North Carolina Math Standards - North Carolina Standard Course of Study - Mathematics

954 standards, Kindergarten through AP.

North Carolina's Standard Course of Study uses an integrated math pathway at high school: Math I, II, III, and IV, alongside Precalculus and Discrete Math. This structure favors integrated problem solving over traditional course separation.

Discrete Mathematics for Computer Science

DCS.F Functions

DCS.F.1 Apply recursively-defined relationships to solve problems.

DCS.F.1.1Implement procedures to find the nth term in an arithmetic or geometric sequence using spreadsheets.

DCS.F.1.2Represent the sum of a sequence using sigma notation.

DCS.F.1.3Implement procedures to find the sum of a finite sequence.

DCS.F.1.4Implement procedures to find the sum of an infinite sequence and determine if the series converges or diverges.

DCS.F.1.5Interpret the solutions to arithmetic and geometric sequences and series problems, in context.

DCS.GT Graph Theory

DCS.GT.1 Understand graph theory to model relationships and solve problems.

DCS.GT.1.1Represent real world situations with a vertex-edge graph, adjacency matrix, and vertex-edge table.

DCS.GT.1.2Test graphs and digraphs for Euler paths, Euler circuits, Hamiltonian paths, or Hamiltonian circuits.

DCS.GT.1.3Interpret a complete digraph to determine rank.

DCS.GT.2 Apply graph theory to solve problems.

DCS.GT.2.1Implement critical path analysis algorithms to determine the minimum project time.

DCS.GT.2.2Implement the brute force method, the nearest-neighbor algorithm, and the cheapest-link algorithm to find solutions to a Traveling Salesperson Problem.

DCS.GT.2.3Implement vertex-coloring techniques to solve problems.

DCS.GT.2.4Implement Kruskal and Prim’s algorithms to determine the weight of the minimum spanning tree of a connected graph.

DCS.L.1 Evaluate mathematical logic to model and solve problems.

DCS.L.1.1Construct truth tables that encode the truth and falsity of two or more statements.

DCS.L.1.2Critique logic arguments (e.g., determine if a statement is valid or whether an argument is a tautology or contradiction).

DCS.L.1.3Check 1s and 0s to determine whether a statement is true or false using Boolean logic circuits.

DCS.L.1.4Judge whether two statements are logically equivalent using truth tables.

DCS.N Number and Quantity

DCS.N.1 Apply operations with matrices and vectors.

DCS.N.1.1Implement procedures of addition, subtraction, multiplication, and scalar multiplication on matrices.

DCS.N.1.2Implement procedures of addition, subtraction, and scalar multiplication on vectors.

DCS.N.1.3Implement procedures to find the inverse of a matrix.

DCS.N.2 Understand matrices to solve problems.

DCS.N.2.1Organize data into matrices to solve problems.

DCS.N.2.2Interpret solutions found using matrix operations including Leslie Models and Markov Chains, in context.

DCS.N.2.3Represent a system of equations as a matrix equation.

DCS.N.2.4Use inverse matrices to solve a system of equations with technology.

DCS.N.3 Understand set theory to solve problems.

DCS.N.3.1Recognize sets, subsets, and proper subsets.

DCS.N.3.2Implement set operations to find unions, intersections, complements and set differences with multiple sets.

DCS.N.3.3Represent properties and relationships among sets using Venn diagrams.

DCS.N.3.4Interpret Venn diagrams to solve problems.

DCS.N.4 Understand statements related to number theory and set theory.

DCS.N.4.1Use the Euclidean Algorithm to determine greatest common factor and least common multiple.

DCS.N.4.2Use the Fundamental Theorem of Arithmetic to solve problems.

DCS.N.4.3Conclude that sets are equal using the properties of set operations.

DCS.N.4.4Explain theorems related to greatest common factor, least common multiple, even numbers, odd numbers, prime numbers, and composite numbers.

DCS.SP Statistics and Probability

DCS.SP.1 Apply combinatorics concepts to solve problems.

DCS.SP.1.1Implement the Fundamental Counting Principle to solve problems.

DCS.SP.1.2Implement procedures to calculate a permutation or combination.

Grade 1

NC.1.G Geometry

NC.1.G.1 Distinguish between defining and non-defining attributes and create shapes with defining attributes by:

NC.1.G.1.aBuilding and drawing triangles, rectangles, squares, trapezoids, hexagons, circles.

NC.1.G.1.bBuilding cubes, rectangular prisms, cones, spheres, and cylinders.

NC.1.G.2 Create composite shapes by:

NC.1.G.2.aMaking a two-dimensional composite shape using rectangles, squares, trapezoids, triangles, and half-circles naming the components of the new shape.

NC.1.G.2.bMaking a three-dimensional composite shape using cubes, rectangular prisms, cones, and cylinders, naming the components of the new shape.

NC.1.G.3 Partition circles and rectangles into two and four equal shares.

NC.1.G.3.aDescribe the shares as halves and fourths, as half of and fourth of.

NC.1.G.3.bDescribe the whole as two of, or four of the shares.

NC.1.G.3.cExplain that decomposing into more equal shares creates smaller shares.

NC.1.MD Measurement and Data

NC.1.MD.1Order three objects by length; compare the lengths of two objects indirectly by using a third object.

NC.1.MD.2 Measure lengths with non-standard units.

NC.1.MD.2.aExpress the length of an object as a whole number of non-standard length units.

NC.1.MD.2.bMeasure by laying multiple copies of a shorter object (the length unit) end to end (iterating) with no gaps or overlaps.

NC.1.MD.3Tell and write time in hours and half-hours using analog and digital clocks.

NC.1.MD.4 Organize, represent, and interpret data with up to three categories.

NC.1.MD.4.aAsk and answer questions about the total number of data points.

NC.1.MD.4.bAsk and answer questions about how many in each category.

NC.1.MD.4.cAsk and answer questions about how many more or less are in one category than in another.

NC.1.MD.5Identify quarters, dimes, and nickels and relate their values to pennies.

NC.1.NBT Number and Operations in Base Ten

NC.1.NBT.1Count to 150, starting at any number less than 150.

NC.1.NBT.2 Understand that the two digits of a two-digit number represent amounts of tens and ones.

NC.1.NBT.2.aUnitize by making a ten from a collection of ten ones.

NC.1.NBT.2.bModel the numbers from 11 to 19 as composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

NC.1.NBT.2.cDemonstrate that the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens, with 0 ones.

NC.1.NBT.3Compare two two-digit numbers based on the value of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

NC.1.NBT.4 Using concrete models or drawings, strategies based on place value, properties of operations, and explaining the reasoning used, add, within 100, in the following situations:

NC.1.NBT.4.aA two-digit number and a one-digit number

NC.1.NBT.4.bA two-digit number and a multiple of 10

NC.1.NBT.5Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

NC.1.NBT.6 Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90, explaining the reasoning, using:

NC.1.NBT.6.aConcrete models and drawings

NC.1.NBT.6.bNumber lines

NC.1.NBT.6.cStrategies based on place value

NC.1.NBT.6.dProperties of operations

NC.1.NBT.6.eThe relationship between addition and subtraction

NC.1.NBT.7Read and write numerals, and represent a number of objects with a written numeral, to 100.

NC.1.OA Operations and Algebraic Thinking

NC.1.OA.1 Represent and solve addition and subtraction word problems, within 20, with unknowns, by using objects, drawings, and equations with a symbol for the unknown number to represent the problem, when solving:

NC.1.OA.1.aAdd to/Take from-Change Unknown

NC.1.OA.1.bPut together/Take Apart-Addend Unknown

NC.1.OA.1.cCompare-Difference Unknown

NC.1.OA.2Represent and solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, by using objects, drawings, and equations with a symbol for the unknown number.

NC.1.OA.3Apply the commutative and associative properties as strategies for solving addition problems.

NC.1.OA.4Solve an unknown-addend problem, within 20, by using addition strategies and/or changing it to a subtraction problem.

NC.1.OA.6 Add and subtract, within 20, using strategies such as:

NC.1.OA.6.aCounting on

NC.1.OA.6.bMaking ten

NC.1.OA.6.cDecomposing a number leading to a ten

NC.1.OA.6.dUsing the relationship between addition and subtraction

NC.1.OA.6.eUsing a number line

NC.1.OA.6.fCreating equivalent but simpler or known sums

NC.1.OA.7Apply understanding of the equal sign to determine if equations involving addition and subtraction are true.

NC.1.OA.8Determine the unknown whole number in an addition or subtraction equation involving three whole numbers.

NC.1.OA.9Demonstrate fluency with addition and subtraction within 10.

Grade 2

NC.2.G Geometry

NC.2.G.1Recognize and draw triangles, quadrilaterals, pentagons, and hexagons, having specified attributes; recognize and describe attributes of rectangular prisms and cubes.

NC.2.G.3 Partition circles and rectangles into two, three, or four equal shares.

NC.2.G.3.aDescribe the shares using the words halves, thirds, half of, a third of, fourths, fourth of, quarter of.

NC.2.G.3.bDescribe the whole as two halves, three thirds, four fourths.

NC.2.G.3.cExplain that equal shares of identical wholes need not have the same shape.

NC.2.MD Measurement and Data

NC.2.MD.1Measure the length of an object in standard units by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

NC.2.MD.10 Organize, represent, and interpret data with up to four categories.

NC.2.MD.10.aDraw a picture graph and a bar graph with a single-unit scale to represent a data set.

NC.2.MD.10.bSolve simple put-together, take-apart, and compare problems using information presented in a picture and a bar graph.

NC.2.MD.2Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

NC.2.MD.3Estimate lengths in using standard units of inches, feet, yards, centimeters, and meters.

NC.2.MD.4Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

NC.2.MD.5Use addition and subtraction, within 100, to solve word problems involving lengths that are given in the same units, using equations with a symbol for the unknown number to represent the problem.

NC.2.MD.6Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points and represent whole-number sums and differences, within 100, on a number line.

NC.2.MD.7Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

NC.2.MD.8 Solve word problems involving:

NC.2.MD.8.aQuarters, dimes, nickels, and pennies within 99, using symbols appropriately.

NC.2.MD.8.bWhole dollar amounts, using the $ symbol appropriately.

NC.2.NBT Number and Operations in Base Ten

NC.2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones.

NC.2.NBT.1.aUnitize by making a hundred from a collection of ten tens.

NC.2.NBT.1.bDemonstrate that the numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds, with 0 tens and 0 ones.

NC.2.NBT.1.cCompose and decompose numbers using various groupings of hundreds, tens, and ones.

NC.2.NBT.2Count within 1,000; skip-count by 5s, 10s, and 100s.

NC.2.NBT.3Read and write numbers, within 1,000, using base-ten numerals, number names, and expanded form.

NC.2.NBT.4Compare two three-digit numbers based on the value of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

NC.2.NBT.5 Demonstrate fluency with addition and subtraction, within 100, by:

NC.2.NBT.5.aFlexibly using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

NC.2.NBT.5.bComparing addition and subtraction strategies, and explaining why they work.

NC.2.NBT.5.cSelecting an appropriate strategy in order to efficiently compute sums and differences.

NC.2.NBT.6Add up to three two-digit numbers using strategies based on place value and properties of operations.

NC.2.NBT.7 Add and subtract, within 1,000, relating the strategy to a written method, using:

NC.2.NBT.7.aConcrete models or drawings

NC.2.NBT.7.bStrategies based on place value

NC.2.NBT.7.cProperties of operations

NC.2.NBT.7.dRelationship between addition and subtraction

NC.2.NBT.8Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

NC.2.OA Operations and Algebraic Thinking

NC.2.OA.1 Represent and solve addition and subtraction word problems, within 100, with unknowns in all positions, by using representations and equations with a symbol for the unknown number to represent the problem, when solving:

NC.2.OA.1.a One-Step problems:

NC.2.OA.1.a.1Add to/Take from-Start Unknown

NC.2.OA.1.a.2Compare-Bigger Unknown

NC.2.OA.1.a.3Compare-Smaller Unknown

NC.2.OA.1.b Two-Step problems involving single digits:

NC.2.OA.1.b.1Add to/Take from- Change Unknown

NC.2.OA.1.b.2Add to/Take From- Result Unknown

NC.2.OA.2Demonstrate fluency with addition and subtraction, within 20, using mental strategies.

NC.2.OA.3 Determine whether a group of objects, within 20, has an odd or even number of members by:

NC.2.OA.3.aPairing objects, then counting them by 2s.

NC.2.OA.3.bDetermining whether objects can be placed into two equal groups.

NC.2.OA.3.cWriting an equation to express an even number as a sum of two equal addends.

NC.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

NC.3.G Geometry

NC.3.G.1 Reason with two-dimensional shapes and their attributes.

NC.3.G.1.aInvestigate, describe, and reason about composing triangles and quadrilaterals and decomposing quadrilaterals.

NC.3.G.1.bRecognize and draw examples and non-examples of types of quadrilaterals including rhombuses, rectangles, squares, parallelograms, and trapezoids.

NC.3.MD Measurement and Data

NC.3.MD.1Tell and write time to the nearest minute. Solve word problems involving addition and subtraction of time intervals within the same hour.

NC.3.MD.2 Solve problems involving customary measurement.

NC.3.MD.2.aEstimate and measure lengths in customary units to the quarter-inch and half-inch, and feet and yards to the whole unit.

NC.3.MD.2.bEstimate and measure capacity and weight in customary units to a whole number: cups, pints, quarts, gallons, ounces, and pounds.

NC.3.MD.2.cAdd, subtract, multiply, or divide to solve one-step word problems involving whole number measurements of length, weight, and capacity in the same customary units.

NC.3.MD.3 Represent and interpret scaled picture and bar graphs:

NC.3.MD.3.aCollect data by asking a question that yields data in up to four categories.

NC.3.MD.3.bMake a representation of data and interpret data in a frequency table, scaled picture graph, and/or scaled bar graph with axes provided.

NC.3.MD.3.cSolve one and two-step “how many more” and “how many less” problems using information from these graphs.

NC.3.MD.5Find the area of a rectangle with whole-number side lengths by tiling without gaps or overlaps and counting unit squares.

NC.3.MD.7 Relate area to the operations of multiplication and addition.

NC.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.

NC.3.MD.7.bMultiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving problems, and represent whole-number products as rectangular areas in mathematical reasoning.

NC.3.MD.7.cUse tiles and/or arrays to illustrate and explain that the area of a rectangle can be found by partitioning it into two smaller rectangles, and that the area of the large rectangle is the sum of the two smaller rectangles.

NC.3.MD.8Solve problems involving perimeters of polygons, including finding the perimeter given the side lengths, and finding an unknown side length.

NC.3.NBT Number and Operations in Base Ten

NC.3.NBT.2 Add and subtract whole numbers up to and including 1,000.

NC.3.NBT.2.aUse estimation strategies to assess reasonableness of answers.

NC.3.NBT.2.bModel and explain how the relationship between addition and subtraction can be applied to solve addition and subtraction problems.

NC.3.NBT.2.cUse expanded form to decompose numbers and then find sums and differences.

NC.3.NBT.3Use concrete and pictorial models, based on place value and the properties of operations, to find the product of a one-digit whole number by a multiple of 10 in the range 10–90.

NC.3.NF Number and Operations – Fractions

NC.3.NF.1 Interpret unit fractions with denominators of 2, 3, 4, 6, and 8 as quantities formed when a whole is partitioned into equal parts;

NC.3.NF.1.aExplain that a unit fraction is one of those parts.

NC.3.NF.1.bRepresent and identify unit fractions using area and length models.

NC.3.NF.2 Interpret fractions with denominators of 2, 3, 4, 6, and 8 using area and length models.

NC.3.NF.2.aUsing an area model, explain that the numerator of a fraction represents the number of equal parts of the unit fraction.

NC.3.NF.2.bUsing a number line, explain that the numerator of a fraction represents the number of lengths of the unit fraction from 0.

NC.3.NF.3 Represent equivalent fractions with area and length models by:

NC.3.NF.3.aComposing and decomposing fractions into equivalent fractions using related fractions: halves, fourths and eighths; thirds and sixths.

NC.3.NF.3.bExplaining that a fraction with the same numerator and denominator equals one whole.

NC.3.NF.3.cExpressing whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

NC.3.NF.4Compare two fractions with the same numerator or the same denominator by reasoning about their size, using area and length models, and using the >, <, and = symbols. Recognize that comparisons are valid only when the two fractions refer to the same whole with denominators: halves, fourths and eighths; thirds and sixths.

NC.3.OA Operations and Algebraic Thinking

NC.3.OA.1 For products of whole numbers with two factors up to and including 10:

NC.3.OA.1.aInterpret the factors as representing the number of equal groups and the number of objects in each group.

NC.3.OA.1.bIllustrate and explain strategies including arrays, repeated addition, decomposing a factor, and applying the commutative and associative properties.

NC.3.OA.2 For whole-number quotients of whole numbers with a one-digit divisor and a one-digit quotient:

NC.3.OA.2.aInterpret the divisor and quotient in a division equation as representing the number of equal groups and the number of objects in each group.

NC.3.OA.2.bIllustrate and explain strategies including arrays, repeated addition or subtraction, and decomposing a factor.

NC.3.OA.3 Represent, interpret, and solve one-step problems involving multiplication and division.

NC.3.OA.3.aSolve multiplication word problems with factors up to and including 10. Represent the problem using arrays, pictures, and/or equations with a symbol for the unknown number to represent the problem.

NC.3.OA.3.bSolve division word problems with a divisor and quotient up to and including 10. Represent the problem using arrays, pictures, repeated subtraction and/or equations with a symbol for the unknown number to represent the problem.

NC.3.OA.6Solve an unknown-factor problem, by using division strategies and/or changing it to a multiplication problem.

NC.3.OA.7 Demonstrate fluency with multiplication and division with factors, quotients and divisors up to and including 10.

NC.3.OA.7.aKnow from memory all products with factors up to and including 10.

NC.3.OA.7.bIllustrate and explain using the relationship between multiplication and division.

NC.3.OA.7.cDetermine the unknown whole number in a multiplication or division equation relating three whole numbers.

NC.3.OA.8Solve two-step word problems using addition, subtraction, and multiplication, representing problems using equations with a symbol for the unknown number.

NC.3.OA.9Interpret patterns of multiplication on a hundreds board and/or multiplication table.

Grade 4

NC.4.G Geometry

NC.4.G.1Draw and identify points, lines, line segments, rays, angles, and perpendicular and parallel lines.

NC.4.G.2Classify quadrilaterals and triangles based on angle measure, side lengths, and the presence or absence of parallel or perpendicular lines.

NC.4.G.3Recognize symmetry in a two-dimensional figure, and identify and draw lines of symmetry.

NC.4.MD Measurement and Data

NC.4.MD.1 Know relative sizes of measurement units. Solve problems involving metric measurement.

NC.4.MD.1.aMeasure to solve problems involving metric units: centimeter, meter, gram, kilogram, Liter, milliliter.

NC.4.MD.1.bAdd, subtract, multiply, and divide to solve one-step word problems involving whole-number measurements of length, mass, and capacity that are given in metric units.

NC.4.MD.2Use multiplicative reasoning to convert metric measurements from a larger unit to a smaller unit using place value understanding, two-column tables, and length models.

NC.4.MD.3 Solve problems with area and perimeter.

NC.4.MD.3.aFind areas of rectilinear figures with known side lengths.

NC.4.MD.3.bSolve problems involving a fixed area and varying perimeters and a fixed perimeter and varying areas.

NC.4.MD.3.cApply the area and perimeter formulas for rectangles in real world and mathematical problems.

NC.4.MD.4 Represent and interpret data using whole numbers.

NC.4.MD.4.aCollect data by asking a question that yields numerical data.

NC.4.MD.4.bMake a representation of data and interpret data in a frequency table, scaled bar graph, and/or line plot.

NC.4.MD.4.cDetermine whether a survey question will yield categorical or numerical data.

NC.4.MD.6 Develop an understanding of angles and angle measurement.

NC.4.MD.6.aUnderstand angles as geometric shapes that are formed wherever two rays share a common endpoint, and are measured in degrees.

NC.4.MD.6.bMeasure and sketch angles in whole-number degrees using a protractor.

NC.4.MD.6.cSolve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems.

NC.4.MD.8Solve word problems involving addition and subtraction of time intervals that cross the hour.

NC.4.NBT Number and Operations in Base Ten

NC.4.NBT.1Explain that in a multi-digit whole number, a digit in one place represents 10 times as much as it represents in the place to its right, up to 100,000.

NC.4.NBT.2Read and write multi-digit whole numbers up to and including 100,000 using numerals, number names, and expanded form.

NC.4.NBT.4Add and subtract multi-digit whole numbers up to and including 100,000 using the standard algorithm with place value understanding.

NC.4.NBT.5Multiply a whole number of up to three digits by a one-digit whole number, and multiply up to two two-digit numbers with place value understanding using area models, partial products, and the properties of operations. Use models to make connections and develop the algorithm.

NC.4.NBT.6Find whole-number quotients and remainders with up to three-digit dividends and one-digit divisors with place value understanding using rectangular arrays, area models, repeated subtraction, partial quotients, properties of operations, and/or the relationship between multiplication and division.

NC.4.NBT.7Compare two multi-digit numbers up to and including 100,000 based on the values of the digits in each place, using >, =, and < symbols to record the results of comparisons.

NC.4.NF Number and Operations – Fractions

NC.4.NF.1Explain why a fraction is equivalent to another fraction by using area and length fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size.

NC.4.NF.2 Compare two fractions with different numerators and different denominators, using the denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. 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 by:

NC.4.NF.2.aReasoning about their size and using area and length models.

NC.4.NF.2.bUsing benchmark fractions 0, , and a whole.

NC.4.NF.2.cComparing common numerator or common denominators.

NC.4.NF.3 Understand and justify decompositions of fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100.

NC.4.NF.3.aUnderstand addition and subtraction of fractions as joining and separating parts referring to the same whole.

NC.4.NF.3.bDecompose a fraction into a sum of unit fractions and a sum of fractions with the same denominator in more than one way using area models, length models, and equations.

NC.4.NF.3.cAdd and subtract fractions, including mixed numbers with like denominators, by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

NC.4.NF.3.dSolve word problems involving addition and subtraction of fractions, including mixed numbers by writing equations from a visual representation of the problem.

NC.4.NF.4 Apply and extend previous understandings of multiplication to:

NC.4.NF.4.aModel and explain how fractions can be represented by multiplying a whole number by a unit fraction, using this understanding to multiply a whole number by any fraction less than one.

NC.4.NF.4.bSolve word problems involving multiplication of a fraction by a whole number.

NC.4.NF.6 Use decimal notation to represent fractions.

NC.4.NF.6.aExpress, model and explain the equivalence between fractions with denominators of 10 and 100.

NC.4.NF.6.bUse equivalent fractions to add two fractions with denominators of 10 or 100.

NC.4.NF.6.cRepresent tenths and hundredths with models, making connections between fractions and decimals.

NC.4.NF.7Compare two decimals to hundredths by reasoning about their size using area and length models, and recording the results of comparisons with the symbols >, =, or <. Recognize that comparisons are valid only when the two decimals refer to the same whole.

NC.4.OA Operations and Algebraic Thinking

NC.4.OA.1Interpret a multiplication equation as a comparison. Multiply or divide to solve word problems involving multiplicative comparisons using models and equations with a symbol for the unknown number. Distinguish multiplicative comparison from additive comparison.

NC.4.OA.3 Solve two-step word problems involving the four operations with whole numbers.

NC.4.OA.3.aUse estimation strategies to assess reasonableness of answers.

NC.4.OA.3.bInterpret remainders in word problems.

NC.4.OA.3.cRepresent problems using equations with a letter standing for the unknown quantity.

NC.4.OA.4 Find all factor pairs for whole numbers up to and including 50 to:

NC.4.OA.4.aRecognize that a whole number is a multiple of each of its factors.

NC.4.OA.4.bDetermine whether a given whole number is a multiple of a given one-digit number.

NC.4.OA.4.cDetermine if the number is prime or composite.

NC.4.OA.5Generate and analyze a number or shape pattern that follows a given rule.

Grade 5

NC.5.G Geometry

NC.5.G.1Graph points in the first quadrant of a coordinate plane, and identify and interpret the x and y coordinates to solve problems.

NC.5.G.3 Classify quadrilaterals into categories based on their properties.

NC.5.G.3.aExplain that attributes belonging to a category of quadrilaterals also belong to all subcategories of that category.

NC.5.G.3.bClassify quadrilaterals in a hierarchy based on properties.

NC.5.MD Measurement and Data

NC.5.MD.1Given a conversion chart, use multiplicative reasoning to solve one-step conversion problems within a given measurement system.

NC.5.MD.2 Represent and interpret data.

NC.5.MD.2.aCollect data by asking a question that yields data that changes over time.

NC.5.MD.2.bMake and interpret a representation of data using a line graph.

NC.5.MD.2.cDetermine whether a survey question will yield categorical or numerical data, or data that changes over time.

NC.5.MD.4Recognize volume as an attribute of solid figures and measure volume by counting unit cubes, using cubic centimeters, cubic inches, cubic feet, and improvised units.

NC.5.MD.5 Relate volume to the operations of multiplication and addition.

NC.5.MD.5.aFind the volume of a rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths.

NC.5.MD.5.bBuild understanding of the volume formula for rectangular prisms with whole-number edge lengths in the context of solving problems.

NC.5.MD.5.cFind volume of solid figures with one-digit dimensions composed of two non-overlapping rectangular prisms.

NC.5.NBT Number and Operations in Base Ten

NC.5.NBT.1 Explain the patterns in the place value system from one million to the thousandths place.

NC.5.NBT.1.aExplain 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.

NC.5.NBT.1.bExplain patterns in products and quotients when numbers are multiplied by 1,000, 100, 10, 0.1, and 0.01 and/or divided by 10 and 100.

NC.5.NBT.3 Read, write, and compare decimals to thousandths.

NC.5.NBT.3.aWrite decimals using base-ten numerals, number names, and expanded form.

NC.5.NBT.3.bCompare two decimals to thousandths based on the value of the digits in each place, using >, =, and < symbols to record the results of comparisons.

NC.5.NBT.5Demonstrate fluency with the multiplication of two whole numbers up to a three-digit number by a two-digit number using the standard algorithm.

NC.5.NBT.6Find quotients with remainders when dividing whole numbers with up to four-digit dividends and two-digit divisors using rectangular arrays, area models, repeated subtraction, partial quotients, and/or the relationship between multiplication and division. Use models to make connections and develop the algorithm.

NC.5.NBT.7 Compute and solve real-world problems with multi-digit whole numbers and decimal numbers.

NC.5.NBT.7.aAdd and subtract decimals to thousandths using models, drawings or strategies based on place value.

NC.5.NBT.7.bMultiply decimals with a product to thousandths using models, drawings, or strategies based on place value.

NC.5.NBT.7.cDivide a whole number by a decimal and divide a decimal by a whole number, using repeated subtraction or area models. Decimals should be limited to hundredths.

NC.5.NBT.7.dUse estimation strategies to assess reasonableness of answers.

NC.5.NF Number and Operations – Fractions

NC.5.NF.1 Add and subtract fractions, including mixed numbers, with unlike denominators using related fractions: halves, fourths and eighths; thirds, sixths, and twelfths; fifths, tenths, and hundredths.

NC.5.NF.1.aUse benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

NC.5.NF.1.bSolve one- and two-step word problems in context using area and length models to develop the algorithm. Represent the word problem in an equation.

NC.5.NF.3 Use fractions to model and solve division problems.

NC.5.NF.3.aInterpret a fraction as an equal sharing context, where a quantity is divided into equal parts.

NC.5.NF.3.bModel and interpret a fraction as the division of the numerator by the denominator.

NC.5.NF.3.cSolve one-step word problems involving division of whole numbers leading to answers in the form of fractions and mixed numbers, with denominators of 2, 3, 4, 5, 6, 8, 10, and 12, using area, length, and set models or equations.

NC.5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction, including mixed numbers.

NC.5.NF.4.aUse area and length models to multiply two fractions, with the denominators 2, 3, 4.

NC.5.NF.4.bExplain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number and when multiplying a given number by a fraction less than 1 results in a product smaller than the given number.

NC.5.NF.4.cSolve one-step word problems involving multiplication of fractions using models to develop the algorithm.

NC.5.NF.7Solve one-step word problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions using area and length models, and equations to represent the problem.

NC.5.OA Operations and Algebraic Thinking

NC.5.OA.2 Write, explain, and evaluate numerical expressions involving the four operations to solve up to two-step problems. Include expressions involving:

NC.5.OA.2.aParentheses, using the order of operations.

NC.5.OA.2.bCommutative, associative and distributive properties.

NC.5.OA.3 Generate two numerical patterns using two given rules.

NC.5.OA.3.aIdentify apparent relationships between corresponding terms.

NC.5.OA.3.bForm ordered pairs consisting of corresponding terms from the two patterns.

NC.5.OA.3.cGraph the ordered pairs on a coordinate plane.

Grade 6

NC.6.EE Expressions and Equations

NC.6.EE.1Write and evaluate numerical expressions, with and without grouping symbols, involving whole-number exponents.

NC.6.EE.2 Write, read, and evaluate algebraic expressions.

NC.6.EE.2.aWrite expressions that record operations with numbers and with letters standing for numbers.

NC.6.EE.2.bIdentify parts of an expression using mathematical terms and view one or more of those parts as a single entity.

NC.6.EE.2.cEvaluate expressions at specific values of their variables using expressions that arise from formulas used in real-world problems.

NC.6.EE.3Apply the properties of operations to generate equivalent expressions without exponents.

NC.6.EE.4Identify when two expressions are equivalent and justify with mathematical reasoning.

NC.6.EE.5Use substitution to determine whether a given number in a specified set makes an equation true.

NC.6.EE.6Use variables to represent numbers and write expressions when solving a real-world or mathematical problem.

NC.6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form:

NC.6.EE.7.ax + p = q in which p, q and x are all nonnegative rational numbers; and,

NC.6.EE.7.bp ∙ x = q for cases in which p, q and x are all nonnegative rational numbers.

NC.6.EE.8 Reason about inequalities by:

NC.6.EE.8.aUsing substitution to determine whether a given number in a specified set makes an inequality true.

NC.6.EE.8.bWriting an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem.

NC.6.EE.8.cRecognizing that inequalities of the form x > c or x < c have infinitely many solutions.

NC.6.EE.8.dRepresenting solutions of inequalities on number line diagrams.

NC.6.EE.9 Represent and analyze quantitative relationships by:

NC.6.EE.9.aUsing variables to represent two quantities in a real-world or mathematical context that change in relationship to one another.

NC.6.EE.9.bAnalyze the relationship between quantities in different representations (context, equations, tables, and graphs).

NC.6.G Geometry

NC.6.G.1 Create geometric models to solve real-world and mathematical problems to:

NC.6.G.1.aFind the area of triangles by composing into rectangles and decomposing into right triangles.

NC.6.G.1.bFind the area of special quadrilaterals and polygons by decomposing into triangles or rectangles.

NC.6.G.2Apply and extend previous understandings of the volume of a right rectangular prism to find the volume of right rectangular prisms with fractional edge lengths. Apply this understanding to the context of solving real-world and mathematical problems.

NC.6.G.3 Use the coordinate plane to solve real-world and mathematical problems by:

NC.6.G.3.aDrawing polygons in the coordinate plane given coordinates for the vertices.

NC.6.G.3.bUsing coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate.

NC.6.G.4Represent right prisms and right pyramids using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

NC.6.NS The Number System

NC.6.NS.1 Use visual models and common denominators to:

NC.6.NS.1.aInterpret and compute quotients of fractions.

NC.6.NS.1.bSolve real-world and mathematical problems involving division of fractions.

NC.6.NS.2Fluently divide using long division with a minimum of a four-digit dividend and interpret the quotient and remainder in context.

NC.6.NS.3Apply and extend previous understandings of decimals to develop and fluently use the standard algorithms for addition, subtraction, multiplication and division of decimals.

NC.6.NS.4 Understand and use prime factorization and the relationships between factors to:

NC.6.NS.4.aFind the unique prime factorization for a whole number.

NC.6.NS.4.bFind the greatest common factor of two whole numbers less than or equal to 100.

NC.6.NS.4.cUse the greatest common factor and the distributive property to rewrite the sum of two whole numbers, each less than or equal to 100.

NC.6.NS.4.dFind the least common multiple of two whole numbers less than or equal to 12 to add and subtract fractions with unlike denominators.

NC.6.NS.5 Understand and use rational numbers to:

NC.6.NS.5.aDescribe quantities having opposite directions or values.

NC.6.NS.5.bRepresent quantities in real-world contexts, explaining the meaning of 0 in each situation.

NC.6.NS.5.c Understand the absolute value of a rational number as its distance from 0 on the number line to:

NC.6.NS.5.c.1Interpret absolute value as magnitude for a positive or negative quantity in a real-world context.

NC.6.NS.5.c.2Distinguish comparisons of absolute value from statements about order.

NC.6.NS.6 Understand rational numbers as points on the number line and as ordered pairs on a coordinate plane.

NC.6.NS.6.a On a number line:

NC.6.NS.6.a.1Recognize opposite signs of numbers as indicating locations on opposite sides of 0 and that the opposite of the opposite of a number is the number itself.

NC.6.NS.6.a.2Find and position rational numbers on a horizontal or vertical number line.

NC.6.NS.6.b On a coordinate plane:

NC.6.NS.6.b.1Understand signs of numbers in ordered pairs as indicating locations in quadrants.

NC.6.NS.6.b.2Recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

NC.6.NS.6.b.3Find and position pairs of rational numbers on a coordinate plane.

NC.6.NS.7 Understand ordering of rational numbers.

NC.6.NS.7.aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.

NC.6.NS.7.bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.

NC.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.

NC.6.NS.9 Apply and extend previous understandings of addition and subtraction.

NC.6.NS.9.a Understand additive inverses when adding and subtracting integers.

NC.6.NS.9.a.1Describe situations in which opposite quantities combine to make 0.

NC.6.NS.9.a.2Understand p + q as the number located a distance q from p, in the positive or negative direction depending on the sign of q. Show that a number and its additive inverse create a zero pair.

NC.6.NS.9.a.3Understand subtraction of integers as adding the additive inverse, p - q = p + (-q). Show that the distance between two integers on the number line is the absolute value of their difference.

NC.6.NS.9.a.4Use models to add and subtract integers from -20 to 20 and describe real-world contexts using sums and differences.

NC.6.RP Ratio and Proportional Relationships

NC.6.RP.1 Understand the concept of a ratio and use ratio language to:

NC.6.RP.1.aDescribe a ratio as a multiplicative relationship between two quantities.

NC.6.RP.1.bModel a ratio relationship using a variety of representations.

NC.6.RP.2Understand that ratios can be expressed as equivalent unit ratios by finding and interpreting both unit ratios in context.

NC.6.RP.3 Use ratio reasoning with equivalent whole-number ratios to solve real-world and mathematical problems by:

NC.6.RP.3.aCreating and using a table to compare ratios.

NC.6.RP.3.bFinding missing values in the tables.

NC.6.RP.3.cUsing a unit ratio.

NC.6.RP.3.dConverting and manipulating measurements using given ratios.

NC.6.RP.3.ePlotting the pairs of values on the coordinate plane.

NC.6.RP.4 Use ratio reasoning to solve real-world and mathematical problems with percents by:

NC.6.RP.4.aUnderstanding and finding a percent of a quantity as a ratio per 100.

NC.6.RP.4.bUsing equivalent ratios, such as benchmark percents (50%, 25%, 10%, 5%, 1%), to determine a part of any given quantity.

NC.6.RP.4.cFinding the whole, given a part and the percent.

NC.6.SP Statistics and Probability

NC.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.

NC.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.

NC.6.SP.3 Understand that both a measure of center and a description of variability should be considered when describing a numerical data set.

NC.6.SP.3.a Determine the measure of center of a data set and understand that it is a single number that summarizes all the values of that data set.

NC.6.SP.3.a.1Understand that a mean is a measure of center that represents a balance point or fair share of a data set and can be influenced by the presence of extreme values within the data set.

NC.6.SP.3.a.2Understand the median as a measure of center that is the numerical middle of an ordered data set.

NC.6.SP.3.bUnderstand that describing the variability of a data set is needed to distinguish between data sets in the same scale, by comparing graphical representations of different data sets in the same scale that have similar measures of center, but different spreads.

NC.6.SP.4 Display numerical data in plots on a number line.

NC.6.SP.4.aUse dot plots, histograms, and box plots to represent data.

NC.6.SP.4.bCompare the attributes of different representations of the same data.

NC.6.SP.5 Summarize numerical data sets in relation to their context.

NC.6.SP.5.a Describe the collected data by:

NC.6.SP.5.a.1Reporting the number of observations in dot plots and histograms.

NC.6.SP.5.a.2Communicating the nature of the attribute under investigation, how it was measured, and the units of measurement.

NC.6.SP.5.b Analyze center and variability by:

NC.6.SP.5.b.1Giving quantitative measures of center, describing variability, and any overall pattern, and noting any striking deviations.

NC.6.SP.5.b.2Justifying the appropriate choice of measures of center using the shape of the data distribution.

Grade 7

NC.7.EE Expressions and Equations

NC.7.EE.1 Apply properties of operations as strategies to:

NC.7.EE.1.aAdd, subtract, and expand linear expressions with rational coefficients.

NC.7.EE.1.bFactor linear expression with an integer GCF.

NC.7.EE.2Understand that equivalent expressions can reveal real-world and mathematical relationships. Interpret the meaning of the parts of each expression in context.

NC.7.EE.3 Solve multi-step real-world and mathematical problems posed with rational numbers in algebraic expressions.

NC.7.EE.3.aApply properties of operations to calculate with positive and negative numbers in any form.

NC.7.EE.3.bConvert between different forms of a number and equivalent forms of the expression as appropriate.

NC.7.EE.4 Use variables to represent quantities to solve real-world or mathematical problems.

NC.7.EE.4.a Construct equations to solve problems by reasoning about the quantities.

NC.7.EE.4.a.1Fluently solve multistep equations with the variable on one side, including those generated by word problems.

NC.7.EE.4.a.2Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

NC.7.EE.4.a.3Interpret the solution in context.

NC.7.EE.4.b Construct inequalities to solve problems by reasoning about the quantities.

NC.7.EE.4.b.1Fluently solve multi-step inequalities with the variable on one side, including those generated by word problems.

NC.7.EE.4.b.2Compare an algebraic solution process for equations and an algebraic solution process for inequalities.

NC.7.EE.4.b.3Graph the solution set of the inequality and interpret in context.

NC.7.G Geometry

NC.7.G.1 Solve problems involving scale drawings of geometric figures by:

NC.7.G.1.aBuilding an understanding that angle measures remain the same and side lengths are proportional.

NC.7.G.1.bUsing a scale factor to compute actual lengths and areas from a scale drawing.

NC.7.G.1.cCreating a scale drawing.

NC.7.G.2Understand the characteristics of angles and side lengths that create a unique triangle, more than one triangle or no triangle. Build triangles from three measures of angles and/or sides.

NC.7.G.4 Understand area and circumference of a circle.

NC.7.G.4.aUnderstand the relationships between the radius, diameter, circumference, and area.

NC.7.G.4.bApply the formulas for area and circumference of a circle to solve problems.

NC.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.

NC.7.G.6 Solve real-world and mathematical problems involving:

NC.7.G.6.aArea and perimeter of two-dimensional objects composed of triangles, quadrilaterals, and polygons.

NC.7.G.6.bVolume and surface area of pyramids, prisms, or three-dimensional objects composed of cubes, pyramids, and right prisms.

NC.7.NS The Number System

NC.7.NS.1Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers, using the properties of operations, and describing real-world contexts using sums and differences.

NC.7.NS.2 Apply and extend previous understandings of multiplication and division.

NC.7.NS.2.aUnderstand that a rational number is any number that can be written as a quotient of integers with a non-zero divisor.

NC.7.NS.2.bApply properties of operations as strategies, including the standard algorithms, to multiply and divide rational numbers and describe the product and quotient in real-world contexts.

NC.7.NS.2.c Use division and previous understandings of fractions and decimals.

NC.7.NS.2.c.1Convert a fraction to a decimal using long division.

NC.7.NS.2.c.2Understand that the decimal form of a rational number terminates in 0s or eventually repeats.

NC.7.NS.3Solve real-world and mathematical problems involving numerical expressions with rational numbers using the four operations.

NC.7.RP Ratio and Proportional Relationships

NC.7.RP.1Compute unit rates associated with ratios of fractions to solve real-world and mathematical problems.

NC.7.RP.2 Recognize and represent proportional relationships between quantities.

NC.7.RP.2.a Understand that a proportion is a relationship of equality between ratios.

NC.7.RP.2.a.1Represent proportional relationships using tables and graphs.

NC.7.RP.2.a.2Recognize whether ratios are in a proportional relationship using tables and graphs.

NC.7.RP.2.a.3Compare two different proportional relationships using tables, graphs, equations, and verbal descriptions.

NC.7.RP.2.bIdentify the unit rate (constant of proportionality) within two quantities in a proportional relationship using tables, graphs, equations, and verbal descriptions.

NC.7.RP.2.cCreate equations and graphs to represent proportional relationships.

NC.7.RP.2.d Use a graphical representation of a proportional relationship in context to:

NC.7.RP.2.d.1Explain the meaning of any point (x,y).

NC.7.RP.2.d.2Explain the meaning of (0, 0) and why it is included.

NC.7.RP.2.d.3Understand that the y-coordinate of the ordered pair (1,r) corresponds to the unit rate and explain its meaning.

NC.7.RP.3Use scale factors and unit rates in proportional relationships to solve ratio and percent problems.

NC.7.SP Statistics and Probability

NC.7.SP.1 Understand that statistics can be used to gain information about a population by:

NC.7.SP.1.aRecognizing that generalizations about a population from a sample are valid only if the sample is representative of that population.

NC.7.SP.1.bUsing random sampling to produce representative samples to support valid inferences.

NC.7.SP.2Generate multiple random samples (or simulated samples) of the same size to gauge the variation in estimates or predictions, and use this data to draw inferences about a population with an unknown characteristic of interest.

NC.7.SP.3 Recognize the role of variability when comparing two populations.

NC.7.SP.3.a Calculate the measure of variability of a data set and understand that it describes how the values of the data set vary with a single number.

NC.7.SP.3.a.1Understand the mean absolute deviation of a data set is a measure of variability that describes the average distance that points within a data set are from the mean of the data set.

NC.7.SP.3.a.2Understand that the range describes the spread of the entire data set.

NC.7.SP.3.a.3Understand that the interquartile range describes the spread of the middle 50% of the data.

NC.7.SP.3.bInformally assess the difference between two data sets by examining the overlap and separation between the graphical representations of two data sets.

NC.7.SP.4Use measures of center and measures of variability for numerical data from random samples to draw comparative inferences about two populations.

NC.7.SP.5Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring.

NC.7.SP.6Collect data to calculate the experimental probability of a chance event, observing its long-run relative frequency. Use this experimental probability to predict the approximate relative frequency.

NC.7.SP.7 Develop a probability model and use it to find probabilities of simple events.

NC.7.SP.7.aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.

NC.7.SP.7.bDevelop a probability model (which may not be uniform) by repeatedly performing a chance process and observing frequencies in the data generated.

NC.7.SP.7.cCompare theoretical and experimental probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

NC.7.SP.8 Determine probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

NC.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.

NC.7.SP.8.bFor an event described in everyday language, identify the outcomes in the sample space which compose the event, when the sample space is represented using organized lists, tables, and tree diagrams.

NC.7.SP.8.cDesign and use a simulation to generate frequencies for compound events.

Grade 8

NC.8.EE Expressions and Equations

NC.8.EE.1Develop and apply the properties of integer exponents to generate equivalent numerical expressions.

NC.8.EE.2 Use square root and cube root symbols to:

NC.8.EE.2.aRepresent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number.

NC.8.EE.2.bEvaluate square roots of perfect squares and cube roots of perfect cubes for positive numbers less than or equal to 400.

NC.8.EE.3Use numbers expressed in scientific notation to estimate very large or very small quantities and to express how many times as much one is than the other.

NC.8.EE.4Perform multiplication and division with numbers expressed in scientific notation to solve real-world problems, including problems where both decimal and scientific notation are used.

NC.8.EE.7 Solve real-world and mathematical problems by writing and solving equations and inequalities in one variable.

NC.8.EE.7.aRecognize linear equations in one variable as having one solution, infinitely many solutions, or no solutions.

NC.8.EE.7.bSolve linear equations and inequalities including multi-step equations and inequalities with the same variable on both sides.

NC.8.EE.8 Analyze and solve a system of two linear equations in two variables in slope-intercept form.

NC.8.EE.8.aUnderstand that solutions to a system of two linear equations correspond to the points of intersection of their graphs because the point of intersection satisfies both equations simultaneously.

NC.8.EE.8.bSolve real-world and mathematical problems leading to systems of linear equations by graphing the equations. Solve simple cases by inspection.

NC.8.F Functions

NC.8.F.1 Understand that a function is a rule that assigns to each input exactly one output.

NC.8.F.1.aRecognize functions when graphed as the set of ordered pairs consisting of an input and exactly one corresponding output.

NC.8.F.1.bRecognize functions given a table of values or a set of ordered pairs.

NC.8.F.2Compare properties of two linear functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

NC.8.F.3Identify linear functions from tables, equations, and graphs.

NC.8.F.4 Analyze functions that model linear relationships.

NC.8.F.4.aUnderstand that a linear relationship can be generalized by y = mx + b.

NC.8.F.4.bWrite an equation in slope-intercept form to model a linear relationship by determining the rate of change and the initial value, given at least two (x,y) values or a graph.

NC.8.F.4.cConstruct a graph of a linear relationship given an equation in slope-intercept form.

NC.8.F.4.dInterpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of the slope and y-intercept of its graph or a table of values.

NC.8.F.5 Qualitatively analyze the functional relationship between two quantities.

NC.8.F.5.aAnalyze a graph determining where the function is increasing or decreasing; linear or non-linear.

NC.8.F.5.bSketch a graph that exhibits the qualitative features of a real-world function.

NC.8.G Geometry

NC.8.G.2 Use transformations to define congruence.

NC.8.G.2.aVerify experimentally the properties of rotations, reflections, and translations that create congruent figures.

NC.8.G.2.bUnderstand 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.

NC.8.G.2.cGiven two congruent figures, describe a sequence that exhibits the congruence between them.

NC.8.G.3Describe the effect of dilations about the origin, translations, rotations about the origin in 90 degree increments, and reflections across x-axis and y-axis on two-dimensional figures using coordinates.

NC.8.G.4 Use transformations to define similarity.

NC.8.G.4.aVerify experimentally the properties of dilations that create similar figures.

NC.8.G.4.bUnderstand 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.

NC.8.G.4.cGiven two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

NC.8.G.5 Use informal arguments to analyze angle relationships.

NC.8.G.5.aRecognize relationships between interior and exterior angles of a triangle.

NC.8.G.5.bRecognize the relationships between the angles created when parallel lines are cut by a transversal.

NC.8.G.5.cRecognize the angle-angle criterion for similarity of triangles.

NC.8.G.5.dSolve real-world and mathematical problems involving angles.

NC.8.G.6Explain the Pythagorean Theorem and its converse.

NC.8.G.7Apply the Pythagorean Theorem and its converse to solve real-world and mathematical problems.

NC.8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

NC.8.G.9Understand how the formulas for the volumes of cones, cylinders, and spheres are related and use the relationship to solve real-world and mathematical problems.

NC.8.NS The Number System

NC.8.NS.1Understand that every number has a decimal expansion. Building upon the definition of a rational number, know that an irrational number is defined as a non-repeating, non-terminating decimal.

NC.8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers and locate them approximately on a number line. Estimate the value of expressions involving:

NC.8.NS.2.aSquare roots and cube roots to the tenths.

NC.8.NS.2.b𝜋 to the hundredths.

NC.8.SP Statistics and Probability

NC.8.SP.1Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Investigate and describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

NC.8.SP.2 Model the relationship between bivariate quantitative data to:

NC.8.SP.2.aInformally fit a straight line for a scatter plot that suggests a linear association.

NC.8.SP.2.bInformally assess the model fit by judging the closeness of the data points to the line.

NC.8.SP.3Use the equation of a linear model to solve problems in the context of bivariate quantitative data, interpreting the slope and y-intercept.

NC.8.SP.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table.

NC.8.SP.4.aConstruct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects.

NC.8.SP.4.bUse relative frequencies calculated for rows or columns to describe possible association between the two variables.

Kindergarten

NC.K.CC Counting and Cardinality

NC.K.CC.1 Know number names and recognize patterns in the counting sequence by:

NC.K.CC.1.aCounting to 100 by ones.

NC.K.CC.1.bCounting to 100 by tens.

NC.K.CC.2Count forward beginning from a given number within the known sequence, instead of having to begin at 1.

NC.K.CC.3Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20, with 0 representing a count of no objects.

NC.K.CC.4 Understand the relationship between numbers and quantities.

NC.K.CC.4.aWhen counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object (one-to-one correspondence).

NC.K.CC.4.bRecognize that the last number named tells the number of objects counted regardless of their arrangement (cardinality).

NC.K.CC.4.cState the number of objects in a group, of up to 5 objects, without counting the objects (perceptual subitizing).

NC.K.CC.5 Count to answer “How many?” in the following situations:

NC.K.CC.5.aGiven a number from 1–20, count out that many objects.

NC.K.CC.5.bGiven up to 20 objects, name the next successive number when an object is added, recognizing the quantity is one more/greater.

NC.K.CC.5.cGiven 20 objects arranged in a line, a rectangular array, and a circle, identify how many.

NC.K.CC.5.dGiven 10 objects in a scattered arrangement, identify how many.

NC.K.CC.6Identify whether the number of objects, within 10, in one group is greater than, less than, or equal to the number of objects in another group, by using matching and counting strategies.

NC.K.CC.7Compare two numbers, within 10, presented as written numerals.

NC.K.G Geometry

NC.K.G.1Describe objects in the environment using names of shapes, and describe the relative positions of objects using positional terms.

NC.K.G.2Correctly name squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres regardless of their orientations or overall size.

NC.K.G.3Identify squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres as two-dimensional or three-dimensional.

NC.K.G.4Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, attributes and other properties.

NC.K.G.5 Model shapes in the world by:

NC.K.G.5.aBuilding and drawing triangles, rectangles, squares, hexagons, circles.

NC.K.G.5.bBuilding cubes, cones, spheres, and cylinders.

NC.K.G.6Compose larger shapes from simple shapes.

NC.K.MD Measurement and Data

NC.K.MD.1Describe measurable attributes of objects; and describe several different measurable attributes of a single object.

NC.K.MD.2Directly compare two objects with a measurable attribute in common, to see which object has “more of”/“less of” the attribute, and describe the difference.

NC.K.MD.3Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

NC.K.NBT Number and Operations in Base Ten

NC.K.NBT.1 Compose and decompose numbers from 11 to 19 into ten ones and some further ones by:

NC.K.NBT.1.aUsing objects or drawings.

NC.K.NBT.1.bRecording each composition or decomposition by a drawing or expression.

NC.K.NBT.1.cUnderstanding that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

NC.K.OA Operations and Algebraic Thinking

NC.K.OA.1 Represent addition and subtraction, within 10:

NC.K.OA.1.aUse a variety of representations such as objects, fingers, mental images, drawings, sounds, acting out situations, verbal explanations, or expressions.

NC.K.OA.1.bDemonstrate understanding of addition and subtraction by making connections among representations.

NC.K.OA.2 Solve addition and subtraction word problems, within 10, using objects or drawings to represent the problem, when solving:

NC.K.OA.2.aAdd to/Take From-Result Unknown

NC.K.OA.2.bPut Together/ Take Apart (Total Unknown and Two Addends Unknown)

NC.K.OA.3Decompose numbers less than or equal to 10 into pairs in more than one way using objects or drawings, and record each decomposition by a drawing or expression.

NC.K.OA.4For any number from 0 to 10, find the number that makes 10 when added to the given number using objects or drawings, and record the answer with a drawing or expression.

NC.K.OA.5Demonstrate fluency with addition and subtraction within 5.

NC.K.OA.6Recognize and combine groups with totals up to 5 (conceptual subitizing).

Math 1

NC.M1.A-APR Algebra: Arithmetic with Polynomial Expressions

NC.M1.A-APR.1Build an understanding that operations with polynomials are comparable to operations with integers by adding and subtracting quadratic expressions and by adding, subtracting, and multiplying linear expressions.

NC.M1.A-APR.3Understand the relationships among the factors of a quadratic expression, the solutions of a quadratic equation, and the zeros of a quadratic function.

NC.M1.A-CED Algebra: Creating Equations

NC.M1.A-CED.1Create equations and inequalities in one variable that represent linear, exponential, and quadratic relationships and use them to solve problems.

NC.M1.A-CED.2Create and graph equations in two variables to represent linear, exponential, and quadratic relationships between quantities.

NC.M1.A-CED.3Create systems of linear equations and inequalities to model situations in context.

NC.M1.A-CED.4Solve for a quantity of interest in formulas used in science and mathematics using the same reasoning as in solving equations.

NC.M1.A-REI Algebra: Reasoning with Equations and Inequalities

NC.M1.A-REI.1Justify a chosen solution method and each step of the solving process for linear and quadratic equations using mathematical reasoning.

NC.M1.A-REI.10Understand that the graph of a two variable equation represents the set of all solutions to the equation.

NC.M1.A-REI.11Build an understanding of why the x-coordinates of the points where the graphs of two linear, exponential, and/or quadratic equations y = (x) and y = g(x) intersect are the solutions of the equation f(x) = g(x) and approximate solutions using graphing technology or successive approximations with a table of values.

NC.M1.A-REI.12Represent the solutions of a linear inequality or a system of linear inequalities graphically as a region of the plane.

NC.M1.A-REI.3Solve linear equations and inequalities in one variable.

NC.M1.A-REI.4Solve for the real solutions of quadratic equations in one variable by taking square roots and factoring.

NC.M1.A-REI.5Explain why replacing one equation in a system of linear equations by the sum of that equation and a multiple of the other produces a system with the same solutions.

NC.M1.A-REI.6Use tables, graphs, or algebraic methods (substitution and elimination) to find approximate or exact solutions to systems of linear equations and interpret solutions in terms of a context.

NC.M1.A-SSE Algebra: Seeing Structure in Expressions

NC.M1.A-SSE.1Interpret expressions that represent a quantity in terms of its context.

NC.M1.A-SSE.1aIdentify and interpret parts of a linear, exponential, or quadratic expression, including terms, factors, coefficients, and exponents.

NC.M1.A-SSE.1bInterpret a linear, exponential, or quadratic expression made of multiple parts as a combination of entities to give meaning to an expression.

NC.M1.A-SSE.3Write an equivalent form of a quadratic expression ax^2 + bx + c, where a is an integer, by factoring to reveal the solutions of the equation or the zeros of the function the expression defines.

NC.M1.F-BF Functions: Building Functions

NC.M1.F-BF.1Write a function that describes a relationship between two quantities.

NC.M1.F-BF.1bBuild a function that models a relationship between two quantities by combining linear, exponential, or quadratic functions with addition and subtraction or two linear functions with multiplication.

NC.M1.F-BF.1aBuild linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two ordered pairs (include reading these from a table).

NC.M1.F-BF.2Translate between explicit and recursive forms of arithmetic and geometric sequences and use both to model situations.

NC.M1.F-IF Functions: Interpreting Functions

NC.M1.F-IF.1Build an understanding 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 by recognizing that:

NC.M1.F-IF.1aIf f is a function and x is an element of its domain, then (x) denotes the output of f corresponding to the input x.

NC.M1.F-IF.1bThe graph of f is the graph of the equation y = (x).

NC.M1.F-IF.2Use function notation to evaluate linear, quadratic, and exponential functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

NC.M1.F-IF.3Recognize that recursively and explicitly defined sequences are functions whose domain is a subset of the integers, the terms of an arithmetic sequence are a subset of the range of a linear function, and the terms of a geometric sequence are a subset of the range of an exponential function.

NC.M1.F-IF.4Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: intercepts; intervals where the function is increasing, decreasing, positive, or negative; and maximums and minimums.

NC.M1.F-IF.5Interpret a function in terms of the context by relating its domain and range to its graph and, where applicable, to the quantitative relationship it describes.

NC.M1.F-IF.6Calculate and interpret the average rate of change over a specified interval for a function presented numerically, graphically, and/or symbolically.

NC.M1.F-IF.7Analyze linear, exponential, and quadratic functions by generating different representations, by hand in simple cases and using technology for more complicated cases, to show key features, including: domain and range; rate of change; intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; and end behavior.

NC.M1.F-IF.8Use equivalent expressions to reveal and explain different properties of a function.

NC.M1.F-IF.8aRewrite a quadratic function to reveal and explain different key features of the function.

NC.M1.F-IF.8bInterpret and explain growth and decay rates for an exponential function.

NC.M1.F-IF.9Compare key features of two functions (linear, quadratic, or exponential) each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).

NC.M1.F-LE Functions: Linear, Quadratic, and Exponential Models

NC.M1.F-LE.1Identify situations that can be modeled with linear and exponential functions, and justify the most appropriate model for a situation based on the rate of change over equal intervals.

NC.M1.F-LE.3Compare the end behavior of linear, exponential, and quadratic functions using graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.

NC.M1.F-LE.5Interpret the parameters a and b in a linear function (x) = ax + b or an exponential function g(x) = ab^x in terms of a context.

NC.M1.G-GPE Geometry: Expressing Geometric Properties with Equations

NC.M1.G-GPE.4Use coordinates to solve geometric problems involving polygons algebraically

NC.M1.G-GPE.4aUse coordinates to compute perimeters of polygons and areas of triangles and rectangles.

NC.M1.G-GPE.4bUse coordinates to verify algebraically that a given set of points produces a particular type of triangle or quadrilateral.

NC.M1.G-GPE.5Use coordinates to prove the slope criteria for parallel and perpendicular lines and use them to solve problems.

NC.M1.G-GPE.5aDetermine if two lines are parallel, perpendicular, or neither.

NC.M1.G-GPE.5bFind the equation of a line parallel or perpendicular to a given line that passes through a given point.

NC.M1.G-GPE.6Use coordinates to find the midpoint or endpoint of a line segment.

NC.M1.G-SRTGeometry: Similarity, Right Triangles, and Trigonometry

NC.M1.N-RN Number and Quantity: The Real Number System

NC.M1.N-RN.2Rewrite algebraic expressions with integer exponents using the properties of exponents.

NC.M1.S-CPStatistics and Probability: Conditional Probability and the Rules for Probability

NC.M1.S-ICStatistics and Probability: Making Inference and Justifying Conclusions

NC.M1.S-ID Statistics and Probability: Interpreting Categorical and Quantitative Data

NC.M1.S-ID.1Use technology to represent data with plots on the real number line (histograms, and box plots).

NC.M1.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. Interpret differences in shape, center, and spread in the context of the data sets.

NC.M1.S-ID.3Examine the effects of extreme data points (outliers) on shape, center, and/or spread.

NC.M1.S-ID.6Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

NC.M1.S-ID.6aFit a least squares regression line to linear data using technology. Use the fitted function to solve problems.

NC.M1.S-ID.6bAssess the fit of a linear function by analyzing residuals.

NC.M1.S-ID.6cFit a function to exponential data using technology. Use the fitted function to solve problems.

NC.M1.S-ID.7Interpret in context the rate of change and the intercept of a linear model. Use the linear model to interpolate and extrapolate predicted values. Assess the validity of a predicted value.

NC.M1.S-ID.8Analyze patterns and describe relationships between two variables in context. Using technology, determine the correlation coefficient of bivariate data and interpret it as a measure of the strength and direction of a linear relationship. Use a scatter plot, correlation coefficient, and a residual plot to determine the appropriateness of using a linear function to model a relationship between two variables.

NC.M1.S-ID.9Distinguish between association and causation.

Math 2

NC.M2.A-APR Algebra: Arithmetic with Polynomial and Rational Expressions

NC.M2.A-APR.1Extend the understanding that operations with polynomials are comparable to operations with integers by adding, subtracting, and multiplying polynomials.

NC.M2.A-CED Algebra: Creating Equations

NC.M2.A-CED.1Create equations and inequalities in one variable that represent quadratic, square root, inverse variation, and right triangle trigonometric relationships and use them to solve problems.

NC.M2.A-CED.2Create and graph equations in two variables to represent quadratic, square root and inverse variation relationships between quantities.

NC.M2.A-CED.3Create systems of linear, quadratic, square root, and inverse variation equations to model situations in context.

NC.M2.A-REI Algebra: Reasoning with Equations and Inequalities

NC.M2.A-REI.1Justify a chosen solution method and each step of the solving process for quadratic, square root and inverse variation equations using mathematical reasoning.

NC.M2.A-REI.11Extend the understanding that the x-coordinates of the points where the graphs of two square root and/or inverse variation equations y = (x) and y = g(x) intersect are the solutions of the equation f(x) = g(x) and approximate solutions using graphing technology or successive approximations with a table of values.

NC.M2.A-REI.2Solve and interpret one variable inverse variation and square root equations arising from a context, and explain how extraneous solutions may be produced.

NC.M2.A-REI.4Solve for all solutions of quadratic equations in one variable.

NC.M2.A-REI.4aUnderstand that the quadratic formula is the generalization of solving ax^2 + bx + c by using the process of completing the square.

NC.M2.A-REI.4bExplain when quadratic equations will have non-real solutions and express complex solutions as a bi for real numbers a and b.

NC.M2.A-REI.7Use tables, graphs, and algebraic methods to approximate or find exact solutions of systems of linear and quadratic equations, and interpret the solutions in terms of a context.

NC.M2.A-SSE Algebra: Seeing Structure in Expressions

NC.M2.A-SSE.1Interpret expressions that represent a quantity in terms of its context.

NC.M2.A-SSE.1aIdentify and interpret parts of a quadratic, square root, inverse variation, or right triangle trigonometric expression, including terms, factors, coefficients, radicands, and exponents.

NC.M2.A-SSE.1bInterpret quadratic and square root expressions made of multiple parts as a combination of single entities to give meaning in terms of a context.

NC.M2.A-SSE.3Write an equivalent form of a quadratic expression by completing the square, where a is an integer of a quadratic expression, ax^2 + bx + c, to reveal the maximum or minimum value of the function the expression defines.

NC.M2.F-BF Functions: Building Functions

NC.M2.F-BF.1Write a function that describes a relationship between two quantities by building quadratic functions with real solution(s) and inverse variation functions given a graph, a description of a relationship, or ordered pairs (include reading these from a table).

NC.M2.F-BF.3Understand the effects of the graphical and tabular representations of a linear, quadratic, square root, and inverse variation function f with k * f(x), f(x) + k, f(x + k) for specific values of k (both positive and negative).

NC.M2.F-IF Functions: Interpreting Functions

NC.M2.F-IF.1Extend the concept of a function to include geometric transformations in the plane by recognizing that:

NC.M2.F-IF.1aThe domain and range of a transformation function f are sets of points in the plane.

NC.M2.F-IF.1bThe image of a transformation is a function of its pre-image.

NC.M2.F-IF.2Extend the use of function notation to express the image of a geometric figure in the plane resulting from a translation, rotation by multiples of 90 degrees about the origin, reflection across an axis, or dilation as a function of its pre-image.

NC.M2.F-IF.4Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: domain and range, rate of change, symmetries, and end behavior.

NC.M2.F-IF.7Analyze quadratic, square root, and inverse variation functions by generating different representations, by hand in simple cases and using technology for more complicated cases, to show key features, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; maximums and minimums; symmetries; and end behavior.

NC.M2.F-IF.8Use equivalent expressions to reveal and explain different properties of a function by developing and using the process of completing the square to identify the zeros, extreme values, and symmetry in graphs and tables representing quadratic functions, and interpret these in terms of a context.

NC.M2.F-IF.9Compare key features of two functions (linear, quadratic, square root, or inverse variation functions) each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).

NC.M2.G-CO NC.M2.G-CO

NC.M2.G-CO.10Prove theorems about triangles and use them to prove relationships in geometric figures including:

NC.M2.G-CO.10aThe sum of the measures of the interior angles of a triangle is 180.

NC.M2.G-CO.10bAn exterior angle of a triangle is equal to the sum of its remote interior angles.

NC.M2.G-CO.10cThe base angles of an isosceles triangle are congruent.

NC.M2.G-CO.10dThe segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length.

NC.M2.G-CO.2Experiment with transformations in the plane.

NC.M2.G-CO.2aRepresent transformations in the plane.

NC.M2.G-CO.2bCompare rigid motions that preserve distance and angle measure (translations, reflections, rotations) to transformations that do not preserve both distance and angle measure (e.g. stretches, dilations).

NC.M2.G-CO.2cUnderstand that rigid motions produce congruent figures while dilations produce similar figures.

NC.M2.G-CO.3Given a triangle, quadrilateral, or regular polygon, describe any reflection or rotation symmetry i.e., actions that carry the figure onto itself. Identify center and angle(s) of rotation symmetry. Identify line(s) of reflection symmetry.

NC.M2.G-CO.4Verify experimentally properties of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

NC.M2.G-CO.5Given a geometric figure and a rigid motion, find the image of the figure. Given a geometric figure and its image, specify a rigid motion or sequence of rigid motions that will transform the pre-image to its image.

NC.M2.G-CO.6Determine whether two figures are congruent by specifying a rigid motion or sequence of rigid motions that will transform one figure onto the other.

NC.M2.G-CO.7Use the properties 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.

NC.M2.G-CO.8Use congruence in terms of rigid motion. Justify the ASA, SAS, and SSS criteria for triangle congruence. Use criteria for triangle congruence (ASA, SAS, SSS, HL) to determine whether two triangles are congruent.

NC.M2.G-CO.9Prove theorems about lines and angles and use them to prove relationships in geometric figures including:

NC.M2.G-CO.9aVertical angles are congruent.

NC.M2.G-CO.9bWhen a transversal crosses parallel lines, alternate interior angles are congruent.

NC.M2.G-CO.9cWhen a transversal crosses parallel lines, corresponding angles are congruent.

NC.M2.G-CO.9dPoints are on a perpendicular bisector of a line segment if and only if they are equidistant from the endpoints of the segment.

NC.M2.G-CO.9eUse congruent triangles to justify why the bisector of an angle is equidistant from the sides of the angle.

NC.M2.G-SRT NC.M2.G-SRT

NC.M2.G-SRT.1Verify experimentally the properties of dilations with given center and scale factor:

NC.M2.G-SRT.12Develop properties of special right triangles (45-45-90 and 30-60-90) and use them to solve problems.

NC.M2.G-SRT.1aWhen a line segment passes through the center of dilation, the line segment and its image lie on the same line. When a line segment does not pass through the center of dilation, the line segment and its image are parallel.

NC.M2.G-SRT.1bThe length of the image of a line segment is equal to the length of the line segment multiplied by the scale factor.

NC.M2.G-SRT.1cThe distance between the center of a dilation and any point on the image is equal to the scale factor multiplied by the distance between the dilation center and the corresponding point on the pre-image.

NC.M2.G-SRT.1dDilations preserve angle measure.

NC.M2.G-SRT.2Understand similarity in terms of transformations.

NC.M2.G-SRT.2aDetermine whether two figures are similar by specifying a sequence of transformations that will transform one figure into the other.

NC.M2.G-SRT.2bUse the properties of dilations to show that two triangles are similar when all corresponding pairs of sides are proportional and all corresponding pairs of angles are congruent.

NC.M2.G-SRT.3Use transformations (rigid motions and dilations) to justify the AA criterion for triangle similarity.

NC.M2.G-SRT.4Use similarity to solve problems and to prove theorems about triangles. Use theorems about triangles to prove relationships in geometric figures.

NC.M2.G-SRT.4aA line parallel to one side of a triangle divides the other two sides proportionally and its converse.

NC.M2.G-SRT.4bThe Pythagorean Theorem

NC.M2.G-SRT.6Verify experimentally that the side ratios in similar right triangles are properties of the angle measures in the triangle, due to the preservation of angle measure in similarity. Use this discovery to develop definitions of the trigonometric ratios for acute angles.

NC.M2.G-SRT.8Use trigonometric ratios and the Pythagorean Theorem to solve problems involving right triangles in terms of a context.

NC.M2.N-CN Number and Quantity: The Complex Number System

NC.M2.N-CN.1Know there is a complex number i such that i^2 = –1, and every complex number has the form a + bi where a and b are real numbers.

NC.M2.N-RN Number and Quantity: The Real Number System

NC.M2.N-RN.1Explain how expressions with rational exponents can be rewritten as radical expressions.

NC.M2.N-RN.2Rewrite expressions with radicals and rational exponents into equivalent expressions using the properties of exponents.

NC.M2.N-RN.3Use the properties of rational and irrational numbers to explain why:

NC.M2.N-RN.3aThe sum or product of two rational numbers is rational.

NC.M2.N-RN.3bThe sum of a rational number and an irrational number is irrational.

NC.M2.N-RN.3cThe product of a nonzero rational number and an irrational number is irrational.

NC.M2.S-CP NC.M2.S-CP

NC.M2.S-CP.1Describe events as subsets of the outcomes in a sample space using characteristics of the outcomes or as unions, intersections and complements of other events.

NC.M2.S-CP.3Develop and understand independence and conditional probability.

NC.M2.S-CP.3aUse a 2-way table to develop understanding of the conditional probability of A given B (written P(A|B)) as the likelihood that A will occur given that B has occurred. That is, P(A|B) is the fraction of event B’s outcomes that also belong to event A.

NC.M2.S-CP.3bUnderstand 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. That is P(A|B)=P(A).

NC.M2.S-CP.4Represent data on two categorical variables by constructing a two-way frequency table of data. Interpret the two-way table as a sample space to calculate conditional, joint and marginal probabilities. Use the table to decide if events are independent.

NC.M2.S-CP.5Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

NC.M2.S-CP.6Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in context.

NC.M2.S-CP.7Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in context.

NC.M2.S-CP.8Apply the general Multiplication Rule P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in context. Include the case where A and B are independent: P(A and B) = P(A) P(B).

NC.M2.S-IC NC.M2.S-IC

NC.M2.S-IC.2Use simulation to determine whether the experimental probability generated by sample data is consistent with the theoretical probability based on known information about the population.

Math 3

NC.M3.A-APR Algebra: Arithmetic with Polynomial and Rational Expressions

NC.M3.A-APR.2Understand and apply the Remainder Theorem.

NC.M3.A-APR.3Understand the relationship among factors of a polynomial expression, the solutions of a polynomial equation and the zeros of a polynomial function.

NC.M3.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).

NC.M3.A-APR.7Understand the similarities between arithmetic with rational expressions and arithmetic with rational numbers.

NC.M3.A-APR.7aAdd and subtract two rational expressions, (x) and b(x), where the denominators of both a(x) and b(x) are linear expressions.

NC.M3.A-APR.7bMultiply and divide two rational expressions.

NC.M3.A-CED Algebra: Creating Equations

NC.M3.A-CED.1Create equations and inequalities in one variable that represent absolute value, polynomial, exponential, and rational relationships and use them to solve problems algebraically and graphically.

NC.M3.A-CED.2Create and graph equations in two variables to represent absolute value, polynomial, exponential and rational relationships between quantities.

NC.M3.A-CED.3Create systems of equations and/or inequalities to model situations in context.

NC.M3.A-REI Algebra: Reasoning with Equations and Inequalities

NC.M3.A-REI.1Justify a solution method for equations and explain each step of the solving process using mathematical reasoning.

NC.M3.A-REI.11Extend an understanding that the x-coordinates of the points where the graphs of two equations y = (x) and y = g(x) intersect are the solutions of the equation f(x) = g(x) and approximate solutions using a graphing technology or successive approximations with a table of values.

NC.M3.A-REI.2Solve and interpret one variable rational equations arising from a context, and explain how extraneous solutions may be produced.

NC.M3.A-SSE Algebra: Seeing Structure in Expressions

NC.M3.A-SSE.1Interpret expressions that represent a quantity in terms of its context.

NC.M3.A-SSE.1aIdentify and interpret parts of a piecewise, absolute value, polynomial, exponential and rational expressions including terms, factors, coefficients, and exponents.

NC.M3.A-SSE.1bInterpret expressions composed of multiple parts by viewing one or more of their parts as a single entity to give meaning in terms of a context.

NC.M3.A-SSE.2Use the structure of an expression to identify ways to write equivalent expressions.

NC.M3.A-SSE.3cWrite an equivalent form of an exponential expression by using the properties of exponents to transform expressions to reveal rates based on different intervals of the domain.

NC.M3.F-BF Functions: Building Functions

NC.M3.F-BF.1Write a function that describes a relationship between two quantities.

NC.M3.F-BF.1aBuild polynomial and exponential functions with real solution(s) given a graph, a description of a relationship, or ordered pairs (include reading these from a table).

NC.M3.F-BF.1bBuild a new function, in terms of a context, by combining standard function types using arithmetic operations.

NC.M3.F-BF.3Extend an understanding of the effects on the graphical and tabular representations of a function when replacing (x) with k * f(x), f(x) + k, f(x + k) to include f(k ∙ x) for specific values of k (both positive and negative).

NC.M3.F-BF.4Find an inverse function.

NC.M3.F-BF.4aUnderstand the inverse relationship between exponential and logarithmic, quadratic and square root, and linear to linear functions and use this relationship to solve problems using tables, graphs, and equations.

NC.M3.F-BF.4bDetermine if an inverse function exists by analyzing tables, graphs, and equations.

NC.M3.F-BF.4cIf an inverse function exists for a linear, quadratic and/or exponential function, f, represent the inverse function, f^1, with a table, graph, or equation and use it to solve problems in terms of a contex.

NC.M3.F-IF Functions: Interpreting Functions

NC.M3.F-IF.1Extend the concept of a function by recognizing that trigonometric ratios are functions of angle measure.

NC.M3.F-IF.2Use function notation to evaluate piecewise defined functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

NC.M3.F-IF.4Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities to include periodicity and discontinuities.

NC.M3.F-IF.7Analyze piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) using different representations to show key features of the graph, by hand in simple cases and using technology for more complicated cases, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; relative maximums and minimums; symmetries; end behavior; period; and discontinuities.

NC.M3.F-IF.9Compare key features of two functions using different representations by comparing properties of two different functions, each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).

NC.M3.F-LE Functions: Linear, Quadratic, and Exponential Models

NC.M3.F-LE.3Compare the end behavior of functions using their rates of change over intervals of the same length to show that a quantity increasing exponentially eventually exceeds a quantity increasing as a polynomial function.

NC.M3.F-LE.4Use logarithms to express the solution to ab^ct = d where a, c, and d are numbers and evaluate the logarithm using technology.

NC.M3.F-TF Functions: Trigonometric Functions

NC.M3.F-TF.1Understand radian measure of an angle as:

NC.M3.F-TF.1aThe ratio of the length of an arc on a circle subtended by the angle to its radius.

NC.M3.F-TF.1bA dimensionless measure of length defined by the quotient of arc length and radius that is a real number.

NC.M3.F-TF.1cThe domain for trigonometric functions.

NC.M3.F-TF.2Build an understanding of trigonometric functions by using tables, graphs and technology to represent the cosine and sine functions.

NC.M3.F-TF.2aInterpret the sine function as the relationship between the radian measure of an angle formed by the horizontal axis and a terminal ray on the unit circle and its y coordinate.

NC.M3.F-TF.2bInterpret the cosine function as the relationship between the radian measure of an angle formed by the horizontal axis and a terminal ray on the unit circle and its x coordinate.

NC.M3.F-TF.5Use technology to investigate the parameters, a, b, and h of a sine function, (x) = a * sin(b * x) + h, to represent periodic phenomena and interpret key features in terms of a context.

NC.M3.G-C Geometry: Circles

NC.M3.G-C.2Understand and apply theorems about circles.

NC.M3.G-C.2aUnderstand and apply theorems about relationships with angles and circles, including central, inscribed and circumscribed angles.

NC.M3.G-C.2bUnderstand and apply theorems about relationships with line segments and circles including, radii, diameter, secants, tangents and chords.

NC.M3.G-C.5Using similarity, demonstrate that the length of an arc, s, for a given central angle is proportional to the radius, r, of the circle. Define radian measure of the central angle as the ratio of the length of the arc to the radius of the circle, s/r. Find arc lengths and areas of sectors of circles.

NC.M3.G-CO Geometry: Congruence

NC.M3.G-CO.10Verify experimentally properties of the centers of triangles (centroid, incenter, and circumcenter).

NC.M3.G-CO.11Prove theorems about parallelograms.

NC.M3.G-CO.11aOpposite sides of a parallelogram are congruent.

NC.M3.G-CO.11bOpposite angles of a parallelogram are congruent.

NC.M3.G-CO.11cDiagonals of a parallelogram bisect each other.

NC.M3.G-CO.11dIf the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

NC.M3.G-CO.14Apply properties, definitions, and theorems of two-dimensional figures to prove geometric theorems and solve problems.

NC.M3.G-GMD Geometry: Geometric Measurement & Dimension

NC.M3.G-GMD.3Use the volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems.

NC.M3.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.

NC.M3.G-GPE Geometry: Expressing Geometric Properties with Equations

NC.M3.G-GPE.1Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

NC.M3.G-MG Geometry: Modeling with Geometry

NC.M3.G-MG.1Apply geometric concepts in modeling situations

NC.M3.G-MG.1aUse geometric and algebraic concepts to solve problems in modeling situations:

NC.M3.G-MG.1bUse geometric shapes, their measures, and their properties, to model real-life objects.

NC.M3.G-MG.1cUse geometric formulas and algebraic functions to model relationships.

NC.M3.G-MG.1dApply concepts of density based on area and volume.

NC.M3.G-MG.1eApply geometric concepts to solve design and optimization problems.

NC.M3.N-CN Number and Quantity: The Complex Number System

NC.M3.N-CN.9Use the Fundamental Theorem of Algebra to determine the number and potential types of solutions for polynomial functions.

NC.M3.S-IC Statistics and Probability: Making Inference and Justifying Conclusions

NC.M3.S-IC.1Understand the process of making inferences about a population based on a random sample from that population.

NC.M3.S-IC.3Recognize the purposes of and differences between sample surveys, experiments, and observational studies and understand how randomization should be used in each.

NC.M3.S-IC.4Use simulation to understand how samples can be used to estimate a population mean or proportion and how to determine a margin of error for the estimate.

NC.M3.S-IC.5Use simulation to determine whether observed differences between samples from two distinct populations indicate that the two populations are actually different in terms of a parameter of interest.

NC.M3.S-IC.6Evaluate articles and websites that report data by identifying the source of the data, the design of the study, and the way the data are graphically displayed.

Math 4

NC.M4.AF Algebra and Functions

NC.M4.AF.1 Apply properties of function composition to build new functions from existing functions.

NC.M4.AF.1.1Execute algebraic procedures to compose two functions.

NC.M4.AF.1.2Execute a procedure to determine the value of a composite function at a given value when the functions are in algebraic, graphical, or tabular representations.

NC.M4.AF.2 Apply properties of trigonometry to solve problems.

NC.M4.AF.2.1Translate trigonometric expressions using the reciprocal and Pythagorean identities.

NC.M4.AF.2.2Implement the Law of Sines and the Law of Cosines to solve problems.

NC.M4.AF.2.3Interpret key features (amplitude, period, phase shift, vertical shifts, midline, domain, range) of models using sine and cosine functions in terms of a context.

NC.M4.AF.3 Apply the properties and key features of logarithmic functions.

NC.M4.AF.3.1Execute properties of logarithms to rewrite expressions and solve equations algebraically.

NC.M4.AF.3.2Implement properties of logarithms to solve equations in contextual situations.

NC.M4.AF.3.3Interpret key features of a logarithmic function using multiple representations.

NC.M4.AF.4 Understand the properties and key features of piecewise functions.

NC.M4.AF.4.1Translate between algebraic and graphical representations of piecewise functions (linear, exponential, quadratic, polynomial, square root).

NC.M4.AF.4.2Construct piecewise functions to model a contextual situation.

NC.M4.AF.5 Understand how to model functions with regression.

NC.M4.AF.5.1Construct regression models of linear, quadratic, exponential, logarithmic, & sinusoidal functions of bivariate data using technology to model data and solve problems.

NC.M4.AF.5.2Compare residuals and residual plots of non-linear models to assess the goodness-of-fit of the model.

NC.M4.N Number and Quantity

NC.M4.N.1 Apply properties and operations with complex numbers.

NC.M4.N.1.1Execute procedures to add and subtract complex numbers.

NC.M4.N.1.2Execute procedures to multiply complex numbers.

NC.M4.N.2 Apply properties and operations with matrices and vectors.

NC.M4.N.2.1Execute procedures of addition, subtraction, multiplication, and scalar multiplication on matrices.

NC.M4.N.2.2Execute procedures of addition, subtraction, and scalar multiplication on vectors

NC.M4.SP Statistics and Probability

NC.M4.SP.1 Create statistical investigations to make sense of real-world phenomena.

NC.M4.SP.1.1Construct statistical questions to guide explorations of data in context.

NC.M4.SP.1.2Design sample surveys and comparative experiments using sampling methods to collect and analyze data to answer a statistical question.

NC.M4.SP.1.3Organize large datasets of real-world contexts (i.e. datasets that include 3 or more measures and have sample sizes >200) using technology (e.g., spreadsheets, dynamic data analysis tools) to determine: types of variables in the data set, possible outcomes for each variable, statistical questions that could be asked of the data, and types of numerical and graphical summaries could be used to make sense of the data.

NC.M4.SP.1.4Interpret non-standard data visualizations from the media or scientific papers to make sense of real-world phenomena.

NC.M4.SP.2 Apply informal and formal statistical inference to make sense of, and make decisions in, meaningful real-world contexts.

NC.M4.SP.2.1Design a simulation to make a sampling distribution that can be used in making informal statistical inferences.

NC.M4.SP.2.2Construct confidence intervals of population proportions in the context of the data.

NC.M4.SP.2.3Implement a one proportion z-test to determine if an observed proportion is significantly different from a hypothesized proportion.

NC.M4.SP.3 Apply probability distributions in making decisions in uncertainty.

NC.M4.SP.3.1Implement discrete probability distributions to model random phenomena and make decisions (e.g., expected value of playing a game, etc.).

NC.M4.SP.3.2Implement the binomial distribution to model situations and make decisions.

NC.M4.SP.3.3Recognize from simulations of sampling distributions of sample means and proportions that a normal distribution can be used as an approximate model in certain situations.

NC.M4.SP.3.4Implement the normal distribution as a probability distribution to determine the likelihood of events occurring.

Precalculus

PC.A.1 Apply properties of solving inequalities that include rational and polynomial expressions in one variable.

PC.A.1.1Implement algebraic (sign analysis) methods to solve rational and polynomial inequalities.

PC.A.1.2Implement graphical methods to solve rational and polynomial inequalities.

PC.A.2 Apply properties of solving equations involving exponential, logarithmic, and trigonometric functions.

PC.A.2.1Use properties of logarithms to rewrite expressions.

PC.A.2.2Implement properties of exponentials and logarithms to solve equations.

PC.A.2.3Implement properties of trigonometric functions to solve equations including • inverse trigonometric functions, • double angle formulas, and • Pythagorean identities.

PC.A.2.4Implement algebraic techniques to rewrite parametric equations in cartesian form by eliminating the parameter.

PC.F.1 Understand key features of sine, cosine, tangent, cotangent, secant and cosecant functions.

PC.F.1.1Interpret algebraic and graphical representations to determine key features of transformed sine and cosine functions. Key features include: amplitude, domain, midline, phase shift, frequency, period, intervals where the function is increasing, decreasing, positive or negative, relative maximums and minimums.

PC.F.1.2Interpret algebraic and graphical representations to determine key features of tangent, cotangent, secant, and cosecant. Key features include: domain, frequency, period, intervals where the function is increasing, decreasing, positive or negative, relative maximums and minimums, and asymptotes.

PC.F.1.3Integrate information to build trigonometric functions with specified amplitude, frequency, period, phase shift, or midline with or without context.

PC.F.1.4Implement graphical and algebraic methods to solve trigonometric equations and inequalities in context with support from technology.

PC.F.2 Apply properties of a unit circle with center (0,0) to determine the values of sine, cosine, tangent, cotangent, secant, and cosecant.

PC.F.2.1Use a unit circle to find values of sine, cosine, and tangent for angles in terms of reference angles.

PC.F.2.2Explain the relationship between the symmetry of a unit circle and the periodicity of trigonometric functions.

PC.F.3 Apply properties of trigonometry to solve problems involving all types of triangles.

PC.F.3.1Implement a strategy to solve equations using inverse trigonometric functions.

PC.F.3.2Implement the Law of Sines and the Law of Cosines to solve problems.

PC.F.3.3Implement the Pythagorean identity to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

PC.F.4 Understand the relationship of algebraic and graphical representations of exponential, logarithmic, rational, power functions, and conic sections to their key features.

PC.F.4.1Interpret algebraic and graphical representations to determine key features of exponential functions. Key features include: domain, range, intercepts, intervals where the function is increasing, decreasing, positive or negative, concavity, end behavior, limits, and asymptotes.

PC.F.4.2Integrate information to build exponential functions to model phenomena involving growth or decay.

PC.F.4.3Interpret algebraic and graphical representations to determine key features of logarithmic functions. Key features include: domain, range, intercepts, intervals where the function is increasing, decreasing, positive or negative, concavity, end behavior, continuity, limits, and asymptotes.

PC.F.4.4Implement graphical and algebraic methods to solve exponential and logarithmic equations in context with support from technology.

PC.F.4.5Interpret algebraic and graphical representations to determine key features of rational functions. Key features include: domain, range, intercepts, intervals where the function is increasing, decreasing, positive or negative, concavity, end behavior, continuity, limits, and asymptotes.

PC.F.4.6Implement graphical and algebraic methods to solve optimization problems given rational and polynomial functions in context with support from technology.

PC.F.4.7Construct graphs of transformations of power, exponential, and logarithmic functions showing key features.

PC.F.4.8Identify the conic section (ellipse, hyperbola, parabola) from its algebraic representation in standard form.

PC.F.4.9Interpret algebraic and graphical representations to determine key features of conic sections (ellipse: center, length of the major and minor axes; hyperbola: vertices, transverse axis; parabola: vertex, axis of symmetry).

PC.F.5 Apply properties of function composition to build new functions from existing functions.

PC.F.5.1Implement algebraic procedures to compose functions.

PC.F.5.2Execute a procedure to determine the value of a composite function at a given value using algebraic, graphical, and tabular representations.

PC.F.5.3Implement algebraic methods to find the domain of a composite function.

PC.F.5.4Organize information to build models involving function composition.

PC.F.5.5Deconstruct a composite function into two functions.

PC.F.5.6Implement algebraic and graphical methods to find an inverse function of an existing function, restricting domains if necessary.

PC.F.5.7Use composition to determine if one function is the inverse of another function.

PC.F.6 Apply mathematical reasoning to build recursive functions to model and solve problems.

PC.F.6.1Use algebraic representations to build recursive functions.

PC.F.6.2Construct a recursive function for a sequence represented numerically.

PC.F.7 Apply mathematical reasoning to build parametric functions and solve problems.

PC.F.7.1Implement algebraic methods to write parametric equations in context.

PC.F.7.2Implement technology to solve contextual problems involving parametric equations.

PC.N Number and Quantity

PC.N.1 Apply properties of complex numbers and the complex number system.

PC.N.1.1Execute the sum and difference algorithms to combine complex numbers.

PC.N.1.2Execute the multiplication algorithm with complex numbers.

PC.N.2 Apply properties and operations with matrices.

PC.N.2.1Execute the sum and difference algorithms to combine matrices of appropriate dimensions.

PC.N.2.2Execute associative and distributive properties to matrices.

PC.N.2.3Execute commutative property to add matrices.

PC.N.2.4Execute properties of matrices to multiply a matrix by a scalar.

PC.N.2.5Execute the multiplication algorithm with matrices.

PC.N.3 Understand properties and operations with vectors.

PC.N.3.1Represent a vector indicating magnitude and direction.

PC.N.3.2Execute sum and difference algorithms to combine vectors.

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.