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12.1
Students will communicate number sense concepts using multiple representations to reason, solve problems, and make connections within mathematics and across disciplines.
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12.1.1
Students will represent and show relationships among complex numbers.
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12.1.1.a
Demonstrate multiple equivalent forms of irrational numbers (e.g., square root of 8 = 8 to the 1/2 power = 2 square root of 2)
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12.1.1.b
Compare, contrast and apply the properties of numbers and the real number system, including rational, irrational, imaginary, and complex numbers
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12.1.2
Students will demonstrate the meaning and effects of arithmetic operations with real numbers.
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12.1.2.a
Use drawings, words, and symbols to explain the effects of such operations as multiplication and division, and computing positive powers and roots on the magnitude of quantities (e.g., if you take the square root of a number, will the result always be smaller than the original number? (e.g., square root of 1/4 = 1/2))
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12.1.2.b
Use drawings, words, and symbols to explain that the distance between two numbers on the number line is the absolute value of their difference
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12.1.3
Students will compute fluently and accurately using appropriate strategies and tools.
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12.1.3.a
Compute accurately with real numbers
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12.1.3.b
Simplify exponential expressions (e.g., powers of -1, 0, ½, 32 * 32 = 3 to the 4th power)
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12.1.3.c
Multiply and divide numbers using scientific notation
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12.1.3.d
Select, apply, and explain the method of computation when problem solving using real numbers (e.g., models, mental computation, paper-pencil, or technology)
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12.1.4
Students will estimate and check reasonableness of answers using appropriate strategies and tools.
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12.1.4.a
Use estimation methods to check the reasonableness of real number computations and decide if the problem calls for an approximation or an exact number (e.g., 10 "pi" (pi) is approximately 3.14, square and cube roots)
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12.1.4.b
Distinguish relevant from irrelevant information, identify missing information and either find what is needed or make appropriate estimates
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12.2
Students will communicate geometric concepts and measurement concepts using multiple representations to reason, solve problems, and make connections within mathematics and across disciplines.
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12.2.1
Students will analyze characteristics, properties, and relationships among geometric shapes and objects.
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12.2.1.a
Identify and explain the necessity of and give examples of definitions and theorems
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12.2.1.b
Analyze properties and relationships among classes of two and three dimensional geometric objects using inductive reasoning and counterexamples
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12.2.1.c
State and prove geometric theorems using deductive reasoning (e.g., parallel lines with transversals, congruent triangles, similar triangles)
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12.2.1.d
Apply geometric properties to solve problems (e.g., parallel lines, line transversals, similar triangles, congruent triangles, proportions)
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12.2.1.e
Identify and apply right triangle relationships (e.g., sine, cosine, tangent, special right triangles, converse of Pythagorean Theorem)
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12.2.1.f
Recognize that there are geometries, other than Euclidean geometry, in which the parallel postulate is not true
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12.2.1.g
Know the definitions and basic properties of a circle and use them to prove basic theorems and solve problems
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12.2.2
Student will use coordinate geometry to analyze and describe relationships in the coordinate plane.
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12.2.2.a
Use coordinate geometry to analyze geometric situations (e.g., parallel lines, perpendicular lines, circle equations)
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12.2.2.b
Apply the midpoint formula
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12.2.2.c
Apply the distance formula
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12.2.2.d
Prove special types of triangles and quadrilaterals (e.g., right triangles, isosceles trapezoid, parallelogram, rectangle, square)
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12.2.3
Students will apply and analyze transformations.
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12.2.3.a
Explain and justify the effects of simple transformations on the ordered pairs of two-dimensional shapes
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12.2.3.b
Perform and describe multiple transformations
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12.2.4
Students will use visualization, spatial reasoning, and geometric modeling to solve problems.
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12.2.4.a
Sketch and draw appropriate representations of geometric objects using ruler, protractor, or technology
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12.2.4.b
Use geometric models to visualize, describe, and solve problems (e.g., find the height of a tree; find the amount of paint needed for a room; scale model)
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12.2.5
Students will apply the units, systems, and formulas to solve problems.
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12.2.5.a
Use strategies to find surface area and volume of complex objects
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12.2.5.b
Apply appropriate units and scales to solve problems involving measurement
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12.2.5.c
Convert between various units of area and volume, such as square feet to square yards
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12.2.5.d
Convert equivalent rates (e.g., feet/second to miles/hour)
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12.2.5.e
Find arc length and area of sectors of a circle
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12.2.5.f
Determine surface area and volume of three-dimensional objects (e.g., spheres, cones, pyramids)
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12.2.5.g
Know that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively
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12.3
Students will communicate algebraic concepts using multiple representations to reason, solve problems, and make connections within mathematics and across disciplines.
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12.3.1
Students will generalize, represent, and analyze relationships using algebraic symbols.
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12.3.1.a
Represent, interpret, and analyze functions with graphs, tables, and algebraic notation and convert among these representations (e.g., linear, non-linear)
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Relations and functions: Graph a function (Algebra - Q.9)
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Relations and functions: Find points on a function graph (Algebra - Q.11)
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Relations and functions: Write a rule for a function table (Algebra - Q.12)
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Linear functions: Slope-intercept form: graph an equation (Algebra - S.5)
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Linear functions: Slope-intercept form: write an equation from a graph (Algebra - S.6)
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Linear functions: Standard form: graph an equation (Algebra - S.11)
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Linear functions: Point-slope form: graph an equation (Algebra - S.14)
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Linear functions: Point-slope form: write an equation from a graph (Algebra - S.15)
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Functions: linear, quadratic, exponential: Identify linear, quadratic, and exponential functions from graphs (Algebra - CC.1)
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Functions: linear, quadratic, exponential: Identify linear, quadratic, and exponential functions from tables (Algebra - CC.2)
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Functions: linear, quadratic, exponential: Write linear, quadratic, and exponential functions (Algebra - CC.3)
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12.3.1.b
Identify domain and range of functions represented in either symbolic or graphical form (e.g., linear, non-linear)
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12.3.1.c
Identify the slope and intercepts of a linear relationship from an equation or graph
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12.3.1.d
Identify characteristics of linear and non-linear functions
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12.3.1.e
Graph linear and non-linear functions
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12.3.1.f
Compare and analyze the rate of change by using ordered pairs, tables, graphs, and equations
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12.3.1.g
Graph and interpret linear inequalities
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12.3.1.h
Represent, interpret, and analyze functions and their inverses
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12.3.1.i
Determine if a relation is a function
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12.3.2
Students will model and analyze quantitative relationships.
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12.3.2.a
Model contextualized problems using various representations (e.g., graphs, tables, one variable equalities, one variable inequalities, linear equations in slope intercept form, inequalities in slope intercept form, system of linear equations with two variables)
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12.3.2.b
Represent a variety of quantitative relationships using linear equations and one variable inequalities
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12.3.2.c
Analyze situations to determine the type of algebraic relationship (e.g., linear, nonlinear)
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Linear functions: Identify linear functions (Algebra - S.1)
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Functions: linear, quadratic, exponential: Identify linear, quadratic, and exponential functions from graphs (Algebra - CC.1)
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Functions: linear, quadratic, exponential: Identify linear, quadratic, and exponential functions from tables (Algebra - CC.2)
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Functions: linear, quadratic, exponential: Write linear, quadratic, and exponential functions (Algebra - CC.3)
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12.3.2.d
Model contextualized problems using various representations for non-linear functions (e.g., quadratic, exponential, square root, and absolute value)
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12.3.3
Students will represent and solve equations and inequalities.
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12.3.3.a
Explain/apply the reflexive, symmetric, and transitive properties of equality
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12.3.3.b
Simplify algebraic expressions involving exponents (e.g., (3x to the 4th power)2)
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12.3.3.c
Add and subtract polynomials
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12.3.3.d
Multiply and divide polynomials (e.g., divide x3 - 8 by x - 2, divide x to the 4th power - 5x3 - 2x by x2)
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12.3.3.e
Factor polynomials
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12.3.3.f
Identify and generate equivalent forms of linear equations
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12.3.3.g
Solve linear equations and inequalities including absolute value
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12.3.3.h
Identify and explain the properties used in solving equations and inequalities
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12.3.3.i
Solve quadratic equations (e.g., factoring, graphing, quadratic formula)
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12.3.3.j
Add, subtract, and simplify rational expressions
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12.3.3.k
Multiply, divide, and simplify rational expressions
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12.3.3.l
Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified values of their variables
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12.3.3.m
Derive and use the formulas for the general term and summation of finite arithmetic and geometric series
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12.3.3.n
Combine functions by composition, as well as by addition, subtraction, multiplication, and division
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12.3.3.o
Solve an equation involving several variables for one variable in terms of the others
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12.3.3.p
Analyze and solve systems of two linear equations in two variables algebraically and graphically
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12.4
Students will communicate data analysis/probability concepts using multiple representations to reason, solve problems, and make connections within mathematics and across disciplines.
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12.4.1
Students will formulate a question and design a survey or an experiment in which data is collected and displayed in a variety of formats, then select and use appropriate statistical methods to analyze the data.
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12.4.1.a
Interpret data represented by the normal distribution and formulate conclusions
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12.4.1.b
Compute, identify, and interpret measures of central tendency (mean, median, mode) when provided a graph or data set
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12.4.1.c
Explain how sample size and transformations of data affect measures of central tendency
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12.4.1.d
Describe the shape and determine spread (variance, standard deviation) and outliers of a data set
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12.4.1.e
Explain how statistics are used or misused in the world
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12.4.1.f
Create scatter plots, analyze patterns, and describe relationships in paired data
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12.4.1.g
Explain the impact of sampling methods, bias, and the phrasing of questions asked during data collection and the conclusions that can rightfully be made
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12.4.1.h
Explain the differences between randomized experiment and observational studies
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12.4.2
Students will develop and evaluate inferences to make predictions.
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12.4.2.a
Compare data sets and evaluate conclusions using graphs and summary statistics
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12.4.2.b
Support inferences with valid arguments
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12.4.2.c
Develop linear equations for linear models to predict unobserved outcomes using regression line and correlation coefficient
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12.4.2.d
Recognize when arguments based on data confuse correlation with causation
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12.4.3
Students will apply and analyze concepts of probability.
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12.4.3.a
Construct a sample space and a probability distribution
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12.4.3.b
Identify dependent and independent events and calculate their probabilities
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12.4.3.c
Use the appropriate counting techniques to determine the probability of an event (e.g., combinations, permutations)
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12.4.3.d
Analyze events to determine if they are mutually exclusive
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12.4.3.e
Determine the relative frequency of a specified outcome of an event to estimate the probability of the outcome
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