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Skills available for Colorado high school math standards

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9-12.1 Number Sense, Properties, and Operations

9-12.2 Patterns, Functions, and Algebraic Structures

9-12.3 Data Analysis, Statistics, and Probability

  • 9-12.3.1 Visual displays and summary statistics condense the information in data sets into usable knowledge

    • 9-12.3.1.a Summarize, represent, and interpret data on a single count or measurement variable

    • 9-12.3.1.b Summarize, represent, and interpret data on two categorical and quantitative variables

      • 9-12.3.1.b.i Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

      • 9-12.3.1.b.ii Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

        • 9-12.3.1.b.ii.1 Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

        • 9-12.3.1.b.ii.2 Informally assess the fit of a function by plotting and analyzing residuals.

        • 9-12.3.1.b.ii.3 Fit a linear function for a scatter plot that suggests a linear association.

    • 9-12.3.1.c Interpret linear models

      • 9-12.3.1.c.i Interpret the slope and the intercept of a linear model in the context of the data.

      • 9-12.3.1.c.ii Using technology, compute and interpret the correlation coefficient of a linear fit.

      • 9-12.3.1.c.iii Distinguish between correlation and causation.

  • 9-12.3.2 Statistical methods take variability into account supporting informed decisions making through quantitative studies designed to answer specific questions

    • 9-12.3.2.a Understand and evaluate random processes underlying statistical experiments

      • 9-12.3.2.a.i Describe statistics as a process for making inferences about population parameters based on a random sample from that population.

      • 9-12.3.2.a.ii Decide if a specified model is consistent with results from a given data-generating process.

    • 9-12.3.2.b Make inferences and justify conclusions from sample surveys, experiments, and observational studies

      • 9-12.3.2.b.i Identify the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

      • 9-12.3.2.b.ii Use data from a sample survey to estimate a population mean or proportion.

      • 9-12.3.2.b.iii Develop a margin of error through the use of simulation models for random sampling.

      • 9-12.3.2.b.iv Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

      • 9-12.3.2.b.v Define and explain the meaning of significance, both statistical (using p-values) and practical (using effect size).

      • 9-12.3.2.b.vi Evaluate reports based on data.

  • 9-12.3.3 Probability models outcomes for situations in which there is inherent randomness

    • 9-12.3.3.a Understand independence and conditional probability and use them to interpret data

      • 9-12.3.3.a.i Describe events as subsets of a sample space using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events.

      • 9-12.3.3.a.ii Explain that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

      • 9-12.3.3.a.iii Using the conditional probability of A given B as P(A and B)/P(B), interpret the independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

      • 9-12.3.3.a.iv Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

      • 9-12.3.3.a.v Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

    • 9-12.3.3.b Use the rules of probability to compute probabilities of compound events in a uniform probability model

      • 9-12.3.3.b.i Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.

      • 9-12.3.3.b.ii Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.

    • 9-12.3.3.c Analyze the cost of insurance as a method to offset the risk of a situation

9-12.4 Shape, Dimension, and Geometric Relationships

  • 9-12.4.1 Objects in the plane can be transformed, and those transformations can be described and analyzed mathematically

    • 9-12.4.1.a Experiment with transformations in the plane

    • 9-12.4.1.b Understand congruence in terms of rigid motions

      • 9-12.4.1.b.i Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure.

      • 9-12.4.1.b.ii Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

      • 9-12.4.1.b.iii Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

      • 9-12.4.1.b.iv Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

    • 9-12.4.1.c Prove geometric theorems

      • 9-12.4.1.c.i Prove theorems about lines and angles.

      • 9-12.4.1.c.ii Prove theorems about triangles.

      • 9-12.4.1.c.iii Prove theorems about parallelograms.

    • 9-12.4.1.d Make geometric constructions

      • 9-12.4.1.d.i Make formal geometric constructions with a variety of tools and methods.

      • 9-12.4.1.d.ii Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

  • 9-12.4.2 Concepts of similarity are foundational to geometry and its applications

    • 9-12.4.2.a Understand similarity in terms of similarity transformations

    • 9-12.4.2.b Prove theorems involving similarity

    • 9-12.4.2.c Define trigonometric ratios and solve problems involving right triangles

      • 9-12.4.2.c.i Explain that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

      • 9-12.4.2.c.ii Explain and use the relationship between the sine and cosine of complementary angles.

      • 9-12.4.2.c.iii Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

    • 9-12.4.2.d Prove and apply trigonometric identities

      • 9-12.4.2.d.i Prove the Pythagorean identity sinĀ²(θ) + cosĀ²(θ) = 1.

      • 9-12.4.2.d.ii Use the Pythagorean identity to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

    • 9-12.4.2.e Understand and apply theorems about circles.

      • 9-12.4.2.e.i Identify and describe relationships among inscribed angles, radii, and chords.

      • 9-12.4.2.e.ii Construct the inscribed and circumscribed circles of a triangle.

      • 9-12.4.2.e.iii Prove properties of angles for a quadrilateral inscribed in a circle.

    • 9-12.4.2.f Find arc lengths and areas of sectors of circles.

      • 9-12.4.2.f.i Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality.

      • 9-12.4.2.f.ii Derive the formula for the area of a sector.

  • 9-12.4.3 Objects in the plane can be described and analyzed algebraically

    • 9-12.4.3.a Express Geometric Properties with Equations.

      • 9-12.4.3.a.i Translate between the geometric description and the equation for a conic section

        • 9-12.4.3.a.i.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem.

        • 9-12.4.3.a.i.2 Complete the square to find the center and radius of a circle given by an equation.

        • 9-12.4.3.a.i.3 Derive the equation of a parabola given a focus and directrix.

      • 9-12.4.3.a.ii Use coordinates to prove simple geometric theorems algebraically

  • 9-12.4.4 Attributes of two- and three-dimensional objects are measurable and can be quantified

    • 9-12.4.4.a Explain volume formulas and use them to solve problems

      • 9-12.4.4.a.i Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

      • 9-12.4.4.a.ii Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

    • 9-12.4.4.b Visualize relationships between two-dimensional and three-dimensional objects

  • 9-12.4.5 Objects in the real world can be modeled using geometric concepts