Problem Solving
All students should be able to:
  • Build new mathematical knowledge through problem solving
  • Solve problems that arise in mathematics and in other contexts
  • Apply and adapt a variety of appropriate strategies to solve problems
  • Monitor and reflect on the process of mathematical problem solving
Problem solving is not just a skill that all students must develop, it is also the means for effectively teaching and learning mathematics. Problem-based instructional tasks should be used in the classroom to teach important mathematics. These tasks should be chosen carefully, addressing real-world problems that allow students to have multiple ways to solve the problems, centered on an important mathematical idea, concept, or skill that is part of a course of study. These tasks should encourage the connection across curricular strands of mathematics. Teachers should choose tasks that require a high level of cognitive demand to promote the development of a deep knowledge of mathematics. Assessments designed to check for understanding should allow for problem solving to be demonstrated. Assessments should focus on the process of solving the problems as well as on correct solutions. (Adapted from Teaching Mathematics through Problem Solving, Schoen, NCTM, 2003) "Problem solving is an integral part of all mathematics learning, and so it should not be an isolated part of the mathematics program. Problem solving in mathematics should involve all [mathematical strands]. The contexts of the problems can vary from familiar experiences involving students' lives or the school day to applications involving the sciences or the world of work. Good problems will integrate multiple topics and will involve significant mathematics." (NCTM, 2000, p. 52)
  Communication(Reading, Writing, Speaking, Listening, Viewing)
All students should be able to:
  • Organize and consolidate their mathematical thinking through communication
  • Communicate their mathematical thinking coherently and clearly to peers, teachers, and others
  • Analyze and evaluate the mathematical thinking and strategies of others
  • Use the language of mathematics to express mathematical ideas precisely
Communication should be addressed throughout curriculum, instruction and assessment. The curriculum materials used in a classroom should reflect this emphasis on communication by providing lessons that promote student-to-student, student-to-teacher, and teacher-to-student communication. Instructional practices should provide opportunities for students to communicate with each other as they study mathematics in the classroom. Teachers should act as facilitators for learning, encouraging student discourse. In doing this, students should be encouraged to explain their thinking and listen to each other as they solve problems. "Students who have opportunities, encouragement, and support for speaking, writing, reading, and listening in mathematics classes reap dual benefits: they communicate to learn mathematics, and they learn to communicate mathematically. à Students need to work with mathematical tasks that are worthwhile topics of discussion. Procedural tasks for which students are expected to have well-developed algorithmic approaches are usually not good candidates for such discourse. Interesting problems that 'go somewhere' mathematically can often be catalysts for rich conversations." (NCTM, 2000, p. 60) The students' ability to communicate is vital to assessing their mathematical understanding. Students' understanding should be assessed through the use of good questions that promote the need for communication among students. Assessments in the mathematics classroom should include open-ended questions as well as peer and self-assessment. Assessments should ask students to describe and explain mathematical concepts and methods in multiple ways (with multiple representations) to demonstrate deep understanding.
  Reasoning and Proof
All students should be able to:
  • Reason in a wide range of mathematical and applied settings
  • Recognize reasoning and proof as fundamental aspects of mathematics
  • Make and investigate mathematical conjectures
  • Develop and evaluate mathematical arguments and proof
  • Select and use various types of reasoning and methods of proof
Reasoning and proof should be addressed throughout curriculum, instruction, and assessment. These skills should be taught as an integral part of classroom instruction in all areas of mathematics. As the context for reasoning and proof, teachers should choose problems rich in mathematical content and accessible and challenging to all students. Students build confidence in their abilities to develop and defend their own arguments as they solve problems in a classroom environment that supports questioning, discussion, and listening. In such a supportive, inquiry-based classroom environment students will use their mathematical knowledge to make conjectures about problems. Students will analyze various approaches to investigate their conjectures. They will develop a carefully reasoned mathematical argument to support their conclusion. This justification of their conjecture will be communicated through interactions with classmates and teacher and validated against conventional arguments. "Reasoning and proof cannot simply be taught in a single unit on logic, for example, or by "doing proofs" in geometry. Proof is a very difficult area for undergraduate mathematics students. Perhaps students at the postsecondary level find proof so difficult because their only experience in writing proofs has been in a high school geometry course, so they have a limited perspective (Moore 1994). Reasoning and proof should be a consistent part of students' mathematical experience in prekindergarten through grade 12. Reasoning mathematically is a habit of mind, and like all habits, it must be developed through consistent use in many contexts." (NCTM, 2000, p. 56)
  Ability to Recognize, Make and Apply Connection
All students should be able to:
  • Recognize and use connections among mathematical ideas
  • Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
  • Recognize and apply mathematics in contexts outside of mathematics
"As the Learning Principle [in NCTM's Principles and Standards] emphasizes, understanding involves making connections" (NCTM 2000, p. 64). A connected and coherent mathematics curriculum helps students make connections across the strands of mathematics. Problem-based instructional tasks provide connections to other disciplines and to the real world. Instruction should emphasize important mathematics across and within the disciplines. Educators should pose questions that encourage students to make connections, including connections to their previous mathematical knowledge.
  Ability to Construct & Apply Multiple Connected Representations
All students should be able to:
  • Create and use representations to organize, record, and communicate mathematical ideas
  • Select, apply, and translate among mathematical representations to solve problems
  • Use representations to model and interpret physical, social, and mathematical phenomena
Teachers should introduce students to multiple connected mathematical representations and help them use those representations effectively. They should highlight ways in which different representations can convey different information and emphasize the importance of selecting representations suited to the particular mathematical tasks at hand. Assessments should allow for students to have choices when representing problems and solutions. Students should be encouraged to evaluate which representation is best to use when solving a problem or investigating a mathematical idea. (Adapted from NCTM, 2000) "Representations should be treated as essential elements in supporting students' understanding of mathematical concepts and relationships; in communicating mathematical approaches, arguments, and understandings to one's self and to others; in recognizing connections among related mathematical concepts; and in applying mathematics to realistic problem situations through modeling. New forms of representation associated with electronic technology create a need for even greater instructional attention to representation." (NCTM, 2000, p. 67)

  Number & Operations (K-8)
 K-2  3-5  6-8
  Count, represent, read, compare, order and conserve (knows that the total number does not change when configured differently) whole numbers.
  • Count, represent, read, compare, order, and conserve whole numbers up to 1000.
  • Write, compare, and order numbers to at least 120 using the words equal to, greater than, less than, greatest, and least when appropriate.
  • Count by tens or hundreds, forwards and backwards, starting at any number from 1 to 1000.
  • Represent numbers to at least 1000 in different way using written words, numerals, or models, and translate among representations.
  • Identify the placement and relationships between digits and their values in numbers up to 1000.
  Develop understandings of addition and subtraction and strategies for basic addition facts and related subtraction facts.
  • Solve and create story problems that match addition or subtraction expressions or equations using physical objects, pictures, or words.
  • Solve simple story problems (result unknown) involving joining, separating, and grouping situations. Solve story problems involving joining, separating, comparing, grouping, and partitioning using a variety of strategies, such as direct modeling with objects or pictures, counting on and counting back, and using related facts and known facts.
  • Add and subtract two-digit numbers efficiently and accurately using a procedure that can be generalized, including the standard algorithm and describe why the procedure works.
  Express numbers as equivalent representations to fluently compose and decompose numbers (putting together and taking apart).
  • Fluently compose (put together) and decompose (take apart) numbers at least to 10.
  • Compose and decompose two- and three-digit numbers based on the values of the digits used to write the number.
  • Solve word problems involving joining, separating, part/whole, comparing, grouping, and partitioning, using a variety of strategies, such as direct modeling, counting up or counting back by 1s or 10s, and deriving or recalling facts. (The unknown can appear in a variety of positions).
  Develop fluency and quick recall of addition facts and related subtraction facts and fluency with multi-digit addition and subtraction.
  • Show the inverse relationship between addition and subtraction by using physical models, diagrams, and/or acting-out situations.
  • Explain and use strategies for understanding addition facts for sums equal to at least 10, and related subtraction facts.
  • Develop and demonstrate quick recall of basic addition facts to 20 and related subtraction facts.
  • Solve word problems involving joining, separating, part/whole, comparing, grouping, and partitioning, using a variety of strategies, such as direct modeling, counting up or counting back by 1s or 10s, and deriving or recalling facts. (The unknown can appear in a variety of positions).
  Estimate the answer to an addition or subtraction problem before computing, and determine whether the computed answer makes sense.
  • Determine whether the computed answer to an addition or subtraction problem is reasonable.
  • Estimate an answer prior to computing. (For example, 23 + 48 is about 70.)
  Develop an understanding of whole number relationships, including grouping in tens and ones and apply place-value concepts.
  • Group and count objects by 2s, 5s, and 10s.
  • Find a number that is 10 more or 10 less than a given number.
  • Group numbers into 10s and 1s in more than one way and explain why the total remains the same.
  • Explain and use strategies for remembering addition and subtraction facts to 20.
  • Use mental strategies, invented algorithms, and traditional algorithms based on knowledge of place value to add and subtract two-digit numbers.
  Understand fractional parts are equal shares or equal portions of a whole unit (a unit can be an object or a collection of things).
  • Understand and represent commonly used fractions, such as 1/4, 1/3, and 1/2.
  Develop an understanding of multiplication and division concepts and strategies for basic multiplication facts and related division facts.
  • Develop concepts of multiplication and division through the use of different representations (e.g. equal-sized groups, arrays, area models, and skip counting on number lines for multiplication, and successive subtraction, partitioning, and sharing for division).
  • Use commutative, associative, and distributive properties to develop strategies and generalizations to solve multiplication problems. These strategies will evolve from simple strategies (e.g. times 0, times 1, doubles, count by fives) to more sophisticated strategies, such as splitting the array.
  • Relate multiplication and division as inverse operations and learn division facts by relating them to the appropriate multiplication facts.
  • Consider the context in which a problem is situated to select the most useful form of the quotient for the solution, and they interpret it appropriately.
  • Be able to make comparisons involving multiplication and division, using such words as "twice as many" or "half as many".
  Develop fluency and quick recall of multiplication facts and related division facts and fluency with multi-digit multiplication and division.
  • Extend their work with multiplication and division strategies to develop fluency and recall of multiplication and division facts.
  • Apply their understanding of models for multiplication (i.e. equal-sized groups, arrays, area models), place value, and properties of operations (in particular, the distributive property) as they develop, discuss, and use efficient, accurate, and generalizable methods to multiply multidigit whole numbers.
  • Apply their understanding of models for division (partitioning, successive subtraction) place value, properties, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multidigit dividends.
  • Develop fluency with efficient procedures for multiplying and dividing whole numbers and use them to solve problems.
  Develop the ability to estimate the results of computation with whole numbers, fractions or decimals and be able to judge reasonableness.
  • Generalize patterns of multiplying and dividing whole numbers by 10, 100, and 1000 and develop understandings of relative size of numbers.
  • Be able to estimate sums and differences with whole numbers up to three digits.
  • Build facility and understand when estimation, mental computation or paper-and-pencil computations are appropriate in a given problem.
  • Select and apply appropriate strategies (mental computation, number sense and estimation) for estimating products and quotients or determining reasonableness of results, depending on the context and numbers involved.
  • Make reasonable estimates of fraction and decimal sums and differences.
  Extend place value concepts to represent and compare both whole numbers and decimals.
  • Extend their understanding of place value to numbers up to 10,000, 100,000 and millions in various contexts and depending on grade level.
  • Understand decimal notation as an extension of the base-ten system of writing whole numbers through place-value patterns and models (place-value charts and base-ten blocks) from tenths to hundredths and thousandths, depending on grade level.
  Use benchmarks to help develop number sense.
  • Use estimation in determining the relative sizes of number including amounts and distances, such as 500 is 5 flats or 5 x 100, or 500 is one-half of 1000.
  • Learn about the position of numbers in the base-ten number system (763 is 7 x 100 plus 6 x 10 plus 3 x 1) and its relationship to benchmarks such as 500, 750, 800 and 1000.
  • Extend common benchmarks such as 10, 25, 50, and 100 to understand and use benchmarks of 500 and 1000.
  • Understand and use common benchmarks such as ½ or 1 to compare fractions.
  Develop an understanding of commonly used fractions, decimals, and percents, including recognizing and generating equivalent representations.
  • Develop an understanding of the meanings and uses of fractions to represent parts of a whole, parts of a set, or points or distances on a number line.
  • Understand that the size of a fractional part is relative to the size of the whole, and use fractions to represent numbers that are equal to, less than, or greater than 1.
  • Solve problems that involve comparing and ordering fractions by using models, benchmark fractions, or strategies involving common numerators or denominators.
  • Understand and use models, including the number line, to identify equivalent fractions including numbers greater than one.
  • Connect and extend their understanding of fractions to modeling, reading and writing decimals (tenths, hundredth and thousandths), that are greater than or less than 1, identifying equivalent decimals, and comparing and ordering decimals.
  • Connect fractions (initially halves, fourths, and tenths, and then fifths, thirds, and eighths) and their equivalent decimals through representations including word names, symbols and models (10 x 10 grids and number lines).
  • Recognize and generate equivalent forms of commonly used fractions, decimals and percents.
  Develop an understanding of and fluency with addition and subtraction of fractions and decimals.
  • Apply their understandings of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators.
  • Apply their understandings of decimal models, place value, and properties to develop strategies to add and subtract fractions and decimals.
  • Develop fluency with standard procedures for adding and subtracting fractions and decimals.
  • Add and subtract fractions and decimals to solve problems and use number sense to determine reasonableness of results.
  Understand, apply, and be computationally fluent with multiplication and division of fractions and decimals.
  • Understand that multiplying two numbers does not necessarily make a bigger number, nor does dividing always result in a smaller number.
  • Understand and explain procedures for multiplying and dividing fractions by using the meanings of fractions, multiplication and division, and the inverse relationship between multiplication and division.
  • Understand and explain procedures for multiplying and dividing decimals by using the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number).
  • Use common procedures to multiply and divide fractions and decimals efficiently and accurately.
  • Convert from one unit to another in the metric system of measurement by using understanding of the relationships among the units and by multiplying and dividing decimals.
  • Convert from one unit to another in the customary system of measurement by using understanding of the relationships among the units and by multiplying and dividing fractions.
  • Multiply and divide fractions and decimals to solve problems, including multi-step problems.
  Understand, apply, and be computationally fluent with rational numbers, including negative numbers.
  • Understand negative numbers in terms of their position on the number line, their role in the system of all rational numbers, and in everyday situations (e.g., situations of owing money or measuring elevations above and below sea level).
  • Understand absolute value in terms of distance on the number line and simplify numerical expressions involving absolute value.
  • By applying properties of arithmetic and considering negative numbers in everyday contexts, explain why the rules for adding, subtracting, multiplying, and dividing with negative numbers make sense.
  • Understand positive integer exponents in terms of repeated multiplication and evaluate simple exponential expressions.
  • Effectively compute with and solve problems using rational numbers, including negative numbers.
  Understand and apply ratio and rate, including percents, and connect ratio and rate to fractions and decimals.
  • Build on understanding of fractions and part-whole relationships to understand ratios (by, for example, analyzing the relative quantities of boys and girls in the classroom or triangles and squares in a drawing).
  • Understand percent as a rate and develop fluency in converting among fractions, decimals, and percents.
  • Understand equivalent ratios as deriving from, and extending, pairs of rows (or columns) in the multiplication table.
  • Understand rate as a way to compare unlike quantities (such as miles per hour or a situation in which 5 pens cost $3.75).
  • Use a variety of strategies to solve problems involving ratio and rate.
  Understand and apply proportional reasoning.
  • Understand that a proportion is an equation that states that two ratios are equivalent.
  • Understand proportional relationships (y = kx or = k), and distinguish proportional relationships from other relationships, including inverse proportionality (xy = k or y = ).
  • Understand that in a proportional relationship of two variables, if one variable doubles or triples, for example, then the other variable also doubles or triples, and if one variable changes additively by a specific amount, a, then the other variable changes additively by the amount ka.
  • Graph proportional relationships and identify the constant of proportionality as the slope of the related line.
  • Use ratios and proportionality to solve a wide variety of percent problems, including problems involving discounts, interest, taxes, tips, and percent increase or decrease.
  • Use proportionality to solve single and multi-step problems in a variety of other contexts.
  Understand, estimate, and represent real numbers, including common irrational numbers and with scientific notation.
  • Recognize that the set of real numbers, which can be represented as the number line, consists of two disjoint sets — the set of rational numbers and the set of irrational numbers.
  • Estimate irrational numbers and represent them as points on the number line.
  • Recognize irrational numbers as non-repeating, non-terminating decimals, including common irrational numbers such as p and non-perfect square roots and cube roots
  • Understand and determine the square roots of perfect squares.
  • Understand and estimate square roots of non-perfect-squares, and determine more precise values using a calculator.
  • Represent, use, and interpret numbers in scientific notation.
  • Use scientific notation and rational and irrational numbers to model and solve problems.

 K-2  3-5  6-8  9-12
  Recognize, describe, create and extend repeating and growing patterns such as physical, geometric and numeric patterns and translate from one representation to another.
  • Recognize, describe, create and extend color, rhythmic, shape, number and letter repeating patterns with simple attributes.
  • Identify a missing element in a pattern.
  • Make a generalization that patterns can translate from one representation to another.
  • Recognize, describe, create and extend repeating and growing patterns.
  • Translate a pattern between sound, symbols, movements and objects.
  • Identify, create, describe, and extend simple number and growing patterns. involving repeated addition and subtraction, skip counting and arrays of objects.
  • Use patterns to solve problems in various contexts.
  Sort, classify, and order objects by size, number and other properties.
  • Sort and a classify objects by a single attribute and explain the sorting rule.
  • Sort and a classify objects by multiple attributes and explain the sorting rule (sort and classify the same set of objects in multiple ways and explain the various sorting rules.).
  • Sort and classify a set of objects using a Venn diagram.
  Demonstrate the use of the commutative and associative properties and mathematical reasoning to solve for the unknown quantity in addition and subtraction problems; justify the solution.
  • Solve, with objects, simple problems involving joining and separating.
  • Develop concepts of addition and subtraction (including commutativity and associativity of addition) using mathematical tools (objects, number line, hundreds chart, etc.), pictures, and mathematical notation.
  • Use commutative and associative properties and mathematical reasoning to solve a variety of addition and subtraction problems involving two or more one-digit numbers; justify the solution.
  Understand equality as meaning "the same as" and use the = symbol appropriately.
  • Recognize the use of symbols to represent mathematical ideas in joining and separating problems.
  • Determine if equations involving addition and subtraction are true.
  • Demonstrate an understanding that the "=" sign means "the same as" by solving open number sentences including those with variables.
  • Write number sentences using mathematical notation ( +, =, -, <, >, ?, and variables) to represent mathematical relationships to solve problems.
  • Solve equations in which the unknown and the equal sign appear in a variety of positions.
  • Use number sentences involving addition and subtraction, and unknowns to represent and solve given problem situations.
  Represent and analyze patterns and relationships involving multiplication and division to introduce multiplicative reasoning.
  • Build a foundation using multiplicative contexts for later understanding of functional relationships with such statements as, "The number of legs is 4 times the number of chairs" or "A quarter is five times the value of a nickel."
  • Make generalizations by reasoning about the structure of the pattern to determine if the patterns are nonnumeric growing, repeating, or multiplicative patterns.
  Identify the commutative, associative, and distributive properties and use them to compute with whole numbers.
  • Explore the commutative and associative properties through models and examples to determine which properties hold for multiplication and division facts and develop increasingly sophisticated strategies based on these properties and the distributive property to solve multiplication problems involving basic facts.
  • Use properties of addition and multiplication to multiply and divide whole numbers and understand why these algorithms work.
  Understand and apply the idea of a variable as an unknown quantity and express mathematical relationships using equations.
  • Use invented notation, standard symbols and variables to express a pattern, generalization, or situation.
  • Develop an understanding of the use of a rule to describe a sequence of numbers or objects.
  • Use patterns, models, and relationships as contexts for writing and solving simple equations and inequalities.
  Represent and analyze patterns and functions, using words, tables, and graphs.
  • Describe patterns verbally and represent them with tables or symbols.
  • Continue to identify, describe, and extend numeric patterns involving all operations and nonnumeric growing or repeating patterns.
  • Identify patterns graphically, numerically, or symbolically and use this information to predict how patterns will continue.
  • Create graphs of simple equations.
  • Be able to use various techniques including words, tables, numbers and symbols for organizing and expressing ideas about relationships and functions.
  Write, interpret, and use mathematical expressions and equations, find equivalent forms, and relate such symbolic representations to verbal, graphical, and tabular representations.
  • Write mathematical expressions, equations, and formulas that correspond to given situations.
  • Understand that variables represent numbers whose exact values are not yet specified, use single letters, words, or phrases as variables, and use variables appropriately.
  • Evaluate expressions (for example, find the value of 3x if x is 7).
  • Understand that expressions in different forms can be equivalent, and rewrite an expression to represent a quantity in a different way (e.g., to make it more compact or to feature different information).
  • Understand that solutions of an equation are the values of the variables that make the equation true.
  • Solve simple one-step equations (i.e., involving a single operation) by using number sense, properties of operation, and the idea of maintaining equality on both sides of an equation.
  • Construct and analyze tables (e.g., to show quantities that are in equivalent ratios), and use equations to describe simple relationships shown in a table (such as 3x = y).
  • Use expressions, equations, and formulas to solve problems, and justify their solutions.
  Understand and apply proportionality.
  • Understand that a proportion is an equation that states that two ratios are equivalent.
  • Understand proportional relationships (y = kx or = k), and distinguish proportional relationships from other relationships, including inverse proportionality (xy = k or y = ).
  • Graph proportional relationships and identify the constant of proportionality as the slope of the related line.
  • Use ratios and proportionality to solve a wide variety of percent problems, including problems involving discounts, interest, taxes, tips, and percent increase or decrease.
  • Use proportionality to solve single and multi-step problems in a variety of other contexts.
  Understand, solve, and apply linear equations and inequalities.
  • Make strategic choices of procedures to solve linear equations and inequalities in one variable and implement them efficiently.
  • Recognize and generate equivalent forms of linear expressions, by using the associative, commutative, and distributive properties.
  • Understand that when properties of equality are used to transform an equation into a new equivalent equation, solutions obtained for the new equation also solve the original equation.
  • Solve more complicated linear equations, including solving for one variable in terms of another.
  • Solve linear inequalities and represent the solution on a number line.
  • Formulate linear equations and inequalities in one variable and use them to solve problems, including in applied settings, and justify the solution using multiple representations.
  Understand and apply linear functions.
  • Understand linear functions and slope of lines in terms of constant rate of change.
  • Understand that the slope of a line is constant, for example by using similar triangles (e.g., as shown in the rise and run of "slope triangles"), and compute the slope of a line using any two points on the line.
  • Build on the concept of proportion, recognizing a proportional relationship ( = k, or y = kx) as a special case of a linear function. In this special case, understand that if one variable doubles or triples, for example, then the other variable also doubles or triples; and understand that if the input, or x-coordinate in this case, changes additively by a specific amount, a, then the output, or y-coordinate in this case, changes additively by the amount ka.
  • Understand that the graph of the equation y = mx + b is a line with y-intercept b and slope m.
  • Translate among verbal, tabular, graphical, and algebraic representations of functions, including recursive representations such as NEXT = NOW +3 (recognizing that tabular and graphical representations often only yield approximate solutions), and describe how such aspects of a linear function as slope, constant rate of change, and intercepts appear in different representations.
  • Use linear functions, and understanding of the slope of a line and constant rate of change, to analyze situations and solve problems.
  Use tables and graphs to analyze systems of linear equations.
  • Use tables and graphs to analyze and (approximately) solve systems of two linear equations in two variables.
  • Relate a system of two linear equations in two variables to a pair of lines in the plane that intersect, are parallel, or are the same.
  • Use systems of linear equations to analyze situations and solve problems.
  Understand, analyze, represent, and apply functions.
    The concept of function is central to the study of algebra (and extends beyond algebra as well). Functions can be used to represent and reason about patterns of change and relationships between quantitative variables, including in real-world situations. Often when modeling or solving problems with functions, students will develop, analyze, and manipulate algebraic expressions and solve equations and inequalities. Students' experiences with functions should include analysis of families of functions (linear, quadratic, other polynomial, exponential, trigonometric, rational, and logarithmic). Students should also study absolute value, square root, cube root, and piecewise functions. Analysis of functions should include: zeros, maximum and minimum, domain and range, global and local behavior, intercepts, rate of change, and inverse functions. Students should be able to recognize, represent, compare/contrast, compose, and transform functions. They should represent functions in multiple ways: symbolically (explicitly and recursively), graphically, numerically, and verbally, and understand the connections among these representations. Students should also understand and analyze relations that are not functions.
  Understand, analyze, solve, and apply equations and inequalities.
    Equations and inequalities can be used to symbolically model situations. Studying equations and inequalities in context helps students develop a deep understanding of the meaning of both the equation or the inequality and the solution. Students should become fluent in connecting the symbolic representation with the situation being represented. Inherent in the study of equations and inequalities is the use of algebraic expressions, and students should understand the difference between equations and expressions. Students should distinguish between an equation and an inequality and compare and contrast their properties and the methods for solving them. Further, discussion about the reasonableness and meaning of a solution is important. Methods for solving equations and inequalities include symbolic, numeric, and graphic. Algebraic properties of real numbers should be used fluently, with a focus on equivalent equations. A particular emphasis is on solving linear and quadratic equations, although much of the work with linear equations should have been completed in middle school. Students should be able to manipulate formulas, including solving for one variable in terms of the others, and they should develop a conceptual understanding of the meaning of the formulas through their context. Once the concept of an equation and its solution is studied, students move to the study of systems of equations, both linear and nonlinear systems. Students should analyze, apply, and choose appropriate methods for solving systems of equations (symbolic, graphic, numeric, and matrix methods).
  Understand, analyze, transform, and apply algebraic expressions.
    Algebraic expressions often arise when modeling situations. Students should understand and use algebraic expressions based on "symbol sense," that is, the ability to connect algebraic forms to numeric, graphic, and contextual interpretations and implications. Students' symbol sense builds on their number sense. Symbol sense allows students to represent situations with algebraic expressions and interpret expressions in terms of the situation. Students with symbol sense should meaningfully manipulate algebraic expressions to obtain equivalent forms by simplifying, factoring, expanding, and using order of operations, laws of exponents, and properties of real numbers.
  Understand, analyze, approximate, and interpret rate of change.
    A key concept in the study of functions is rate of change. Rate of change is the rate at which one variable changes with respect to another. Situations involving rate of change may include the speed of a car, the number of people per year by which a population increases, and slope of a line. Rate of change should be analyzed in multiple ways, including numeric, symbolic (recursive and explicit), and graphic representations. Students should approximate and interpret rate of change based on graphs, numerical data, and real-world situations. The study of rate of change focuses on slope and lays the groundwork for calculus. Students should distinguish between a constant rate of change and a non-constant rate of change. In addition, some students may investigate rate of change in terms of finite differences tables.
  Understand and apply recursion and iteration.
    Recursion and iteration are powerful mathematical tools for solving problems related to sequential (i.e., step-by-step) change, such as population change from year to year or the growth of money over time due to compound interest. To iterate means to repeat, so iteration is the process of repeating a procedure or computation over and over again. Recursion is the method of describing a given step in a sequence in terms of the previous step(s). Students should be able to represent recursive relationships with informal notation, subscript notation, and function notation. They should understand and use a recursive view of functions, including for deeper understanding of key ideas. For example, NEXT = NOW + 3 could represent a linear function with slope 3, and S(n + 1) = 3S(n) could represent an exponential function with constant multiplier 3. Students should understand and apply finite arithmetic and geometric sequences and series, including an analysis with both recursive and explicit formulas. They should use recursion and iteration to represent and solve problems. Skydiving is an exciting but dangerous sport. Many precautions are taken to ensure the safety of the skydivers. The basic fact underlying these precautions is that acceleration due to the force of gravity is 32 feet per second per second (written as 32 ft/sec2). Thus, each second that the skydiver is falling, her speed increases by 32 ft/sec (ignoring air resistance and other complicating factors; focus only on the force of gravity). Determine both the recursive and explicit formulas that model the total distance fallen by a skydiver after each second before her parachute opens. Describe the method(s) you used to find these formulas. What type of function is represented by these formulas? How do you know this? Compare the different representations (table, graph, explicit form, and recursive form) of your function to other types of functions you know. (See student investigation sheet and problem-based instructional task lesson plan - See Skydiving Activity.

  Data Analysis/Statistics & Probability
 K-2  3-5  6-8  9-12
  Collect, sort, organize, and represent data to ask and answer questions relevant to the  K-2 environment.
  • Collect and organize data in lists, tables, and simple graphs.
  • Collect, organize, represent, and interpret data in bar-type graphs, picture graphs, frequency tables, and line plots.
  • Use interviews, surveys, and observations to collect data that answers questions about themselves and their surroundings.
  Compare different representations of the same data using these types of graphs: bar graphs, frequency tables, line plots, and picture graphs.
  • Represent a collection of data using tallies, tables, picture graphs and bar graphs.
  • Compare a single data set using different types of graphs.
  Use information displayed on graphs to answer questions and make predictions, inferences and generalizations such as likely or unlikely events.
  • Answer simple questions relating to the information displayed on a graph, table, or list.
  • Use interviews, surveys, and observations to collect data that answers questions about themselves and their surroundings.
  • Analyze information by asking and answering questions about the data.
  • Contrast different sets of data displayed on the same type of graph to draw conclusions and make generalizations.
  • Use information from data to make observations and inferences, draw conclusions, and make predictions.
  Represent and analyze data using tallies, pictographs, tables, line plots, bar graphs, circle graphs and line graphs.
  • Recognize the differences representing categorical and numerical data.
  • Construct and analyze frequency tables, bar graphs, picture graphs, and line plots and use them to address a question.
  • Compare different representations of the same data and evaluate how well each representation shows important aspects of the data.
  • Use their understanding of whole numbers, fractions, and decimals to construct and analyze circle graphs and line graphs.
  • Apply their understanding of place value to develop and use stem-and-leaf plots.
  Describe the distribution of the data using mean, median, mode or range.
  • Learn to compare related data sets, noting the similarities and differences between the two sets and develop the idea of a "average" value.
  • Learn to select and use measures of center: mean, median and mode and apply them to describing data sets.
  • Build an understanding of what the measures of center tells them about the data and to see this value in the context of other characteristics of the data such as the range.
  • Begin to conceptually explore the meaning of mean as the balance point for the data set.
  Propose and justify conclusions and predictions based on data.
  • Learn how to describe data, make a prediction to describe the data, and then justify their predictions.
  • Learn to collect data using observations, surveys and experiments and propose conjectures.
  • Design simple experiments to examine their conjectures and justify their conclusions.
  • Design investigations to address a question and consider how data collection methods affect the nature of the data set.
  • Examine the role of sample size has in predictions about data.
  Predict the probability of simple experiments and test predictions.
  • Examine the probability of experiments that have only a few outcomes, such as game spinners (i.e., how likely is it that the spinner will land on a particular color?), by first predicting the probability of the desired event and then exploring the outcome through experimental probability.
  • Learn to represent the probability of a certain event as 1 and the probability of an impossible event as 0.
  • Learn to use common fractions to represent events that are neither certain nor impossible.
  Describe events as likely or unlikely and discuss the degree of likelihood using words like certain, equally likely and impossible.
  • Understand probability as the measurement of the likelihood of events.
  • Learn to estimate the probability of events as certain, equally likely or impossible by designing simple experiments to collect data and draw conclusions.
  Understand, interpret, determine, and apply measures of center and graphical representations of data.
  • Extend prior work with mode, median, and mean as measures of center.
  • Compute the mean for small data sets and explore its meaning as a balance point for a data set.
  • Extend prior work with bar graphs, line graphs, line plots, histograms, circle graphs, and stem and leaf plots as graphical representations of data to include box-and-whisker plots and scatterplots.
  • Create and interpret box-and-whisker plots and scatterplots.
  Understand and represent simple probabilistic situations.
  • Represent the probability of events that are impossible, unlikely, likely, and certain using rational numbers from 0 to 1.
  • List all possible outcomes of a given experiment or event.
  Use proportions and percentages to analyze data and chance.
  • Use proportions to make estimates relating to a population on the basis of a sample.
  • Apply percentages to make and interpret histograms and circle graphs.
  • Explore situations in which all outcomes of an experiment are equally likely, and thus the theoretical probability of an event is the number of outcomes corresponding to the event divided by total number of possible outcomes.
  • Use theoretical probability and proportions to make approximate predictions.
  Analyze and summarize data sets, including initial analysis of variability.
  • Select, determine, explain, and interpret appropriate measures of center for given data sets (mean, median, mode).
  • Select, create, explain, and interpret appropriate graphical representations for given data sets (bar graphs, circle graphs, line graphs, histograms, line plots, stem and leaf plots, box-and-whisker plots, scatterplots).
  • Summarize and compare data sets using appropriate numerical statistics and graphical representations.
  • Compare the information provided by the mean and the median and investigate the different effects that changes in the data values have on these measures of center.
  • Understand that a measure of center alone does not thoroughly describe a data set because very different data sets can share the same measure of center, and thus consider and describe the variability of the data (e.g, range and interquartile range).
  • Informally determine a line of best fit for a scatterplot to make predictions and estimates.
  • Formulate questions, gather data relevant to the questions, organize and analyze the data to help answer the questions, including informal analysis of randomness and bias.
  Understand, compute, and estimate simple probabilities using counting strategies and simulation.
  • Understand and apply the Multiplication Principle of Counting in simple situations.
  • Compute probabilities for compound events, using such methods as organized lists, tree diagrams (counting trees), area models, and counting principles.
  • Estimate the probability of simple and compound events through experimentation and simulation.
  • Use a variety of experiments to explore the relationship between experimental and theoretical probabilities and the effect of sample size on this relationship.
  Understands and interprets descriptive statistics.
  • Descriptive statistics involves describing and summarizing data. For example, for univariate (one-variable) data like test scores, we might describe the range of the data, the mean score, or the standard deviation of the scores. For bivariate (two-variable) data like weight before and after a diet program, we might fit a line to the (before, after) data points and use the line to estimate and predict weight.
  • As societal demands for statistical reasoning increase, high school students need rich experiences to create, choose, understand, and interpret statistical models, with and without technology. These experiences should be built on their understanding of data analysis developed in the middle grades.
  • Students should collect, represent, and analyze numerical and categorical data, and both univariate and bivariate data. Representations of data should include histograms, box plots, scatterplots, bar graphs, line graphs, stem and leaf plots, frequency distributions, and relative frequency distributions. Students should compare and contrast these different representations, and choose appropriate representations.
  • Analysis of data should include graphical representations, measures of center and variability, transformations of univariate data, outliers, regression, and correlation. Students should describe and analyze distributions of data in terms of center, spread, and shape. Much of this material should be studied in middle school.
  • In high school, students should reinforce their previous knowledge of data analysis from the middle grades and focus on extending that knowledge to standard deviation, linear regression, and correlation.
  Understands and applies the basic ideas of probability.
  • Probability is the study of chance and likelihood. The study of probability should include the essential ideas needed for making inferences from data. Experiments should be conducted to develop the idea of sample space and events. Counting concepts and methods, including permutations, combinations, and the multiplication principle of counting, should be applied to probability. The probability of an event, when the outcomes are equally likely, should be understood as the ratio of the size of the event (number of outcomes corresponding to the event) to the size of the sample space (number of possible outcomes).
  • The rules for probability of events and compound events should be addressed in terms of the students' experiments and simulations. Special emphasis should be given to the addition rule, because of the need to consider the intersection, the analysis of which leads to the ideas of independent events, multiplication rule, mutually exclusive, and conditional probability.
  • The notion of random variables and probability distribution of a discrete random variable should be introduced through simple experiments. Students should compare and contrast experimental and theoretical probabilities. The analysis of probability distributions should include the expected value and measures of variability. Various types of probability distributions should be studied, including binomial and normal distributions.
  Understands and interprets inferential statistics.
  • Statistical inference is the process of using a sample to draw conclusions (make inferences) that go beyond the sample. Since the sample does not contain all the information about the population or all the results from all possible experiments, probability is used to help describe the limitations of the inference. For example, in an experiment to test the relative effectiveness of two drugs, the inference that one is better than the other is qualified by using probability to describe how likely it is that the results of the experiment are due to chance rather than effectiveness of the drugs.
  • Students should understand that information from a sample can be used to estimate information about a population. Instruction should start with activities involving concrete experiments and simulations and should address issues of randomness, rare events, sources of bias, and sample size. Building on the use of simulation, students should understand key ideas such as sampling distribution and rare event, and use these ideas to analyze and interpret published statistical reports, such as in newspaper articles.
  • Making inferences from data is often part of a statistical study. A statistical study is a survey, observational study, or experiment that applies statistical thinking. Statistical thinking involves formulating questions, collecting data relevant to those questions, analyzing the data, and drawing appropriate conclusions. Students should apply statistical thinking to design, conduct, and analyze simple statistical studies. In doing so, they may use descriptive statistics, statistical inference, and probability.

  Geometry & Measurement
 K-2  3-5  6-8  9-12
  Recognize and describe shapes and structures in the physical environment.
  • Identify, name, sort, and describe two- and three-dimensional shapes (including circles, triangles, rectangles, squares, cubes, and spheres), and real-world approximations of the shapes, regardless of size or orientation.
  Compose and decompose geometric shapes, including plane and solid figures to develop a foundation for understanding area, volume, fractions, and proportions.
  • Compose (combine) and decompose (take apart) two- and three-dimensional figures and analyze the results.
  • Compose and decompose two- and three-dimensional shapes to develop a foundation of fractional relationships and proportions.
  • Cover two-dimensional objects with shapes to develop a foundation for area.
  • Fill three-dimensional objects to develop a foundation for volume.
  Identify, name, sort, and describe two- and three-dimensional geometric figures regardless of size or orientation.
  • Describe characteristics of two- and three-dimensional objects (number of corners, edges, and sides, length of sides, etc.).
  Describe and specify space and location with simple relationships and with coordinate systems.
  • Describe the location of one object relative to another object using words such as in, out, over, under, above, below, between, next to, behind, and in front of.
  • Locate points on maps and simple coordinate grids with letters and numbers.
  • Represent points and simple figures on maps using simple coordinate grids with letters and numbers.
  Experience and recognize slides, flips, turns and symmetry to analyze mathematical situations.
  • Identify shapes that have been rotated (turned), reflected (flipped), translated (slid), and enlarged. Describe the direction of the translation (left, right, up, down).
  Use attributes of geometric figures to solve spatial problems.
  • Describe and represent shapes from different perspective.
  • Explore relationships of different attributes.
  • Describe geometric shapes in the environment and specify their location.
  Identify attributes that are measurable, such as length, weight, time and capacity, and use these attributes to order objects and make direct comparisons.
  • Identify attributes that are measurable such as length, volume, weight, and area. Use these attributes and appropriate language to make direct comparisons. (Taller, shorter, longer, same length; heavier, lighter, same weight; holds more, holds less, holds the same amount).
  • Recognize temporal concepts such as before, after, sooner, later, morning, afternoon, evening.
  • Use a seriated set of objects to order and compare lengths.
  • Recognize that objects used to measure an attribute (length, weight, capacity) must have that attribute and must be consistent in size.
  • Determines the relationship between the size of the unit and the number of units needed to make a measurement.
  Estimate, measure and compute measurable attributes while solving problems.
  • Select appropriate measurement tools and units (standard and non-standard) to solve problems.
  Estimate and measure length using standard (customary and metric) and non-standard units with comprehension.
  • Understand the necessity for identical units (standard or non-standard) for accurate measurements.
  • Use a variety of non-standard units to measure length without gaps or overlaps.
  • Use non-standard units to compare objects according to their capacities or weights.
  • Associate the time of day with everyday events.
  • Name standard units of time (day, week, month).
  • Use both analog and digital clock to tell time to the hour and half hour.
  • Estimate and measure length using metric and customary units.
  • Select appropriate measurement tools and units (standard and non-standard) to solve problems.
  • Use both analog and digital clock to tell time to the nearest five-minute interval.
  • Describe the relationship among standard units of time: minutes, hours days, weeks, months and years.
  Describe, analyze and classify two-dimensional and three-dimensional shapes.
  • Describe, analyze, and compare two-dimensional shapes by their sides and angles and connect these attributes to definitions of shapes.
  • Relate two-dimensional shapes to three-dimensional shapes and analyze properties of polyhedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces.
  • Classify two- and three-dimensional shapes according to their attributes and develop definitions of classes of shapes such as parallelograms and prisms.
  Explore congruence and similarity.
  • Understand attributes and properties of two-dimensional space through building, drawing and analyzing two-dimensional shapes and use the attributes and properties to solve problems, including applications involving congruence and symmetry.
  • Apply congruence to other contexts such as three-dimensional shapes and repeating the congruent shapes to build a similar shape.
  • Explore similar shapes to determine that angle measure is the same and the related sides are proportional, that is, related by the same multiplicative or scale factor.
  Predict and describe the results of sliding (translation), flipping (reflection), and turning (rotation) two-dimensional shapes.
  • Investigate, describe, and reason about decomposing, combining, and transforming polygons to make other polygons.
  • Investigate and describe line and rotational symmetry.
  • Extend their understanding of two-dimensional space by using transformations to design and analyze simple tilings and tessellations.
  Use ordered pairs on a coordinate grid to describe points or paths (first quadrant).
  • Learn how to use two numbers to name points on a coordinate grid and know this ordered pair corresponds to a particular point on the grid.
  • Make and use coordinate systems to specify locations and to describe paths.
  • Explore methods for measuring the distance between two locations on the grid along horizontal and vertical lines.
  Use geometric models to solve problems, such as determining perimeter, area, volume, and surface area.
  • Develop measurement concepts and skills through experiences in analyzing attributes and properties of two- and three-dimensional objects.
  • Form an understanding of perimeter as a measurable attribute and quantify perimeter by finding the total distance or length around the shape.
  • Recognize area as an attribute of two-dimensional regions and that they can quantify area by finding the total number of same-sized units of area that cover the shape without gaps or overlaps.
  • Connect area measure to the area model that has been used to represent multiplication, and use this connection to justify the formula for the area of a rectangle.
  • Develop, understand and use formulas to find the area of rectangles, related triangles and parallelograms and learn to measure the necessary attributes of shapes.
  • Recognize volume as an attribute of three-dimensional space and understand they can quantify volume by finding the total number of same-sized units of volume that fill the space without gaps or overlaps.
  • Decompose three-dimensional shapes to develop strategies for determining surface area.
  • Develop strategies to determine the volumes of prisms by layering.
  Select and apply appropriate standard (customary and metric) units and tools to measure length, area, volume, weight, time, temperature, and the size of angles.
  • Select appropriate units, strategies, and tools to solve problems that involve estimating and measuring perimeter, area and volume.
  • Develop facility in measuring with fractional parts of linear units.
  • Understand that a square that is 1 unit on a side is the standard unit for measuring area.
  • Understand that a cube that is 1 unit on an edge is the standard unit for measuring volume.
  • Select and apply appropriate units, strategies and tools to solve problems that involve estimating and measuring weight, time and temperature.
  • Measure and classify angles.
  Select and use benchmarks ( inch, 2 liters, 5 pounds, etc.) to estimate measurements.
  • Develop strategies for estimating measurements using appropriate benchmarks, both standard units such as 1 foot and nonstandard units such as the length a book.
  • Learn to use strategies involving multiplicative reasoning to estimate measurements (i.e. estimating their teacher's height to be one and a quarter times the student's own height).
  • Estimate angle measure using a right angle as the benchmark.
  Understand, determine, and apply area of polygons.
  • Use physical models, such as geoboards, to develop and make sense of area formulas.
  • Use knowledge of area of simpler shapes to help find area of more complex shapes.
  • Understand and apply formulas to find area of triangles and quadrilaterals.
  • Solve problems related to and using area, including in real-world settings.
  Understand and apply similarity, with connections to proportion.
  • Understand that two objects are similar if they have the same shape (i.e., corresponding angles are congruent) but not necessarily the same size.
  • Understand similarity in terms of a scale factor between corresponding lengths in similar objects (i.e., similar objects are related by transformations of magnifying or shrinking).
  • Understand that relationships of lengths within similar objects are preserved (i.e., ratios of corresponding sides in similar objects are equal).
  • Understand that congruent figures are similar with a scale factor of 1.
  • Use understanding of similarity to solve problems in a variety of contexts.
  Understand, determine, and apply surface area and volume of prisms and cylinders and circumference and area of circles.
  • Find the area of more complex two-dimensional shapes, such as pentagons, hexagons, or irregular shaped regions, by decomposing the complex shapes into simpler shapes, such as triangles.
  • Understand that the ratio of the circumference to the diameter of a circle is constant and equal to p, and use this fact to develop a formula for the circumference of a circle.
  • Understand that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram.
  • Develop and justify strategies for determining the surface area of prisms and cylinders by determining the areas of shapes that comprise the surface.
  • By decomposing prisms and cylinders by slicing them, develop and understand formulas for their volumes (Volume = Area of base x Height).
  • Select appropriate two-and three-dimensional shapes to model real-world situations and solve a variety of problems (including multi-step problems) involving surface area, area and circumference of circles, and volume of prisms and cylinders.
  Analyze two-dimensional space and figures by using distance, angle, coordinates, and transformations.
  • Explore and explain the relationships among angles when a transversal cuts parallel lines.
  • Use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and apply this fact about triangles to find unknown measures of angles.
  • Understand and explain how particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines (e.g., "slope triangles").
  • Use reasoning about similar triangles to solve a variety of problems, including those that involve determining heights and distances.
  • Explain why the Pythagorean Theorem is valid by using a variety of methods — for example, by decomposing a square in different ways.
  • Apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane and to measure lengths and analyze polygons.
  • Understand and apply transformations — reflection, translation, rotation, and dilation, and understand similarity, congruence, and symmetry in terms of transformations.
  Visualize, represent, and describe three-dimensional shapes.
  • Recognize and draw two-dimensional representations of three-dimensional figures, including nets, front-side-top views, and perspective drawings.
  • Identify and describe three-dimensional shapes, including prisms, pyramids, cylinders, and spheres.
  • Examine, build, compose, and decompose three-dimensional objects, using a variety of tools, including paper-and-pencil, geometric models, and dynamic geometry software.
  • Use visualization and three-dimensional shapes to solve problems, especially in real-world settings.
  Represent and solve geometric problems by specifying locations using coordinates.
  • Rectangular coordinates are the focus of the study of coordinate geometry in the core curriculum. However, students should recognize that the location of a point can be described in other ways, such as by using angle and distance (as in polar coordinates or bearings) or using latitude and longitude. The study of coordinate geometry includes investigating conjectures, modeling, and solving problems. By using inductive and deductive reasoning with coordinates, properties of geometric objects can be conjectured and proven.
  • Coordinates can be used to describe points, lines, and other two- and three-dimensional figures. Transformations of these objects can be described using coordinate rules. Analysis of the relationships of geometric objects includes the use of formulas for distance, midpoint, and slope, and the Pythagorean theorem. Students should find and analyze equations that represent lines, circles and parabolas. (Students should be introduced to the other conic sections—ellipses and hyperbolas). In three dimensions, students should be able to plot points using rectangular coordinates.
  Understand and apply the basic principles of transformational geometry.
  • Transformations, such as reflections and rotations, are mappings that move points. Students should be familiar with three classes of transformations: (1) transformations that preserve distance (called isometries or rigid motions, such as reflections, rotations, translations), (2) transformations that preserve shape (such as size transformations, dilations, or similarity transformations), and (3) transformations that change distance and shape (e.g., shears). Students should recognize similarity and congruence in terms of certain transformations.
  • Students should be able to identify, create, describe, and justify transformations using multiple representations. They should be able to find and describe an image under a given transformation or composition of transformations. Students should also be able to identify the transformations that produce a given image. Transformations should be represented algebraically (using coordinate rules, matrices, vectors), and those representations should be used to analyze and reason about transformations.
  Understand and apply properties and relationships of geometric figures.
  • Students should be able to visualize, describe, reason about, prove, and apply properties and relationships of two- and three-dimensional objects. Specific geometric skills students should demonstrate include visualizing, drawing, geometric modeling, making and testing conjectures, and using inductive and deductive reasoning.
  • Properties and relationships of geometric objects should be examined and justified, including similarity, congruence, and measurement. Objects should be represented with drawings, coordinates, and matrices; and transformations of the objects should be investigated.
  • The primary focus should be on two-dimensional objects, their properties and relationships. Particular emphasis should be given to properties of angles, lines, polygons, and circles. In three dimensions, students should be able to visualize, draw, and determine measurements of simple three-dimensional shapes.
  • Measurement skills and concepts should be included in the study of geometry, including finding perimeter, area, volume and surface area (much of which is studied in middle school). Estimation, appropriate units, dimensional analysis, and judgments about accuracy should be part of the study of measurement.
  Use trigonometry based on triangles and circles to solve problems about length and angle measures.
  • Students should study trigonometry with respect to right triangles, general triangles, circles, and periodic relationships. Included in the study of right triangle trigonometry are the trigonometric ratios, the Pythagorean theorem and its converse, and the two special-case triangles, 30°—60°—90° and 45°—45°—90°. Trigonometry should be extended beyond right triangles to general triangles using the Law of Sines and Law of Cosines.
  • Examining right triangles in relation to the unit circle extends analysis to general periodic relationships. Degree and radian measure should be studied. The analysis of trigonometric functions includes: domain and range, period, amplitude, and vertical and horizontal shifts. Students should be able to recognize and model relevant periodic phenomenon with trigonometric functions.
  • Students should use trigonometry to solve problems. Students should reason about, reason with, and apply fundamental trigonometric relationships, including sin2 x + cos2x = 1, tanx = sinx/cosx, and cosx = sin (90 — x).
  Uses diagrams consisting of vertices and edges (vertex-edge graphs) to model and solve problems.
  • Vertex-edge graphs are diagrams consisting of vertices (points) and edges (line segments or arcs) connecting some of the vertices. The term "vertex-edge graph" is used to distinguish this type of graph from other graphs, such as function graphs or data plots. Nevertheless, vertex-edge graphs are often simply called graphs, especially in college mathematics courses. Vertex-edge graphs are also sometimes called networks, discrete graphs, or finite graphs. Whatever term is used, a vertex-edge graph shows relationships and connections among objects, such as in a road network, a telecommunications network, or a family tree.
  • Within the context of school geometry, which is fundamentally the study of shape, vertex-edge graphs represent, in a sense, the situation of no shape. That is, vertex-edge graphs are geometric models consisting of vertices and edges in which shape is not essential, only the connections among vertices are essential. These graphs are widely used in business and industry to solve problems about networks, paths, and relationships among a finite number of objects (such as, analyzing a computer network; optimizing the route used for snowplowing, garbage collection, or visiting business clients; scheduling committee meetings to avoid conflicts; or planning a large construction project to finish on time).
  • Students should understand, analyze, and apply vertex-edge graphs to model and solve problems related to paths, circuits, networks, and relationships among a finite number of elements, in real-world and abstract settings. Important vertex-edge graph topics for the high school curriculum include: Euler and Hamilton paths and circuits, the traveling salesman problem (TSP), minimum spanning trees, critical paths, shortest paths, and vertex coloring. These topics can be compared and contrasted in terms of algorithms, optimization, properties, and types of problems that can be solved. Students should represent and analyze vertex-edge graphs using adjacency matrices. Some students may also analyze and interpret powers of an adjacency matrix. This important material on vertex-edge graphs may be addressed as part of instruction in geometry or when teaching matrices or in separate mini-units.

  Quantitative Literacy
  Understand and apply number operations and properties.
  • As problem situations become more sophisticated in high school, the need arises to build on and expand students' understanding of number, operations on numbers, properties of numbers, and different number systems. Students should solve problems with solutions involving natural numbers, integers, rational numbers, real numbers, and complex numbers. Students should understand and apply the properties of the operations within the number systems.
  • Students should understand and apply matrices and the operations of matrix addition and multiplication. They should compare and contrast the properties of matrix operations with the properties of operations on real numbers.
  • Students should use mental, pencil-paper, and technology-based computation techniques. Given a variety of situations, students should determine the reasonableness of computation results. Students should develop skills in estimating, approximating, and judging the size and appropriateness of numbers.
  Understand and apply some basic mathematics of decision making in a democratic society.
  • Two fundamental aspects of life in a modern democratic society are voting and the Internet. Social Decision Making as described here includes a mathematical analysis of voting. Some of the mathematics of the Internet is included in the next topic.
  • To be informed and productive citizens in a democratic society, students should understand and apply basic voting methods, such as majority, plurality, runoff, approval, the Borda method (in which points are assigned to preferences), and the Condorcet method (in which each pair of candidates is run off head to head). Understanding these voting methods, and the issues associated with all voting methods, can help ensure fairer elections when there are more than two candidates. This important topic may only take a couple days in the entire high school curriculum, and could be taught in a social studies class. Related to the idea of social decision making, some students may also learn about mathematical concepts and methods of fair division and apportionment.
  Understand and apply some basic mathematics of information processing and the Internet.
  • We live in a society in which the Internet is ubiquitous. To be informed consumers and citizens in the information-dense modern world permeated by the Internet, students should have a basic mathematical understanding of some of the issues of information processing on the Internet. For example, when making an online purchase, mathematics is used to help you find what you want, encrypt your credit card number so that you can safely buy it, send your order accurately to the vendor, and, if your order is immediately downloaded, as when purchasing software, music, or video, ensure that your download occurs quickly and error-free.
  • Students should understand and apply elementary set theory and logic, as used in Internet searches. Students should also understand and apply basic number theory, including modular arithmetic, as used in cryptography. These topics are not only fundamental to information processing on the Internet, but they are also important mathematical topics in their own right with applications in many other areas. These topics may be included as part of instruction in other areas, such as number and operations or proof, or they could be included as separate mini-units. Some students may also learn about error-detecting and error-correcting codes and data compression through Huffman codes.
  Understand and apply the mathematics of systematic counting.
  • Systematic counting is sometimes more formally called combinatorics. This includes mathematical concepts and methods needed to solve counting problems, such as determining how many computer passwords are possible using two letters and four digits, how many different license plate numbers or telephone numbers are possible, or how many different pizzas you can order if you choose three toppings from six available toppings.
  • Students should understand and apply basic counting methods including systematic listing, tree diagrams, and the multiplication principle of counting. They should understand the importance of ordering and repetition when attempting to count the number of possible choices from a collection. They should understand and apply permutations, combinations, and combinatorial reasoning.