 times and getting 3 heads) and without replacement (e.g., number of possible delegations of 3 out of 23 students).
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12.4.4.12 C.2
Apply the multiplication rule of counting in complex situations, recognize the difference between situations with replacement and without replacement, and recognize the difference between ordered and unordered counting situations.
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12.4.4.12 C.3
Justify solutions to counting problems.
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12.4.4.12 C.4
Recognize and explain relationships involving combinations and Pascal's Triangle, and apply those methods to situations involving probability.
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12.4.4.12 D
Discrete Mathematics-Vertex-Edge Graphs and Algorithms
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12.4.4.12 D.1
Use vertex-edge graphs and algorithmic thinking to represent and solve practical problems.
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12.4.4.12 D.1.1
Circuits that include every edge in a graph
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12.4.4.12 D.1.2
Circuits that include every vertex in a graph
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12.4.4.12 D.1.3
Scheduling problems (e.g., when project meetings should be scheduled to avoid conflicts) using graph coloring
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12.4.4.12 D.1.4
Applications to science (e.g., who-eats-whom graphs, genetic trees, molecular structures)
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12.4.4.12 D.2
Explore strategies for making fair decisions.
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12.4.4.12 D.2.1
Combining individual preferences into a group decision (e.g., determining winner of an election or selection process)
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12.4.4.12 D.2.2
Determining how many Student Council representatives each class (9th, 10th, 11th, and 12th grade) gets when the classes have unequal sizes (apportionment)
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12.4.5
Mathematical Processes
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12.4.5 A
Problem Solving
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12.4.5 A.1
Learn mathematics through problem solving, inquiry, and discovery.
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12.4.5 A.2
Solve problems that arise in mathematics and in other contexts.
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12.4.5 A.2.1
Open-ended problems
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12.4.5 A.2.2
Non-routine problems
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12.4.5 A.2.3
Problems with multiple solutions
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12.4.5 A.2.4
Problems that can be solved in several ways
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12.4.5 A.3
Select and apply a variety of appropriate problem-solving strategies (e.g., "try a simpler problem" or "make a diagram") to solve problems.
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12.4.5 A.4
Pose problems of various types and levels of difficulty.
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12.4.5 A.5
Monitor their progress and reflect on the process of their problem solving activity.
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12.4.5 A.6
Distinguish relevant from irrelevant information, and identify missing information.
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12.4.5 B
Communication
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12.4.5 B.1
Use communication to organize and clarify their mathematical thinking.
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12.4.5 B.1.1
Reading and writing
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12.4.5 B.1.2
Discussion, listening, and questioning
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12.4.5 B.2
Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing.
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12.4.5 B.3
Analyze and evaluate the mathematical thinking and strategies of others.
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12.4.5 B.4
Use the language of mathematics to express mathematical ideas precisely.
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12.4.5 C
Connections
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12.4.5 C.1
Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry).
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12.4.5 C.2
Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point).
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12.4.5 C.3
Recognize that mathematics is used in a variety of contexts outside of mathematics.
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12.4.5 C.4
Apply mathematics in practical situations and in other disciplines.
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12.4.5 C.5
Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards).
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12.4.5 C.6
Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
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12.4.5 D
Reasoning
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12.4.5 D.1
Recognize that mathematical facts, procedures, and claims must be justified.
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12.4.5 D.2
Use reasoning to support their mathematical conclusions and problem solutions.
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12.4.5 D.3
Select and use various types of reasoning and methods of proof.
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12.4.5 D.4
Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions.
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12.4.5 D.5
Make and investigate mathematical conjectures.
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12.4.5 D.5.1
Counterexamples as a means of disproving conjectures
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12.4.5 D.5.2
Verifying conjectures using informal reasoning or proofs.
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12.4.5 D.6
Evaluate examples of mathematical reasoning and determine whether they are valid.
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12.4.5 E
Representations
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12.4.5 E.1
Create and use representations to organize, record, and communicate mathematical ideas.
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12.4.5 E.1.1
Concrete representations (e.g., base-ten blocks or algebra tiles)
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12.4.5 E.1.2
Pictorial representations (e.g., diagrams, charts, or tables)
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12.4.5 E.1.3
Symbolic representations (e.g., a formula)
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12.4.5 E.1.4
Graphical representations (e.g., a line graph)
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12.4.5 E.2
Select, apply, and translate among mathematical representations to solve problems.
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Numbers: Number lines (Algebra - A.4)
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Variable expressions and equations: Write variable expressions (Algebra - I.1)
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Variable expressions and equations: Write variable equations (Algebra - I.3)
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Solve equations: Model and solve equations using algebra tiles (Algebra - J.1)
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Solve equations: Write and solve equations that represent diagrams (Algebra - J.2)
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Single-variable inequalities: Write inequalities from graphs (Algebra - K.2)
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Single-variable inequalities: Write compound inequalities from graphs (Algebra - K.13)
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Charts and graphs: Interpret bar graphs, line graphs, and histograms (Algebra - N.1)
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Charts and graphs: Create bar graphs, line graphs, and histograms (Algebra - N.2)
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Charts and graphs: Circle graphs (Algebra - N.3)
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Charts and graphs: Interpret stem-and-leaf plots (Algebra - N.4)
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Number sequences: Write variable expressions for arithmetic sequences (Algebra - P.5)
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Number sequences: Write variable expressions for geometric sequences (Algebra - P.6)
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Relations and functions: Relations: convert between tables, graphs, mappings, and lists of points (Algebra - Q.1)
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Relations and functions: Write a function rule: word problems (Algebra - Q.10)
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Relations and functions: Write a rule for a function table (Algebra - Q.12)
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Direct and inverse variation: Write direct variation equations (Algebra - R.4)
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Direct and inverse variation: Write and solve direct variation equations (Algebra - R.5)
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Direct and inverse variation: Write inverse variation equations (Algebra - R.7)
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Direct and inverse variation: Write and solve inverse variation equations (Algebra - R.8)
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Linear functions: Write equations in standard form (Algebra - S.9)
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Polynomials: Model polynomials with algebra tiles (Algebra - Z.2)
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Functions: linear, quadratic, exponential: Write linear, quadratic, and exponential functions (Algebra - CC.3)
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12.4.5 E.3
Use representations to model and interpret physical, social, and mathematical phenomena.
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Numbers: Number lines (Algebra - A.4)
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Variable expressions and equations: Write variable expressions (Algebra - I.1)
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Variable expressions and equations: Write variable equations (Algebra - I.3)
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Solve equations: Model and solve equations using algebra tiles (Algebra - J.1)
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Solve equations: Write and solve equations that represent diagrams (Algebra - J.2)
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Single-variable inequalities: Write inequalities from graphs (Algebra - K.2)
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Single-variable inequalities: Write compound inequalities from graphs (Algebra - K.13)
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Charts and graphs: Interpret bar graphs, line graphs, and histograms (Algebra - N.1)
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Charts and graphs: Create bar graphs, line graphs, and histograms (Algebra - N.2)
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Charts and graphs: Circle graphs (Algebra - N.3)
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Charts and graphs: Interpret stem-and-leaf plots (Algebra - N.4)
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Number sequences: Write variable expressions for arithmetic sequences (Algebra - P.5)
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Number sequences: Write variable expressions for geometric sequences (Algebra - P.6)
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Relations and functions: Relations: convert between tables, graphs, mappings, and lists of points (Algebra - Q.1)
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Relations and functions: Write a function rule: word problems (Algebra - Q.10)
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Relations and functions: Write a rule for a function table (Algebra - Q.12)
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Direct and inverse variation: Write direct variation equations (Algebra - R.4)
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Direct and inverse variation: Write and solve direct variation equations (Algebra - R.5)
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Direct and inverse variation: Write inverse variation equations (Algebra - R.7)
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Direct and inverse variation: Write and solve inverse variation equations (Algebra - R.8)
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Linear functions: Write equations in standard form (Algebra - S.9)
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Systems of linear equations: Solve a system of equations by graphing: word problems (Algebra - U.3)
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Systems of linear equations: Solve a system of equations using substitution: word problems (Algebra - U.9)
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Systems of linear equations: Solve a system of equations using elimination: word problems (Algebra - U.11)
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Systems of linear equations: Solve a system of equations using augmented matrices: word problems (Algebra - U.13)
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Systems of linear equations: Solve a system of equations using any method: word problems (Algebra - U.15)
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Polynomials: Model polynomials with algebra tiles (Algebra - Z.2)
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Functions: linear, quadratic, exponential: Write linear, quadratic, and exponential functions (Algebra - CC.3)
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12.4.5 F
Technology
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12.4.5 F.1
Use technology to gather, analyze, and communicate mathematical information.
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12.4.5 F.2
Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information.
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12.4.5 F.3
Use graphing calculators and computer software to investigate properties of functions and their graphs.
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12.4.5 F.4
Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions).
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12.4.5 F.5
Use computer software to make and verify conjectures about geometric objects.
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12.4.5 F.6
Use computer-based laboratory technology for mathematical applications in the sciences (cf. science standards).
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