Analyze and generalize mathematical patterns including sequences, series, and recursive patterns.bm1

Analyze, interpret and translate among representations of patterns including tables, charts, graphs, matrices and vectors.bm2

Study and employ mathematical models of patterns to make inferences, predictions and decisions.bm3

Explore patterns (graphic, numeric, etc.) characteristic of families of functions; explore structural patterns within systems of objects, operations or relations.bm4

Use patterns and reasoning to solve problems and explore new content.bm5

Identify and describe the nature of change and begin to use the more formal language such as rate of change, continuity, limit, distribution and deviation.bm6

Develop a mathematical concept of function and recognize that functions display characteristic patterns of change (e.g., linear, quadratic, exponential) bm7

Expand their understanding of function to include non-linear functions, composition of functions, inverses of functions, and piecewise- and recursively- defined functions.bm8

Represent functions using symbolism such as matrices, vectors and functional representation (f(x)).bm9

Differentiate and analyze classes of functions including linear, power, quadratic, exponential, circular, and trigonometric functions, and realize that many different situations can be modeled by a particular type of function.bm10

Increase their use of functions and mathematical models to solve problems in context.bm11

Select and use appropriate tools; make accurate measurements using both metric and common units, and measure angles in degrees and radians.bm12

Continue to make and apply measurements of length, mass, weight, time, temperature, area, volume, angle; classify objects according to their dimensions.bm13

Interpret measurements and explain how changes in one measure may affect other measures.bm14

Use proportional reasoning and indirect measurements, including applications of trigonometric ratios to measure inaccessible distances and to determine derived measures such as density.bm15

Use proportional reasoning and indirect measurements, including applications of trigonometric ratios to measure inaccessible distances and to determine derived measures such as density.bm16

Locate and describe objects in terms of their position, including polar coordinates, three-dimensional Cartesian coordinates, vectors and limits.bm17

Locate and describe objects in terms of their position, including polar coordinates, three-dimensional Cartesian coordinates, vectors and limits.bm18

Locate and describe objects in terms of their orientation and relative position, including displacement vectors, phase shift, maxima, minima and inflection points; give precise mathematical descriptions of symmetries.bm19

Give precise mathematical descriptions of transformations and describe the effects of transformations on size, shape, position and orientation.bm20

Describe the locus of a point by a rule or mathematical expression; trace the locus of a moving point.bm21

Use concepts of position, direction and orientation to describe the physical world and to solve problems.bm22

Use shape to identify plane and solid figures, graphs, loci, functions and data distributions.bm23

Determine necessary and sufficient conditions for the existence of a particular shape and apply those conditions to analyze shapes.bm24

Use transformational, coordinate or synthetic methods to verify (prove) the generalizations they have made about properties of classes of shapes.bm25

Draw and construct shapes in two and three dimensions and analyze and justify the steps of their constructions.bm26

Study transformations of shapes using isometries, size transformations and coordinate mappings.bm27

Compare and analyze shapes and formally establish the relationships among them, including congruence, similarity, parallelism, perpendicularity, and incidence.bm28

Use shape, shape properties and shape relationships to describe the physical world and to solve problems.bm29

Collect and explore data through observation, measurement, surveys, sampling techniques and simulations.bm30

Organize data using tables, charts, graphs, spreadsheets and data bases.bm31

Present data using the most appropriate representation and give a rationale for their choice; show how certain representations may skew the data or bias the presentation.bm32

Identify what data are needed to answer a particular question or solve a given problem and design and implement strategies to obtain, organize and present those data.bm33

Critically read data from tables, charts, or graphs and explain the source of the data and what the data represent.bm34

Describe the shape of a data distribution and determine measures of central tendency, variability and correlation.bm35

Use data and their characteristics to draw and support conclusions.bm36

Critically question the sources of data; the techniques used to collect, organize and present data; the inferences drawn from the data; and the sources of bias and measures taken to eliminate such bias.bm37

Formulate questions and problems and gather and interpret data to answer those questions.bm38

Make and test hypotheses.bm39

Design investigations to model and solve problems; also employ confidence intervals and curve fitting in analyzing the data.bm40

Formulate and communicate arguments and conclusions based on data and evaluate their arguments and those of others.bm41

Make predictions and decisions based on data, including interpolations and extrapolations.bm42

Employ investigations, mathematical models, and simulations to make inferences and predictions to answer questions and solve problems.bm43

Develop an understanding of irrational, real and complex numbers.bm44

Use the a+bi and polar forms of complex numbers.bm45

Develop an understanding of the properties of the real and complex number systems and of the properties of special numbers p, i, e. and conjugates.bm46

Apply their understanding of number systems to model, and solve mathematical and applied problems.bm47

Compare and order real numbers and compare rational approximations to exact values.bm48

Express numerical comparisons as ratios and rates.bm49

Extend the relationships of primes, factors, multiples and divisibility in an algebraic setting.bm50

Express number relationships using positive and negative rational exponents, logarithms and radicals.bm51

Apply their understanding of number relationships in solving problems.bm52

Give decimal representations of rational and irrational numbers and coordinate and vector representations of complex numbers.bm53

Develop an understanding of more complex representations of numbers, including exponential and logarithmic expressions, and select an appropriate representation to facilitate problem solving.bm54

Determine when to use rational approximations and the exact values of numbers such as e, p and the irrationals.bm55

Apply estimation to increasingly complex situations.bm56

Select appropriate representations for numbers, including representations of rational and irrational numbers and coordinate and vector representations of complex numbers in order to simplify and solve problems.bm57

Identify important variables in a context, symbolize them and express their relationships algebraically.bm58

Represent algebraic concepts and relationships with matrices, spreadsheets, diagrams, graphs, tables, physical models, vectors, equations and inequalities; and translate among the various representations.bm59

Solve linear equations and inequalities algebraically and non-linear equations using graphing, symbol-manipulating or spreadsheet technology; and solve linear and non-linear systems using appropriate methods.bm60

Analyze problems that can be modeled by functions, determine strategies for solving the problems and evaluate the adequacy of the solutions in the context of the problems.bm61

Explore problems that reflect the contemporary uses of mathematics in significant contexts and use the power of technology and algebraic and analytic reasoning to experience the ways mathematics is used in society.bm62

Use manipulatives to model operations with numbers; develop their own methods of recording operations; and relate their models and recordings to standard symbolic expressions and algorithms.bm63

Develop and apply the appropriate method of computation from among mental computation, estimation, paper-and-pencil or calculators; explain why they are choosing a method and how they know which operations to perform in a given situation.bm64

Describe the properties of operations with numbers, algebraic expressions, vectors and matrices, and make generalizations about the properties of given mathematical systems.bm65

Apply operations efficiently and accurately in solving problems.bm66

Derive and use formulas for calculating (the number of) permutations and combinations.bm67

Use sets and set relationships to represent algebraic and geometric concepts.bm68

Use vertex-edge graphs to solve network problems such as finding circuits, critical paths, minimum spanning tress and adjacency matrices.bm69

Analyze and use discrete ideas, such as induction, iteration and recurrence relations.bm70

Describe and analyze efficient algorithms to accomplish a task or solve a problem in a variety of contexts, including practical, mathematical and computer-related situations.bm71

Use discrete mathematics concepts as described above to model situations and solve problems; and look for whether or not there is a solution (existence problems), determine how many solutions there are (counting problems) and decide upon a best solution (optimization problems). bm72

Develop an understanding of randomness and chance variation and describe chance and certainty in the language of probability.bm73

Give a mathematical definition of probability and determine the probabilities of more complex events, and generate and interpret probability distributions.bm74

Analyze events to determine their dependence or independence and calculate probabilities of compound events.bm75

Use sampling and simulations to determine empirical probabilities and, when appropriate, compare them to the corresponding theoretical probabilities; understand and apply the law of large numbers.bm76

Conduct probability experiments and simulations, to model and solve problems, including compound events.bm77