South Carolina Math Standards - SC CCR

1065 standards, Kindergarten through AP.

South Carolina's College and Career Ready Standards use domain based organization, with K-5 domains like ATO (Arrays, Tools, Operations), NSBT (Number Sense, Base Ten), and NSF (Number Sense, Fractions). Grades 6-8 shift to domains like EEI (Equations and Expressions), NS (Number Systems), and RP (Ratios and Proportional Relationships).

Kindergarten

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K.ATO Algebraic Thinking and Operations
K.ATO.1Model situations that involve addition and subtraction within 10 using objects, fingers, mental images, drawings, acting out situations, verbal explanations, expressions, and equations.
K.ATO.2Solve real-world/story problems using objects and drawings to find sums up to 10 and differences within 10.
K.ATO.3Compose and decompose numbers up to 10 using objects, drawings, and equations.
K.ATO.4Create a sum of 10 using objects and drawings when given one of two addends 1 – 9.
K.ATO.5Add and subtract fluently within 5.
K.ATO.6Describe simple repeating patterns using AB, AAB, ABB, and ABC type patterns.
K.G Geometry
K.G.1Describe positions of objects by appropriately using terms, including below, above, beside, between, inside, outside, in front of, or behind.
K.G.2Identify and describe a given shape and shapes of objects in everyday situations to include two-dimensional shapes (i.e., triangle, square, rectangle, hexagon, and circle) and three-dimensional shapes (i.e., cone, cube, cylinder, and sphere).
K.G.3Classify shapes as two-dimensional/flat or three-dimensional/solid and explain the reasoning used.
K.G.4Analyze and compare two- and three-dimensional shapes of different sizes and orientations using informal language.
K.G.5Draw two-dimensional shapes (i.e., square, rectangle, triangle, hexagon, and circle) and create models of three-dimensional shapes (i.e., cone, cube, cylinder, and sphere).
K.MDA Measurement and Data Analysis
K.MDA.1Identify measureable attributes (length, weight) of an object.
K.MDA.2Compare objects using words such as shorter/longer, shorter/taller, and lighter/heavier.
K.MDA.3Sort and classify data into 2 or 3 categories with data not to exceed 20 items in each category.
K.MDA.4Represent data using object and picture graphs and draw conclusions from the graphs.
K.NS Number Sense
K.NS.1Count forward by ones and tens to 100.
K.NS.2Count forward by ones beginning from any number less than 100.
K.NS.3Read numbers from 0 – 20 and represent a number of objects 0 – 20 with a written numeral.
K.NS.4Understand the relationship between number and quantity. Connect counting to cardinality by demonstrating an understanding that:
K.NS.4aThe last number said tells the number of objects in the set (cardinality).
K.NS.4bThe number of objects is the same regardless of their arrangement or the order in which they are counted (conservation of number).
K.NS.4cEach successive number name refers to a quantity that is one more and each previous number name refers to a quantity that is one less.
K.NS.5Count a given number of objects from 1 – 20 and connect this sequence in a one-to-one manner.
K.NS.6Recognize a quantity of up to ten objects in an organized arrangement (subitizing).
K.NS.7Determine whether the number of up to ten objects in one group is more than, less than, or equal to the number of up to ten objects in another group using matching and counting strategies.
K.NS.8Compare two written numerals up to 10 using more than, less than or equal to.
K.NS.9Identify first through fifth and last positions in a line of objects.
K.NSBT Number Sense and Base Ten
K.NSBT.1Compose and decompose numbers from 11 – 19 separating ten ones from the remaining ones using objects and drawings.

Grade 1

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1.ATO Algebraic Thinking and Operations
1.ATO.1Solve real-world/story problems using addition (as a joining action and as a part-part-whole action) and subtraction (as a separation action, finding parts of the whole, and as a comparison) through 20 with unknowns in all positions.
1.ATO.2Solve real-world/story problems that include three whole number addends whose sum is less than or equal to 20.
1.ATO.3Apply Commutative and Associative Properties of Addition to find the sum (through 20) of two or three addends.
1.ATO.4Understand subtraction as an unknown addend problem.
1.ATO.5Recognize how counting relates to addition and subtraction.
1.ATO.6Demonstrate:
1.ATO.6aAddition and subtraction through 20.
1.ATO.6bFluency with addition and related subtraction facts through 10.
1.ATO.7Understand the meaning of the equal sign as a relationship between two quantities sameness) and determine if equations involving addition and subtraction are true.
1.ATO.8Determine the missing number in addition and subtraction equations within 20.
1.ATO.9Create, extend and explain using pictures and words for:
1.ATO.9aRepeating patterns (e.g., AB, AAB, ABB, and ABC type patterns).
1.ATO.9bGrowing patterns (between 2 and 4 terms/figures).
1.G Geometry
1.G.1Distinguish between a two-dimensional shape’s defining (e.g., number of sides) and non-defining attributes (e.g., color).
1.G.2Combine two-dimensional shapes (i.e., square, rectangle, triangle, hexagon, rhombus, and trapezoid) or three-dimensional shapes (i.e., cube, rectangular prism, cone, and cylinder) in more than one way to form a composite shape.
1.G.3Partition two-dimensional shapes (i.e., square, rectangle, circle) into two or four equal parts.
1.G.4Identify and name two-dimensional shapes (i.e., square, rectangle, triangle, hexagon, rhombus, trapezoid, and circle).
1.MDA Measurement and Data Analysis
1.MDA.1Order three objects by length using indirect comparison.
1.MDA.2Use nonstandard physical models to show the length of an object as the number of same size units of length with no gaps or overlaps.
1.MDA.3Use analog and digital clocks to tell and record time to the hour and half hour.
1.MDA.4Collect, organize, and represent data with up to 3 categories using object graphs, picture graphs, t-charts and tallies.
1.MDA.5Draw conclusions from given object graphs, picture graphs, t-charts, tallies, and bar graphs.
1.MDA.6Identify a penny, nickel, dime and quarter and write the coin values using a ȼ symbol.
1.NSBT Number Sense and Base Ten
1.NSBT.1Extend the number sequence to:
1.NSBT.1aCount forward by ones to 120 starting at any number.
1.NSBT.1bCount by fives and tens to 100, starting at any number.
1.NSBT.1cRead, write and represent numbers to 100 using concrete models, standard form, and equations in expanded form.
1.NSBT.1dRead and write in word form numbers zero through nineteen, and multiples of ten through ninety.
1.NSBT.2Understand place value through 99 by demonstrating that:
1.NSBT.2aTen ones can be thought of as a bundle (group) called a “ten”.
1.NSBT.2bThe tens digit in a two-digit number represents the number of tens and the ones digit represents the number of ones.
1.NSBT.2cTwo-digit numbers can be decomposed in a variety of ways (e.g., 52 can be decomposed as 5 tens and 2 ones or 4 tens and 12 ones, etc.) and record the decomposition as an equation.
1.NSBT.3Compare two two-digit numbers based on the meanings of the tens and ones digits, using the words greater than, equal to, or less than.
1.NSBT.4Add through 99 using concrete models, drawings, and strategies based on place value to:
1.NSBT.4aAdd a two-digit number and a one-digit number, understanding that sometimes it is necessary to compose a ten (regroup).
1.NSBT.4bAdd a two-digit number and a multiple of 10.
1.NSBT.5Determine the number that is 10 more or 10 less than a given number through 99 and explain the reasoning verbally and with multiple representations, including concrete models.
1.NSBT.6Subtract a multiple of 10 from a larger multiple of 10, both in the range 10 to 90, using concrete models, drawings, and strategies based on place value.

Grade 2

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2.ATO Algebraic Thinking and Operations
2.ATO.1Solve one- and two-step real-world/story problems using addition (as a joining action and as a part-part-whole action) and subtraction (as a separation action, finding parts of the whole, and as a comparison) through 99 with unknowns in all positions.
2.ATO.2Demonstrate fluency with addition and related subtraction facts through 20.
2.ATO.3Determine whether a number through 20 is odd or even using pairings of objects, counting by twos, or finding two equal addends to represent the number (e.g., 3 + 3 = 6).
2.ATO.4Use repeated addition to find the total number of objects arranged in a rectangular array with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
2.G Geometry
2.G.1Identify triangles, quadrilaterals, hexagons, and cubes. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.
2.G.2Partition a rectangle into rows and columns of same-size squares to form an array and count to find the total number of parts.
2.G.3Partition squares, rectangles and circles into two or four equal parts, and describe the parts using the words halves, fourths, a half of, and a fourth of. Understand that when partitioning a square, rectangle or circle into two or four equal parts, the parts become smaller as the number of parts increases.
2.MDA Measurement and Data Analysis
2.MDA.1Select and use appropriate tools (e.g., rulers, yardsticks, meter sticks, measuring tapes) to measure the length of an object.
2.MDA.10Draw conclusions from t-charts, object graphs, picture graphs, and bar graphs.
2.MDA.2Measure the same object or distance using a standard unit of one length and then a standard unit of a different length and explain verbally and in writing how and why the measurements differ.
2.MDA.3Estimate and measure length/distance in customary units (i.e., inch, foot, yard) and metric units (i.e., centimeter, meter).
2.MDA.4Measure to determine how much longer one object is than another, using standard length units.
2.MDA.5Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, and represent whole-number sums and differences through 99 on a number line diagram.
2.MDA.6Use analog and digital clocks to tell and record time to the nearest five-minute interval using a.m. and p.m.
2.MDA.7Solve real-world/story problems involving dollar bills using the $ symbol or involving quarters, dimes, nickels, and pennies using the ¢ symbol.
2.MDA.8Generate data by measuring objects in whole unit lengths and organize the data in a line plot using a horizontal scale marked in whole number units.
2.MDA.9Collect, organize, and represent data with up to four categories using picture graphs and bar graphs with a single-unit scale.
2.NSBT Number Sense and Base Ten
2.NSBT.1Understand place value through 999 by demonstrating that:
2.NSBT.1a100 can be thought of as a bundle (group) of 10 tens called a “hundred”.
2.NSBT.1bThe hundreds digit in a three-digit number represents the number of hundreds, the tens digit represents the number of tens, and the ones digit represents the number of ones.
2.NSBT.1cThree-digit numbers can be decomposed in multiple ways (e.g., 524 can be decomposed as 5 hundreds, 2 tens and 4 ones or 4 hundreds, 12 tens, and 4 ones, etc.).
2.NSBT.2Count by tens and hundreds to 1,000 starting with any number.
2.NSBT.3Read, write and represent numbers through 999 using concrete models, standard form, and equations in expanded form.
2.NSBT.4Compare two numbers with up to three digits using words and symbols (i.e., >, =, or <).
2.NSBT.5Add and subtract fluently through 99 using knowledge of place value and properties of operations.
2.NSBT.6Add up to four two-digit numbers using strategies based on knowledge of place value and properties of operations.
2.NSBT.7Add and subtract through 999 using concrete models, drawings, and symbols which convey strategies connected to place value understanding.
2.NSBT.8Determine the number that is 10 or 100 more or less than a given number through 1,000 and explain the reasoning verbally and in writing.

Grade 3

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3.ATO Algebraic Thinking and Operations
3.ATO.1Use concrete objects, drawings and symbols to represent multiplication facts of two single-digit whole numbers and explain the relationship between the factors (i.e., 0 – 10) and the product.
3.ATO.2Use concrete objects, drawings and symbols to represent division without remainders and explain the relationship among the whole number quotient (i.e., 0 – 10), divisor (i.e., 0 – 10), and dividend.
3.ATO.3Solve real-world problems involving equal groups, area/array, and number line models using basic multiplication and related division facts. Represent the problem situation using an equation with a symbol for the unknown.
3.ATO.4Determine the unknown whole number in a multiplication or division equation relating three whole numbers when the unknown is a missing factor, product, dividend, divisor, or quotient.
3.ATO.5Apply properties of operations (i.e., Commutative Property of Multiplication, Associative Property of Multiplication, and Distributive Property) as strategies to multiply and divide and explain the reasoning.
3.ATO.6Understand division as a missing factor problem.
3.ATO.7Demonstrate fluency with basic multiplication and related division facts of products and dividends through 100.
3.ATO.8Solve two-step real-world problems using addition, subtraction, multiplication and division of whole numbers and having whole number answers. Represent these problems using equations with a letter for the unknown quantity.
3.ATO.9Identify a rule for an arithmetic pattern (e.g., patterns in the addition table or multiplication table).
3.G Geometry
3.G.1Understand that shapes in different categories (e.g., rhombus, rectangle, square, and other 4-sided shapes) may share attributes (e.g., 4-sided figures) and the shared attributes can define a larger category (e.g., quadrilateral). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.
3.G.2Partition two-dimensional shapes into 2, 3, 4, 6, or 8 parts with equal areas and express the area of each part using the same unit fraction. Recognize that equal parts of identical wholes need not have the same shape.
3.G.3Use a right angle as a benchmark to identify and sketch acute and obtuse angles.
3.G.4Identify a three-dimensional shape (i.e., right rectangular prism, right triangular prism, pyramid) based on a given two-dimensional net and explain the relationship between the shape and the net.
3.MDA Measurement and Data Analysis
3.MDA.1Use analog and digital clocks to determine and record time to the nearest minute, using a.m. and p.m.; measure time intervals in minutes; and solve problems involving addition and subtraction of time intervals within 60 minutes.
3.MDA.2Estimate and measure liquid volumes (capacity) in customary units (i.e., c., pt., qt., gal.) and metric units (mL, L) to the nearest whole unit.
3.MDA.3Collect, organize, classify, and interpret data with multiple categories and draw a scaled picture graph and a scaled bar graph to represent the data.
3.MDA.4Generate data by measuring length to the nearest inch, half-inch and quarter-inch and organize the data in a line plot using a horizontal scale marked off in appropriate units.
3.MDA.5Understand the concept of area measurement.
3.MDA.5aRecognize area as an attribute of plane figures.
3.MDA.5bMeasure area by building arrays and counting standard unit squares.
3.MDA.5cDetermine the area of a rectilinear polygon and relate to multiplication and addition.
3.MDA.6Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
3.NSBT Number Sense and Base Ten
3.NSBT.1Use place value understanding to round whole numbers to the nearest 10 or 100.
3.NSBT.2Add and subtract whole numbers fluently to 1,000 using knowledge of place value and properties of operations.
3.NSBT.3Multiply one-digit whole numbers by multiples of 10 in the range 10 – 90, using knowledge of place value and properties of operations.
3.NSBT.4Read and write numbers through 999,999 in standard form and equations in expanded form.
3.NSBT.5Compare and order numbers through 999,999 and represent the comparison using the symbols >, =, or <.
3.NSF Number Sense – Fractions
3.NSF.1Develop an understanding of fractions (i.e., denominators 2, 3, 4, 6, 8, 10) as numbers.
3.NSF.1aA fraction 1/b (called a unit fraction) is the quantity formed by one part when a whole is partitioned into b equal parts.
3.NSF.1bA fraction a/b is the quantity formed by a parts of size 1/b.
3.NSF.1cA fraction is a number that can be represented on a number line based on counts of a unit fraction.
3.NSF.1dA fraction can be represented using set, area, and linear models.
3.NSF.2Explain fraction equivalence (i.e., denominators 2, 3, 4, 6, 8, 10) by demonstrating an understanding that:
3.NSF.2aTwo fractions are equal if they are the same size, based on the same whole, or at the same point on a number line.
3.NSF.2bFraction equivalence can be represented using set, area, and linear models.
3.NSF.2cWhole numbers can be written as fractions (e.g., 4= 4/1 and 1= 4/4).
3.NSF.2dFractions with the same numerator or same denominator can be compared by reasoning about their size based on the same whole.
3.NSF.3Develop an understanding of mixed numbers (i.e., denominators 2, 3, 4, 6, 8, 10) as iterations of unit fractions on a number line.

Grade 4

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4.ATO Algebraic Thinking and Operations
4.ATO.1Interpret a multiplication equation as a comparison (e.g. interpret 35 = 5x7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5.) Represent verbal statements of multiplicative comparisons as multiplication equations.
4.ATO.2Solve real-world problems using multiplication (product unknown) and division (group size unknown, number of groups unknown).
4.ATO.3Solve multi-step, real-world problems using the four operations. Represent the problem using an equation with a variable as the unknown quantity.
4.ATO.4Recognize that a whole number is a multiple of each of its factors. Find all factors for a whole number in the range 1 – 100 and determine whether the whole number is prime or composite.
4.ATO.5Generate a number or shape pattern that follows a given rule and determine a term that appears later in the sequence.
4.G Geometry
4.G.1Draw points, lines, line segments, rays, angles (i.e., right, acute, obtuse), and parallel and perpendicular lines. Identify these in two-dimensional figures.
4.G.2Classify quadrilaterals based on the presence or absence of parallel or perpendicular lines.
4.G.3Recognize right triangles as a category, and identify right triangles.
4.G.4Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
4.MDA Measurement and Data Analysis
4.MDA.1Convert measurements within a single system of measurement, customary (i.e., in., ft., yd., oz., lb., sec., min., hr.) or metric (i.e., cm, m, km, g, kg, mL, L) from a larger to a smaller unit.
4.MDA.2Solve real-world problems involving distance/length, intervals of time within 12 hours, liquid volume, mass, and money using the four operations.
4.MDA.3Apply the area and perimeter formulas for rectangles.
4.MDA.4Create a line plot to display a data set (i.e., generated by measuring length to the nearest quarter-inch and eighth-inch) and interpret the line plot.
4.MDA.5Understand the relationship of an angle measurement to a circle.
4.MDA.6Measure and draw angles in whole number degrees using a protractor.
4.MDA.7Solve addition and subtraction problems to find unknown angles in real-world and mathematical problems.
4.MDA.8Determine the value of a collection of coins and bills greater than $1.00.
4.NSBT Number Sense and Base Ten
4.NSBT.1Understand that, in a multi-digit whole number, a digit represents ten times what the same digit represents in the place to its right.
4.NSBT.2Recognize math periods and number patterns within each period to read and write in standard form large numbers through 999,999,999.
4.NSBT.3Use rounding as one form of estimation and round whole numbers to any given place value.
4.NSBT.4Fluently add and subtract multi-digit whole numbers using strategies to include a standard algorithm.
4.NSBT.5Multiply up to a four-digit number by a one-digit number and multiply a two-digit number by a two-digit number using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using rectangular arrays, area models and/or equations.
4.NSBT.6Divide up to a four-digit dividend by a one-digit divisor using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.
4.NSF Number Sense and Operations – Fractions
4.NSF.1Explain why a fraction (i.e., denominators 2, 3, 4, 5, 6, 8, 10, 12, 25, 100), a/b, is equivalent to a fraction, (nxa)/(nxb), by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
4.NSF.2Compare two given fractions (i.e., denominators 2, 3, 4, 5, 6, 8, 10, 12, 25, 100) by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2 and represent the comparison using the symbols >, =, or <.
4.NSF.3Develop an understanding of addition and subtraction of fractions (i.e., denominators 2, 3, 4, 5, 6, 8, 10, 12, 25, 100) based on unit fractions.
4.NSF.3aCompose and decompose a fraction in more than one way, recording each composition and decomposition as an addition or subtraction equation.
4.NSF.3bAdd and subtract mixed numbers with like denominators.
4.NSF.3cSolve real-world problems involving addition and subtraction of fractions referring to the same whole and having like denominators.
4.NSF.4Apply and extend an understanding of multiplication by multiplying a whole number and a fraction (i.e., denominators 2, 3, 4, 5, 6, 8, 10, 12, 25, 100).
4.NSF.4aUnderstand a fraction a/b as a multiple of 1/b.
4.NSF.4bUnderstand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.
4.NSF.4cSolve real-world problems involving multiplication of a fraction by a whole number (i.e., use visual fraction models and equations to represent the problem).
4.NSF.5Express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100 and use this technique to add two fractions with respective denominators of 10 and 100.
4.NSF.6Write a fraction with a denominator of 10 or 100 using decimal notation, and read and write a decimal number as a fraction.
4.NSF.7Compare and order decimal numbers to hundredths, and justify using concrete and visual models.

Grade 5

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5.ATO Algebraic Thinking and Operations
5.ATO.1Evaluate numerical expressions involving grouping symbols (i.e., parentheses, brackets, braces).
5.ATO.2Translate verbal phrases into numerical expressions and interpret numerical expressions as verbal phrases.
5.ATO.3Investigate the relationship between two numerical patterns.
5.ATO.3aGenerate two numerical patterns given two rules and organize in tables.
5.ATO.3bTranslate the two numerical patterns into two sets of ordered pairs.
5.ATO.3cGraph the two sets of ordered pairs on the same coordinate plane.
5.ATO.3dIdentify the relationship between the two numerical patterns.
5.G Geometry
5.G.1Define a coordinate system.
5.G.1aThe x- and y- axes are perpendicular number lines that intersect at 0 (the origin).
5.G.1bAny point on the coordinate plane can be represented by its coordinates.
5.G.1cThe first number in an ordered pair is the x-coordinate and represents the horizontal distance from the origin.
5.G.1dThe second number in an ordered pair is the y-coordinate and represents the vertical distance from the origin.
5.G.2Plot and interpret points in the first quadrant of the coordinate plane to represent real-world and mathematical situations.
5.G.3Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
5.G.4Classify two-dimensional figures in a hierarchy based on their attributes.
5.MDA Measurement and Data Analysis
5.MDA.1Convert measurements within a single system of measurement: customary (i.e., in., ft., yd., oz., lb., sec., min., and hr.) or metric (i.e., mm, cm, m, km, g, kg, mL, L) from a larger to a smaller unit and a smaller to a larger unit.
5.MDA.2Create a line plot consisting of unit fractions and use operations on fractions to solve problems related to the line plot.
5.MDA.3Understand the concept of volume measurement.
5.MDA.3aRecognize volume as an attribute of right rectangular prisms.
5.MDA.3bRelate volume measurement to the operations of multiplication and addition by packing right rectangular prisms and then counting the layers of standard unit cubes.
5.MDA.3cDetermine the volume of right rectangular prisms using the formula derived from packing right rectangular prisms and counting the layers of standard unit cubes.
5.MDA.4Differentiate among perimeter, area and volume and identify which application is appropriate for a given situation.
5.NSBT Number Sense and Base Ten
5.NSBT.1Understand that, in a multi-digit whole number, a digit in one place represents 10 times what the same digit represents in the place to its right, and represents 1/10 times what the same digit represents in the place to its left.
5.NSBT.2Use whole number exponents to explain:
5.NSBT.2aPatterns in the number of zeroes of the product when multiplying a number by powers of 10.
5.NSBT.2bPatterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.
5.NSBT.3Read and write decimals in standard and expanded form. Compare two decimal numbers to the thousandths using the symbols >, =, or <.
5.NSBT.4Round decimals to any given place value within thousandths.
5.NSBT.5Fluently multiply multi-digit whole numbers using strategies to include a standard algorithm.
5.NSBT.6Divide up to a four-digit dividend by a two-digit divisor, using strategies based on place value, the properties of operations, and the relationship between multiplication and division.
5.NSBT.7Add, subtract, multiply, and divide decimal numbers to hundredths using concrete area models and drawings.
5.NSF Number Sense and Operations – Fractions
5.NSF.1Add and subtract fractions with unlike denominators (including mixed numbers) using a variety of models, including an area model and number line.
5.NSF.2Solve real-world problems involving addition and subtraction of fractions with unlike denominators.
5.NSF.3Understand the relationship between fractions and division of whole numbers by interpreting a fraction as the numerator divided by the denominator (i.e., a/b=a÷b).
5.NSF.4Extend the concept of multiplication to multiply a fraction or whole number by a fraction.
5.NSF.4aRecognize the relationship between multiplying fractions and finding the areas of rectangles with fractional side lengths.
5.NSF.4bInterpret multiplication of a fraction by a whole number and a whole number by a fraction and compute the product.
5.NSF.4cInterpret multiplication in which both factors are fractions less than one and compute the product.
5.NSF.5Justify the reasonableness of a product when multiplying with fractions.
5.NSF.5aEstimate the size of the product based on the size of the two factors.
5.NSF.5bExplain why multiplying a given number by a number greater than 1 (e.g., improper fractions, mixed numbers, whole numbers) results in a product larger than the given number.
5.NSF.5cExplain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number.
5.NSF.5dExplain why multiplying the numerator and denominator by the same number has the same effect as multiplying the fraction by 1.
5.NSF.6Solve real-world problems involving multiplication of a fraction by a fraction, improper fraction and a mixed number.
5.NSF.7Extend the concept of division to divide unit fractions and whole numbers by using visual fraction models and equations.
5.NSF.7aInterpret division of a unit fraction by a non-zero whole number and compute the quotient.
5.NSF.7bInterpret division of a whole number by a unit fraction and compute the quotient.
5.NSF.8Solve real-world problems involving division of unit fractions and whole numbers, using visual fraction models and equations.

Grade 6

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6.DS Data Analysis and Statistics
6.DS.1Differentiate between statistical and non-statistical questions.
6.DS.2Use center (mean, median, mode), spread (range, interquartile range, mean absolute value), and shape (symmetrical, skewed left, skewed right) to describe the distribution of a set of data collected to answer a statistical question.
6.DS.3Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
6.DS.4Select and create an appropriate display for numerical data, including dot plots, histograms, and box plots.
6.DS.5Describe numerical data sets in relation to their real-world context.
6.DS.5aState the sample size.
6.DS.5bDescribe the qualitative aspects of the data (e.g., how it was measured, units of measurement).
6.DS.5cGive measures of center (median, mean).
6.DS.5dFind measures of variability (interquartile range, mean absolute deviation) using a number line.
6.DS.5eDescribe the overall pattern (shape) of the distribution.
6.DS.5fJustify the choices for measure of center and measure of variability based on the shape of the distribution.
6.DS.5gDescribe the impact that inserting or deleting a data point has on the measures of center (median, mean) for a data set.
6.EEI Expressions, Equations, and Inequalities
6.EEI.1Write and evaluate numerical expressions involving whole-number exponents and positive rational number bases using the Order of Operations.
6.EEI.2Extend the concepts of numerical expressions to algebraic expressions involving positive rational numbers.
6.EEI.2aTranslate between algebraic expressions and verbal phrases that include variables.
6.EEI.2bInvestigate and identify parts of algebraic expressions using mathematical terminology, including term, coefficient, constant, and factor.
6.EEI.2cEvaluate real-world and algebraic expressions for specific values using the Order of Operations. Grouping symbols should be limited to parentheses, braces, and brackets. Exponents should be limited to whole-numbers.
6.EEI.3Apply mathematical properties (e.g., commutative, associative, distributive) to generate equivalent expressions.
6.EEI.4Apply mathematical properties (e.g., commutative, associative, distributive) to justify that two expressions are equivalent.
6.EEI.5Understand that if any solutions exist, the solution set for an equation or inequality consists of values that make the equation or inequality true.
6.EEI.6Write expressions using variables to represent quantities in real-world and mathematical situations. Understand the meaning of the variable in the context of the situation.
6.EEI.7Write and solve one-step linear equations in one variable involving nonnegative rational numbers for real-world and mathematical situations.
6.EEI.8Extend knowledge of inequalities used to compare numerical expressions to include algebraic expressions in real-world and mathematical situations.
6.EEI.8aWrite an inequality of the form x>c or x<c and graph the solution set on a number line.
6.EEI.8bRecognize that inequalities have infinitely many solutions.
6.EEI.9Investigate multiple representations of relationships in real-world and mathematical situations.
6.EEI.9aWrite an equation that models a relationship between independent and dependent variables.
6.EEI.9bAnalyze the relationship between independent and dependent variables using graphs and tables.
6.EEI.9cTranslate among graphs, tables, and equations.
6.GM Geometry and Measurement
6.GM.1Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
6.GM.2Use visual models (e.g., model by packing) to discover that the formulas for the volume of a right rectangular prism (V=lwh, V=Bh) are the same for whole or fractional edge lengths. Apply these formulas to solve real-world and mathematical problems.
6.GM.3Apply the concepts of polygons and the coordinate plane to real-world and mathematical situations.
6.GM.3aGiven coordinates of the vertices, draw a polygon in the coordinate plane.
6.GM.3bFind the length of an edge if the vertices have the same x-coordinates or same y-coordinates.
6.GM.4Unfold three-dimensional figures into two-dimensional rectangles and triangles (nets) to find the surface area and to solve real-world and mathematical problems.
6.NS The Number System
6.NS.1Compute and represent quotients of positive fractions using a variety of procedures (e.g., visual models, equations, and real-world situations).
6.NS.2Fluently divide multi-digit whole numbers using a standard algorithmic approach.
6.NS.3Fluently add, subtract, multiply and divide multi-digit decimal numbers using a standard algorithmic approach.
6.NS.4Find common factors and multiples using two whole numbers.
6.NS.4aCompute the greatest common factor (GCF) of two numbers both less than or equal to 100.
6.NS.4bCompute the least common multiple (LCM) of two numbers both less than or equal to 12.
6.NS.4cExpress sums of two whole numbers, each less than or equal to 100, using the distributive property to factor out a common factor of the original addends.
6.NS.5Understand that the positive and negative representations of a number are opposites in direction and value. Use integers to represent quantities in real-world situations and explain the meaning of zero in each situation.
6.NS.6Extend the understanding of the number line to include all rational numbers and apply this concept to the coordinate plane.
6.NS.6aUnderstand the concept of opposite numbers, including zero, and their relative locations on the number line.
6.NS.6bUnderstand that the signs of the coordinates in ordered pairs indicate their location on an axis or in a quadrant on the coordinate plane.
6.NS.6cRecognize when ordered pairs are reflections of each other on the coordinate plane across one axis, both axes, or the origin.
6.NS.6dPlot rational numbers on number lines and ordered pairs on coordinate planes.
6.NS.7Understand and apply the concepts of comparing, ordering, and finding absolute value to rational numbers.
6.NS.7aInterpret statements using equal to (=) and not equal to (≠).
6.NS.7bInterpret statements using less than (<), greater than (>), and equal to (=) as relative locations on the number line.
6.NS.7cUse concepts of equality and inequality to write and to explain real-world and mathematical situations.
6.NS.7dUnderstand that absolute value represents a number’s distance from zero on the number line and use the absolute value of a rational number to represent real-world situations.
6.NS.7eRecognize the difference between comparing absolute values and ordering rational numbers. For negative rational numbers, understand that as the absolute value increases, the value of the negative number decreases.
6.NS.8Extend knowledge of the coordinate plane to solve real-world and mathematical problems involving rational numbers.
6.NS.8aPlot points in all four quadrants to represent the problem.
6.NS.8bFind the distance between two points when ordered pairs have the same x-coordinates or same y-coordinates.
6.NS.8cRelate finding the distance between two points in a coordinate plane to absolute value using a number line.
6.NS.9Investigate and translate among multiple representations of rational numbers (fractions, decimal numbers, and percentages). Fractions should be limited to those with denominators of 2, 3, 4, 5, 8, 10, and 100.
6.RP Ratios and Proportional Relationships
6.RP.1Interpret the concept of a ratio as the relationship between two quantities, including part to part and part to whole.
6.RP.2Investigate relationships between ratios and rates.
6.RP.2aTranslate between multiple representations of ratios (i.e., a/b, a:b, a to b, visual models).
6.RP.2bRecognize that a rate is a type of ratio involving two different units.
6.RP.2cConvert from rates to unit rates.
6.RP.3Apply the concepts of ratios and rates to solve real-world and mathematical problems.
6.RP.3aCreate a table consisting of equivalent ratios and plot the results on the coordinate plane.
6.RP.3bUse multiple representations, including tape diagrams, tables, double number lines, and equations, to find missing values of equivalent ratios.
6.RP.3cUse two tables to compare related ratios.
6.RP.3dApply concepts of unit rate to solve problems, including unit pricing and constant speed.
6.RP.3eUnderstand that a percentage is a rate per 100 and use this to solve problems involving wholes, parts, and percentages.
6.RP.3fSolve one-step problems involving ratios and unit rates (e.g., dimensional analysis).

Grade 7

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7.DSP Data Analysis, Statistics, and Probability
7.DSP.1Investigate concepts of random sampling.
7.DSP.1aUnderstand that a sample is a subset of a population and both possess the same characteristics.
7.DSP.1bDifferentiate between random and non-random sampling.
7.DSP.1cUnderstand that generalizations from a sample are valid only if the sample is representative of the population.
7.DSP.1dUnderstand that random sampling is used to gather a representative sample and supports valid inferences about the population.
7.DSP.2Draw inferences about a population by collecting multiple random samples of the same size to investigate variability in estimates of the characteristic of interest.
7.DSP.3Visually compare the centers, spreads, and overlap of two displays of data (i.e., dot plots, histograms, box plots) that are graphed on the same scale and draw inferences about this data.
7.DSP.4Compare the numerical measures of center (mean, median, mode) and variability (range, interquartile range, mean absolute deviation) from two random samples to draw inferences about the populations.
7.DSP.5Investigate the concept of probability of chance events.
7.DSP.5aDetermine probabilities of simple events.
7.DSP.5bUnderstand that probability measures likelihood of a chance event occurring.
7.DSP.5cUnderstand that the probability of a chance event is a number between 0 and 1.
7.DSP.5dUnderstand that a probability closer to 1 indicates a likely chance event.
7.DSP.5eUnderstand that a probability close to 1/2 indicates that a chance event is neither likely nor unlikely.
7.DSP.5fUnderstand that a probability closer to 0 indicates an unlikely chance event.
7.DSP.6Investigate the relationship between theoretical and experimental probabilities for simple events.
7.DSP.6aDetermine approximate outcomes using theoretical probability.
7.DSP.6bPerform experiments that model theoretical probability.
7.DSP.6cCompare theoretical and experimental probabilities.
7.DSP.7Apply the concepts of theoretical and experimental probabilities for simple events.
7.DSP.7aDifferentiate between uniform and non-uniform probability models (distributions).
7.DSP.7bDevelop both uniform and non-uniform probability models.
7.DSP.7cPerform experiments to test the validity of probability models.
7.DSP.8Extend the concepts of simple events to investigate compound events.
7.DSP.8aUnderstand that the probability of a compound event is between 0 and 1.
7.DSP.8bIdentify the outcomes in a sample space using organized lists, tables, and tree diagrams.
7.DSP.8cDetermine probabilities of compound events using organized lists, tables, and tree diagrams.
7.DSP.8dDesign and use simulations to collect data and determine probabilities.
7.DSP.8eCompare theoretical and experimental probabilities for compound events.
7.EEI Expressions, Equations, and Inequalities
7.EEI.1Apply mathematical properties (e.g., commutative, associative, distributive) to simplify and to factor linear algebraic expressions with rational coefficients.
7.EEI.2Recognize that algebraic expressions may have a variety of equivalent forms and determine an appropriate form for a given real-world situation.
7.EEI.3Extend previous understanding of Order of Operations to solve multi-step real-world and mathematical problems involving rational numbers. Include fraction bars as a grouping symbol.
7.EEI.4Apply the concepts of linear equations and inequalities in one variable to real-world and mathematical situations.
7.EEI.4aWrite and fluently solve linear equations of the form ax+b=c and a(x+b)=c where a, b, and c are rational numbers.
7.EEI.4bWrite and solve multi-step linear equations that include the use of the distributive property and combining like terms. Exclude equations that contain variables on both sides.
7.EEI.4cWrite and solve two-step linear inequalities. Graph the solution set on a number line and interpret its meaning.
7.EEI.4dIdentify and justify the steps for solving multi-step linear equations and two-step linear inequalities.
7.EEI.5Understand and apply the laws of exponents (i.e., product rule, quotient rule, power to a power, product to a power, quotient to a power, zero power property) to simplify numerical expressions that include whole-number exponents.
7.GM Geometry and Measurement
7.GM.1Determine the scale factor and translate between scale models and actual measurements (e.g., lengths, area) of real-world objects and geometric figures using proportional reasoning.
7.GM.2Construct triangles and special quadrilaterals using a variety of tools (e.g., freehand, ruler and protractor, technology).
7.GM.2aConstruct triangles given all measurements of either angles or sides.
7.GM.2bDecide if the measurements determine a unique triangle, more than one triangle, or no triangle.
7.GM.2cConstruct special quadrilaterals (i.e., kite, trapezoid, isosceles trapezoid, rhombus, parallelogram, rectangle) given specific parameters about angles or sides.
7.GM.3Describe two-dimensional cross-sections of three-dimensional figures, specifically right rectangular prisms and right rectangular pyramids.
7.GM.4Investigate the concept of circles.
7.GM.4aDemonstrate an understanding of the proportional relationships between diameter, radius, and circumference of a circle.
7.GM.4bUnderstand that the constant of proportionality between the circumference and diameter is equivalent to π.
7.GM.4cExplore the relationship between circumference and area using a visual model.
7.GM.4dUse the formulas for circumference and area of circles appropriately to solve real-world and mathematical problems.
7.GM.5Write equations to solve problems involving the relationships between angles formed by two intersecting lines, including supplementary, complementary, vertical, and adjacent.
7.GM.6Apply the concepts of two- and three-dimensional figures to real-world and mathematical situations.
7.GM.6aUnderstand that the concept of area is applied to two-dimensional figures such as triangles, quadrilaterals, and polygons.
7.GM.6bUnderstand that the concepts of volume and surface area are applied to three-dimensional figures such as cubes, right rectangular prisms, and right triangular prisms.
7.GM.6cDecompose cubes, right rectangular prisms, and right triangular prisms into rectangles and triangles to derive the formulas for volume and surface area.
7.GM.6dUse the formulas for area, volume, and surface area appropriately.
7.NS The Number System
7.NS.1Extend prior knowledge of operations with positive rational numbers to add and to subtract all rational numbers and represent the sum or difference on a number line.
7.NS.1aUnderstand that the additive inverse of a number is its opposite and their sum is equal to zero.
7.NS.1bUnderstand that the sum of two rational numbers (p+q) represents a distance from p on the number line equal to |q| where the direction is indicated by the sign of q.
7.NS.1cTranslate between the subtraction of rational numbers and addition using the additive inverse, p-q=p+(-q).
7.NS.1dDemonstrate that the distance between two rational numbers on the number line is the absolute value of their difference.
7.NS.1eApply mathematical properties (e.g., commutative, associative, distributive, or the properties of identity and inverse elements) to add and subtract rational numbers.
7.NS.2Extend prior knowledge of operations with positive rational numbers to multiply and to divide all rational numbers.
7.NS.2aUnderstand that the multiplicative inverse of a number is its reciprocal and their product is equal to one.
7.NS.2bUnderstand sign rules for multiplying rational numbers.
7.NS.2cUnderstand sign rules for dividing rational numbers and that a quotient of integers (with a non-zero divisor) is a rational number.
7.NS.2dApply mathematical properties (e.g., commutative, associative, distributive, or the properties of identity and inverse elements) to multiply and divide rational numbers.
7.NS.2eUnderstand that some rational numbers can be written as integers and all rational numbers can be written as fractions or decimal numbers that terminate or repeat.
7.NS.3Apply the concepts of all four operations with rational numbers to solve real-world and mathematical problems.
7.NS.4Understand and apply the concepts of comparing and ordering to rational numbers.
7.NS.4aInterpret statements using less than (<), greater than (>), less than or equal to (≤), greater than or equal to (≥), and equal to (=) as relative locations on the number line.
7.NS.4bUse concepts of equality and inequality to write and explain real-world and mathematical situations.
7.NS.5Extend prior knowledge to translate among multiple representations of rational numbers (fractions, decimal numbers, and percentages). Exclude the conversion of repeating decimal numbers to fractions.
7.RP Ratios and Proportional Relationships
7.RP.1Compute unit rates, including those involving complex fractions, with like or different units.
7.RP.2Identify and model proportional relationships given multiple representations, including tables, graphs, equations, diagrams, verbal descriptions, and real-world situations.
7.RP.2aDetermine when two quantities are in a proportional relationship.
7.RP.2bRecognize or compute the constant of proportionality.
7.RP.2cUnderstand that the constant of proportionality is the unit rate.
7.RP.2dUse equations to model proportional relationships.
7.RP.2eInvestigate the graph of a proportional relationship and explain the meaning of specific points (e.g., origin, unit rate) in the context of the situation.
7.RP.3Solve real-world and mathematical problems involving ratios and percentages using proportional reasoning (e.g., multi-step dimensional analysis, percent increase/decrease, tax).

Grade 8

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8.DSP Data Analysis, Statistics, and Probability
8.DSP.1Investigate bivariate data.
8.DSP.1aCollect bivariate data.
8.DSP.1bGraph the bivariate data on a scatter plot.
8.DSP.1cDescribe patterns observed on a scatter plot, including clustering, outliers, and association (positive, negative, no correlation, linear, nonlinear).
8.DSP.2Draw an approximate line of best fit on a scatter plot that appears to have a linear association and informally assess the fit of the line to the data points.
8.DSP.3Apply concepts of an approximate line of best fit in real-world situations.
8.DSP.3aFind an approximate equation for the line of best fit using two appropriate data points.
8.DSP.3bInterpret the slope and intercept.
8.DSP.3cSolve problems using the equation.
8.DSP.4Investigate bivariate categorical data in two-way tables.
8.DSP.4aOrganize bivariate categorical data in a two-way table.
8.DSP.4bInterpret data in two-way tables using relative frequencies.
8.DSP.4cExplore patterns of possible association between the two categorical variables.
8.DSP.5Organize data in matrices with rational numbers and apply to real-world and mathematical situations.
8.DSP.5aUnderstand that a matrix is a way to organize data.
8.DSP.5bRecognize that a m x n matrix has m rows and n columns.
8.DSP.5cAdd and subtract matrices of the same size.
8.DSP.5dMultiply a matrix by a scalar.
8.EEI Expressions, Equations, and Inequalities
8.EEI.1Understand and apply the laws of exponents (i.e. product rule, quotient rule, power to a power, product to a power, quotient to a power, zero power property, negative exponents) to simplify numerical expressions that include integer exponents.
8.EEI.2Investigate concepts of square and cube roots.
8.EEI.2aFind the exact and approximate solutions to equations of the form x^2=p and x^3=p where p is a positive rational number.
8.EEI.2bEvaluate square roots of perfect squares.
8.EEI.2cEvaluate cube roots of perfect cubes.
8.EEI.2dRecognize that square roots of non-perfect squares are irrational.
8.EEI.3Explore the relationship between quantities in decimal and scientific notation.
8.EEI.3aExpress very large and very small quantities in scientific notation in the form a x 10^b = p where 1≤a&lt;10 and b is an integer.
8.EEI.3bTranslate between decimal notation and scientific notation.
8.EEI.3cEstimate and compare the relative size of two quantities in scientific notation.
8.EEI.4Apply the concepts of decimal and scientific notation to solve real-world and mathematical problems.
8.EEI.4aMultiply and divide numbers expressed in both decimal and scientific notation.
8.EEI.4bSelect appropriate units of measure when representing answers in scientific notation.
8.EEI.4cTranslate how different technological devices display numbers in scientific notation.
8.EEI.5Apply concepts of proportional relationships to real-world and mathematical situations.
8.EEI.5aGraph proportional relationships.
8.EEI.5bInterpret unit rate as the slope of the graph.
8.EEI.5cCompare two different proportional relationships given multiple representations, including tables, graphs, equations, diagrams, and verbal descriptions.
8.EEI.6Apply concepts of slope and y-intercept to graphs, equations, and proportional relationships.
8.EEI.6aExplain why the slope, m, is the same between any two distinct points on a non-vertical line using similar triangles.
8.EEI.6bDerive the slope-intercept form (y=mx+b) for a non-vertical line.
8.EEI.6cRelate equations for proportional relationships (y=kx) with the slope-intercept form (y=mx+b) where b=0.
8.EEI.7Extend concepts of linear equations and inequalities in one variable to more complex multi-step equations and inequalities in real-world and mathematical situations.
8.EEI.7aSolve linear equations and inequalities with rational number coefficients that include the use of the distributive property, combining like terms, and variables on both sides.
8.EEI.7bRecognize the three types of solutions to linear equations: one solution (x=a), infinitely many solutions (a=a), or no solutions (a=b).
8.EEI.7cGenerate linear equations with the three types of solutions.
8.EEI.7dJustify why linear equations have a specific type of solution.
8.EEI.8Investigate and solve real-world and mathematical problems involving systems of linear equations in two variables with integer coefficients and solutions.
8.EEI.8aGraph systems of linear equations and estimate their point of intersection.
8.EEI.8bUnderstand and verify that a solution to a system of linear equations is represented on a graph as the point of intersection of the two lines.
8.EEI.8cSolve systems of linear equations algebraically, including methods of substitution and elimination, or through inspection.
8.EEI.8dUnderstand that systems of linear equations can have one solution, no solution, or infinitely many solutions.
8.F Functions
8.F.1Explore the concept of functions.
8.F.1aUnderstand that a function assigns to each input exactly one output.
8.F.1bRelate inputs (x-values or domain) and outputs (y-values or range) to independent and dependent variables.
8.F.1cTranslate among the multiple representations of a function, including mappings, tables, graphs, equations, and verbal descriptions.
8.F.1dDetermine if a relation is a function using multiple representations, including mappings, tables, graphs, equations, and verbal descriptions.
8.F.1eGraph a function from a table of values. Understand that the graph and table both represent a set of ordered pairs of that function.
8.F.2Compare multiple representations of two functions, including mappings, tables, graphs, equations, and verbal descriptions, in order to draw conclusions.
8.F.3Investigate the differences between linear and nonlinear functions using multiple representations (i.e. tables, graphs, equations, and verbal descriptions).
8.F.3aDefine an equation in slope-intercept form (y=mx+b) as being a linear function.
8.F.3bRecognize that the graph of a linear function has a constant rate of change.
8.F.3cProvide examples of nonlinear functions.
8.F.4Apply the concepts of linear functions to real-world and mathematical situations.
8.F.4aUnderstand that the slope is the constant rate of change and the y-intercept is the point where x = 0.
8.F.4bDetermine the slope and the y-intercept of a linear function given multiple representations, including two points, tables, graphs, equations, and verbal descriptions.
8.F.4cConstruct a function in slope-intercept form that models a linear relationship between two quantities.
8.F.4dInterpret the meaning of the slope and the y-intercept of a linear function in the context of the situation.
8.F.4eExplore the relationship between linear functions and arithmetic sequences.
8.F.5Apply the concepts of linear and nonlinear functions to graphs in real-world and mathematical situations.
8.F.5aAnalyze and describe attributes of graphs of functions (e.g., constant, increasing/decreasing, linear/nonlinear, maximum/minimum, discrete/continuous).
8.F.5bSketch the graph of a function from a verbal description.
8.F.5cWrite a verbal description from the graph of a function with and without scales.
8.GM Geometry and Measurement
8.GM.1Investigate the properties of rigid transformations (rotations, reflections, translations) using a variety of tools (e.g., grid paper, reflective devices, graphing paper, technology).
8.GM.1aVerify that lines are mapped to lines, including parallel lines.
8.GM.1bVerify that corresponding angles are congruent.
8.GM.1cVerify that corresponding line segments are congruent.
8.GM.2Apply the properties of rigid transformations (rotations, reflections, translations).
8.GM.2aRotate geometric figures 90, 180, and 270 degrees, both clockwise and counterclockwise, about the origin.
8.GM.2bReflect geometric figures with respect to the x-axis and/or y-axis.
8.GM.2cTranslate geometric figures vertically and/or horizontally.
8.GM.2dRecognize that two-dimensional figures are only congruent if a series of rigid transformations can be performed to map the pre-image to the image.
8.GM.2eGiven two congruent figures, describe the series of rigid transformations that justifies this congruence.
8.GM.3Investigate the properties of transformations (rotations, reflections, translations, dilations) using a variety of tools (e.g., grid paper, reflective devices, graphing paper, dynamic software).
8.GM.3aUse coordinate geometry to describe the effect of transformations on two-dimensional figures.
8.GM.3bRelate scale drawings to dilations of geometric figures.
8.GM.4Apply the properties of transformations (rotations, reflections, translations, dilations).
8.GM.4aDilate geometric figures using scale factors that are positive rational numbers.
8.GM.4bRecognize that two-dimensional figures are only similar if a series of transformations can be performed to map the pre-image to the image.
8.GM.4cGiven two similar figures, describe the series of transformations that justifies this similarity.
8.GM.4dUse proportional reasoning to find the missing side lengths of two similar figures.
8.GM.5Extend and apply previous knowledge of angles to properties of triangles, similar figures, and parallel lines cut by a transversal.
8.GM.5aDiscover that the sum of the three angles in a triangle is 180 degrees.
8.GM.5bDiscover and use the relationship between interior and exterior angles of a triangle.
8.GM.5cIdentify congruent and supplementary pairs of angles when two parallel lines are cut by a transversal.
8.GM.5dRecognize that two similar figures have congruent corresponding angles.
8.GM.6Use models to demonstrate a proof of the Pythagorean Theorem and its converse.
8.GM.7Apply the Pythagorean Theorem to model and solve real-world and mathematical problems in two and three dimensions involving right triangles.
8.GM.8Find the distance between any two points in the coordinate plane using the Pythagorean Theorem.
8.GM.9Solve real-world and mathematical problems involving volumes of cones, cylinders, and spheres and the surface area of cylinders.
8.NS The Number System
8.NS.1Explore the real number system and its appropriate usage in real-world situations.
8.NS.1aRecognize the differences between rational and irrational numbers.
8.NS.1bUnderstand that all real numbers have a decimal expansion.
8.NS.1cModel the hierarchy of the real number system, including natural, whole, integer, rational, and irrational numbers.
8.NS.2Estimate and compare the value of irrational numbers by plotting them on a number line.
8.NS.3Extend prior knowledge to translate among multiple representations of rational numbers (fractions, decimal numbers, and percentages). Include the conversion of repeating decimal numbers to fractions.

Algebra 1

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A1.AAPR Arithmetic with Polynomials and Rational Expressions
A1.AAPR.1Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. (Limit to linear; quadratic.)
A1.ACE Creating Equations
A1.ACE.1Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. (Limit to linear; quadratic; exponential with integer exponents.)
A1.ACE.2Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.)
A1.ACE.4Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines.
A1.AREI Reasoning with Equations and Inequalities
A1.AREI.1Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original.
A1.AREI.10Explain that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.
A1.AREI.11Solve an equation of the form f(x)=g(x) graphically by identifying the x-coordinate(s) of the point(s) of intersection of the graphs of y=f(x) and y=g(x). (Limit to linear; quadratic; exponential.)
A1.AREI.12Graph the solutions to a linear inequality in two variables.
A1.AREI.3Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
A1.AREI.4Solve mathematical and real-world problems involving quadratic equations in one variable. (Note: A1.AREI.4a and 4b are not Graduation Standards.)
A1.AREI.4aUse the method of completing the square to transform any quadratic equation in x into an equation of the form (x-h)^2=k that has the same solutions. Derive the quadratic formula from this form.
A1.AREI.4bSolve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a+bi for real numbers a and b. (Limit to non-complex roots.)
A1.AREI.5Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation.
A1.AREI.6Solve systems of linear equations algebraically and graphically focusing on pairs of linear equations in two variables. (Note: A1.AREI.6a and 6b are not Graduation Standards.)
A1.AREI.6aSolve systems of linear equations using the substitution method.
A1.AREI.6bSolve systems of linear equations using linear combination.
A1.ASE Structure and Expressions
A1.ASE.1Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. (Limit to linear; quadratic; exponential.)
A1.ASE.2Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions.
A1.ASE.3Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
A1.ASE.3aFind the zeros of a quadratic function by rewriting it in equivalent factored form and explain the connection between the zeros of the function, its linear factors, the x-intercepts of its graph, and the solutions to the corresponding quadratic equation.
A1.FBF Building Functions
A1.FBF.3Describe the effect of the transformations kf(x), f(x)+k, f(x+k), and combinations of such transformations on the graph of y=f(x) for any real number k. Find the value of k given the graphs and write the equation of a transformed parent function given its graph. (Limit to linear; quadratic; exponential with integer exponents; vertical shift and vertical stretch.)
A1.FIF Interpreting Functions
A1.FIF.1Extend previous knowledge of a function to apply to general behavior and features of a function.
A1.FIF.1aUnderstand 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.
A1.FIF.1bRepresent a function using function notation and explain that f(x) denotes the output of function f that corresponds to the input x.
A1.FIF.1cUnderstand that the graph of a function labeled as f is the set of all ordered pairs (x,y) that satisfy the equation y=f(x).
A1.FIF.2Evaluate functions and interpret the meaning of expressions involving function notation from a mathematical perspective and in terms of the context when the function describes a real-world situation.
A1.FIF.4Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.)
A1.FIF.5Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. (Limit to linear; quadratic; exponential.)
A1.FIF.6Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. (Limit to linear; quadratic; exponential.)
A1.FIF.7Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form y=(a^x)+k.)
A1.FIF.8Translate between different but equivalent forms of a function equation to reveal and explain different properties of the function. (Limit to linear; quadratic; exponential.) (Note: A1.FIF.8a is not a Graduation Standard.)
A1.FIF.8aUse the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
A1.FIF.9Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. (Limit to linear; quadratic; exponential.)
A1.FLQE Linear, Quadratic, and Exponential
A1.FLQE.1Distinguish between situations that can be modeled with linear functions or exponential functions by recognizing situations in which one quantity changes at a constant rate per unit interval as opposed to those in which a quantity changes by a constant percent rate per unit interval. (Note: A1.FLQE.1a is not a Graduation Standard.)
A1.FLQE.1aProve that linear functions grow by equal differences over equal intervals and that exponential function grow by equal factors over equal intervals.
A1.FLQE.2Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables. (Limit to linear; exponential.)
A1.FLQE.3Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function.
A1.FLQE.5Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.)
A1.NQ Quantities
A1.NQ.1Use units of measurement to guide the solution of multi-step tasks. Choose and interpret appropriate labels, units, and scales when constructing graphs and other data displays.
A1.NQ.2Label and define appropriate quantities in descriptive modeling contexts.
A1.NQ.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities in context.
A1.NRNS Real Number System
A1.NRNS.1Rewrite expressions involving simple radicals and rational exponents in different forms.
A1.NRNS.2Use the definition of the meaning of rational exponents to translate between rational exponent and radical forms.
A1.NRNS.3Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
A1.SPID Interpreting Data
A1.SPID.6Using technology, create scatterplots and analyze those plots to compare the fit of linear, quadratic, or exponential models to a given data set. Select the appropriate model, fit a function to the data set, and uses the function to solve problems in the context of the data.
A1.SPID.7Create a linear function to graphically model data from a real-world problem and interpret the meaning of the slope and intercept(s) in the context of the given problem.
A1.SPID.8Using technology, compute and interpret the correlation coefficient of a linear fit.

Algebra 2

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A2.AAPR Arithmetic with Polynomials and Rational Expressions
A2.AAPR.1Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations.
A2.AAPR.3Graph polynomials identifying zeros when suitable factorizations are available and indicating end behavior. Write a polynomial function of least degree corresponding to a given graph. (Limit to polynomials with degrees 3 or less.)
A2.ACE Creating Equations
A2.ACE.1Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable.
A2.ACE.2Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales.
A2.ACE.3Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.)
A2.ACE.4Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines.
A2.AREI Reasoning with Equations and Inequalities
A2.AREI.11Solve an equation of the form f(x)=g(x) graphically by identifying the x-coordinate(s) of the point(s) of intersection of the graphs of y=f(x) and y=g(x).
A2.AREI.2Solve simple rational and radical equations in one variable and understand how extraneous solutions may arise.
A2.AREI.4Solve mathematical and real-world problems involving quadratic equations in one variable.
A2.AREI.4bSolve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a+bi for real numbers a and b. (Note: A2.AREI.4b is not a Graduation Standard.)
A2.AREI.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.)
A2.ASE Structure and Expressions
A2.ASE.1Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions.
A2.ASE.2Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions.
A2.ASE.3Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (Note: A2.ASE.3b and 3c are not Graduation Standards.)
A2.ASE.3bDetermine the maximum or minimum value of a quadratic function by completing the square.
A2.ASE.3cUse the properties of exponents to transform expressions for exponential functions.
A2.FBF Building Functions
A2.FBF.1Write a function that describes a relationship between two quantities. (Note: IA.FBF.1a is not a Graduation Standard.)
A2.FBF.1aWrite a function that models a relationship between two quantities using both explicit expressions and a recursive process and by combining standard forms using addition, subtraction, multiplication and division to build new functions.
A2.FBF.1bCombine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations.
A2.FBF.2Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
A2.FBF.3Describe the effect of the transformations kf(x), f(x)+k, f(x+k), and combinations of such transformations on the graph of y=f(x) for any real number k. Find the value of k given the graphs and write the equation of a transformed parent function given its graph.
A2.FIF Interpreting Functions
A2.FIF.3Define functions recursively and recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
A2.FIF.4Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity.
A2.FIF.5Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes.
A2.FIF.6Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context.
A2.FIF.7Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases.
A2.FIF.8Translate between different but equivalent forms of a function equation to reveal and explain different properties of the function. (Note: A2.FIF.8b is not a Graduation Standard.)
A2.FIF.8bInterpret expressions for exponential functions by using the properties of exponents.
A2.FIF.9Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal.
A2.FLQE Linear, Quadratic, and Exponential
A2.FLQE.1Distinguish between situations that can be modeled with linear functions or exponential functions by recognizing situations in which one quantity changes at a constant rate per unit interval as opposed to those in which a quantity changes by a constant percent rate per unit interval. (Note: A2.FLQE.1b is not a Graduation Standard.)
A2.FLQE.1bRecognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
A2.FLQE.2Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables.
A2.FLQE.5Interpret the parameters in a linear or exponential function in terms of the context.
A2.NCNS Complex Number System
A2.NCNS.1Know there is a complex number i such that i^2 =−1, and every complex number has the form a+bi with a and b real.
A2.NCNS.7Solve quadratic equations in one variable that have complex solutions.

Calculus

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C.D Derivatives
C.D.1Understand the concept of the derivative of a function geometrically, numerically, analytically, and verbally.
C.D.1aInterpret the value of the derivative of a function as the slope of the corresponding tangent line.
C.D.1bInterpret the value of the derivative as an instantaneous rate of change in a variety of real-world contexts such as velocity and population growth.
C.D.1cApproximate the derivative graphically by finding the slope of the tangent line drawn to a curve at a given point and numerically by using the difference quotient.
C.D.1dUnderstand and explain graphically and analytically the relationship between differentiability and continuity.
C.D.1eExplain graphically and analytically the relationship between the average rate of change and the instantaneous rate of change.
C.D.1fUnderstand the definition of the derivative and use this definition to determine the derivatives of various functions.
C.D.2Apply the rules of differentiation to functions.
C.D.2aKnow and apply the derivatives of constant, power, trigonometric, inverse trigonometric, exponential, and logarithmic functions.
C.D.2bUse the constant multiple, sum, difference, product, quotient, and chain rules to find the derivatives of functions.
C.D.2cUnderstand and apply the methods of implicit and logarithmic differentiation.
C.D.3Apply theorems and rules of differentiation to solve mathematical and real-world problems.
C.D.3aExplain geometrically and verbally the mathematical and real-world meanings of the Extreme Value Theorem and the Mean Value Theorem.
C.D.3bWrite an equation of a line tangent to the graph of a function at a point.
C.D.3cExplain the relationship between the increasing/decreasing behavior of f and the signs of f’. Use the relationship to generate a graph of f given the graph of f’, and vice versa, and to identify relative and absolute extrema of f.
C.D.3dExplain the relationships among the concavity of the graph of f, the increasing/decreasing behavior of f’ and the signs of f’’. Use those relationships to generate graphs of f, f′, and f′′ given any one of them and identify the points of inflection of f.
C.D.3eSolve a variety of real-world problems involving related rates, optimization, linear approximation, and rates of change.
C.I Integrals
C.I.1Understand the concept of the integral of a function geometrically, numerically, analytically, and contextually.
C.I.1aExplain how the definite integral is used to solve area problems.
C.I.1bApproximate definite integrals by calculating Riemann sums using left, right, and mid-point evaluations, and using trapezoidal sums.
C.I.1cInterpret the definite integral as a limit of Riemann sums.
C.I.1dExplain the relationship between the integral and derivative as expressed in both parts of the Fundamental Theorem of Calculus. Interpret the relationship in terms of rates of change.
C.I.2Apply theorems and rules of integration to solve mathematical and real-world problems.
C.I.2aApply the Fundamental Theorems of Calculus to solve mathematical and real-world problems.
C.I.2bExplain graphically and verbally the properties of the definite integral. Apply these properties to evaluate basic definite integrals.
C.I.2cEvaluate integrals using substitution.
C.LC Limits and Continuity
C.LC.1Understand the concept of a limit graphically, numerically, analytically, and contextually.
C.LC.1aEstimate and verify limits using tables, graphs of functions, and technology.
C.LC.1bCalculate limits, including one-sided limits, algebraically using direct substitution, simplification, rationalization, and the limit laws for constant multiples, sums, differences, products, and quotients.
C.LC.1cCalculate infinite limits and limits at infinity. Understand that infinite limits and limits at infinity provide information regarding the asymptotes of certain functions, including rational, exponential and logarithmic functions.
C.LC.2Understand the definition and graphical interpretation of continuity of a function.
C.LC.2aApply the definition of continuity of a function at a point to solve problems.
C.LC.2bClassify discontinuities as removable, jump, or infinite. Justify that classification using the definition of continuity.
C.LC.2cUnderstand the Intermediate Value Theorem and apply the theorem to prove the existence of solutions of equations arising in mathematical and real-world problems.

Foundations in Algebra

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FA.ACE Creating Equations
FA.ACE.1Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. (Limit to linear; quadratic; exponential with integer exponents.)
FA.ACE.2Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.)
FA.ACE.4Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines.
FA.AREI Reasoning with Equations and Inequalities
FA.AREI.1Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original.
FA.AREI.10Explain that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.
FA.AREI.11Solve an equation of the form f(x)=g(x) graphically by identifying the x-coordinate(s) of the point(s) of intersection of the graphs of y=f(x) and y=g(x). (Limit to linear; quadratic; exponential.)
FA.AREI.12Graph the solutions to a linear inequality in two variables.
FA.AREI.3Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
FA.AREI.5Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation.
FA.AREI.6Solve systems of linear equations algebraically and graphically focusing on pairs of linear equations in two variables. (Note: FA.AREI.6a and 6b are not Graduation Standards.)
FA.AREI.6aSolve systems of linear equations using the substitution method.
FA.AREI.6bSolve systems of linear equations using linear combination.
FA.ASE Structure and Expressions
FA.ASE.1Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. (Limit to linear; quadratic; exponential.)
FA.FBF Building Functions
FA.FBF.3Describe the effect of the transformations kf(x), f(x)+k, f(x+k), and combinations of such transformations on the graph of y=f(x) for any real number k. Find the value of k given the graphs and write the equation of a transformed parent function given its graph. (Limit to linear; quadratic; exponential with integer exponents; vertical shift and vertical stretch.)
FA.FIF Interpreting Functions
FA.FIF.1Extend previous knowledge of a function to apply to general behavior and features of a function.
FA.FIF.1aUnderstand 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.
FA.FIF.1bRepresent a function using function notation and explain that f(x) denotes the output of function f that corresponds to the input x.
FA.FIF.1cUnderstand that the graph of a function labeled as f is the set of all ordered pairs (x,y) that satisfy the equation y=f(x).
FA.FIF.2Evaluate functions and interpret the meaning of expressions involving function notation from a mathematical perspective and in terms of the context when the function describes a real-world situation.
FA.FIF.4Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. (Limit to linear; quadratic; exponential.)
FA.FIF.5Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. (Limit to linear; quadratic; exponential.)
FA.FIF.7Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Limit to linear; quadratic; exponential only in the form y=a^x+k.)
FA.FIF.8Translate between different but equivalent forms of a function equation to reveal and explain different properties of the function. (Limit to linear; quadratic; exponential.) (Note: FA.FIF.8a is not a Graduation Standard.)
FA.FIF.8aUse the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
FA.FIF.9Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. (Limit to linear; quadratic; exponential.)
FA.FLQE Linear, Quadratic, and Exponential
FA.FLQE.1Distinguish between situations that can be modeled with linear functions or exponential functions by recognizing situations in which one quantity changes at a constant rate per unit interval as opposed to those in which a quantity changes by a constant percent rate per unit interval. (Note: FA.FLQE.1a is not a Graduation Standard.)
FA.FLQE.1aProve that linear functions grow by equal differences over equal intervals and those exponential functions grow by equal factors over equal intervals.
FA.FLQE.3Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function.
FA.FLQE.5Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.)
FA.NQ Quantities
FA.NQ.1Use units of measurement to guide the solution of multi-step tasks. Choose and interpret appropriate labels, units, and scales when constructing graphs and other data displays.
FA.NQ.2Label and define appropriate quantities in descriptive modeling contexts.
FA.NQ.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities in context.
FA.NRNS Real Number System
FA.NRNS.1Rewrite expressions involving simple radicals and rational exponents in different forms.
FA.NRNS.2Use the definition of the meaning of rational exponents to translate between rational exponent and radical forms.
FA.NRNS.3Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
FA.SPID Interpreting Data
FA.SPID.5Analyze bivariate categorical data using two-way tables and identify possible associations between the two categories using marginal, joint, and conditional frequencies.
FA.SPID.6Using technology, create scatterplots and analyze those plots to compare the fit of linear, quadratic, or exponential models to a given data set. Select the appropriate model, fit a function to the data set, and uses the function to solve problems in the context of the data.
FA.SPID.7Create a linear function to graphically model data from a real-world problem and interpret the meaning of the slope and intercept(s) in the context of the given problem.
FA.SPID.8Using technology, compute and interpret the correlation coefficient of a linear fit.
FA.SPMD Using Probability to Make Decisions
FA.SPMD.4Use probability to evaluate outcomes of decisions by finding expected values and determine if decisions are fair.
FA.SPMD.5Use probability to evaluate outcomes of decisions. Use probabilities to make fair decisions.
FA.SPMD.6Analyze decisions and strategies using probability concepts.
FA.SPMJ Making Inferences and Justifying Conclusions
FA.SPMJ.1Understand statistics and sampling distributions as a process for making inferences about population parameters based on a random sample from that population.
FA.SPMJ.2Distinguish between experimental and theoretical probabilities. Collect data on a chance event and use the relative frequency to estimate the theoretical probability of that event. Determine whether a given probability model is consistent with experimental results.

Geometry

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G.GCI Circles
G.GCI.1Prove that all circles are similar.
G.GCI.2Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems.
G.GCI.3Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle.
G.GCI.4Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction.
G.GCI.5Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems.
G.GCO Congruence
G.GCO.1Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects.
G.GCO.10Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following:
G.GCO.10aOpposite sides of a parallelogram are congruent.
G.GCO.10bOpposite angles of a parallelogram are congruent.
G.GCO.10cDiagonals of a parallelogram bisect each other.
G.GCO.10dRectangles are parallelograms with congruent diagonals.
G.GCO.10eA parallelograms is a rhombus if and only if the diagonals are perpendicular.
G.GCO.11Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships.
G.GCO.2Represent translations, reflections, rotations, and dilations of objects in the plane by using paper folding, sketches, coordinates, function notation, and dynamic geometry software, and use various representations to help understand the effects of simple transformations and their compositions.
G.GCO.3Describe rotations and reflections that carry a regular polygon onto itself and identify types of symmetry of polygons, including line, point, rotational, and self-congruence, and use symmetry to analyze mathematical situations.
G.GCO.4Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G.GCO.5Predict and describe the results of transformations on a given figure using geometric terminology from the definitions of the transformations, and describe a sequence of transformations that maps a figure onto its image.
G.GCO.6Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other.
G.GCO.7Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions.
G.GCO.8Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following:
G.GCO.8aVertical angles are congruent.
G.GCO.8bWhen a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary.
G.GCO.8cAny point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment.
G.GCO.8dPerpendicular lines form four right angles.
G.GCO.9Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following:
G.GCO.9aMeasures of interior angles of a triangle sum to 180°.
G.GCO.9bBase angles of isosceles triangles are congruent.
G.GCO.9cThe segment joining midpoints of two sides of a triangle is parallel to the third side and half the length.
G.GCO.9dThe medians of a triangle meet at a point.
G.GGMD G.GGMD
G.GGMD.1Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems.
G.GGMD.2Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle.
G.GGMD.3Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications.
G.GGMD.4Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems.
G.GGPE Expressing Geometric Properties with Equations
G.GGPE.1Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula.
G.GGPE.4Use coordinates to prove simple geometric theorems algebraically.
G.GGPE.5Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope.
G.GGPE.6Given two points, find the point on the line segment between the two points that divides the segment into a given ratio.
G.GGPE.7Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates.
G.GM Modeling
G.GM.1Use geometric shapes, their measures, and their properties to describe real-world objects.
G.GM.2Use geometry concepts and methods to model real-world situations and solve problems using a model.
G.GMDGeometric Measurement and Dimension
G.GSRT Similarity, Right Triangles, and Trigonometry
G.GSRT.1Understand dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G.GSRT.2Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other.
G.GSRT.3Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results.
G.GSRT.4Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following:
G.GSRT.4aA line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion.
G.GSRT.4bIf a line divides two sides of a triangle proportionally, then it is parallel to the third side.
G.GSRT.4cThe square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides.
G.GSRT.5Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
G.GSRT.6Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle.
G.GSRT.7Explain and use the relationship between the sine and cosine of complementary angles.
G.GSRT.8Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem.
G.SPID Interpreting Data
G.SPID.1Select and create an appropriate display, including dot plots, histograms, and box plots, for data that includes only real numbers.
G.SPID.2Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets that include all real numbers.
G.SPID.3Summarize and represent data from a single data set. Interpret differences in shape, center, and spread in the context of the data set, accounting for possible effects of extreme data points (outliers).

Intermediate Algebra

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IA.AAPR Arithmetic with Polynomials and Rational Expressions
IA.AAPR.1Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations.
IA.ACE Creating Equations
IA.ACE.1Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable.
IA.ACE.2Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales.
IA.ACE.4Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines.
IA.AREI Reasoning with Equations and Inequalities
IA.AREI.11Solve an equation of the form f(x)=g(x) graphically by identifying the x-coordinate(s) of the point(s) of intersection of the graphs of y=f(x) and y=g(x).
IA.AREI.2Solve simple rational and radical equations in one variable and understand how extraneous solutions may arise.
IA.AREI.4Solve mathematical and real-world problems involving quadratic equations in one variable. (Note: IA.AREI.4a and 4b are not Graduation Standards.)
IA.AREI.4aUse the method of completing the square to transform any quadratic equation in x into an equation of the form (x−h)^2=k that has the same solutions. Derive the quadratic formula from this form.
IA.AREI.4bSolve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a+bi for real numbers a and b.
IA.ASE Structure and Expressions
IA.ASE.1Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions.
IA.ASE.2Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions.
IA.ASE.3Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (Note: IA.ASE.3b is not a Graduation Standard.)
IA.ASE.3aFind the zeros of a quadratic function by rewriting it in equivalent factored form and explain the connection between the zeros of the function, its linear factors, the x-intercepts of its graph, and the solutions to the corresponding quadratic equation.
IA.ASE.3bDetermine the maximum or minimum value of a quadratic function by completing the square.
IA.FBF Building Functions
IA.FBF.1Write a function that describes a relationship between two quantities. (Note: IA.FBF.1a is not a Graduation Standard.)
IA.FBF.1aWrite a function that models a relationship between two quantities using both explicit expressions and a recursive process and by combining standard forms using addition, subtraction, multiplication and division to build new functions.
IA.FBF.1bCombine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations.
IA.FBF.2Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
IA.FBF.3Describe the effect of the transformations kf(x), f(x)+k, f(x+k), and combinations of such transformations on the graph of y=f(x) for any real number k. Find the value of k given the graphs and write the equation of a transformed parent function given its graph.
IA.FIF Interpreting Functions
IA.FIF.3Define functions recursively and recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
IA.FIF.4Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity.
IA.FIF.5Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes.
IA.FIF.6Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context.
IA.FIF.7Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases.
IA.FIF.8Translate between different but equivalent forms of a function equation to reveal and explain different properties of the function. (Note: IA.FIF.8b is not a Graduation Standard.) b. Interpret expressions for exponential functions by using the properties of exponents.
IA.FIF.9Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal.
IA.FLQE Linear, Quadratic, and Exponential
IA.FLQE.2Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables.
IA.FLQE.5Interpret the parameters in a linear or exponential function in terms of the context.
IA.NCNS Complex Number System
IA.NCNS.1Know there is a complex number i such that i^2=−1, and every complex number has the form a+bi with a and b real.
IA.NCNS.7Solve quadratic equations in one variable that have complex solutions.

Pre-Calculus

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PC.AAPR Arithmetic with Polynomials and Rational Expressions
PC.AAPR.2Know and apply the Division Theorem and the Remainder Theorem for polynomials.
PC.AAPR.3Graph polynomials identifying zeros when suitable factorizations are available and indicating end behavior. Write a polynomial function of least degree corresponding to a given graph.
PC.AAPR.4Prove polynomial identities and use them to describe numerical relationships.
PC.AAPR.5Apply the Binomial Theorem to expand powers of binomials, including those with one and with two variables. Use the Binomial Theorem to factor squares, cubes, and fourth powers of binomials.
PC.AAPR.6Apply algebraic techniques to rewrite simple rational expressions in different forms; using inspection, long division, or, for the more complicated examples, a computer algebra system.
PC.AAPR.7Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
PC.AREI Reasoning with Equations and Inequalities
PC.AREI.11Solve an equation of the form f(x)=g(x) graphically by identifying the x-coordinate(s) of the point(s) of intersection of the graphs of y=f(x) and y=g(x).
PC.AREI.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions.
PC.AREI.8Represent a system of linear equations as a single matrix equation in a vector variable.
PC.AREI.9Using technology for matrices of dimension 3 × 3 or greater, find the inverse of a matrix if it exists and use it to solve systems of linear equations.
PC.ASE Structure and Expressions
PC.ASE.1Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions.
PC.ASE.2Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions.
PC.ASE.4Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems including applications to finance.
PC.FBF Building Functions
PC.FBF.1Write a function that describes a relationship between two quantities.
PC.FBF.1bCombine functions using the operations addition, subtraction, multiplication, and division to build new functions that describe the relationship between two quantities in mathematical and real-world situations.
PC.FBF.3Describe the effect of the transformations kf(x), f(x)+k, f(x+k), and combinations of such transformations on the graph of y=f(x) for any real number k. Find the value of k given the graphs and write the equation of a transformed parent function given its graph.
PC.FBF.4Understand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, as f and g are inverse functions if and only if f(x)=y and g(y)=x, for all values of x in the domain of f and all values of y in the domain of g, and find inverse functions for one-to-one function or by restricting the domain.
PC.FBF.4aUse composition to verify one function is an inverse of another.
PC.FBF.4bIf a function has an inverse, find values of the inverse function from a graph or table.
PC.FBF.5Understand and verify through function composition that exponential and logarithmic functions are inverses of each other and use this relationship to solve problems involving logarithms and exponents.
PC.FIF Interpreting Functions
PC.FIF.4Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity.
PC.FIF.5Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes.
PC.FIF.6Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context.
PC.FIF.7Graph functions from their symbolic representations. Indicate key features including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. Graph simple cases by hand and use technology for complicated cases. (Note: PC.FIF.7a – d are not Graduation Standards.)
PC.FIF.7aGraph rational functions, identifying zeros and asymptotes when suitable factorizations are available and showing end behavior.
PC.FIF.7bGraph radical functions over their domain show end behavior.
PC.FIF.7cGraph exponential and logarithmic functions, showing intercepts and end behavior.
PC.FIF.7dGraph trigonometric functions, showing period, midline, and amplitude.
PC.FLQE Linear, Quadratic, and Exponential
PC.FLQE.4Express a logarithm as the solution to the exponential equation, ab^(ct)=d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
PC.FT Trigonometry
PC.FT.1Understand that the radian measure of an angle is the length of the arc on the unit circle subtended by the angle.
PC.FT.2Define sine and cosine as functions of the radian measure of an angle in terms of the x- and y-coordinates of the point on the unit circle corresponding to that angle and explain how these definitions are extensions of the right triangle definitions.
PC.FT.2aDefine the tangent, cotangent, secant, and cosecant functions as ratios involving sine and cosine.
PC.FT.2bWrite cotangent, secant, and cosecant functions as the reciprocals of tangent, cosine, and sine, respectively.
PC.FT.3Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number.
PC.FT.4Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
PC.FT.5Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
PC.FT.6Define the six inverse trigonometric functions using domain restrictions for regions where the function is always increasing or always decreasing.
PC.FT.7Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
PC.FT.8Justify the Pythagorean, even/odd, and cofunction identities for sine and cosine using their unit circle definitions and symmetries of the unit circle and use the Pythagorean identity to find sin A, cos A, or tan A, given sin A, cos A, or tan A, and the quadrant of the angle.
PC.FT.9Justify the sum and difference formulas for sine, cosine, and tangent and use them to solve problems.
PC.GCI Circles
PC.GCI.5Derive the formulas for the length of an arc and the area of a sector in a circle, and apply these formulas to solve mathematical and real-world problems.
PC.GGPE Expressing Geometric Properties with Equations
PC.GGPE.2Use the geometric definition of a parabola to derive its equation given the focus and directrix.
PC.GGPE.3Use the geometric definition of an ellipse and of a hyperbola to derive the equation of each given the foci and points whose sum or difference of distance from the foci are constant.
PC.GSRT Similarity, Right Triangles, and Trigonometry
PC.GSRT.10Prove the Laws of Sines and Cosines and use them to solve problems.
PC.GSRT.11Use the Law of Sines and the Law of Cosines to solve for unknown measures of sides and angles of triangles that arise in mathematical and real-world problems.
PC.GSRT.9Derive the formula A=(1/2)ab sin C for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
PC.NCNS Complex Number System
PC.NCNS.2Use the relation i^2=−1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
PC.NCNS.3Find the conjugate of a complex number in rectangular and polar forms and use conjugates to find moduli and quotients of complex numbers.
PC.NCNS.4Graph complex numbers on the complex plane in rectangular and polar form and explain why the rectangular and polar forms of a given complex number represent the same number.
PC.NCNS.5Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
PC.NCNS.6Determine the modulus of a complex number by multiplying by its conjugate and determine the distance between two complex numbers by calculating the modulus of their difference.
PC.NCNS.7Solve quadratic equations in one variable that have complex solutions.
PC.NCNS.8Extend polynomial identities to the complex numbers and use DeMoivre’s Theorem to calculate a power of a complex number.
PC.NCNS.9Know the Fundamental Theorem of Algebra and explain why complex roots of polynomials with real coefficients must occur in conjugate pairs.
PC.NVMQ Vector and Matrix Quantities
PC.NVMQ.1Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes.
PC.NVMQ.10Multiply a vector by a matrix of appropriate dimension to produce another vector. Work with matrices as transformations of vectors.
PC.NVMQ.11Apply 2×2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
PC.NVMQ.2Represent and model with vector quantities. Use the coordinates of an initial point and of a terminal point to find the components of a vector.
PC.NVMQ.3Represent and model with vector quantities. Solve problems involving velocity and other quantities that can be represented by vectors.
PC.NVMQ.4Perform operations on vectors.
PC.NVMQ.4aAdd and subtract vectors using components of the vectors and graphically.
PC.NVMQ.4bGiven the magnitude and direction of two vectors, determine the magnitude of their sum and of their difference.
PC.NVMQ.5Multiply a vector by a scalar, representing the multiplication graphically and computing the magnitude of the scalar multiple.
PC.NVMQ.6Use matrices to represent and manipulate data. (Note: This Graduation Standard is covered in Grade 8.)
PC.NVMQ.7Perform operations with matrices of appropriate dimensions including addition, subtraction, and scalar multiplication.
PC.NVMQ.8Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
PC.NVMQ.9Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

Probability and Statistics

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PS.SPCR Conditional Probability and Rules of Probability
PS.SPCR.1Describe events as subsets of a sample space and
PS.SPCR.1aUse Venn diagrams to represent intersections, unions, and complements.
PS.SPCR.1bRelate intersections, unions, and complements to the words and, or, and not.
PS.SPCR.1cRepresent sample spaces for compound events using Venn diagrams.
PS.SPCR.2Use the multiplication rule to calculate probabilities for independent and dependent events. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
PS.SPCR.3Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
PS.SPCR.4Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.
PS.SPCR.5Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
PS.SPCR.6Calculate the conditional probability of an event A given event B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
PS.SPCR.7Apply the Addition Rule and the Multiplication Rule to determine probabilities, including conditional probabilities, and interpret the results in terms of the probability model.
PS.SPCR.8Use permutations and combinations to solve mathematical and real-world problems, including determining probabilities of compound events. Justify the results.
PS.SPID Interpreting Data
PS.SPID.1Select and create an appropriate display, including dot plots, histograms, and box plots, for data that includes only real numbers.
PS.SPID.10Create residual plots and analyze those plots to compare the fit of linear, quadratic, and exponential models to a given data set. Select the appropriate model and use it for interpolation.
PS.SPID.2Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets that include all real numbers.
PS.SPID.3Summarize and represent data from a single data set. Interpret differences in shape, center, and spread in the context of the data set, accounting for possible effects of extreme data points (outliers).
PS.SPID.4Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
PS.SPID.5Analyze bivariate categorical data using two-way tables and identify possible associations between the two categories using marginal, joint, and conditional frequencies.
PS.SPID.6Using technology, create scatterplots and analyze those plots to compare the fit of linear, quadratic, or exponential models to a given data set. Select the appropriate model, fit a function to the data set, and uses the function to solve problems in the context of the data.
PS.SPID.7Find linear models using median fit and regression methods to make predictions. Interpret the slope and intercept of a linear model in the context of the data.
PS.SPID.8Compute using technology and interpret the correlation coefficient of a linear fit.
PS.SPID.9Differentiate between correlation and causation when describing the relationship between two variables. Identify potential lurking variables which may explain an association between two variables.
PS.SPMD Using Probability to Make Decisions
PS.SPMD.1Develop the probability distribution for a random variable defined for a sample space in which a theoretical probability can be calculated and graph the distribution.
PS.SPMD.2Calculate the expected value of a random variable as the mean of its probability distribution. Find expected values by assigning probabilities to payoff values. Use expected values to evaluate and compare strategies in real-world scenarios.
PS.SPMD.3Construct and compare theoretical and experimental probability distributions and use those distributions to find expected values.
PS.SPMD.4Use probability to evaluate outcomes of decisions by finding expected values and determine if decisions are fair.
PS.SPMD.5Use probability to evaluate outcomes of decisions. Use probabilities to make fair decisions.
PS.SPMD.6Analyze decisions and strategies using probability concepts.
PS.SPMJ Making Inferences and Justifying Conclusions
PS.SPMJ.1Understand statistics and sampling distributions as a process for making inferences about population parameters based on a random sample from that population.
PS.SPMJ.2Distinguish between experimental and theoretical probabilities. Collect data on a chance event and use the relative frequency to estimate the theoretical probability of that event. Determine whether a given probability model is consistent with experimental results.
PS.SPMJ.3Plan and conduct a survey to answer a statistical question. Recognize how the plan addresses sampling technique, randomization, measurement of experimental error and methods to reduce bias.
PS.SPMJ.4Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
PS.SPMJ.5Distinguish between experiments and observational studies. Determine which of two or more possible experimental designs will best answer a given research question and justify the choice based on statistical significance.
PS.SPMJ.6Evaluate claims and conclusions in published reports or articles based on data by analyzing study design and the collection, analysis, and display of the data.

AP Calculus Learning Objectives

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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

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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.