By the end of grade 3:
a. Formulate a problem, determine information required to solve the problem, choose methods for obtaining this information, and set limits for acceptable solutions.
b. Demonstrate that there may be multiple ways to solve a problem and explain why this is so.
c. Understand that there is no one right way to solve mathematical problems but that different methods (e.g., working backward from a solution, using a similar problem type, identifying a pattern) have different advantages and disadvantages.
d. Transfer strategies from a prior problem to a new situation.
e. Use trial and error and the process of elimination to solve problems.
f. Verify the correctness and reasonableness of simple mathematical results.

By the end of grade 5:
a. Know how to select and use mathematical tools and methods (such as manipulatives, mental math, calculator, computer, and paper-and-pencil techniques) as a part of the problem-solving process.
b. Develop and apply a variety of problem-solving strategies (for example, make an organized list, guess-and-check) and justify choice of strategies.
c. Interpret results in the context of the problem being solved (for example, when determining the number of buses necessary to transport students, the remainder must be rounded up).
d. Differentiate between relevant and irrelevant information
e. Understand how to break a complex problem into simpler parts.

By the end of grade 8:
a. Pose, explore, and solve a variety of problems, including those that are non-routine or have a variety of possible strategies or solutions or both, in order to build new mathematical knowledge.
b. Develop, modify, and apply an increasing variety of problem-solving techniques to solve problems (for example, working backward, information organizers, or solving a similar but simpler problem).
c. Reflect on different strategies used when a task is complete.

By the end of grade 12:
a. Explore the validity and efficiency of various problem-posing and problem-solving strategies; develop alternative strategies and generalizations as needed.
b. Monitor progress toward solutions.
c. Generalize strategies and reflect on their proficiency and merit.

By the end of grade 3:
a. Make and describe connections linking conceptual and procedural knowledge using a variety of strategies (manipulative, pictorial, symbolic).

By the end of grade 5:
a. Describe the connections and translate between various representations of equivalent numbers (such as 3/3 = 1, 10% of a dollar = 1 dime).
b. Given a rule or generalization, determine whether the example fits
c. Draw logical conclusions about mathematical situations using informal inductive and deductive reasoning (e.g., observing that the angles of several triangles add up to 180 degrees and concluding that the angles of all triangles add up to 180 degrees; concluding that since all rectangles have 4 90-degree corners, a square must be a rectangle).
d. Interpret statements made with the precise language of logic (such as all, every, none, some).
e. Independently apply mathematical concepts to other content areas such as science, geography, and music.

By the end of grade 8:
a. Make and investigate mathematical conjectures.
b. Use "ifthen" statements to construct simple valid arguments.
c. Use inductive and deductive reasoning appropriately.
d. Apply proportional and spatial thinking.
e. Develop and evaluate mathematical arguments and informal proofs.
f. Use mathematical ideas from one area of mathematics (e.g., an equation or formula from algebra) to explain an idea from another area of mathematics (e.g., the area of a triangle) and to demonstrate how mathematical ideas are a coherent whole.
g. Apply mathematical skills and processes to other disciplines and to everyday situations.

By the end of grade 12:
a. Construct, follow and evaluate arguments, judging their validity using reasoning and logic.
b. Use a variety of methods of proofs (for example, direct, indirect, informal) to validate conjectures.
c. Relate procedures in one representation of a problem to procedures in an equivalent representation.
d. Use the connections among mathematical topics to develop multiple approaches to problems in other subject areas.
e. Demonstrate how graphs can be used to model real-world situations and to determine solutions to numerous problems involving algebraic functions.

By the end of grade 3:
a. Listen to and read about mathematical strategies and solutions.
b. Recognize that certain words give clues to specific operations (e.g., sum means addition, difference means subtraction, of means multiplication and per means division).

By the end of grade 5:
a. Demonstrate an ability to understand others’ strategies or explanations.

By the end of grade 8:
a. Analyze, evaluate, and explain mathematical arguments and conclusions presented by others.

By the end of grade 12:
a. Set-up and sketch problems from a given set of information.

By the end of grade 3:
a. Communicate strategies and solutions using everyday language and correct mathematical terms (e.g., sum, product) and symbols (e.g., +, =, >).
b. Communicate mathematical ideas using concrete, pictorial and symbolic representations.
c. Demonstrate that some ways of representing a problem are more helpful than others.

By the end of grade 5:
a. Identify, communicate, and model key mathematical concepts and situations using oral, written, concrete, pictorial, and graphic methods, making certain that the situation is represented clearly and accurately.
b. Explain and justify mathematical ideas, strategies, and solutions to others, using the correct mathematical vocabulary.

By the end of grade 8:
a. Demonstrate a repertoire of mathematical representations and use them purposefully, flexibly, and appropriately.
b . Use mathematical language, notation, and symbols as a precise means of expressing problem situations and mathematical ideas.

By the end of grade 12:
a. Formulate questions, conjectures, and generalizations about data, information, and problem situations.
b. Present complete and convincing arguments and justifications in various formats.
c. Use technology (such as graphics calculators, spreadsheets, graphing programs) to present information and ideas.
d. Use properties, models, known facts, and relationships to explain and defend thinking.

By the end of grade 3:
a. Represent and examine mathematical situations using concrete materials.
b. Use a variety of standard tools (e.g., rulers, clocks, measuring tapes, thermometers) and non-standard objects (e.g., counters, sticks, bolts), to measure mathematical and physical objects in the environment.

By the end of grade 5:
a. Use calculators or software to verify estimations and in problem-solving situations.
b. Use technology such as spreadsheets, science probe, or calculators to gather, analyze, and display mathematical data and information.

By the end of grade 8:
a. Use a variety of tools and technologies, including computers, scientific calculators, angle rulers, protractors, compass, graphing calculators, science probes, and digital cameras to evaluate and validate problem solutions.
b. Recognize situations when calculator use is not appropriate (for example, when solving a simple quadratic equation which could be factored) or when it yields misleading results (for example, when a non-linear curve appears linear).

By the end of grade 12:
a. Use graphing calculators or computer software effectively and efficiently to define and solve various types of problems.

By the end of grade 3:
a. Connect physical, verbal, and symbolic representations of whole numbers, fractions and mixed numbers.
b. Use drawings, diagrams, and models to show the concept of fractions as part of a whole and part of a set.
c. Explain how numbers are used in various ways, including counting, ordering, representing quantities, measuring, labeling, and indicating location.
d. Apply place-value concepts and numeration to describe, compare, count, order, and group numbers.
e. Explain the connections between operations.
f. Use concrete objects to count, order, group, and demonstrate one-to-one correspondence with whole numbers beyond 100.
g. Identify patterns in number sequences (identify even and odd numbers, count by 2s, 3s, 5s, 10s, and 25s).
h. Read, write, and order numbers to 10,000.

By the end of grade 5:
a. Model and connect physical, verbal, and symbolic representations of fractions, decimals, percentages, whole numbers, and mixed numbers.
b. Order fractions, decimals, and whole numbers using physical, verbal, and symbolic representations.
c. Recognize the relationship among fractions, decimals, and percentages.
d. Use concepts of negative numbers in concrete situations (such as on a number line, with temperature).
e. Identify and describe different uses for the same numerical representation (for example, 1/4 can represent a fraction, a division problem, or a ratio) and different representations for the same number (for example, 2,343 is the same as 2,000 + 300 + 40 + 3; and 1 equals 16/16).
f. Use, model, and identify place value and describe its relationship to magnitude.
g. Demonstrate that mathematical operations can represent a variety of problem situations (for example, multiplication can represent repeated addition and a model for finding area).
h. Explain the relative effect of operations with fractions and decimals (for example, what happens to 10 when you divide by _ or multiply by .75).
i. Explain, derive, compare, and use properties of operations and relationships among operations.
j. Explain and apply number theory concepts (such as primes, multiples, and composites).
k. Read, write, and order numbers to a million.

By the end of grade 8:
a. Describe how percent, ratio, and proportion apply to mathematical situations (such as rate, similar triangles).
b. Recognize and apply multiple representations of rational and irrational numbers, exponents, absolute values, and scientific notation; compare these numbers accurately, find their approximate locations on a number line, and choose appropriate forms of these numbers.
c. Demonstrate an understanding of numbers that represent large and small values, including the use of benchmarks to comprehend their magnitude; and recognize, understand, and appropriately use various representations for large numbers.
d. Represent and explain the effect of operations on positive and negative numbers.
e. Add, subtract, multiply, and divide fractions, decimals, percents, integers, and nonnegative whole number exponents.
f. Understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.
g. Use factors, multiples, and prime factorization to solve problems.
h. Recognize and use the associative and commutative properties of addition and multiplication, and the distributive property of multiplication over addition to simplify computations with rational numbers.

By the end of grade 12:
a. Understands the concept of infinity.
b. Demonstrate a conceptual understanding of irrational numbers and be able to solve problems involving rational numbers (e.g., area of circles, working with radicals).
c. Understands the concept of complex numbers.

By the end of grade 3:
a. Demonstrate proficiency with and memorize addition and subtraction facts through 20 and multiplication facts through 10.
b. Add and subtract single- and multi-digit whole numbers with regrouping.
c. Apply addition and subtraction in a variety of situations (such as computing perimeter, extending functions).
d. Demonstrate the concept of multiplication as repeated addition and arrays; demonstrate the concept of division as repeated subtraction and as sharing.
e. Understand and appropriately use the vocabulary of estimation (such as about, near, between).
f. Use a variety of mental computational methods, strategies, and estimation skills to find solutions and to determine the reasonableness of calculated answers, including those involving concrete and abstract items and situations, such as time and money.
g. Determine the value of a set of host country currency and U.S. currency.
h. Read, write, add, and subtract with decimal notation in situations involving money.

By the end of grade 5:
a. Demonstrate proficiency with and memorize multiplication and division facts through 12.
b. Select and use the most efficient computational methods, choosing among concrete materials, paper and pencil, estimation, mental computation, and calculators.
c. Multiply multi-digit whole numbers by single-digit numbers.
d. Divide two-digit whole numbers by single-digit numbers.
e. Create and solve practical problems involving addition, subtraction,
multiplication, and division of whole numbers, fractions, and mixed numbers.
f. Develop, analyze, and compare algorithms for computing with fractions, decimals, percents, and integers and compute with them efficiently and accurately, including in multi-step problems that require application of order of operations.
g. Know and convert among fractions, decimals, and percents for 1/10, 1/5, 1/2, 1/4, and 1/100.
h. Apply beginning number theory including identifying and using multiples, factors, divisibility, properties of identity (zero and one), and prime and composite numbers.
i. Apply, explain, and assess the appropriateness of a variety of estimation strategies (such as rounding to compatible numbers).
j. Use various forms of estimation, including rounding, to determine the reasonableness of calculated answers; determine if an estimate is too high or too low.
k. Use a variety of strategies to make change and solve problems using U.S. and host country's currency, and to convert between host country and U.S. currency.

By the end of grade 8:
a. Solve problems using rates and understand rate as a unit of measure.
b. Use algorithms for computing with fractions, decimals, percents, and integers and compute them efficiently and accurately with and without a calculator.
c. Use multi-step computational procedures with rational numbers.
d. Develop, analyze, and explain methods to solve problems involving proportions and percents (such as scaling, finding equivalent ratios).
e. Compute circumference, area, surface area, and volume of geometric figures; find missing dimensions of right triangles using the Pythagorean theorem.
f. Estimate the value of tips, discounts, and taxes using host country and U.S. currency.
g. Explain and apply the rules of divisibility, square numbers, prime factorization, and the properties of zero with the order of operations.
h. Determine what a reasonable degree of accuracy would be in particular situations (e.g., great precision is required in scientific experiments, but much less is required in estimating a grocery bill).

By the end of grade 12:
a. Manipulate algebraic procedures with real and complex numbers.
b. Apply exponents to solve practical problems.
c. Estimate the value of irrational numbers.
d. Compute permutations and combinations.
e. Assess the error resulting from estimation and rounding, using both customary and metric units.
f. Estimate algebraic solutions.
g. Determine when to use exact value solutions and distinguish between exact value and approximate values.

By the end of grade 3:
a. Estimate before measuring to determine the reasonableness of a solution.
b. Estimate and measure length, time, temperature, and weight to the nearest unit using customary, metric, and nonstandard measurement.
c. Compare and order measurable characteristics (for example, time, temperature, length, weight, capacity, area, perimeter) of different objects on the same dimensions.
d. Tell time to the minute with both analog and digital clocks.
e. Determine elapsed time to the hour using AM and PM.
f. Distinguish between perimeter and area of rectangles.
g. Recognize the need for a uniform unit of measure.

By the end of grade 5:
a. Select and use appropriate instruments and customary and metric units for measuring quantities, including perimeter, volume, area, weight, time, and temperature, with specified accuracy.
b. Match tools with the attribute they measure (for example, rulers measure length, thermometers measure temperature).
c. Understand and apply formulas for finding perimeter, volume of simple solids (excluding cylinders), and area.
d. Add and subtract measurements (e.g., 12 m. – 6.2 m.).
e. Identify and use equivalent measurements as required by the situation (for example, 60 minutes = 1 hour, 7 days = 1 week).
f. Identify the approximate size of basic standard units of measurement and the relationship between them (for example, there are 100 centimeters in a meter).
g. Solve calendar problems involving days, weeks, months, and years.
h. Determine and compare elapsed time using AM and PM and a 24-hour clock.

By the end of grade 8:
a. Estimate and measure angles and use formulas to find perimeter, area, and circumference of plane figures and the volume and surface area of prisms, pyramids, and cylinders to a specified degree of accuracy.
b. Select and use appropriate units and tools to measure length, area, volume, angle, and weight to appropriate levels of precision.
c. Convert measurements within and between monetary systems and within and between metric and customary systems and demonstrate an understanding of the relationship between units in metric and customary systems.
d. Use measurements, in conjunction with formulae, geometric ideas and relationships (e.g., congruence, simultaneity, Pythagorean theorem) and mathematical relationships (scale, etc.) to solve problems.
e. Apply information about time zones and elapsed time to solve problems.

By the end of grade 12:
a. Select and use a variety of methods and tools to construct and compare plane figures of given measures.
b. Apply indirect methods, such as ratios and trigonometry, to find missing dimensions.

By the end of grade 3:
a. Recognize, reproduce, extend, create, and describe repeating and increasing patterns and sequences using a variety of materials.
b. Use tables, graphic organizers, verbal rules, and open sentences to describe patterns and other relationships.
c. Generate and solve simple functions by identifying and applying addition and subtraction patterns.
d. Generate, write, and solve open sentences using informal methods (such as using manipulatives, drawing, or acting out the solution).
e. Use concrete objects and symbols to model the concepts of variables, expressions, equations, and inequalities (for example, find the missing number, symbol, or operation sign).
f. Identify and describe numeric patterns and make predictions based on them (e.g., 1 bicycle = 2 wheels, so 6 bicycles = how many wheels).

By the end of grade 5:
a. Use patterns and their extensions to make predictions and solve problems.
b. Use rules and variables to describe patterns, functions, and other relationships and to solve equations.
c. Generate and solve simple functions by identifying and applying multiplication and division patterns.
d. Find solutions to inequalities from a given set of possibilities.
e. Solve simple equations using methods such as inverse operations, mental math, and guess-and-check.
f. Use concrete objects and combinations of symbols and numbers to create expressions that model mathematical situations.
g. Understand the basic characteristics of a 2-dimensional coordinate system.

By the end of grade 8:
a. Analyze, create, and generalize numeric and visual patterns.
b. Describe relationships between symbolic expressions and graphs on the coordinate plane, with particular attention to the horizontal and vertical intercepts, points of intersection, and slope of linear relations.
c. Analyze functional relationships to explain how a change in one quantity results in a change in another (for example, the relationship among length, area, and volume).
d. Find the value of a variable by evaluating formulas and algebraic expressions for given values (for example, if an object has a length of 4 and an area of 28, what is the object's width?).
e. Rewrite formulas in terms of the missing variable (for example, "if an object has a length of 4 and an area of 28, what is the object's width?" can be expressed as 4W=28 or as W=28/4).
f. Create expressions, equations, and inequalities to represent problem situations and to solve problems involving linear relationships.
g. Solve linear equations and inequalities.
h. Demonstrate a basic understanding of rate of change, including connections between slope of a line and constant rate of change and their meaning in context.

By the end of grade 12:
a. Define functions and their properties and find the inverse of a function; understand the relationship between a function and its inverse.
b. Create and solve linear and quadratic equations and inequalities.
c. Add, subtract, multiply, divide, and simplify rational and irrational expressions; add, subtract, multiply and divide polynomials.
d. Identify, graph, and describe the graphs of basic families of functions including linear, absolute value, quadratic and exponential, and explain why a variety of phenomena can be modeled by the same type of function.
e. Solve systems of equations and inequalities.
f. Formulate expressions, equations, inequalities, systems of equations, systems of inequalities, and matrices to model problem situations.
g. Use matrices to organize and manipulate data, including matrix addition, subtraction, multiplication, and scalar multiplication.

By the end of grade 3:
a. Use comparative directional and positional words (such as above, inside, left, horizontal, middle).
b. Describe, name, and label related geometric two- and three-dimensional shapes (such as circle and sphere, square and cube, triangle and pyramid, rectangle and prism).
c. Draw two-dimensional geometric shapes and construct rectangles, squares, and triangles using tools (such as geoboards, grid paper, ruler, compass), including representation of side, top, and bottom views of the object.
d. Construct three-dimensional geometric shapes, including boxes and triangular prisms.
e. Identify and describe geometric figures in the environment.
f. Identify and create examples of line symmetry.
g. Order simple geometric figures by size.
h. Estimate and determine the perimeter and area of geometric figures using manipulatives; demonstrate conservation of area.

By the end of grade 5:
a. Locate and describe objects in terms of their position with and without compass directions; identify coordinates for a given point or locate points of given coordinates on a single quadrant grid.
b. Compare, contrast, and describe plane and solid figures and shapes using their attributes (such as number of sides, parallel or perpendicular sides, number of vertices, classification of angles).
c. Sketch and identify line segments, midpoint, intersections, and parallel and perpendicular lines.
d. Identify, draw, and measure, using a protractor, right, obtuse, and acute angles and their parts, including rays, points, and vertices.
e. Identify and model geometric figures that are congruent, similar, or symmetrical or some combination of these properties.
f. Identify the diameter, radius, chord, and circumference of a circle.
g. Determine area and perimeter, finding both using a variety of methods.
h. Analyze and model transformations of geometric figures and rotations of line segments, describing the motions as slides, flips, or rotations.

By the end of grade 8:
a. Describe the relationship between an equation and its graph.
b. Use coordinate geometry to represent and interpret relationships defined by equations and formulas (for example, distance, mid-point), translating among ordered pairs, graphs, and equations.
b. Model, classify, compare, and sketch a variety of two- and three-dimensional regular and irregular figures.
c. Apply properties of equality and proportionality to solve problems involving congruent or similar shapes.
d. Describe and apply geometric properties and relationships (such as congruence, perpendicularity).
e. Describe and determine circumference, perimeter, area, surface area, angle measure, and volume.
f. Explain and apply the Pythagorean theorem.
g. Draw and describe the results of transformations, including translations, rotations, reflections, and dilations (shrinking or enlarging), using proper notation.

By the end of grade 12:
a.Use the y = sinx, y = cosx and y = tanx.equations to solve problems involving right triangles.
b. Use coordinate geometry to graph linear and quadratic equations, determine slopes of lines, identify parallel and perpendicular lines, and find possible solutions to sets of equations.
c. Create two-dimensional representations of three-dimensional objects (e.g., draw a basic cube).
d. Construct geometric models, transformations, and scale drawings using a variety of methods and tools (such as paper folding or protractor).
e. Identify congruent and similar figures; apply this information to solve problems.

By the end of grade 3:
a. Pose a question and collect data by observing, measuring, surveying, and counting.
b. Construct, read, interpret, and label graphs, including pictographs, simple bar and line graphs, and pie charts.
c. Interpret data by looking for patterns and relationships, determining range, considering cause and effect, then drawing conclusions and answering related questions.

By the end of grade 5:
a. Solve problems that involve systematically collecting, organizing, and analyzing data.
b. Discuss the appropriateness of different types of data displays, and use a variety of displays (such as tables, histograms, graphs).
c. Interpret data, using the arithmetic mean, median, mode, range, and make convincing arguments based on data analysis and previous experiences.

By the end of grade 8:
a. Formulate and solve problems by collecting, organizing, analyzing (including comparing and contrasting), and displaying data in a variety of ways (including stem and leaf plots, histograms, whisker plots, surveys, circle graphs) by hand and with technology.
b. Find, describe, and interpret the arithmetic mean, median, and mode as measures of the center of a data set; select the appropriate measure in particular situations to most accurately and fairly represent the data.
c. Describe ways to define a sample group; analyze a sample to make inferences about a population.
d. Design and conduct a simulation to study a problem and communicate the results.

By the end of grade 12:
a. Use regression functions to match curves to data.
b. Use regression curves to interpret and analyze data.

By the end of grade 3:
a. Predict and measure the outcome of events, and explain why the results of an experiment may not match predicted outcomes.
b. Use concepts of certainty, fairness, and chance to discuss the probability of actual events.

By the end of grade 5:
a. Make predictions based on intuitive, experimental, and theoretical probabilities.
b. Conduct simple probability experiments using concrete materials (e.g. tossing one or more coins, spinning a spinner of even or uneven divisions, drawing objects from a container with and without replacement) and represent the results using fractions and probability.

By the end of grade 8:
a. Determine theoretical probability using a variety of methods, including creating a sample space; compare theoretical expectations to experimental results.
b. Design, conduct, and analyze the results of probability experiments.

By the end of grade 12:
a. Describe the normal curve in general terms and use its properties to answer questions about sets of data.
b. Find the probability of simple events, compound events, and independent events using a variety of methods including the fundamental counting principle.