Skills
K-12

Number & Operations (K-8)
K-2  3-5  6-8
 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.

Algebra
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
 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.