Stage J - Mathematics

Descriptors

6A - Students who meet the standard can demonstrate knowledge and use of numbers and their many representations in a broad range of theoretical and practical settings. (Representations)

1. Represent numbers in equivalent forms (e.g., exponential/logarithmic, radical/rational exponents).
2. Graph or interpret the graph of a complex number in rectangular and vector forms.
3. Represent numerical intervals using correct notation.

6B - Students who meet the standard can investigate, represent, and solve problems using number facts, operations and their properties, algorithms, and relationships. (Operations and properties)

• Compare and contrast the properties of numbers and number systems, including the complex numbers as solutions to quadratic equations that do not have real solutions. **
• Simplify expressions using the field properties, order properties, and properties of equality for the set of real numbers.
• Use the field properties and properties of equality for the set of complex numbers.
• Determine the opposite, reciprocal, absolute values, and positive integral powers of a complex number.
• Identify, represent, and solve problems with numbers expressed in exponential, logarithmic, and scientific notations using technology.
• Solve problems using exponents and logarithms.
• Solve problems using complex numbers and their various representations.
• Explain that vectors and matrices are systems that have some of the properties of the real-number system. **
• Solve problems using matrices.
• Develop fluency in operations with real numbers, vectors, and matrices using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases. *

6C - Students who meet the standard can compute and estimate using mental mathematics, paper-and-pencil methods, calculators, and computers. (Choice of method)

Determine the level of accuracy needed for computations involving measurement and irrational numbers.

1. Use the correct number of digits in computation to achieve an appropriate unit or level of accuracy when solving problems.
2. Estimate an appropriate answer for a given term of a sequence.
3. Describe the role of rounding error in calculations.

6D - Students who meet the standard can solve problems using comparison of quantities, ratios, proportions, and percents.

1. Explain the connection of percents to growth patterns, error, and probability.
2. Set up and solve proportions for direct, inverse, and compound variations of quantities involving powers and multiple variables.

7A - Students who meet the standard can measure and compare quantities using appropriate units, instruments, and methods. (Performance and conversion of measurements)

1. Convert angle measures between degrees and radians.
2. Set up and solve measurement conversions using multiple rates and conversion factors.

7B - Students who meet the standard can estimate measurements and determine acceptable levels of accuracy. (Estimation)

1. Analyze precision, accuracy, and approximate error in measurement situations.
2. Determine a reasonable estimate of measure for more complex problem situations.
3. Solve problems to a desired interval of accuracy.
4. Apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations.

7C - Students who meet the standard can select and use appropriate technology, instruments, and formulas to solve problems, interpret results, and communicate findings. (Progression from selection of appropriate tools and methods to application of measurements to solve problems)

1. Solve practical problems using non-linear scales.

8A - Students who meet the standard can describe numerical relationships using variables and patterns. (Representations and algebraic manipulations)

1. Generalize patterns using explicitly-defined and recursively-defined sequences. **
2. Translate between explicit and recursive forms of sequences where possible.
3. Represent relationships arising from various contexts using symbolic expressions, including iterative and recursive forms.
4. Symbolize growth patterns using variables.
5. Explain the differences and similarities of different forms of growth formulas.
6. Describe the limiting process using numerical analysis, graphs, and algebra.
7. Simplify algebraic expressions using exponential, logarithmic, and rational number techniques, including more advanced factoring.

8B - Students who meet the standard can interpret and describe numerical relationships using tables, graphs, and symbols. (Connections of representations including the rate of change)

1. Fit an equation to data using a calculator.
2. Interpret the overall relationship of two variables and connect it to one of the function families (linear, exponential, logarithmic or power) from a graph.
3. Relate the effect of transformations on graphs and equations.
4. Analyze functions by investigating domain, range, rates of change, intercepts, zeros, asymptotes, and local and global behavior. **
5. Describe the properties and features of any non-degenerate conic section from an equation or graph.
6. Describe and perform transformations, such as arithmetically combining, composing, and inverting commonly used functions using technology, to perform operations on more complicated symbolic expressions.
7. Relate the situation to the graph and the function values for direct, inverse, and joint variations.
8. Relate functions to their inverses and their reflections over the line y = x.
9. Write an equation for conic sections from a graph.
10. Analyze functions and their graphs for symmetries.
11. Use a variety of symbolic representations for functions and relations, including piecewise functions.

8C - Students who meet the standard can solve problems using systems of numbers and their properties. (Problem solving)

1. Describe and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions. *
2. Identify and explain the relationship between arithmetic/geometric sequences and linear/exponential functions.
3. Describe the relationship of a mathematical model of a problem to the real problem.
4. Apply sequences and their properties to solve real problems.
5. Model and solve real problems using mathematical functions and relations.
6. Identify essential quantitative relationships in a situation and determine the class or classes of functions (e.g., power, exponential, logarithmic) that might model the relationships. **
7. Explain and apply relationships of x, y, and t in parametric equations.

8D - Students who meet the standard can use algebraic concepts and procedures to represent and solve problems. (Connection of 8A, 8B, 8C to solve problems)

1. Solve problems using linear programming.
2. Solve problems using equations of exponential and logarithmic growth.
3. Solve problems using direct, inverse, and mixed variation.
4. Apply solutions of real problems to similar situations with appropriate adaptation.
5. Solve problems using rational equations and inequalities.
6. Set up and solve problems of non-linear growth.

9A - Students who meet the standard can demonstrate and apply geometric concepts involving points, lines, planes, and space. (Properties of single figures, coordinate geometry and constructions)

1. Analyze geometric situations using Cartesian coordinates and other coordinate systems such as navigational, polar, or spherical systems. **
2. Represent transformations of an object in the plane using function notation and matrices.
3. Represent and describe with the language of geometry real-life objects, paths and regions in space.
4. Apply properties of two- and three-dimensional models to solve problems.

9B - Students who meet the standard can identify, describe, classify and compare relationships using points, lines, planes, and solids. (Connections between and among multiple geometric figures)

1. Solve problems using relationships between and among figures.
2. Represent and describe with the language of geometry intersections and cross sections of three-dimensional objects.

9C - Students who meet the standard can construct convincing arguments and proofs to solve problems. (Justifications of conjectures and conclusions)

1. Prove conjectures about geometric figures on the plane or in space using coordinate geometry.
2. Extend the ideas of formal and informal proof to non-geometric situations.
3. Develop formal and informal proofs for three-dimensional figures.

9D - Students who meet the standard can use trigonometric ratios and circular functions to solve problems.

1. Solve problems using the Laws of Sines and Law of Cosines.
2. Relate vector representation and trigonometric functions.
3. Solve problems using vectors.
4. Relate circular functions, arcs, and radian measure to triangle trigonometry and degree measure.
5. Simplify expressions and solve problems using trigonometric identities.
6. Solve trigonometric equations using circular functions.
7. Rotate conic sections using trigonometric functions.
8. Identify key characteristics of graphs of trigonometric functions and their inverses.
9. Graph trigonometric functions using translations and dilations.
10. Graph a given trigonometric function using its characteristics (e.g., period, amplitude).

10A - Students who meet the standard can organize, describe and make predictions from existing data. (Data Analysis)

1. Describe the differences among various kinds of studies and which types of inferences can legitimately be drawn from each. **
2. Recognize how linear transformations of univariate data affect shape, center, and spread.
3. Describe how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference. **
4. Present results and conclusions from given data using basic statistics (e.g., measures of central tendencies, standard deviation).
5. Interpolate, extrapolate, and make predictions from given information.
6. Evaluate survey results for conformity to simple distributions.

10B - Students who meet the standard can formulate questions, design data collection methods, gather and analyze data, and communicate findings. (Data Collection)

1. Explore the variability of sample statistics from a known population and construct sampling distributions using simulations.**
2. Describe how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference.**
3. Create a survey from a critical question and decide which sampling technique to use for the survey.
4. Evaluate surveys for clarity, bias, return rate, and specialized audiences.

10C - Students who meet the standard can determine, describe and apply the probabilities of events. (Probability, including counting techniques)

1. Determine the theoretical probability for a chance event using the binomial probability model.
2. Describe the normal curve and use its properties to answer questions about sets of data that are assumed to be normally distributed.
3. Identify patterns from a sample space.
4. Describe a simulation for a more advanced experiment.
5. Carry out a simulation to estimate probabilities, and if possible, compare it to the theoretical probability.
6. Compute and interpret the expected value of random variables in simple cases. *
7. Apply advanced counting techniques to determine probability.

* National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, Va: National Council of Teachers of Mathematics, 2000.
** Adapted from: National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, Va: National Council of Teachers of Mathematics, 2000.

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