2.1.3.B. Use whole numbers and fractions to represent quantities.

2.1.3.C. Represent equivalent forms of the same number through the use of concrete objects, drawings, word names and symbols.

2.1.3.D. Use drawings, diagrams or models to show the concept of fraction as part of a whole.

2.1.3.E. Count, compare and make change using a collection of coins and one-dollar bills.

2.1.3.F.Apply number patterns (even and odd) and compare values of numbers on the hundred board.

2.1.3.G. Use concrete objects to count, order and group.

2.1.3.H. Demonstrate an understanding of one-to-one correspondence.

2.1.3.I. Apply place-value concepts and numeration to counting, ordering and grouping.

2.1.3.J. Estimate, approximate, round or use exact numbers as appropriate.

2.1.3.K. Describe the inverse relationship between addition and subtraction.

2.1.5.A. Use expanded notation to represent whole numbers or decimals.

2.1.5.C. Demonstrate that mathematical operations can represent a variety of problem situations.

2.1.5.D. Use models to represent fractions and decimals.

2.1.5.E. Explain the concepts of prime and composite numbers.

2.1.5.F. Use simple concepts of negative numbers (e.g., on a number line, in counting, in temperature).

2.1.8.A.Represent and use numbers in equivalent forms (e.g., integers, fractions, decimals, percents, exponents, scientific notation, square roots).

2.1.8.B. Simplify numerical expressions involving exponents, scientific notation and using order of operations.

2.1.8.C.Distinguish between and order rational and irrational numbers.

2.1.8.D. Apply ratio and proportion to mathematical problem situations involving distance, rate, time and similar triangles.

2.1.8.E. Simplify and expand algebraic expressions using exponential forms.

2.1.8.F. Use the number line model to demonstrate integers and their applications.

2.1.11.A. Use operations (e.g., opposite, reciprocal, absolute value, raising to a power, finding roots, finding logarithms).

2.2.3.A. Apply addition and subtraction in everyday situations using concrete objects.

2.2.3.B. Solve single- and double-digit addition and subtraction problems with regrouping in vertical form.

2.2.3.C. Demonstrate the concept of multiplication as repeated addition and arrays.

2.2.3.D. Demonstrate the concept of division as repeated subtraction and as sharing.

2.2.3.E. Use estimation skills to arrive at conclusions.

2.2.3.F. Determine the reasonableness of calculated answers.

2.2.3.G. Explain addition and subtraction algorithms with regrouping.

2.2.5.A. Create and solve word problems involving addition, subtraction, multiplication and division of whole numbers.

2.2.5.B. Develop and apply algorithms to solve word problems that involve addition, subtraction, and/or multiplication with decimals with and without regrouping.

2.2.5.C. Develop and apply algorithms to solve word problems that involve addition, subtraction, and/or multiplication with fractions and mixed numbers that include like and unlike denominators.

2.2.5.D. Demonstrate the ability to round numbers.

2.2.5.E. Determine through estimations the reasonableness of answers to problems involving addition, subtraction, multiplication and division of whole numbers.

2.2.5.F. Demonstrate skills for using fraction calculators to verify conjectures, confirm computations and explore complex problem-solving situations.

2.2.5.G. Apply estimation strategies to a variety of problems including time and money.

2.2.5.H. Explain multiplication and division algorithms.

2.2.8.A. Complete calculations by applying the order of operations.

2.2.8.B. Add, subtract, multiply and divide different kinds and forms of rational numbers including integers, decimal fractions, percents and proper and improper fractions.

2.2.8.C, Estimate the value of irrational numbers.

2.2.8.D. Estimate amount of tips and discounts using ratios, proportions and percents.

2.2.8.E. Determine the appropriateness of overestimating or underestimating in computation.

2.2.8.F. Identify the difference between exact value and approximation and determine which is appropriate for a given situation.

2.2.11.A. Develop and use computation concepts, operations and procedures with real numbers in problem-solving situations.

2.2.11.B. Use estimation to solve problems for which an exact answer is not needed.

2.2.11.C. Construct and apply mathematical models, including lines and curves of best fit, to estimate values of related quantities.

2.2.11.D. Describe and explain the amount of error that may exist in a computation using estimates.

2.2.11.E. Recognize that the degree of precision needed in calculating a number depends on how the results will be used and the instruments used to generate the measure.

2.2.11.F. Demonstrate skills for using computer spreadsheets and scientific and graphing calculators.

2.3.3.A. Compare measurable characteristics of different objects on the same dimensions (e.g., time, temperature, area, length, weight, capacity, perimeter).

2.3.3.B. Determine the measurement of objects with non-standard and standard units (e.g., US customary and metric).

2.3.3.C. Determine and compare elapsed times.

2.3.3.D. Tell time (analog and digital) to the minute.

2.3.3.E. Determine the appropriate unit of measure.

2.3.3.F. Use concrete objects to determine area and perimeter.

2.3.3.G. Estimate and verify measurements.

1. Demonstrate that a single object has different attributes that can be measured in different ways (e.g., length, mass, weight, time, area, temperature, capacity, perimeter)

2.3.5.A. Select and use appropriate instruments and units for measuring quantities (e.g., perimeter, volume, area, weight, time, temperature).

2.3.5.B. Select and use standard tools to measure the size of figures with specified accuracy, including length, width, perimeter and area.

2.3.5.C. Estimate, refine and verify specified measurements of objects.

2.3.5.D. Convert linear measurements within the same system.

2.3.8.A. Develop formulas and procedures for determining measurements (e.g., area, volume, distance).

2.3.8.C. Measure angles in degrees and determine relations of angles.

2.3.8.D. Estimate, use and describe measures of distance, rate, perimeter, area, volume, weight, mass and angles.

2.3.8.F. Use scale measurements to interpret maps or drawings.

2.3.11.A. Select and use appropriate units and tools to measure to the degree of accuracy required in particular measurement situations.

2.3.11.B. Measure and compare angles in degrees and radians.

2.4.3.A. Make, check and verify predictions about the quantity, size and shape of objects and groups of objects.

2.4.3.B. Use measurements in everyday situations (e.g., determine the geography of the school building).

2.5.3.A. Compare quantities and magnitudes of numbers.

2.4.5.B. Use models, number facts, properties and relationships to check and verify predictions and explain reasoning.

2.4.5.C.

Draw inductive and deductive conclusions within mathematical contexts.

2.4.5.D. Distinguish between relevant and irrelevant information in a mathematical problem.

2.4.5.E. Interpret statements made with precise language of logic (e.g., "all", "or", "every", "none", "some", "or", "many").

2.4.8.A. Make conjectures based on logical reasoning and test conjectures by using counter-examples.

2.4.8.B. Combine numeric relationships to arrive at a conclusion.

2.4.8.C. Use if...then statements to construct simple, valid arguments.

2.4.8.D. Construct, use and explain algorithmic procedures for computing and estimating with whole numbers, fractions, decimals and integers.

2.4.8.E. Distinguish between inductive and deductive reasoning.

2.4.11.A. Use direct proofs, indirect proofs or proof by contradiction to validate conjectures.

2.4.11.B. Construct valid arguments from stated facts.

2.4.11.C. Determine the validity of an argument.

2.4.8.D. Use truth tables to reveal the logic of mathematical statements.

2.5.3.A. Use appropriate problem-solving strategies (e.g., guess and check, working backwards).

2.5.3.B. Determine when sufficient information is present to solve a problem and explain how to solve a problem.

2.5.3.C. Select and use an appropriate method, materials and strategy to solve problems, including mental mathematics, paper and pencil and concrete objects.

2.5.5.A. Develop a plan to analyze a problem, identify the information needed to solve the problem, carry out the plan, check whether an answer makes sense and explain how the problem was solved.

2.5.5.B. Use appropriate mathematical terms, vocabulary, language symbols and graphs to explain clearly and logically solutions to problems.

2.5.5.C. Show ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams and models.

2.5.5.D. Connect, extend and generalize problem solutions to other concepts, problems and circumstances in mathematics.

2.5.5.E. Select, use and justify the methods, materials and strategies used to solve problems.

2.5.8.A. Invent, select, use and justify the appropriate methods, materials and strategies to solve problems.

2.5.8.B. Verify and interpret results using precise mathematical language, notation and representations, including numerical tables and equations, simple algebraic equations and formulas, charts, graphs and diagrams.

2.5.8.C. Justify strategies and defend approaches used and conclusions reached.

2.5.11.A. Select and use appropriate mathematical concepts and techniques from different areas of mathematics and apply them to solving non-routine and multi-step problems.

2.5.11.B. Use symbols, mathematical terminology, standard notation, mathematical rules, graphing and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas and results.

2.5.11.C. Present mathematical procedures and results clearly, systematically, succinctly and correctly.

2.5.11.D. Conclude a solution process with a summary of results and evaluate the degree to which the results obtained represent an acceptable response to the initial problem and why the reasoning is valid.

2.6.3.A. Gather, organize and display data using pictures, tallies, charts, bar graphs and pictographs.

2.6.3.B. Formulate and answer questions based on data shown on graphs.

2.6.3.C. Predict the likely number of times a condition will occur based on analyzed data.

2.6.3.D. Form and justify an opinion on whether a given statement is reasonable based on a comparison to data.

2.6.5.A. Organize and display data using pictures, tallies, tables, charts, bar graphs and circle graphs.

2.6.5.B. Describe data sets using mean, median, mode and range.

2.6.5.C. Sort data using Venn diagrams.

2.6.5.D. Predict the likely number of times a condition will occur based on analyzed data.

2.6.5.E. Construct and defend simple conclusions based on data.

2.6.8.A. Compare and contrast different plots of data using values of mean, median, mode, quartiles and range.

2.6.8.B. Explain effects of sampling procedures and missing or incorrect information on reliability.

2.6.8.C. Fit a line to the scatter plot of two quantities and describe any correlation of the variables.

2.6.8.D. Design and carry out a random sampling procedure.

2.6.8.E. Analyze and display data in stem-and-leaf and box-and-whisker plots.

2.6.8.F. Use scientific and graphing calculators and computer spreadsheets to organize and analyze data.

2.6.8.G. Determine the validity of the sampling method described in studies published in local or national newspapers.

2.6.11.A. Design and conduct an experiment using random sampling. Describe the data as an example of a distribution using statistical measures of center and spread. Organize and represent the results with graphs. (Use standard deviation, variance and t-tests.)

2.6.11.B. Use appropriate technology to organize and analyze data taken from the local community.

2.6.11.C. Determine the regression equation of best fit (e.g., linear, quadratic, exponential).

2.6.11.D. Make predictions using interpolation, extrapolation, regression and estimation using technology to verify them.

2.6.11.E. Determine the validity of the sampling method described in a given study.

2.6.11.F. Determine the degree of dependence of two quantities specified by a two-way table.

2.6.11.G. Describe questions of experimental design, control groups, treatment groups, cluster sampling and reliability.

2.6.11.H. Use sampling techniques to draw inferences about large populations.

2.7.3.A. Predict and measure the likelihood of events and recognize that the results of an experiment may not match predicted outcomes.

2.7.3.B. Design a fair and an unfair spinner.

2.7.3.D. Analyze data using the concepts of largest, smallest, most often, least often and middle.

2.7.5.A. Perform simulations with concrete devices (e.g., dice, spinner) to predict the chance of an event occurring.

2.7.5.B. Determine the fairness of the design of a spinner.

2.7.5.C. Express probabilities as fractions and decimals.

2.7.5.D. Compare predictions based on theoretical probability and experimental results.

2.7.5.E. Calculate the probability of a simple event.

2.7.5.F. Determine patterns generated as a result of an experiment.

2.7.5.G. Determine the probability of an event involving "and", "or" or "not".

2.7.5.H. Predict and determine why some outcomes are certain, more likely, less likely, equally likely or impossible

2.7.5.I. Find all possible combinations and arrangements involving a limited number of variables.

2.7.5.J. Develop a tree diagram and list the elements.

2.7.8.A. Determine the number of combinations and permutations for an event.

2.7.8.B. Present the results of an experiment using visual representations (e.g., tables, charts, graphs).

2.7.8.C. Analyze predictions (e.g., election polls).

2.7.8.D. Compare and contrast results from observations and mathematical models.

2.7.8.E. Make valid inferences, predictions and arguments based on probability.

2.7.11.A. Compare odds and probability.

2.7.11.B. Apply probability and statistics to perform an experiment involving a sample and generalize its results to the entire population.

2.7.11.C. Draw and justify a conclusion regarding the validity of a probability or statistical argument.

2.7.11.D. Use experimental and theoretical probability distributions to make judgments about the likelihood of various outcomes in uncertain situations.

2.7.11.E. Solve problems involving independent, simple and compound events.

2.8.3.A. Recognize, describe, extend, create and replicate a variety of patterns including attribute, activity, number and geometric patterns.

2.8.3.B. Use concrete objects and trial and error to solve number sentences and check if solutions are sensible and accurate.

2.8.3.C. Substitute a missing addend in a number sentence.

2.8.3.D. Create a story to match a given combination of symbols and numbers.

2.8.3.E. Use concrete objects and symbols to model the concepts of variables, expressions, equations and inequalities.

2.8.3.F. Explain the meaning of solutions and symbols.

2.8.3.G. Use a table or a chart to display information.

2.8.3.H. Describe and interpret the data shown in tables and charts.

2.8.3.I. Demonstrate simple function rules.

2.8.3.J. Analyze simple functions and relationships and locate points on a simple grid.

2.8.5.A. Recognize, reproduce, extend, create and describe patterns, sequences and relationships verbally, numerically, symbolically and graphically, using a variety of materials.

2.8.5.B. Connect patterns to geometric relations and basic number skills.

2.8.5.C. Form rules based on patterns (e.g., an equation that relates pairs in a sequence).

2.8.5.D. Use concrete objects and combinations of symbols and numbers to create expressions that model mathematical situations.

2.8.5.E. Explain the use of combinations of symbols and numbers in expressions, equations and inequalities.

2.8.5.F. Describe a realistic situation using information given in equations, inequalities, tables or graphs.

2.8.5.G. Select and use appropriate strategies, including concrete materials, to solve number sentences and explain the method of solution.

2.8.5.H. Locate and identify points on a coordinate system.

2.8.8.A. Apply simple algebraic patterns to basic number theory and to spatial relations

2.8.8.B. Discover, describe and generalize patterns, including linear, exponential and simple quadratic relationships.

2.8.8.C. Create and interpret expressions, equations or inequalities that model problem situations.

2.8.8.D. Use concrete objects to model algebraic concepts.

2.8.8.E. Select and use a strategy to solve an equation or inequality, explain the solution and check the solution for accuracy.

2.8.8.F. Solve and graph equations and inequalities using scientific and graphing calculators and computer spreadsheets.

2.8.8.H. Graph a linear function from a rule or table.

2.8.8.I. Generate a table or graph from a function and use graphing calculators and computer spreadsheets to graph and analyze functions.

2.8.11.A. Analyze a given set of data for the existence of a pattern and represent the pattern algebraically and graphically.

2.8.11.B. Give examples of patterns that occur in data from other disciplines.

2.8.11.C. Use patterns, sequences and series to solve routine and non-routine problems.

2.8.11.D. Formulate expressions, equations, inequalities, systems of equations, systems of inequalities and matrices to model routine and non-routine problem situations.

2.8.11.E. Use equations to represent curves (e.g., lines, circles, ellipses, parabolas, hyperbolas).

2.8.11.F. Identify whether systems of equations and inequalities are consistent or inconsistent.

2.8.11.G. Analyze and explain systems of equations, systems of inequalities and matrices.

2.8.11.H. Select and use an appropriate strategy to solve systems of equations and inequalities using graphing calculators, symbol manipulators, spreadsheets and other software.

2.8.11.I. Use matrices to organize and manipulate data, including matrix addition, subtraction, multiplication and scalar multiplication.

2.8.11.J. Demonstrate the connection between algebraic equations and inequalities and the geometry of relations in the coordinate plane.

2.8.11.K. Select, justify and apply an appropriate technique to graph a linear function in two variables, including slope-intercept, x- and y-intercepts, graphing by transformations and the use of a graphing calculator.

2.8.11.L. Write the equation of a line when given the graph of the line, two points on the line, or the slope of the line and a point on the line.

2.8.11.M. Given a set of data points, write an equation for a line of best fit.

2.8.11.N. Solve linear, quadratic and exponential equations both symbolically and graphically.

2.8.11.O. Determine the domain and range of a relation, given a graph or set of ordered pairs.

2.8.11.P. Analyze a relation to determine whether a direct or inverse variation exists and represent it algebraically and graphically.

2.8.11.Q. Represent functional relationships in tables, charts and graphs.

2.8.11.R. Create and interpret functional models.

2.8.11.S. Analyze properties and relationships of functions (e.g., linear, polynomial, rational, trigonometric, exponential, logarithmic).

2.8.11.T. Analyze and categorize functions by their characteristics.

2.9.3.A Name and label geometric shapes in two and three dimensions (e.g., circle/sphere, square/cube, triangle/pyramid, rectangle/prism).

2.9.3.B. Build geometric shapes using concrete objects (e.g., manipulatives).

2.9.3.C. Draw two- and three-dimensional geometric shapes and construct rectangles, squares and triangles on the geoboard and on graph paper satisfying specific criteria.

2.9.3.D. Find and describe geometric figures in real life.

2.9.3.E. Identify and draw lines of symmetry in geometric figures.

2.9.3.F. Identify symmetry in nature.

2.9.3.G. Fold paper to demonstrate the reflections about a line.

2.9.3.H.Show relationships between and among figures using reflections.

2.9.3.I. Predict how shapes can be changed by combining or dividing them.

2.9.5.A. Give formal definitions of geometric figures.

2.9.5.B.Classify and compare triangles and quadrilaterals according to sides or angles.

2.9.5.C.Identify and measure circles, their diameters and their radii.

2.9.5.D. Describe in words how geometric shapes are constructed.

2.9.5.E. Construct two- and three-dimensional shapes and figures using manipulatives, geoboards and computer software.

2.9.5.F. Find familiar solids in the environment and describe them.

2.9.5.G. Create an original tessellation.

2.9.5.H. Describe the relationship between the perimeter and area of triangles, quadrilaterals and circles.

2.9.5.I. Represent and use the concepts of line, point and plane.

2.9.5.K. Analyze simple transformations of geometric figures and rotations of line segments.

2.9.8.A. Construct figures incorporating perpendicular and parallel lines, the perpendicular bisector of a line segment and an angle bisector using computer software.

2.9.8.B. Draw, label, measure and list the properties of complementary, supplementary and vertical angles.

2.9.8.C. Classify familiar polygons as regular or irregular up to a decagon.

2.9.8.D. Identify, name, draw and list all properties of squares, cubes, pyramids, parallelograms, quadrilaterals, trapezoids, polygons, rectangles, rhombi, circles, spheres, triangles, prisms and cylinders.

2.9.8.F. Distinguish between similar and congruent polygons.

2.9.8.H. Use simple geometric figures (e.g., triangles, squares) to create, through rotation, transformational figures in three dimensions.

2.9.8.I. Generate transformations using computer software.

2.9.8.J. Analyze geometric patterns (e.g., tessellations, sequences of shapes) and develop descriptions of the patterns.

2.9.11.A. Construct geometric figures using dynamic geometry tools (e.g., Geometer's Sketchpad, Cabri Geometre).

2.9.11.B. Prove that two triangles or two polygons are congruent or similar using algebraic, coordinate and deductive proofs.

2.9.11.C. Identify and prove the properties of quadrilaterals involving opposite sides and angles, consecutive sides and angles and diagonals using deductive proofs.

2.9.11.D. Identify corresponding parts in congruent triangles to solve problems.

2.9.11.E. Solve problems involving inscribed and circumscribed polygons.

2.9.11.F. Use the properties of angles, arcs, chords, tangents and secants to solve problems involving circles.

2.9.11.G. Solve problems using analytic geometry.

2.9.11.H. Construct a geometric figure and its image using various transformations.

2.10.3.A. Identify right angles in the environment.

2.10.3.B. Model right angles and right triangles using concrete objects.

2.10.5.A. Identify and compare parts of right triangles, including right angles, acute angles, hypotenuses and legs.

2.10.8.A.Compute measures of sides and angles using proportions, the Pythagorean Theorem and right triangle relationships.

2.10.11.A. Use graphing calculators to display periodic and circular functions; describe properties of the graphs.

2.11.3.A. Identify whole number quantities and measurements from least to most and greatest value.

2.11.3.B. Identify least and greatest values represented in bar graphs and pictographs.

2.11.3.C. Categorize rates of change as faster and slower.

2.11.5.A. Make comparisons of numbers (e.g., more, less, same, least, most, greater than, less than).

2.11.5.B. Identify least and greatest values represented in bar and circle graphs.

2.11.5.C. Identify maximum and minimum.

2.11.5.D. Describe the relationship between rates of change and time.

2.11.5.E. Estimate areas and volumes as the sums of areas of tiles and volumes of cubes.

2.11.8.A. Analyze graphs of related quantities for minimum and maximum values and justify the findings.

2.11.8.B. Describe the concept of unit rate, ratio and slope in the context of rate of change.

2.11.11.A. Determine maximum and minimum values of a function over a specified interval.

2.11.11.B. Interpret maximum and minimum values in problem situations.

2.11.11.C. Graph and interpret rates of growth/decay.

2.11.11.D. Determine sums of finite sequences of numbers and infinite geometric series.