Common Core State Standards for Mathematics

Know number names and the count sequence.

Count to 100 by ones and by tens.

Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no...

Count to tell the number of objects.

Understand the relationship between numbers and quantities; connect counting to cardinality.

When counting objects, say the number names in the standard order, pairing each object with one and only one number name...

Understand that the last number name said tells the number of objects counted. The number of objects is the same regardl...

Understand that each successive number name refers to a quantity that is one larger.

Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, o...

Compare numbers.

Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in ano...

Compare two numbers between 1 and 10 presented as written numerals.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Describe objects in the environment using names of shapes, and describe the relative positions of these objects using te...

Correctly name shapes regardless of their orientations or overall size.

Identify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional ("solid").

Analyze, compare, create, and compose shapes.

Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to d...

Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.

Compose simple shapes to form larger shapes.

Describe and compare measurable attributes.

Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single ...

Directly compare two objects with a measurable attribute in common, to see which object has "more of"/"less of" the attr...

Classify objects and count the number of objects in each category.

Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

Work with numbers 11—19 to gain foundations for place value.

Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and...

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situ...

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to repr...

Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and reco...

For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or draw...

Fluently add and subtract within 5.

Reason with shapes and their attributes.

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g...

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-...

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, an...

Measure lengths indirectly and by iterating length units.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the le...

Tell and write time.

Tell and write time in hours and half-hours using analog and digital clocks.

Represent and interpret data.

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of ...

Extend the counting sequence.

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of obj...

Understand place value.

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as spe...

10 can be thought of as a bundle of ten ones — called a "ten."

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (an...

Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with t...

Use place value understanding and properties of operations to add and subtract.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple...

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoni...

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), usin...

Represent and solve problems involving addition and subtraction.

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting to...

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using ...

Understand and apply properties of operations and the relationship between addition and subtraction.

Apply properties of operations as strategies to add and subtract.

Understand subtraction as an unknown-addend problem.

Add and subtract within 20.

Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as countin...

Work with addition and subtraction equations.

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or fals...

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.

Reason with shapes and their attributes.

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces...

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thir...

Measure and estimate lengths in standard units.

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and m...

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how th...

Estimate lengths using units of inches, feet, centimeters, and meters.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard...

Relate addition and subtraction to length.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g.,...

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbe...

Work with time and money.

Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.

Represent and interpret data.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve ...

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measu...

Understand place value.

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals...

100 can be thought of as a bundle of ten tens — called a "hundred."

The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine...

Count within 1000; skip-count by 5s, 10s, and 100s.

Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to r...

Use place value understanding and properties of operations to add and subtract.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relatio...

Add up to four two-digit numbers using strategies based on place value and properties of operations.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of opera...

Mentally add 10 or 100 to a given number 100—900, and mentally subtract 10 or 100 from a given number 100—900.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Represent and solve problems involving addition and subtraction.

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taki...

Add and subtract within 20.

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-dig...

Work with equal groups of objects to gain foundations for multiplication.

Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counti...

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; w...

Reason with shapes and their attributes.

Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., hav...

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition ...

Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters ...

Represent and interpret data.

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-s...

Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by ...

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Recognize area as an attribute of plane figures and understand concepts of area measurement.

A square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to ...

A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units...

Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

Relate area to the operations of multiplication and addition.

Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be...

Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world an...

Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum...

Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and ad...

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area meas...

Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the s...

Use place value understanding and properties of operations to perform multi-digit arithmetic.

Use place value understanding to round whole numbers to the nearest 10 or 100.

Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, an...

Multiply one-digit whole numbers by multiples of 10 in the range 10—90 (e.g., 9 × 80, 5 × 60) using strategies based on ...

Develop understanding of fractions as numbers.

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a ...

Understand a fraction as a number on the number line; represent fractions on a number line diagram.

Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it ...

Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting inte...

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalen...

Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comp...

Represent and solve problems involving multiplication and division.

Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.

Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56...

Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and meas...

Determine the unknown whole number in a multiplication or division equation relating three whole numbers.

Understand properties of multiplication and the relationship between multiplication and division.

Apply properties of operations as strategies to multiply and divide.

Understand division as an unknown-factor problem.

Multiply and divide within 100.

Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (...

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing ...

Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using ...

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify t...

Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or...

Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded...

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min,...

Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects...

Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

Represent and interpret data.

Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving ...

Geometric measurement: understand concepts of angle and measure angles.

Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts ...

An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the f...

An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the wh...

Generalize place value understanding for multi-digit whole numbers.

Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place ...

Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-dig...

Use place value understanding to round multi-digit whole numbers to any place.

Use place value understanding and properties of operations to perform multi-digit arithmetic.

Fluently add and subtract multi-digit whole numbers using the standard algorithm.

Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using stra...

Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies base...

Extend understanding of fraction equivalence and ordering.

Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention t...

Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or num...

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decompositio...

Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction,...

Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denomina...

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

Understand a fraction a/b as a multiple of 1/b.

Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.

Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and ...

Understand decimal notation for fractions, and compare decimal fractions.

Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two...

Use decimal notation for fractions with denominators 10 or 100.

Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two...

Use the four operations with whole numbers to solve problems.

Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as man...

Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations wit...

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, includ...

Gain familiarity with factors and multiples.

Find all factor pairs for a whole number in the range 1—100. Recognize that a whole number is a multiple of each of its ...

Generate and analyze patterns.

Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not ex...

Graph points on the coordinate plane to solve real-world and mathematical problems.

Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines...

Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and int...

Classify two-dimensional figures into categories based on their properties.

Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that c...

Classify two-dimensional figures in a hierarchy based on properties.

Convert like measurement units within a given measurement system.

Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m...

Represent and interpret data.

Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractio...

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to mea...

A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units...

Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving ...

Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that...

Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with w...

Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by ...

Understand the place value system.

Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place t...

Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns i...

Read, write, and compare decimals to thousandths.

Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100...

Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record t...

Use place value understanding to round decimals to any place.

Perform operations with multi-digit whole numbers and with decimals to hundredths.

Fluently multiply multi-digit whole numbers using the standard algorithm.

Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies ba...

Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on pl...

Use equivalent fractions as a strategy to add and subtract fractions.

Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivale...

Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unli...

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving divisi...

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequ...

Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fractio...

Interpret multiplication as scaling (resizing), by:

Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performi...

Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given numbe...

Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models...

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit...

Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

Interpret division of a whole number by a unit fraction, and compute such quotients.

Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers b...

Write and interpret numerical expressions.

Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating t...

Analyze patterns and relationships.

Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form...

Apply and extend previous understandings of arithmetic to algebraic expressions.

Write and evaluate numerical expressions involving whole-number exponents.

Write, read, and evaluate expressions in which letters stand for numbers.

Write expressions that record operations with numbers and with letters standing for numbers.

Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one o...

Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-wo...

Apply the properties of operations to generate equivalent expressions.

Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which va...

Reason about and solve one-variable equations and inequalities.

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if...

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand t...

Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases i...

Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical pr...

Represent and analyze quantitative relationships between dependent and independent variables.

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an e...

Solve real-world and mathematical problems involving area, surface area, and volume.

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or ...

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropria...

Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side j...

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface...

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g....

Compute fluently with multi-digit numbers and find common factors and multiples.

Fluently divide multi-digit numbers using the standard algorithm.

Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whol...

Apply and extend previous understandings of numbers to the system of rational numbers.

Understand that positive and negative numbers are used together to describe quantities having opposite directions or val...

Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar fro...

Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that th...

Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize tha...

Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position...

Understand ordering and absolute value of rational numbers.

Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.

Write, interpret, and explain statements of order for rational numbers in real-world contexts.

Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value a...

Distinguish comparisons of absolute value from statements about order.

Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use...

Understand ratio concepts and use ratio reasoning to solve problems.

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context o...

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalen...

Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, ...

Solve unit rate problems including those involving unit pricing and constant speed.

Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems...

Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or divid...

Develop understanding of statistical variability.

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts fo...

Understand that a set of data collected to answer a statistical question has a distribution which can be described by it...

Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a m...

Summarize and describe distributions.

Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

Summarize numerical data sets in relation to their context, such as by:

Reporting the number of observations.

Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.

Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute de...

Relating the choice of measures of center and variability to the shape of the data distribution and the context in which...

Use properties of operations to generate equivalent expressions.

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coeff...

Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how th...

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whol...

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequa...

Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rationa...

Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rationa...

Draw, construct, and describe geometrical figures and describe the relationships between them.

Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale ...

Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constru...

Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right r...

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation...

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve s...

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects...

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational nu...

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent add...

Describe situations in which opposite quantities combine to make 0.

Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether...

Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance betw...

Apply properties of operations as strategies to add and subtract rational numbers.

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational...

Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to s...

Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non...

Apply properties of operations as strategies to multiply and divide rational numbers.

Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates i...

Solve real-world and mathematical problems involving the four operations with rational numbers.

Analyze proportional relationships and use them to solve real-world and mathematical problems.

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured...

Recognize and represent proportional relationships between quantities.

Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or g...

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of ...

Represent proportional relationships by equations.

Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special at...

Use proportional relationships to solve multistep ratio and percent problems.

Use random sampling to draw inferences about a population.

Understand that statistics can be used to gain information about a population by examining a sample of the population; g...

Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate...

Draw informal comparative inferences about two populations.

Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring...

Use measures of center and measures of variability for numerical data from random samples to draw informal comparative i...

Investigate chance processes and develop, use, and evaluate probability models.

Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event...

Approximate the probability of a chance event by collecting data on the chance process that produces it and observing it...

Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed f...

Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine proba...

Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the samp...

Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an even...

Design and use a simulation to generate frequencies for compound events.

Work with radicals and integer exponents.

Know and apply the properties of integer exponents to generate equivalent numerical expressions.

Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a po...

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small qu...

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific n...

Understand the connections between proportional relationships, lines, and linear equations.

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportion...

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in t...

Analyze and solve linear equations and pairs of simultaneous linear equations.

Solve linear equations in one variable.

Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show wh...

Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressi...

Analyze and solve pairs of simultaneous linear equations.

Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of t...

Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. ...

Solve real-world and mathematical problems leading to two linear equations in two variables.

Define, evaluate, and compare functions.

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set o...

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tabl...

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functi...

Use functions to model relationships between quantities.

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial val...

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function...

Understand congruence and similarity using physical models, transparencies, or geometry software.

Verify experimentally the properties of rotations, reflections, and translations:

Lines are taken to lines, and line segments to line segments of the same length.

Angles are taken to angles of the same measure.

Parallel lines are taken to parallel lines.

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a seque...

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequenc...

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created ...

Understand and apply the Pythagorean Theorem.

Explain a proof of the Pythagorean Theorem and its converse.

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical proble...

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical pro...

Know that there are numbers that are not rational, and approximate them by rational numbers.

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal exp...

Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately o...

Investigate patterns of association in bivariate data.

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two ...

Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots th...

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slop...

Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and rel...

Perform arithmetic operations on polynomials

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of add...

Understand the relationship between zeros and factors of polynomials

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), ...

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph o...

Use polynomial identities to solve problems

Prove polynomial identities and use them to describe numerical relationships.

(+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, whe...

Rewrite rational expressions

Rewrite simple rational expressions in different forms; write a(x /b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), ...

(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtrac...

Create equations that describe numbers or relationships

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear ...

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate a...

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solut...

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Understand solving equations as a process of reasoning and explain the reasoning

Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, ...

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may aris...

Solve equations and inequalities in one variable

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Solve quadratic equations in one variable.

Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² =...

Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic f...

Solve systems of equations

Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a m...

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations i...

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphi...

(+) Represent a system of linear equations as a single matrix equation in a vector variable.

(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matr...

Represent and solve equations and inequalities graphically

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plan...

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the so...

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a str...

Interpret the structure of expressions

Interpret expressions that represent a quantity in terms of its context

Interpret parts of an expression, such as terms, factors, and coefficients.

Interpret complicated expressions by viewing one or more of their parts as a single entity.

Use the structure of an expression to identify ways to rewrite it.

Write expressions in equivalent forms to solve problems

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by t...

Factor a quadratic expression to reveal the zeros of the function it defines.

Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Use the properties of exponents to transform expressions for exponential functions.

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to sol...

Build a function that models a relationship between two quantities

Write a function that describes a relationship between two quantities

Determine an explicit expression, a recursive process, or steps for calculation from a context.

Combine standard function types using arithmetic operations.

(+) Compose functions.

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, an...

Build new functions from existing functions

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (bo...

Find inverse functions.

Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inver...

(+) Verify by composition that one function is the inverse of another.

(+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

(+) Produce an invertible function from a non-invertible function by restricting the domain.

(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems inv...

Understand the concept of a function and use function notation

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of...

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notati...

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

Interpret functions that arise in applications in terms of the context

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms o...

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified...

Analyze functions using different representations

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology ...

Graph linear and quadratic functions and show intercepts, maxima, and minima.

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing e...

Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing p...

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties o...

Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmet...

Use the properties of exponents to interpret expressions for exponential functions.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tabl...

Construct and compare linear, quadratic, and exponential models and solve problems

Distinguish between situations that can be modeled with linear functions and with exponential functions.

Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal...

Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to anothe...

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description o...

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linear...

For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is ...

Interpret expressions for functions in terms of the situation they model

Interpret the parameters in a linear or exponential function in terms of a context.

Extend the domain of trigonometric functions using the unit circle

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers...

(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use t...

(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Model periodic phenomena with trigonometric functions

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasi...

(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using...

Prove and apply trigonometric identities

Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), ...

(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Understand and apply theorems about circles

Prove that all circles are similar.

Identify and describe relationships among inscribed angles, radii, and chords.

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral insc...

(+) Construct a tangent line from a point outside a given circle to the circle.

Find arc lengths and areas of sectors of circles

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and d...

Experiment with transformations in the plane

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined n...

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as fu...

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it on...

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parall...

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph pape...

Understand congruence in terms of rigid motions

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a ...

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corre...

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms o...

Prove geometric theorems

Prove theorems about triangles.

Prove theorems about parallelograms.

Prove theorems about lines and angles.

Make geometric constructions

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective de...

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Explain volume formulas and use them to solve problems

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, py...

(+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid fi...

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Visualize relationships between two-dimensional and three-dimensional objects

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objec...

Translate between the geometric description and the equation for a conic section

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find th...

Derive the equation of a parabola given a focus and directrix.

(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distanc...

Use coordinates to prove simple geometric theorems algebraically

Use coordinates to prove simple geometric theorems algebraically.

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the e...

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formul...

Apply geometric concepts in modeling situations

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human t...

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic...

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints...

Understand similarity in terms of similarity transformations

Verify experimentally the properties of dilations given by a center and a scale factor:

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing thr...

The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar...

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove theorems involving similarity

Prove theorems about triangles.

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Define trigonometric ratios and solve problems involving right triangles

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to d...

Explain and use the relationship between the sine and cosine of complementary angles.

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Apply trigonometry to general triangles

(+) Prove the Laws of Sines and Cosines and use them to solve problems.

(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right tri...

(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendic...

Perform arithmetic operations with complex numbers.

Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.

Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply co...

(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

Represent complex numbers and their operations on the complex plane.

(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers),...

(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex pla...

(+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a ...

Use complex numbers in polynomial identities and equations.

Solve quadratic equations with real coefficients that have complex solutions.

(+) Extend polynomial identities to the complex numbers.

(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Reason quantitatively and use units to solve problems.

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units c...

Define appropriate quantities for the purpose of descriptive modeling.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Extend the properties of exponents to rational exponents.

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponen...

Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Use properties of rational and irrational numbers.

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational ...

Represent and model with vector quantities.

(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line seg...

(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a termina...

(+) Solve problems involving velocity and other quantities that can be represented by vectors.

Perform operations on vectors.

(+) Add and subtract vectors.

Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vec...

Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

Understand vector subtraction v - w as v + (-w), where -w is the additive inverse of w, with the same magnitude as w and...

(+) Multiply a vector by a scalar.

Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar mu...

Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ? ...

Perform operations on matrices and use matrices in applications.

(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role...

(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vecto...

(+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in ter...

(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

(+) Add, subtract, and multiply matrices of appropriate dimensions.

(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative op...

Understand independence and conditional probability and use them to interpret data

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes...

Understand that two events A and B are independent if the probability of A and B occurring together is the product of th...

Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying ...

Construct and interpret two-way frequency tables of data when two categories are associated with each object being class...

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situati...

Use the rules of probability to compute probabilities of compound events in a uniform probability model

Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the a...

Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.

(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and inte...

(+) Use permutations and combinations to compute probabilities of compound events and solve problems.

Understand and evaluate random processes underlying statistical experiments

Understand statistics as a process for making inferences about population parameters based on a random sample from that ...

Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how rand...

Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of ...

Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parame...

Evaluate reports based on data.

Summarize, represent, and interpret data on a single count or measurement variable

Represent data with plots on the real number line (dot plots, histograms, and box plots).

Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquart...

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of e...

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percenta...

Summarize, represent, and interpret data on two categorical and quantitative variables

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context...

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

Informally assess the fit of a function by plotting and analyzing residuals.

Fit a linear function for a scatter plot that suggests a linear association.

Interpret linear models

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Compute (using technology) and interpret the correlation coefficient of a linear fit.

Distinguish between correlation and causation.

Calculate expected values and use them to solve problems

(+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; ...

(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabiliti...

(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assig...

Use probability to evaluate outcomes of decisions

(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

Find the expected payoff for a game of chance.

Evaluate and compare strategies on the basis of expected values.

(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hocke...