North Dakota Math Standards

725 standards, Kindergarten through AP.

Kindergarten

K.CC Counting and Cardinality
K.CC.1Count to 100 by ones and by tens. Count backward from 20 by ones.
K.CC.2Count forward beginning from a given number within 100. Count backward from a given number within 10.
K.CC.3Write numbers sequentially from 0 to 20. Write a given number from 0 to 20.
K.CC.4 Understand the relationship between numbers and quantities up to 20; connect counting to cardinality.
K.CC.4.aUse one to one correspondence when counting objects.
K.CC.4.bUnderstand that the last number name said tells the number of objects counted, regardless of their arrangement or order in which they were counted.
K.CC.4.cUnderstand that each successive number name refers to a quantity that is one more.
K.CC.5 Count to answer “how many?” questions.
K.CC.5.aTell how many objects up to 20 are in an arranged pattern (e.g., a line or an array) or up to 10 objects in a scattered configuration.
K.CC.5.bRepresent a number of objects up to 20 with a written numeral.
K.CC.5.cGiven a number from 1-20, count out that many objects.
K.CC.6Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, using groups of up to 10 objects.
K.CC.7Compare two numbers between 1 and 10 presented as written numerals.
K.G Geometry
K.G.1Describe objects in the environment using names of shapes and solids (squares, circles, triangles, rectangles, cubes, and spheres).
K.G.2Correctly name shapes and solids (squares, circles, triangles, rectangles, cubes, and spheres) regardless of their orientations or overall size.
K.G.3Identify shapes and solids (squares, circles, triangles, rectangles, cubes, and spheres) as two-dimensional or three-dimensional.
K.G.4Compare and classify two-dimensional shapes (squares, circles, triangles, rectangles) of different sizes and orientations, using informal language to describe their similarities, differences, and attributes.
K.G.5No content for this standard code.
K.G.6Compose a new shape by combining two or more simple shapes.
K.MD Measurement and Data
K.MD.1Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.
K.MD.2Compare two objects with a common measurable attribute and describe the difference.
K.MD.3Classify objects into given categories limiting the number in each category to 10 or less. Count the numbers of objects in each category and sort the categories by count.
K.NBT Number and Operations in Base Ten
K.NBT.1Compose and decompose numbers from 11 to 19 using a group of ten ones and additional ones. Record each composition or decomposition with a drawing or equation.
K.OA Operations and Algebraic Thinking
K.OA.1Represent addition and subtraction in a variety of ways.
K.OA.2Use an appropriate strategy to solve word problems that involve adding and subtracting within 10.
K.OA.3Decompose numbers less than or equal to 10 into multiple combinations of two parts. Record each decomposition with a drawing or equation.
K.OA.4Find the number that makes 10 when added to a given number from 1 to 9. Record with a drawing or equation.
K.OA.5Fluently add and subtract within 5.

Grade 1

1.G Geometry
1.G.1Distinguish between defining attributes versus non-defining attributes. Use defining attributes to build and draw two-dimensional shapes (squares, circles, triangles, rectangles, trapezoids, rhombuses, pentagons, hexagons, octagons).
1.G.2Compose a new shape or solid from two-dimensional shapes and/or three-dimensional solids (squares, circles, triangles, rectangles, trapezoids, rhombuses, pentagons, hexagons, octagons, cubes, spheres, cylinders, cones, triangular prisms, and rectangular prisms).
1.G.3Partition circles and rectangles into two equal shares. Describe the shares using the word halves, and use the phrase half of. Describe the whole as two of the shares.
1.MD Measurement and Data
1.MD.1Order three objects by length. Compare the lengths of two objects indirectly by using a third object.
1.MD.2Demonstrate understanding that the length measurement of an object is the number of same-size length units that span the object with no gaps or overlaps. Measure and express the length of an object using whole non-standards units.
1.MD.3Tell and write time to the hour and half-hour (including o’clock and half past) using analog and digital clocks.
1.MD.4Organize, represent, and interpret data with up to three categories. Ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.
1.MD.5Identify and tell the value of a dollar bill, quarter, dime, nickel, and penny.
1.MD.6Count and tell the value of combinations of dimes and pennies up to one dollar.
1.NBT Number and Operations in Base Ten
1.NBT.1Count forward and backward within 120, starting at any given number. Read and write numerals within 120. Represent a number of objects up to 120 with a written numeral.
1.NBT.2 Demonstrate understanding that the two digits of a two-digit number represent amounts of tens and ones, including:
1.NBT.2.a10 can be thought of as a bundle of ten ones — called a “ten.”
1.NBT.2.bThe numbers from 11 to 19 are composed of a ten and additional ones.
1.NBT.2.cMultiples of 10 up to 90 represent a number of tens and 0 ones.
1.NBT.3Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
1.NBT.4 Demonstrate understanding of place value when adding two-digit numbers within 100. Use concrete models or drawing strategies based on place value, properties of operations, and/or the relationship between addition and subtraction to add and subtract within 100. Relate the strategy to a written method and explain the reasoning used.
1.NBT.4.aAdd a two-digit number and a one-digit number.
1.NBT.4.bAdd a two-digit number and a multiple of 10.
1.NBT.5Mentally add or subtract 10 to or from a given two-digit number. Explain the reasoning used.
1.NBT.6Use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction to subtract multiples of 10 in the range of 10-90 from multiples of 10 in the same range resulting in a positive or zero difference. Use a written method to explain the strategy.
1.OA Operations and Algebraic Thinking
1.OA.1Use strategies to add and subtract within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.
1.OA.2Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20.
1.OA.3Apply properties of operations as strategies to add and subtract.
1.OA.4Demonstrate understanding of subtraction as an unknown-addend problem.
1.OA.5Relate counting to addition and subtraction.
1.OA.6Use strategies to add and subtract within 20. Fluently add and subtract within 10.
1.OA.7Demonstrate understanding of the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.
1.OA.8Determine the unknown whole number in an addition or subtraction equation that uses three whole numbers.

Grade 2

2.G Geometry
2.G.1Identify trapezoids, rhombuses, pentagons, hexagons, octagons, parallelograms, quadrilaterals, cubes, spheres, cylinders, cones, triangular prisms, rectangular prisms. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.
2.G.2Partition a rectangle into rows and columns of same-size squares and count to find the total number.
2.G.3Partition circles and rectangles into two, three, or four equal shares. Describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that identical wholes can be equally divided in different ways. Demonstrate understanding that partitioning shapes into more equal shares creates smaller shares.
2.MD Measurement and Data
2.MD.1Select and use appropriate tools to measure the length of an object.
2.MD.10Draw picture graphs and bar graphs with single-unit scales to represent data sets with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.
2.MD.2Measure the length of an object using two different standard units of measurement. Describe how the two measurements relate to the size of the units chosen.
2.MD.3Estimate lengths using units of inches, feet, centimeters, and meters.
2.MD.4Measure to determine how much longer one object is than another, expressing the difference with a standard unit of measurement.
2.MD.5No content for this standard code.
2.MD.6Represent whole numbers on a number line diagram with equally spaced points. Represent whole-number sums and differences within 100 on a number line diagram.
2.MD.7Tell and write time to the nearest five minutes (including quarter after and quarter to) with a.m. and p.m. using analog and digital clocks.
2.MD.8Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and symbols appropriately.
2.MD.9Generate data by measuring lengths of objects to the nearest whole standard unit. Show the measurements by making a line plot, using a horizontal scale marked off in whole-number units.
2.NBT Number and Operations in Base Ten
2.NBT.1 Demonstrate understanding that the three digits of a three-digit number represent amounts of hundreds, tens, and ones, including:
2.NBT.1.a100 can be thought of as a bundle of ten tens called a “hundred”.
2.NBT.1.bMultiples of 100 represent a number of hundreds, 0 tens, and 0 ones.
2.NBT.2Count forward and backward from any given number within 1000. Skip-count by 5s, 10s, and 100s.
2.NBT.3Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
2.NBT.4Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, recording the results of comparisons with the symbols >, =, and <.
2.NBT.5Use strategies based on place value, properties of operations, and/or the relationship between addition and subtraction to fluently add and subtract within 100.
2.NBT.6Use strategies based on place value and properties of operations to add up to four two-digit numbers.
2.NBT.7Demonstrate understanding of place value within 1000 when adding and subtracting three-digit numbers. Use concrete models or drawings and strategies based on place value, properties of operation, and/or the relationship between addition and subtraction to add and subtract within 1000. Use a written method to explain the strategy.
2.NBT.8Mentally add or subtract 10 or 100 to or from a given number between 100 and 900.
2.OA Operations and Algebraic Thinking
2.OA.1Use strategies to add and subtract within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.
2.OA.2Use mental strategies to fluently add and subtract within 20.
2.OA.3Determine whether a given number of objects up to 20 is odd or even. Write an equation to represent an even number using two equal addends or groups of 2.
2.OA.4Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns. Write an equation to express the total as a sum of equal addends.

Grade 3

3.G Geometry
3.G.1Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals. Draw examples of quadrilaterals that do not belong to any of these subcategories.
3.G.2Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.
3.MD Measurement and Data
3.MD.1Tell and write time to the nearest minute and measure time intervals in minutes. Solve elapsed time word problems on the hour and the half hour, using a variety of strategies.
3.MD.2Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units.
3.MD.3Draw scaled picture graphs and scaled bar graphs to represent data sets with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs.
3.MD.4Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked in appropriate units—whole numbers, halves, or quarters.
3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement.
3.MD.5.aA square with a side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.
3.MD.5.bA plane figure, which can be covered without gaps or overlaps by n unit squares, is said to have an area of n square units.
3.MD.6Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).
3.MD.7 Relate area to the operations of multiplication and addition.
3.MD.7.aFind the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
3.MD.7.bMultiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
3.MD.7.cUse 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 of a b and a ח c. Use area models to represent the distributive property in mathematical reasoning.
3.MD.7.dRecognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
3.MD.8Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths. Find an unknown side length. Exhibit rectangles with the same perimeter and different area or with the same area and different perimeters.
3.NBT Number and Operations in Base Ten
3.NBT.1Use place value understanding to round whole numbers to the nearest 10 or 100.
3.NBT.2Using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction, fluently add and subtract within 1000.
3.NBT.3Using strategies based on place value and properties of operations, multiply one-digit whole numbers by multiples of 10 in the range 10-90.
3.NF Number and Operations – Fractions
3.NF.1Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts. Understand a fraction a/b as the quantity formed by “a” parts of size 1/b.
3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
3.NF.2.aRepresent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
3.NF.2.bRepresent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.3.aUnderstand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
3.NF.3.bRecognize and generate simple equivalent fractions. Explain why the fractions are equivalent using a visual fraction model.
3.NF.3.cRecognize fractions, a/1 or a/a, that are equivalent to whole numbers. Express whole numbers as fractions, a/1 or a/a.
3.NF.3.dCompare two fractions with the same numerator or the same denominator by reasoning about their size.
3.NF.3.eRecognize that comparisons are valid only when the two fractions refer to the same whole.
3.NF.3.fRecord the results of comparisons with the symbols >, =, or <, and justify the conclusions by using a visual fraction model.
3.OA Operations and Algebraic Thinking
3.OA.1Interpret and model products of whole numbers.
3.OA.2Interpret and model whole-number quotients of whole numbers, as the number in a group or the number of groups.
3.OA.3Using drawings and equations with a symbol for an unknown number, solve multiplication and division word problems within 100 in situations involving equal groups, arrays, and measurement quantities.
3.OA.4Determine the unknown whole number in a multiplication or division equation relating three whole numbers.
3.OA.5Apply properties of operations as strategies to multiply and divide (without the use of formal terms).
3.OA.6Understand division as an unknown-factor problem.
3.OA.7Using mental strategies, fluently multiply and divide within 100.
3.OA.8Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies.
3.OA.9Identify arithmetic patterns, and explain them using properties of operations.

Grade 4

4.G Geometry
4.G.1Draw and label points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
4.G.2Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of specified size. Recognize right triangles as a category, and identify right triangles.
4.G.3Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures. Draw lines of symmetry.
4.MD Measurement and Data
4.MD.1Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb., oz.; l, ml; hr., min., sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.
4.MD.2Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Using diagrams such as number line diagrams that feature a measurement scale, to represent measurement quantities.
4.MD.3Apply the area and perimeter formulas for rectangles in real world and mathematical problems.
4.MD.4Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.
4.MD.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint. Understand concepts of angle measurement.
4.MD.5.aAn angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.
4.MD.5.bAn angle that turns through n one-degree angles is said to have an angle measure of n degrees.
4.MD.6Measure angles in whole-number degrees using a protractor. Using a protractor and ruler, draw angles of a specified measure.
4.MD.7Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems.
4.NBT Number and Operations in Base Ten
4.NBT.1Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
4.NBT.2Read and write multi-digit whole numbers to the one millions place using base-ten numerals, word form, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
4.NBT.3Use place value and/or understanding of numbers to round multi-digit whole numbers to any place.
4.NBT.4Fluently add and subtract multi-digit whole numbers to the one millions place using strategies flexibly, including the standard algorithm.
4.NBT.5Using strategies based on place value and the properties of operations, multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
4.NBT.6Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
4.NF Number and Operations – Fractions
4.NF.1Using visual fraction models, explain why a fraction a/b is equivalent to a fraction (n X a)/(n x b). Use this principle to recognize and generate equivalent fractions. Attention should focus on how the number and size of the parts differ even though the two fractions themselves are the same size.
4.NF.2By creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2, compare two fractions with different numerators and different denominators. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
4.NF.3 Understand a fraction a/b with a > 1 as a sum of unit fractions 1/b.
4.NF.3.aUnderstand addition and subtraction of fractions as joining and separating parts referring to the same whole.
4.NF.3.bDecompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition with an equation. Justify decompositions by using a visual fraction model or other strategies.
4.NF.3.cAdd and subtract mixed numbers with like denominators.
4.NF.3.dUsing visual fraction models and equations, solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.
4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
4.NF.4.aUnderstand a fraction a/b as a multiple of 1/b.
4.NF.4.bUnderstand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.
4.NF.4.cUsing visual fraction models and equations, solve word problems involving multiplication of a fraction by a whole number.
4.NF.5Express a fraction with denominator 10 as an equivalent fraction with denominator 100. Use this technique to add two fractions with respective denominators 10 and 100.
4.NF.6Use decimal notation for fractions with denominators 10 or 100.
4.NF.7Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions.
4.OA Operations and Algebraic Thinking
4.OA.1Interpret a multiplication equation as a comparison. Represent verbal statements of multiplicative comparisons as multiplication equations.
4.OA.2Use drawings and equations with a symbol for the unknown number (variable) to represent the problem. Multiply or divide to solve word problems involving multiplicative comparison, distinguishing multiplicative comparison from additive comparison.
4.OA.3Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity (variable). Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
4.OA.4Find all factor pairs for a whole number in the range 1-36. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-36 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-36 is prime or composite.
4.OA.5Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.

Grade 5

5.G Geometry
5.G.1Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (x-coordinate and x-axis, y-coordinate and y-axis).
5.G.2Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane. Interpret coordinate values of points in the context of the situation.
5.G.3Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
5.G.4Classify two-dimensional figures in a hierarchy based on properties.
5.MD Measurement and Data
5.MD.1Convert among different-sized standard measurement units within a given measurement system. Use these conversions in solving multi-step, real world problems.
5.MD.2Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.
5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
5.MD.3.aA cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.
5.MD.3.bA solid figure, which can be packed without gaps or overlaps using n unit cubes, is said to have a volume of n cubic units.
5.MD.4Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft., and improvised units.
5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
5.MD.5.aFind the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes. Show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base.
5.MD.5.bRepresent threefold whole-number products as volumes to represent the associative property of multiplication.
5.MD.5.cApply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.
5.MD.5.dRecognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problem
5.NBT Number and Operations in Base Ten
5.NBT.1Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
5.NBT.2Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
5.NBT.3 Read, write, and compare decimals to thousandths.
5.NBT.3.aRead and write decimals to thousandths using base-ten numerals, word form, and expanded form.
5.NBT.3.bCompare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
5.NBT.4Use place value understanding to round decimals to any place.
5.NBT.5Fluently multiply multi-digit whole numbers using strategies flexibly, including the standard algorithm.
5.NBT.6Using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division, find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
5.NBT.7Using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction, add, subtract, multiply, and divide decimals to hundredths. Relate the strategy to a written method and explain the reasoning used.
5.NF Number and Operations – Fractions
5.NF.1Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.
5.NF.2Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, by using visual fraction models and equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.
5.NF.3Interpret a fraction as division of the numerator by the denominator (a/b = a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers by using visual fraction models and equations to represent the problem.
5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.4.aInterpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.
5.NF.4.bFind the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles. Represent fraction products as rectangular areas.
5.NF.5 Interpret multiplication as scaling (resizing), by:
5.NF.5.aComparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
5.NF.5.bExplaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case). Explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number.
5.NF.5.cRelating the principle of fraction equivalence a/b = (n×a) / (n×b) to the effect of multiplying a/b by 1.
5.NF.6Solve real world problems involving multiplication of fractions and mixed numbers using visual fraction models and equations to represent the problem.
5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
5.NF.7.aInterpret division of a unit fraction by a non-zero whole number, and compute such quotients.
5.NF.7.bInterpret division of a whole number by a unit fraction, and compute such quotients.
5.NF.7.cSolve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions using visual fraction models and equations to represent the problem.
5.OA Operations and Algebraic Thinking
5.OA.1Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
5.OA.2Write simple expressions that record calculations with numbers. Interpret numerical expressions without evaluating them.
5.OA.3Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns. Graph the ordered pairs on a coordinate plane. Use the graph to verify relationships.
5.OA.4Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

Grade 6

6.EE Expressions and Equations
6.EE.1Write and evaluate numerical expressions involving whole-number exponents.
6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.2.aWrite expressions that record operations with numbers and with letters standing for numbers.
6.EE.2.bIdentify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient, difference, quantity, etc.); view one or more parts of an expression as a single entity.
6.EE.2.cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).
6.EE.3Apply the properties of operations to generate equivalent expressions.
6.EE.4Identify when two expressions are equivalent.
6.EE.5Understand solving an equation or inequality as a process of answering a question: Which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.EE.6Use variables to represent numbers and write expressions when solving a real world or mathematical problem. Understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.7Solve real world and mathematical problems by writing and solving equations of the form x+p=q and px = q for cases in which p, q and x are all nonnegative rational numbers.
6.EE.8Write a statement of inequality of the form x > c or the form x < c to represent a constraint or condition in a real world or mathematical problem. Recognize that inequalities of the form x > c or the form x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
6.EE.9Use variables to represent two quantities in a real world problem that change in relationship to one another. Write an equation to express one quantity (dependent variable) in terms of the other quantity (independent variable). Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
6.G Geometry
6.G.1Based on prior knowledge of area of rectangles, decompose or compose triangles to find the area of a triangle. Using knowledge of area of triangles and rectangles, compose and/or decompose triangles, special quadrilaterals, and polygons to find their areas. Apply these techniques in the context of solving real world mathematical problems.
6.G.2Using cubes of an appropriate size, pack a right rectangular prism having fractional edge lengths to find its volume. Then show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = Bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real world and mathematical problems.
6.G.3Draw polygons in the coordinate plane given coordinates for the vertices. Use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real world and mathematical problems.
6.G.4Represent three-dimensional figures using nets made up of rectangles and triangles (right prisms and pyramids whose bases are triangles and rectangles). Use the nets to find the surface area of these figures. Apply these techniques in the context of solving real world and mathematical problems.
6.NS The Number System
6.NS.1Use visual fraction models and equations to interpret and compute quotients of fractions. Use models and equations to solve word problems involving division of fractions by fractions.
6.NS.2Fluently divide multi-digit numbers using strategies flexibly, including the standard algorithm.
6.NS.3Fluently add, subtract, multiply, and divide multi-digit decimals using strategies flexibly, including the standard algorithm for each operation.
6.NS.4Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1 to 100 with a common factor as a multiple of a sum of two whole numbers with no common factor.
6.NS.5Understand that rational numbers are used together to describe quantities having opposite directions or values (may include temperature above/below zero, elevation above/below sea level, debits/credits, positive/negative electric charge, etc.). Use rational numbers to represent quantities in real world contexts, explaining the meaning of 0 in each situation.
6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes from previous grades to represent points on the line and in the plane with negative number coordinates.
6.NS.6.aRecognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line. Recognize that the opposite of the opposite of a number is the number itself, for example: –(–3)=3, and that 0 is its own opposite.
6.NS.6.bUnderstand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane. Recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
6.NS.6.cFind and position integers and other rational numbers on a horizontal or vertical number line diagram. Find and position pairs of integers and other rational numbers on a coordinate plane.
6.NS.7 Understand ordering and absolute value of rational numbers.
6.NS.7.aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.
6.NS.7.bWrite, interpret, and explain statements of order for rational numbers in real world contexts.
6.NS.7.cUnderstand the absolute value of a rational number as its distance from 0 on the number line. Interpret absolute value as magnitude for a positive or negative quantity in a real world situation.
6.NS.7.dDistinguish comparisons of absolute value from statements about order.
6.NS.8Solve real world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
6.RP Ratios and Proportional Relationships
6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
6.RP.2Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.
6.RP.3 Use tables of equivalent ratios, tape diagrams, double number line diagrams, and equations to reason about ratios and rates in real world and mathematical problems.
6.RP.3.aMake tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
6.RP.3.bSolve unit rate problems including those involving unit pricing and constant speed.
6.RP.3.cFind a percent of a quantity as a rate per 100. Solve problems involving finding the whole, given a part and the percent.
6.RP.3.dUse ratio reasoning to convert measurement units. Manipulate and transform units appropriately when multiplying or dividing quantities.
6.SP Statistics and Probability
6.SP.1Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.
6.SP.2Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
6.SP.3Recognize that a measure of center for a numerical data set summarizes all of its values using a single number, while a measure of spread (variation) describes how its values vary with a single number.
6.SP.4Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
6.SP.5 Summarize numerical data sets in relation to their context by:
6.SP.5.aReporting the number of observations.
6.SP.5.bDescribing the nature of the attribute being investigated, including how it was measured and its units of measurement.
6.SP.5.cCalculating quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data was gathered.
6.SP.5.dRelating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

Grade 7

7.EE Expressions and Equations
7.EE.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients with an emphasis on writing equivalent expressions.
7.EE.2Understand that rewriting an expression in different forms in a problem context can clarify the problem and how the quantities in it are related.
7.EE.3Solve multi-step real-life and mathematical problems posed with rational numbers in any form (positive and negative, fractions, decimals, and integers), using tools strategically. Apply properties of operations to calculate with numbers in any form. Convert between forms as appropriate. Assess the reasonableness of answers using mental computation and estimation strategies.
7.EE.4 Use variables to represent quantities in a real world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
7.EE.4.aSolve word problems leading to equations of the form px+q=r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare the algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.
7.EE.4.bSolve word problems leading to inequalities of the form x+q>r or px+q<r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.
7.G Geometry
7.G.1Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7.G.2Draw geometric shapes from given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. Use a variety of methods such as freehand, with ruler and protractor, and with technology.
7.G.3Describe the cross-sections (two-dimensional figures that result from slicing three-dimensional figures, as in plane sections) of right rectangular prisms and right rectangular pyramids.
7.G.4Know the formulas for the area and circumference of a circle and use them to solve problems. Informally derive the relationship between the circumference and area of a circle.
7.G.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve equations for an unknown angle in a figure.
7.G.6Solve real world and mathematical problems involving area of two-dimensional figures composed of polygons and/or circles, including composite figures. Use nets to solve real world and mathematical problems involving surface area of prisms and cylinders, including composite solids. Solve real world and mathematical problems involving volumes of right prisms, including composite solids.
7.NS The Number system
7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
7.NS.1.aDescribe situations in which opposite quantities combine to make 0.
7.NS.1.bUnderstand p + q as the number located a distance |q| from p on a number line, in the direction indicated by the sign of q. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real world contexts.
7.NS.1.cUnderstand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real world contexts.
7.NS.1.dApply properties of operations as strategies to fluently add and subtract rational numbers.
7.NS.2 Apply and extend previous understandings of multiplication, division, and fractions to multiply and divide rational numbers.
7.NS.2.aeading to products such as (−1)(−1)=1 and the rules for multiplying rational numbers. Interpret products of rational numbers by describing real world contexts.
7.NS.2.bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = -p/q = p/-q Interpret products of rational numbers by describing real world contexts.
7.NS.2.cApply properties of operations as strategies to fluently multiply and divide rational numbers.
7.NS.2.dConvert a rational number to a decimal using long division. Know that the decimal form of a rational number terminates or eventually repeats.
7.NS.3Solve real world and mathematical problems involving the four operations with rational numbers.
7.RP Ratios and Proportional Relationships
7.RP.1Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
7.RP.2 Recognize and represent proportional relationships between quantities.
7.RP.2.aDecide whether two quantities are in a proportional relationship by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
7.RP.2.bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
7.RP.2.cRepresent proportional relationships by equations.
7.RP.2.dExplain what a point (x,y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0,0) and (1,r) where r is the unit rate.
7.RP.3Use proportional relationships to solve multi-step ratio and percent problems.
7.SP Statistics and Probability
7.SP.1Understand that statistics can be used to gain information about a population by examining a sample of the population. Understand that generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
7.SP.2Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.
7.SP.3Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.
7.SP.4Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.
7.SP.5Understand that the probability of a chance event is a number from 0 through 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
7.SP.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency. Predict the approximate relative frequency given the probability.
7.SP.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies. If there is a discrepancy, explain possible sources.
7.SP.7.aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.
7.SP.7.bDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.
7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
7.SP.8.aUnderstand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
7.SP.8.bRepresent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (such as “rolling double sixes”), identify the outcomes in the sample space which compose the event.
7.SP.8.cDesign and use a simulation to generate frequencies for compound events.

Grade 8

8.EE Expressions and Equations
8.EE.1Develop, know and apply the properties of integer exponents to generate equivalent numeric and algebraic expressions.
8.EE.2Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Classify radicals as rational or irrational.
8.EE.3Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.
8.EE.4Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (such as use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
8.EE.5Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
8.EE.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. Derive the equation y=mx for a line through the origin and the equation = mx + b for a line intercepting the vertical axis at b.
8.EE.7 Solve linear equations in one variable.
8.EE.7.aGive examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
8.EE.7.bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
8.EE.8 Analyze and solve pairs of simultaneous linear equations.
8.EE.8.aUnderstand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
8.EE.8.bSolve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
8.EE.8.cSolve real world and mathematical problems leading to two linear equations in two variables.
8.F Functions
8.F.1Understand that a function is a rule that assigns to each input exactly one output. Understand that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
8.F.2Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, and/or by verbal descriptions).
8.F.3Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line. Give examples of functions that are not linear.
8.F.4Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
8.F.5Describe qualitatively the functional relationship between two quantities by analyzing a graph (may include where the function is increasing or decreasing, linear or nonlinear, etc.). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
8.G Geometry
8.G.1 Understand the properties of rotations, reflections, and translations by experimentation:
8.G.1.aLines are transformed onto lines, and line segments onto line segments of the same length.
8.G.1.bAngles are transformed onto angles of the same measure.
8.G.1.cParallel lines are transformed onto parallel lines.
8.G.2Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. Given two congruent figures, describe a sequence of transformations that exhibits the congruence between them.
8.G.3Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.
8.G.4Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations. Given two similar two-dimensional figures, describe a sequence of transformations that exhibits the similarity between them.
8.G.5 Use informal arguments to establish facts about:
8.G.5.aThe angle sum and exterior angles of triangles
8.G.5.bThe angles created when parallel lines are cut by a transversal
8.G.5.cThe angle-angle criterion for similarity of triangles
8.G.6Explain a proof of the Pythagorean Theorem and its converse.
8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real world and mathematical problems in two and three dimensions.
8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
8.G.9Know the formulas for the volume of cones, cylinders and spheres. Use the formulas to solve real world and mathematical problems.
8.NS The Number System
8.NS.1Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually. Convert a decimal expansion which repeats eventually into a rational number.
8.NS.2Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (such as π^2).
8.SP Statistics and Probability
8.SP.1Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
8.SP.2Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
8.SP.3Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept(s).
8.SP.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

HS

HS.A-APR HS.A-APR
HS.A-APR.1Add, subtract, and multiply polynomials. Understand that polynomials form a system comparable to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.
HS.A-APR.2Apply the Remainder Theorem.
HS.A-APR.3Identify zeros of polynomials when suitable factorizations are available. Use the zeros to construct a rough graph of the function defined by the polynomial.
HS.A-APR.6Rewrite 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), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
HS.A-APR.7Add, subtract, multiply, and divide rational expressions. Understand that rational expressions form a system comparable to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.
HS.A-CED HS.A-CED
HS.A-CED.1Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
HS.A-CED.2Create equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.
HS.A-CED.3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
HS.A-CED.4Rearrange formulas to isolate a quantity of interest, using the same reasoning as in solving equations.
HS.A-REI HS.A-REI
HS.A-REI.1Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
HS.A-REI.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.
HS.A-REI.11Using graphs, technology, tables, or successive approximations, show that the solution(s) to the equation f(x) = g(x) are the x-value(s) that result in the y-values of f(x) and g(x) being the same.
HS.A-REI.12Graph the solutions to a linear inequality in two variables as a half-plane. Graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
HS.A-REI.2Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
HS.A-REI.3Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
HS.A-REI.4 Solve quadratic equations in one variable.
HS.A-REI.4.aUse the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. (+) Derive the quadratic formula from this form.
HS.A-REI.4.bSolve quadratic equations by inspection (e.g., for x = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a +/- bi for real numbers a and b.
HS.A-REI.5This standard has been moved/removed by the committee
HS.A-REI.6Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables.
HS.A-REI.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
HS.A-REI.8(+)Represent a system of linear equations as a single matrix equation.
HS.A-REI.9(+)Find the inverse of a matrix, if it exists, and use it to solve systems of linear equations (using technology for matrices of dimension 3 3 or greater).
HS.A-SSE HS.A-SSE
HS.A-SSE.1 Interpret expressions that represent a quantity in terms of its context.
HS.A-SSE.1.aInterpret parts of an expression, such as terms, factors, and coefficients.
HS.A-SSE.1.bInterpret complicated expressions by examining one or more of their parts as a single entity.
HS.A-SSE.2Use the structure of an expression to identify ways to rewrite it.
HS.A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
HS.A-SSE.3.aFactor a quadratic expression to reveal the zeros of the function it defines.
HS.A-SSE.3.bComplete the square in a quadratic expression to produce an equivalent expression.
HS.A-SSE.3.cUse the properties of exponents to transform exponential expressions.
HS.A-SSE.4This standard has been moved/removed by the committee
HS.F-BF HS.F-BF
HS.F-BF.1 Write a function that describes a relationship between two quantities.
HS.F-BF.1.aDetermine an explicit expression, a recursive process, or steps for calculation from a context.
HS.F-BF.1.bCombine standard function types using arithmetic operations.
HS.F-BF.1.cCompose functions.
HS.F-BF.2Write arithmetic and geometric sequences both recursively and with an explicit formula and convert between the two forms. Use sequences to model situations.
HS.F-BF.3Identify the effect on the graph of replacing f(x) by f(x) + k, f(x + k), k f(x), and f(x + k), for specific values of k (both positive and negative); find the value of k given the graphs. Recognize even and odd functions from their graphs.
HS.F-BF.4 Find inverse functions.
HS.F-BF.4.aWrite an equation for the inverse given a function has an inverse.
HS.F-BF.4.bVerify by composition that one function is the inverse of another.
HS.F-BF.4.cRead values of an inverse function from a graph or a table, given that the function has an inverse.
HS.F-BF.4.dProduce an invertible function from a non-invertible function by restricting the domain.
HS.F-BF.5Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
HS.F-IF HS.F-IF
HS.F-IF.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
HS.F-IF.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
HS.F-IF.3Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
HS.F-IF.4Use tables, graphs, verbal descriptions, and equations to interpret and sketch the key features of a function modeling the relationship between two quantities.
HS.F-IF.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
HS.F-IF.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
HS.F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
HS.F-IF.7.aGraph linear and quadratic functions and show intercepts, maxima, and minima where appropriate.
HS.F-IF.7.bGraph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
HS.F-IF.7.cGraph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
HS.F-IF.7.d(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
HS.F-IF.7.eGraph exponential and logarithmic functions, showing intercepts and end behavior.
HS.F-IF.7.fGraph f(x) = sin x and f(x) = cos x as representations of periodic phenomena.
HS.F-IF.7.g(+) Graph trigonometric functions, showing period, midline, phase shift and amplitude.
HS.F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
HS.F-IF.8.aUse the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
HS.F-IF.8.bUse the properties of exponents to interpret expressions for exponential functions.
HS.F-IF.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
HS.F-LE HS.F-LE
HS.F-LE.1Identify situations that can be modeled with linear, quadratic, and exponential functions. Justify the most appropriate model for a situation based on the rate of change over equal intervals. Include situations in which a quantity grows or decays.
HS.F-LE.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a table, a description, or two input-output pairs given their relationship.
HS.F-LE.3Compare the end behavior of linear, quadratic, and exponential functions using graphs and/or tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing as a linear or quadratic function.
HS.F-LE.4Use logarithms to express the solution to ab^ct = d where a, c, and d are real numbers and b is a positive real number. Evaluate the logarithm using technology when appropriate.
HS.F-LE.5Interpret the parameters in a linear, quadratic, or exponential function in context.
HS.F-TF HS.F-TF
HS.F-TF.1Understand that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
HS.F-TF.2Extend right triangle trigonometry to the four quadrants. (+) Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
HS.F-TF.3Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6. (+) Use the unit circle to express the values of sine, cosine, and tangent for π - x, π + x, and 2π -x, in terms of their values for x, where x is any real number.
HS.F-TF.4(+)Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
HS.F-TF.5(+)Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
HS.F-TF.6(+)Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
HS.F-TF.7(+)Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
HS.F-TF.8Prove the Pythagorean identity sin^2(ϴ) + cos^2(ϴ) = 1 and use it to find n(ϴ), cos(ϴ), or tan(ϴ) given sin(ϴ), cos(ϴ), or tan(ϴ)and the quadrant of the angle.
HS.F-TF.9(+)Know and apply the addition and subtraction formulas for sine, cosine, and tangent.
HS.G-C HS.G-C
HS.G-C.1Understand and apply theorems about relationships with line segments and circles including radii, diameter, secants, tangents, and chords.
HS.G-C.2Understand and apply theorems about relationships with angles formed by radii, diameter, secants, tangents, and chords. Understand and apply properties of angles for a quadrilateral inscribed in a circle.
HS.G-C.3Construct the incenter and circumcenter of a triangle. Relate the incenter and circumcenter to the inscribed and circumscribed circles.
HS.G-C.4(+)Construct a tangent line from a point outside a given circle to the circle.
HS.G-C.5Explain and use the formulas for arc length and area of sectors of circles.
HS.G-CO HS.G-CO
HS.G-CO.1Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, and plane.
HS.G-CO.10Prove and apply theorems about triangle properties.
HS.G-CO.11Prove and apply theorems about parallelograms.
HS.G-CO.12Make basic geometric constructions with a variety of tools and methods.
HS.G-CO.13(+)Apply basic constructions to create polygons such as equilateral triangles, squares, and regular hexagons inscribed in circles.
HS.G-CO.2Represent transformations in the plane. Describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
HS.G-CO.3Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
HS.G-CO.4Develop or verify experimentally the characteristics of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
HS.G-CO.5Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
HS.G-CO.6Use geometric descriptions of rigid motions to predict the effect of a given rigid motion on a given figure. Use the definition of congruence in terms of rigid motions to decide if two figures are congruent.
HS.G-CO.7Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
HS.G-CO.8Prove two triangles are congruent using the congruence theorems such as ASA, SAS, and SSS.
HS.G-CO.9Prove and apply theorems about lines and angles.
HS.G-GMD HS.G-GMD
HS.G-GMD.1Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
HS.G-GMD.2Calculate the surface area for prisms, cylinders, pyramids, cones, and spheres to solve problems.
HS.G-GMD.3Know and apply volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems.
HS.G-GMD.4Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
HS.G-GPE HS.G-GPE
HS.G-GPE.1Derive the equation of a circle of given center and radius. Derive the equation of a parabola given a focus and directrix. (+) Derive the equations of ellipses and hyperbolas given foci, using the fact that the sum or difference of distances from the foci is constant.
HS.G-GPE.2Convert between the standard and general form equations of conic sections.
HS.G-GPE.3Identify key features of conic sections given their equations. Apply properties of conic sections in real world situations.
HS.G-GPE.4Use coordinates to verify simple geometric theorems algebraically. Use coordinates to verify algebraically that a given set of points produces a particular type of triangle or quadrilateral.
HS.G-GPE.5Develop and verify the slope criteria for parallel and perpendicular lines. Apply the slope criteria for parallel and perpendicular lines to solve geometric problems using algebra.
HS.G-GPE.6Use coordinates to find the midpoint or endpoint of a line segment. (+) Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
HS.G-GPE.7Use coordinates to compute perimeters of polygons and areas of triangles, parallelograms, trapezoids and kites.
HS.G-MG HS.G-MG
HS.G-MG.1Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
HS.G-MG.2Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
HS.G-MG.3Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
HS.G-SRT HS.G-SRT
HS.G-SRT.1Verify experimentally the properties of dilations given by a center and a scale factor.
HS.G-SRT.10(+)Solve unknown sides and angles of non-right triangles using the Laws of Sines and Cosines.
HS.G-SRT.11(+)Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in context.
HS.G-SRT.2Given two figures, use transformations to decide if they are similar. Apply the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
HS.G-SRT.3Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
HS.G-SRT.4Prove similarity theorems about triangles.
HS.G-SRT.5Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
HS.G-SRT.6Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine, and tangent of an acute angle in a right triangle.
HS.G-SRT.7Explain and use the relationship between the sine and cosine of complementary angles.
HS.G-SRT.8Use special right triangles (30-60а-90 and 45а-45-90а), trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
HS.G-SRT.9(+)Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
HS.N-CN HS.N-CN
HS.N-CN.1Know there is an imaginary number i, such that i² = −1, and every complex number has the form a + bi where a and b are real. Understand the hierarchal relationships among subsets of the complex number system.
HS.N-CN.2Use the definition i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
HS.N-CN.3Use conjugates to find quotients of complex numbers.
HS.N-CN.4(+)Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers). Find moduli (absolute value) of a complex number. Explain why the rectangular and polar forms of a given complex number represent the same number.
HS.N-CN.5(+)Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
HS.N-CN.6(+)This standard has been moved/removed by the committee
HS.N-CN.7Solve quadratic equations with real coefficients that have complex solutions.
HS.N-CN.8(+)Extend polynomial identities to the complex numbers.
HS.N-CN.9(+)Apply the Fundamental Theorem of Algebra to determine the number of zeros for polynomial functions. Find all solutions to a polynomial equation.
HS.N-Q HS.N-Q
HS.N-Q.1Use units as a way to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the origin in graphs and data displays.
HS.N-Q.2Define appropriate quantities for the purpose of descriptive modeling.
HS.N-Q.3Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
HS.N-RN HS.N-RN
HS.N-RN.1Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
HS.N-RN.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.
HS.N-RN.3Demonstrate that the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational.
HS.N-RN.4Perform basic operations on radicals and simplify radicals to write equivalent expressions.
HS.N-VM HS.N-VM
HS.N-VM.1(+)Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
HS.N-VM.10(+)Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
HS.N-VM.11(+)Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Understand a matrix as a transformation of vectors.
HS.N-VM.12(+)Understand a 2 2 matrix as a transformation of the plane. Interpret the absolute value of the determinant in terms of area.
HS.N-VM.2(+)Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
HS.N-VM.3(+)Solve problems involving velocity and other quantities that can be represented by vectors.
HS.N-VM.4 (+)Add and subtract vectors.
HS.N-VM.4.aAdd vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
HS.N-VM.4.bGiven two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
HS.N-VM.4.cUnderstand that vector subtraction v – w is defined as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order and use the components to perform vector subtraction.
HS.N-VM.5 (+)Multiply a vector by a scalar.
HS.N-VM.5.aRepresent scalar multiplication graphically by scaling vectors and possibly reversing their direction. Use the components to perform scalar multiplication (e.g., as c(v_x , v_y) = (cv_x, cv_y)).
HS.N-VM.5.bmultiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
HS.N-VM.6Use matrices to represent and manipulate data.
HS.N-VM.7Multiply matrices by scalars to produce new matrices.
HS.N-VM.8Add, subtract, and multiply matrices of appropriate dimensions.
HS.N-VM.9Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
HS.S-CP HS.S-CP
HS.S-CP.1Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
HS.S-CP.2Understand that event A is independent from event B if the probability of event A does not change in response to the occurrence of event B. Apply the formula P(A and B) = P(A) * P(B) given that event A and B are independent.
HS.S-CP.3Understand that the conditional probability of an event A given B is the probability that event A will occur given the knowledge that event B has already occurred. Apply the formula P(A given B) = P(A and B)/P(B) given a conditional probability situation.
HS.S-CP.4Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
HS.S-CP.5Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
HS.S-CP.6Find the conditional probability of A given B and interpret the answer in terms of the model.
HS.S-CP.7Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.
HS.S-CP.8Apply the general Multiplication Rule in a uniform probability model, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model.
HS.S-CP.9Use permutations and combinations to determine the number of outcomes in terms of the model. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
HS.S-IC HS.S-IC
HS.S-IC.1Understand the process of making inferences about population parameters based on a random sample from that population.
HS.S-IC.2Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
HS.S-IC.3Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
HS.S-IC.4(+)Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
HS.S-IC.5This standard has been moved/removed by the committee
HS.S-IC.6 Evaluate reports based on data.
HS.S-IC.6.aEvaluate articles, reports or websites based on data published in the media by identifying the source of the data, the design of the study, and the way the data are analyzed and displayed.
HS.S-IC.6.bIdentify and explain misleading use of data; recognize when claims based on data confuse correlation and causation.
HS.S-IC.6.cRecognize and describe how graphs and data can be distorted to support different points of view.
HS.S-ID HS.S-ID
HS.S-ID.1Represent data with plots on the real number line (dot plots, histograms, and box plots).
HS.S-ID.2Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
HS.S-ID.3Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
HS.S-ID.4Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, or tables to estimate areas under the normal curve.
HS.S-ID.5Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
HS.S-ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
HS.S-ID.6.aFit a function to the data (with or without technology). Use functions fitted to data to solve problems in the context of the data.
HS.S-ID.6.b(+) Informally assess the fit of a function by plotting and analyzing residuals.
HS.S-ID.7Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpolate and extrapolate the linear model to predict values.
HS.S-ID.8Compute (using technology) and interpret the correlation coefficient of a linear fit.
HS.S-ID.9Distinguish between correlation and causation.
HS.S-MD HS.S-MD
HS.S-MD.1(+)Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space. Graph the corresponding probability distribution using the same graphical displays as for data distributions.
HS.S-MD.2(+)Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
HS.S-MD.3(+)Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.
HS.S-MD.4(+)Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.
HS.S-MD.5 (+)Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
HS.S-MD.5.aFind the expected payoff for a game of chance.
HS.S-MD.5.bEvaluate and compare strategies on the basis of expected values.
HS.S-MD.6(+)Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
HS.S-MD.7(+)Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

AP Calculus Learning Objectives

CB.Calc.LIM Limits
CB.Calc.LIM-1 Reasoning with definitions, theorems, and properties can be used to justify claims about limits.
CB.Calc.LIM-1.ARepresent limits analytically using correct notation.
CB.Calc.LIM-1.BInterpret limits expressed in analytic notation.
CB.Calc.LIM-1.CEstimate limits of functions.
CB.Calc.LIM-1.DDetermine the limits of functions using limit theorems.
CB.Calc.LIM-1.EDetermine the limits of functions using equivalent expressions for the function or the squeeze theorem.
CB.Calc.LIM-2 Reasoning with definitions, theorems, and properties can be used to justify claims about continuity.
CB.Calc.LIM-2.AJustify conclusions about continuity at a point using the definition.
CB.Calc.LIM-2.BDetermine intervals over which a function is continuous.
CB.Calc.LIM-2.CDetermine values of x or solve for parameters that make discontinuous functions continuous, if possible.
CB.Calc.LIM-2.DInterpret the behavior of functions using limits involving infinity.
CB.Calc.LIM-3 The derivative of a function is defined as the limit of a difference quotient.
CB.Calc.LIM-3.AInterpret a limit as a definition of a derivative.
CB.Calc.LIM-4 L'Hospital's Rule allows us to determine the limits of some indeterminate forms.
CB.Calc.LIM-4.ADetermine limits of functions that result in indeterminate forms.
CB.Calc.LIM-5 Definite integrals can be approximated using geometric and numerical methods.
CB.Calc.LIM-5.AApproximate a definite integral using geometric and numerical methods.
CB.Calc.LIM-5.BInterpret the limiting case of the Riemann sum as a definite integral.
CB.Calc.LIM-5.CRepresent the limiting case of the Riemann sum as a definite integral.
CB.Calc.LIM-6 Improper integrals can be determined using limits of definite integrals.
CB.Calc.LIM-6.AEvaluate an improper integral or determine that the integral diverges.
CB.Calc.LIM-7 The behavior of a convergent or divergent series can be determined or estimated.
CB.Calc.LIM-7.ADetermine whether a series converges or diverges.
CB.Calc.LIM-7.BApproximate the sum of a series.
CB.Calc.LIM-8 Power series and Taylor series can represent functions.
CB.Calc.LIM-8.ARepresent a function at a point as a Taylor polynomial.
CB.Calc.LIM-8.BApproximate function values using a Taylor polynomial.
CB.Calc.LIM-8.CDetermine the error bound associated with a Taylor polynomial approximation.
CB.Calc.LIM-8.DDetermine the radius of convergence and interval of convergence for a power series.
CB.Calc.LIM-8.ERepresent a function as a Taylor series or a Maclaurin series.
CB.Calc.LIM-8.FInterpret Taylor series and Maclaurin series.
CB.Calc.LIM-8.GRepresent a given function as a power series.
CB.Calc.CHA Change
CB.Calc.CHA-1 The derivative of a function is the instantaneous rate of change of the function.
CB.Calc.CHA-1.AInterpret the rate of change at an instant in terms of average rates of change over intervals containing that instant.
CB.Calc.CHA-2 Derivatives can be computed using definitions and rules.
CB.Calc.CHA-2.ADetermine average rates of change using difference quotients.
CB.Calc.CHA-2.BRepresent the derivative of a function as the limit of a difference quotient.
CB.Calc.CHA-2.CDetermine the equation of a line tangent to a curve at a given point.
CB.Calc.CHA-2.DEstimate derivatives.
CB.Calc.CHA-3 Derivatives can be used to model rates of change in applied contexts.
CB.Calc.CHA-3.AInterpret the meaning of a derivative in context.
CB.Calc.CHA-3.BCalculate rates of change in applied contexts.
CB.Calc.CHA-3.CInterpret rates of change in applied contexts.
CB.Calc.CHA-3.DCalculate related rates in applied contexts.
CB.Calc.CHA-3.EInterpret related rates in applied contexts.
CB.Calc.CHA-3.FApproximate a value on a curve using the equation of a tangent line.
CB.Calc.CHA-3.GCalculate derivatives of parametric functions.
CB.Calc.CHA-3.HCalculate derivatives of vector-valued functions.
CB.Calc.CHA-4 Integrals can be used to model accumulation and net change in applied contexts.
CB.Calc.CHA-4.AInterpret the meaning of areas associated with the graph of a rate of change in context.
CB.Calc.CHA-4.BDetermine the average value of a function using definite integrals.
CB.Calc.CHA-4.CDetermine values for positions and rates of change using definite integrals in problems involving rectilinear motion.
CB.Calc.CHA-4.DInterpret the meaning of a definite integral in accumulation problems.
CB.Calc.CHA-4.EDetermine net change using definite integrals in applied contexts.
CB.Calc.CHA-5 Definite integrals can be used to calculate areas, volumes, and lengths.
CB.Calc.CHA-5.ACalculate areas in the plane using the definite integral.
CB.Calc.CHA-5.BCalculate volumes of solids with known cross sections using definite integrals.
CB.Calc.CHA-5.CCalculate volumes of solids of revolution using definite integrals.
CB.Calc.CHA-5.DCalculate areas of regions defined by polar curves using definite integrals.
CB.Calc.CHA-6 Arc length can be calculated using definite integrals.
CB.Calc.CHA-6.ADetermine the length of a curve in the plane defined by a function, using a definite integral.
CB.Calc.CHA-6.BDetermine the length of a curve in the plane defined by parametric functions, using a definite integral.
CB.Calc.FUN Analysis of Functions
CB.Calc.FUN-1 Existence theorems can be used to draw conclusions about the behavior of functions.
CB.Calc.FUN-1.AExplain the behavior of a function on an interval using the Intermediate Value Theorem.
CB.Calc.FUN-1.BJustify conclusions about functions by applying the Mean Value Theorem over an interval.
CB.Calc.FUN-1.CJustify conclusions about functions by applying the Extreme Value Theorem.
CB.Calc.FUN-2 A function's derivative, when it exists, can be used to determine whether the function is increasing or decreasing.
CB.Calc.FUN-2.AExplain the relationship between differentiability and continuity.
CB.Calc.FUN-3 Derivatives can be calculated for various types of functions.
CB.Calc.FUN-3.ACalculate derivatives of familiar functions.
CB.Calc.FUN-3.BCalculate derivatives of products and quotients of differentiable functions.
CB.Calc.FUN-3.CCalculate derivatives of compositions of differentiable functions.
CB.Calc.FUN-3.DCalculate derivatives of implicitly defined functions.
CB.Calc.FUN-3.ECalculate derivatives of inverse and inverse trigonometric functions.
CB.Calc.FUN-3.FDetermine higher order derivatives of a function.
CB.Calc.FUN-3.GCalculate derivatives of functions written in polar coordinates.
CB.Calc.FUN-4 A function's derivative can be used to understand function behavior.
CB.Calc.FUN-4.AJustify conclusions about the behavior of a function based on the behavior of its derivatives.
CB.Calc.FUN-4.BCalculate minimum and maximum values in applied contexts or analysis of functions.
CB.Calc.FUN-4.CInterpret minimum and maximum values calculated in applied contexts.
CB.Calc.FUN-4.DDetermine critical points of implicit relations.
CB.Calc.FUN-4.EJustify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives.
CB.Calc.FUN-5 The Fundamental Theorem of Calculus connects differentiation and integration.
CB.Calc.FUN-5.ARepresent accumulation functions using definite integrals.
CB.Calc.FUN-6 Recognizing and applying techniques for antidifferentiation.
CB.Calc.FUN-6.ACalculate a definite integral using areas and properties of definite integrals.
CB.Calc.FUN-6.BEvaluate definite integrals analytically using the Fundamental Theorem of Calculus.
CB.Calc.FUN-6.CDetermine antiderivatives of functions and indefinite integrals, using knowledge of derivatives.
CB.Calc.FUN-6.DUse substitution and rearrangements into equivalent forms, such as long division and completing the square, for integrands requiring these techniques.
CB.Calc.FUN-6.EUse integration by parts for integrands requiring this technique.
CB.Calc.FUN-6.FUse integration by linear partial fractions for integrands requiring this technique.
CB.Calc.FUN-7 Solving differential equations allows us to model rates of change.
CB.Calc.FUN-7.AInterpret verbal statements of problems as differential equations involving a derivative expression.
CB.Calc.FUN-7.BVerify solutions to differential equations.
CB.Calc.FUN-7.CEstimate solutions to differential equations.
CB.Calc.FUN-7.DDetermine general solutions to differential equations.
CB.Calc.FUN-7.EDetermine particular solutions to differential equations.
CB.Calc.FUN-7.FInterpret the meaning of a differential equation and its variables in context.
CB.Calc.FUN-7.GDetermine general and particular solutions for problems involving differential equations in context.
CB.Calc.FUN-7.HInterpret the meaning of the logistic growth model in context.
CB.Calc.FUN-8 Solutions to differential equations can model motion in the plane.
CB.Calc.FUN-8.ADetermine a particular solution given a rate vector and initial conditions.
CB.Calc.FUN-8.BDetermine values for positions and rates of change in problems involving planar motion.

AP Statistics Learning Objectives

CB.Stats.VAR Variation and Distribution
CB.Stats.VAR-1 Exploring data through asking and answering statistical questions.
CB.Stats.VAR-1.AIdentify questions to be answered, based on variation in one-variable data.
CB.Stats.VAR-1.BIdentify variables in a set of data.
CB.Stats.VAR-1.CClassify types of variables.
CB.Stats.VAR-1.DIdentify questions to be answered about possible relationships in data.
CB.Stats.VAR-1.EIdentify questions to be answered about data collection methods.
CB.Stats.VAR-1.FIdentify questions suggested by patterns in data.
CB.Stats.VAR-1.GIdentify questions suggested by variation in statistics for samples collected from the same population.
CB.Stats.VAR-1.HIdentify questions suggested by variation in the shapes of distributions of samples taken from the same population.
CB.Stats.VAR-1.IIdentify questions suggested by probabilities of errors in statistical inference.
CB.Stats.VAR-1.JIdentify questions suggested by variation between observed and expected counts in categorical data.
CB.Stats.VAR-1.KIdentify questions suggested by variation in scatter plots.
CB.Stats.VAR-2 Normal distributions and comparing data.
CB.Stats.VAR-2.ACompare a data distribution to the normal distribution model.
CB.Stats.VAR-2.BDetermine proportions and percentiles from a normal distribution.
CB.Stats.VAR-2.CCompare measures of relative position in data sets.
CB.Stats.VAR-3 Designing and analyzing experiments.
CB.Stats.VAR-3.AIdentify the components of an experiment.
CB.Stats.VAR-3.BDescribe elements of a well-designed experiment.
CB.Stats.VAR-3.CCompare experimental designs and methods.
CB.Stats.VAR-3.DExplain why a particular experimental design is appropriate.
CB.Stats.VAR-3.EInterpret the results of a well-designed experiment.
CB.Stats.VAR-4 Probability rules and calculations.
CB.Stats.VAR-4.ACalculate probabilities for events and their complements.
CB.Stats.VAR-4.BInterpret probabilities for events.
CB.Stats.VAR-4.CExplain why two events are (or are not) mutually exclusive.
CB.Stats.VAR-4.DCalculate conditional probabilities.
CB.Stats.VAR-4.ECalculate probabilities for independent events and for the union of two events.
CB.Stats.VAR-5 Discrete random variables and probability distributions.
CB.Stats.VAR-5.ARepresent the probability distribution for a discrete random variable.
CB.Stats.VAR-5.BInterpret a probability distribution.
CB.Stats.VAR-5.CCalculate parameters for a discrete random variable.
CB.Stats.VAR-5.DInterpret parameters for a discrete random variable.
CB.Stats.VAR-5.ECalculate parameters for linear combinations of random variables.
CB.Stats.VAR-5.FDescribe the effects of linear transformations of parameters of random variables.
CB.Stats.VAR-6 Normal distributions and hypothesis testing for proportions.
CB.Stats.VAR-6.ACalculate the probability that a particular value lies in a given interval of a normal distribution.
CB.Stats.VAR-6.BDetermine the interval associated with a given area in a normal distribution.
CB.Stats.VAR-6.CDetermine the appropriateness of using the normal distribution to approximate probabilities for unknown distributions.
CB.Stats.VAR-6.DIdentify the null and alternative hypotheses for a population proportion.
CB.Stats.VAR-6.EIdentify an appropriate testing method for a population proportion.
CB.Stats.VAR-6.FVerify the conditions for making statistical inferences when testing a population proportion.
CB.Stats.VAR-6.GCalculate an appropriate test statistic and p-value for a population proportion.
CB.Stats.VAR-6.HIdentify the null and alternative hypotheses for a difference of two population proportions.
CB.Stats.VAR-6.IIdentify an appropriate testing method for the difference of two population proportions.
CB.Stats.VAR-6.JVerify the conditions for making statistical inferences when testing a difference of two population proportions.
CB.Stats.VAR-6.KCalculate an appropriate test statistic for the difference of two population proportions.
CB.Stats.VAR-7 t-distributions and hypothesis testing for means and regression.
CB.Stats.VAR-7.ADescribe t-distributions.
CB.Stats.VAR-7.BIdentify an appropriate testing method for a population mean with unknown sigma, including the mean difference between values in matched pairs.
CB.Stats.VAR-7.CIdentify the null and alternative hypotheses for a population mean with unknown sigma, including the mean difference between values in matched pairs.
CB.Stats.VAR-7.DVerify the conditions for the test for a population mean, including the mean difference between values in matched pairs.
CB.Stats.VAR-7.ECalculate an appropriate test statistic for a population mean, including the mean difference between values in matched pairs.
CB.Stats.VAR-7.FIdentify an appropriate selection of a testing method for a difference of two population means.
CB.Stats.VAR-7.GIdentify the null and alternative hypotheses for a difference of two population means.
CB.Stats.VAR-7.HVerify the conditions for the significance test for the difference of two population means.
CB.Stats.VAR-7.ICalculate an appropriate test statistic for a difference of two means.
CB.Stats.VAR-7.JIdentify the appropriate selection of a testing method for a slope of a regression model.
CB.Stats.VAR-7.KIdentify appropriate null and alternative hypotheses for a slope of a regression model.
CB.Stats.VAR-7.LVerify the conditions for the significance test for the slope of a regression model.
CB.Stats.VAR-7.MCalculate an appropriate test statistic for the slope of a regression model.
CB.Stats.VAR-8 Chi-square distributions and tests.
CB.Stats.VAR-8.ADescribe chi-square distributions.
CB.Stats.VAR-8.BIdentify the null and alternative hypotheses in a test for a distribution of proportions in a set of categorical data.
CB.Stats.VAR-8.CIdentify an appropriate testing method for a distribution of proportions in a set of categorical data.
CB.Stats.VAR-8.DCalculate expected counts for the chi-square test for goodness of fit.
CB.Stats.VAR-8.EVerify the conditions for making statistical inferences when testing goodness of fit for a chi-square distribution.
CB.Stats.VAR-8.FCalculate the appropriate statistic for the chi-square test for goodness of fit.
CB.Stats.VAR-8.GDetermine the p-value for the chi-square test for goodness of fit.
CB.Stats.VAR-8.HCalculate expected counts for two-way tables of categorical data.
CB.Stats.VAR-8.IIdentify the null and alternative hypotheses for a chi-square test for homogeneity or independence.
CB.Stats.VAR-8.JIdentify an appropriate testing method for comparing distributions in two-way tables of categorical data.
CB.Stats.VAR-8.KVerify the conditions for making statistical inferences when testing a chi-square distribution for independence or homogeneity.
CB.Stats.VAR-8.LCalculate the appropriate statistic for a chi-square test for homogeneity or independence.
CB.Stats.VAR-8.MDetermine the p-value for a chi-square test for independence or homogeneity.
CB.Stats.UNC Uncertainty and Inference
CB.Stats.UNC-1 Representing and describing categorical and quantitative data.
CB.Stats.UNC-1.ARepresent categorical data using frequency or relative frequency tables.
CB.Stats.UNC-1.BDescribe categorical data represented in frequency or relative tables.
CB.Stats.UNC-1.CRepresent categorical data graphically.
CB.Stats.UNC-1.DDescribe categorical data represented graphically.
CB.Stats.UNC-1.ECompare multiple sets of categorical data.
CB.Stats.UNC-1.FClassify types of quantitative variables.
CB.Stats.UNC-1.GRepresent quantitative data graphically.
CB.Stats.UNC-1.HDescribe the characteristics of quantitative data distributions.
CB.Stats.UNC-1.ICalculate measures of center and position for quantitative data.
CB.Stats.UNC-1.JCalculate measures of variability for quantitative data.
CB.Stats.UNC-1.KExplain the selection of a particular measure of center and/or variability for describing a set of quantitative data.
CB.Stats.UNC-1.LRepresent summary statistics for quantitative data graphically.
CB.Stats.UNC-1.MDescribe summary statistics of quantitative data represented graphically.
CB.Stats.UNC-1.NCompare graphical representations for multiple sets of quantitative data.
CB.Stats.UNC-1.OCompare summary statistics for multiple sets of quantitative data.
CB.Stats.UNC-1.PCompare numerical and graphical representations for two categorical variables.
CB.Stats.UNC-1.QCalculate statistics for two categorical variables.
CB.Stats.UNC-1.RCompare statistics for two categorical variables.
CB.Stats.UNC-1.SRepresent bivariate quantitative data using scatterplots.
CB.Stats.UNC-2 Probability and simulation.
CB.Stats.UNC-2.AEstimate probabilities using simulation.
CB.Stats.UNC-3 Random variables and sampling distributions.
CB.Stats.UNC-3.AEstimate probabilities of binomial random variables using data from a simulation.
CB.Stats.UNC-3.BCalculate probabilities for a binomial distribution.
CB.Stats.UNC-3.CCalculate parameters for a binomial distribution.
CB.Stats.UNC-3.DInterpret probabilities and parameters for a binomial distribution.
CB.Stats.UNC-3.ECalculate probabilities for geometric random variables.
CB.Stats.UNC-3.FCalculate parameters of a geometric distribution.
CB.Stats.UNC-3.GInterpret probabilities and parameters for a geometric distribution.
CB.Stats.UNC-3.HEstimate sampling distributions using simulation.
CB.Stats.UNC-3.IExplain why an estimator is or is not unbiased.
CB.Stats.UNC-3.JCalculate estimates for a population parameter.
CB.Stats.UNC-3.KDetermine parameters of a sampling distribution for sample proportions.
CB.Stats.UNC-3.LDetermine whether a sampling distribution for a sample proportion can be described as approximately normal.
CB.Stats.UNC-3.MInterpret probabilities and parameters for a sampling distribution for a sample proportion.
CB.Stats.UNC-3.NDetermine parameters of a sampling distribution for a difference in sample proportions.
CB.Stats.UNC-3.ODetermine whether a sampling distribution for a difference of sample proportions can be described as approximately normal.
CB.Stats.UNC-3.PInterpret probabilities and parameters for a sampling distribution for a difference in proportions.
CB.Stats.UNC-3.QDetermine parameters of a sampling distribution for a sample mean.
CB.Stats.UNC-3.RDetermine whether a sampling distribution of a sample mean can be described as approximately normal.
CB.Stats.UNC-3.SInterpret probabilities and parameters for a sampling distribution for a sample mean.
CB.Stats.UNC-3.TDetermine parameters of a sampling distribution for a difference in sample means.
CB.Stats.UNC-3.UDetermine whether a sampling distribution of a difference in sample means can be described as approximately normal.
CB.Stats.UNC-3.VInterpret probabilities and parameters for a sampling distribution for a difference in sample means.
CB.Stats.UNC-4 Confidence intervals for proportions and means.
CB.Stats.UNC-4.AIdentify an appropriate confidence interval procedure for a population proportion.
CB.Stats.UNC-4.BVerify the conditions for calculating confidence intervals for a population proportion.
CB.Stats.UNC-4.CDetermine the margin of error for a given sample size and an estimate for the sample size that will result in a given margin of error for a population proportion.
CB.Stats.UNC-4.DCalculate an appropriate confidence interval for a population proportion.
CB.Stats.UNC-4.ECalculate an interval estimate based on a confidence interval for a population proportion.
CB.Stats.UNC-4.FInterpret a confidence interval for a population proportion.
CB.Stats.UNC-4.GJustify a claim based on a confidence interval for a population proportion.
CB.Stats.UNC-4.HIdentify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population proportion.
CB.Stats.UNC-4.IIdentify an appropriate confidence interval procedure for a comparison of population proportions.
CB.Stats.UNC-4.JVerify the conditions for calculating confidence intervals for a difference between population proportions.
CB.Stats.UNC-4.KCalculate an appropriate confidence interval for a comparison of population proportions.
CB.Stats.UNC-4.LCalculate an interval estimate based on a confidence interval for a difference of proportions.
CB.Stats.UNC-4.MInterpret a confidence interval for a difference of proportions.
CB.Stats.UNC-4.NJustify a claim based on a confidence interval for a difference of proportions.
CB.Stats.UNC-4.OIdentify an appropriate confidence interval procedure for a population mean, including the mean difference between values in matched pairs.
CB.Stats.UNC-4.PVerify the conditions for calculating confidence intervals for a population mean, including the mean difference between values in matched pairs.
CB.Stats.UNC-4.QDetermine the margin of error for a given sample size for a one-sample t-interval.
CB.Stats.UNC-4.RCalculate an appropriate confidence interval for a population mean, including the mean difference between values in matched pairs.
CB.Stats.UNC-4.SInterpret a confidence interval for a population mean, including the mean difference between values in matched pairs.
CB.Stats.UNC-4.TJustify a claim based on a confidence interval for a population mean, including the mean difference between values in matched pairs.
CB.Stats.UNC-4.UIdentify the relationships between sample size, width of a confidence interval, confidence level, and margin of error for a population mean.
CB.Stats.UNC-4.VIdentify an appropriate confidence interval procedure for a difference of two population means.
CB.Stats.UNC-4.WVerify the conditions for calculating confidence intervals for the difference of two population means.
CB.Stats.UNC-4.XDetermine the margin of error for the difference of two population means.
CB.Stats.UNC-4.YCalculate an appropriate confidence interval for a difference of two population means.
CB.Stats.UNC-4.ZInterpret a confidence interval for a difference of population means.
CB.Stats.UNC-5 Errors in statistical inference.
CB.Stats.UNC-5.AIdentify Type I and Type II errors.
CB.Stats.UNC-5.BCalculate the probability of a Type I and Type II errors.
CB.Stats.UNC-5.CIdentify factors that affect the probability of errors in significance testing.
CB.Stats.UNC-5.DInterpret Type I and Type II errors.
CB.Stats.DAT Data Analysis
CB.Stats.DAT-1 Bivariate data analysis and linear regression.
CB.Stats.DAT-1.ADescribe the characteristics of a scatter plot.
CB.Stats.DAT-1.BDetermine the correlation for a linear relationship.
CB.Stats.DAT-1.CInterpret the correlation for a linear relationship.
CB.Stats.DAT-1.DCalculate a predicted response value using a linear regression model.
CB.Stats.DAT-1.ERepresent differences between measured and predicted responses using residual plots.
CB.Stats.DAT-1.FDescribe the form of association of bivariate data using residual plots.
CB.Stats.DAT-1.GEstimate parameters for the least-squares regression line model.
CB.Stats.DAT-1.HInterpret coefficients for the least-squares regression line model.
CB.Stats.DAT-1.IIdentify influential points in regression.
CB.Stats.DAT-1.JCalculate a predicted response using a least-squares regression line for a transformed data set.
CB.Stats.DAT-2 Studies, sampling, and sources of bias.
CB.Stats.DAT-2.AIdentify the type of a study.
CB.Stats.DAT-2.BIdentify appropriate generalizations and determinations based on observational studies.
CB.Stats.DAT-2.CIdentify a sampling method, given a description of a study.
CB.Stats.DAT-2.DExplain why a particular sampling method is or is not appropriate for a given situation.
CB.Stats.DAT-2.EIdentify potential sources of bias in sampling methods.
CB.Stats.DAT-3 Interpreting and justifying conclusions from significance tests.
CB.Stats.DAT-3.AInterpret the p-value of a significance test for a population proportion.
CB.Stats.DAT-3.BJustify a claim about the population based on the results of a significance test for a population proportion.
CB.Stats.DAT-3.CInterpret the p-value of a significance test for a difference of population proportions.
CB.Stats.DAT-3.DJustify a claim about the population based on the results of a significance test for a difference of population proportions.
CB.Stats.DAT-3.EInterpret the p-value of a significance test for a population mean, including the mean difference between values in matched pairs.
CB.Stats.DAT-3.FJustify a claim about the population based on the results of a significance test for a population mean.
CB.Stats.DAT-3.GInterpret the p-value of a significance test for a difference of population means.
CB.Stats.DAT-3.HJustify a claim about the population based on the results of a significance test for a difference of two population means in context.
CB.Stats.DAT-3.IInterpret the p-value for the chi-square test for goodness of fit.
CB.Stats.DAT-3.JJustify a claim about the population based on the results of a chi-square test for goodness of fit.
CB.Stats.DAT-3.KInterpret the p-value for the chi-square test for homogeneity or independence.
CB.Stats.DAT-3.LJustify a claim about the population based on the results of a chi-square test for homogeneity or independence.
CB.Stats.DAT-3.MInterpret the p-value of a significance test for the slope of a regression model.
CB.Stats.DAT-3.NJustify a claim about the population based on the results of a significance test for the slope of a regression model.

Standards on this page are sourced from published state frameworks. Goblins aligns tutoring to these standards.