Alabama Math Standards - Alabama Mathematics Standards

1027 standards, Kindergarten through AP.

Alabama's Course of Study Mathematics, revised in 2019, includes accelerated pathways in grades 7 and 8 alongside standard progression. High school features distinctive courses like Applications of Finite Mathematics and Mathematical Modeling in addition to traditional sequences.

Kindergarten

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K.1Count forward orally from 0 to 100 by ones and by tens. Count backward orally from 10 to 0 by ones.
K.10Decompose numbers less than or equal to 10 into pairs of smaller numbers in more than one way, by using concrete objects or drawings, and record each decomposition by a drawing or equation.
K.11For any number from 0 to 10, find the number that makes 10 when added to the given number, by using concrete objects or drawings, and record the answer with a drawing or equation.
K.12Fluently add and subtract within 5.
K.13Duplicate and extend simple patterns using concrete objects.
K.14Compose and decompose numbers from 11 to 19 by using concrete objects or drawings to demonstrate understanding that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.
K.15 Classify objects into given categories of 10 or fewer; count the number of objects in each category and sort the categories by count.
K.15.aCategorize data on Venn diagrams, pictographs, and "yes-no" charts using real objects, symbolic representations, or pictorial representations.
K.16Identify and describe measurable attributes (length, weight, height) of a single object using vocabulary such as long/short, heavy/light, or tall/short.
K.17Directly compare two objects with a measurable attribute in common to see which object has “more of” or “less of” the attribute and describe the difference.
K.18Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.
K.19Correctly name shapes regardless of their orientations or overall sizes.
K.2Count to 100 by ones beginning with any given number between 0 and 99.
K.20Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).
K.21Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (number of sides and vertices or “corners”), and other attributes.
K.22Model shapes in the world by building them from sticks, clay balls, or other components and by drawing them.
K.23Use simple shapes to compose larger shapes.
K.3 Write numerals from 0 to 20.
K.3.aRepresent 0 to 20 using concrete objects when given a written numeral from 0 to 20 (with 0 representing a count of no objects).
K.4 Connect counting to cardinality using a variety of concrete objects.
K.4.aSay the number names in consecutive order when counting objects.
K.4.bIndicate that the last number name said tells the number of objects counted in a set.
K.4.cIndicate that the number of objects in a set is the same regardless of their arrangement or the order in which they were counted.
K.4.dExplain that each successive number name refers to a quantity that is one larger.
K.5 Count to answer “how many?” questions.
K.5.aCount using no more than 20 concrete objects arranged in a line, a rectangular array, or a circle.
K.5.bCount using no more than 10 concrete objects in a scattered configuration.
K.5.cDraw the number of objects that matches a given numeral from 0 to 20.
K.6Orally identify whether the number of objects in one group is greater/more than, less/fewer than, or equal/the same as the number of objects in another group, in groups containing up to 10 objects, by using matching, counting, or other strategies.
K.7Compare two numbers between 0 and 10 presented as written numerals (without using inequality symbols).
K.8Represent addition and subtraction up to 10 with concrete objects, fingers, pennies, mental images, drawings, claps or other sounds, acting out situations, verbal explanations, expressions, or equations.
K.9Solve addition and subtraction word problems, and add and subtract within 10, by using concrete objects or drawings to represent the problem.
K.DAData Analysis
K.FCFoundations of Counting
K.GGeometry
K.MMeasurement
K.OATOperations and Algebraic Thinking
K.ONOperations with Numbers

Grade 1

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1.1 Use addition and subtraction to solve word problems within 20 by using concrete objects, drawings, and equations with a symbol for the unknown number to represent the problem.
1.1.aAdd to with change unknown to solve word problems within 20.
1.1.bTake from with change unknown to solve word problems within 20.
1.1.cPut together/take apart with addend unknown to solve word problems within 20.
1.1.dCompare quantities, with difference unknown, bigger unknown, and smaller unknown while solving word problems within 20.
1.10 Extend the number sequence from 0 to 120.
1.10.aCount forward and backward by ones, starting at any number less than 120.
1.10.bRead numerals from 0 to 120.
1.10.cWrite numerals from 0 to 120.
1.10.dRepresent a number of objects from 0 to 120 with a written numeral.
1.11 Explain that the two digits of a two-digit number represent amounts of tens and ones.
1.11.aIdentify a bundle of ten ones as a “ten.”
1.11.bIdentify the numbers from 11 to 19 as composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
1.11.cIdentify the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 as one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
1.12Compare pairs of two-digit numbers based on the values of the tens and ones digits, recording the results of comparisons with the symbols >, =, and < and orally with the words “is greater than,” “is equal to,” and “is less than.”
1.13 Add within 100, using concrete models or drawings and strategies based on place value.
1.13.aAdd a two-digit number and a one-digit number.
1.13.bAdd a two-digit number and a multiple of 10.
1.13.cDemonstrate that in adding two-digit numbers, tens are added to tens, ones are added to ones, and sometimes it is necessary to compose a ten.
1.13.dRelate the strategy for adding a two-digit number and a one-digit number to a written method and explain the reasoning used.
1.14Given a two-digit number, mentally find 10 more or 10 less than the number without having to count, and explain the reasoning used.
1.15Subtract multiples of 10 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Relate the strategy to a written method and explain the reasoning used.
1.16 Organize, represent, and interpret data with up to three categories.
1.16.aAsk and answer questions about the total number of data points in organized data.
1.16.bSummarize data on Venn diagrams, pictographs, and "yes-no" charts using real objects, symbolic representations, or pictorial representations.
1.16.cDetermine “how many” in each category using up to three categories of data.
1.16.dDetermine “how many more” or “how many less” are in one category than in another using data organized into two or three categories.
1.17Order three objects by length; compare the lengths of two objects indirectly by using a third object.
1.18Determine the length of an object using non-standard units with no gaps or overlaps, expressing the length of the object with a whole number.
1.19Tell and write time to the hours and half hours using analog and digital clocks.
1.2Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20 by using concrete objects, drawings, or equations with a symbol for the unknown number to represent the problem.
1.20Identify pennies and dimes by name and value.
1.21 Build and draw shapes which have defining attributes.
1.21.aDistinguish between defining attributes and non-defining attributes.
1.22Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.
1.23 Partition circles and rectangles into two and four equal shares and describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.
1.23.aDescribe “the whole” as two of or four of the shares of circles and rectangles partitioned into two or four equal shares.
1.23.bExplain that decomposing into more equal shares creates smaller shares of circles and rectangles.
1.3Apply properties of operations as strategies to add and subtract.
1.4Explain subtraction as an unknown-addend problem.
1.5Relate counting to addition and subtraction. Example: counting on 2 to add 2
1.6 Add and subtract within 20.
1.6.aDemonstrate fluency with addition and subtraction facts with sums or differences to 10 by counting on.
1.6.bDemonstrate fluency with addition and subtraction facts with sums or differences to 10 by making ten.
1.6.cDemonstrate fluency with addition and subtraction facts with sums or differences to 10 by decomposing a number leading to a ten.
1.6.dDemonstrate fluency with addition and subtraction facts with sums or differences to 10 by using the relationship between addition and subtraction.
1.6.eDemonstrate fluency with addition and subtraction facts with sums or differences to 10 by creating equivalent but easier or known sums.
1.7Explain that the equal sign means “the same as.” Determine whether equations involving addition and subtraction are true or false.
1.8Solve for the unknown whole number in various positions in an addition or subtraction equation, relating three whole numbers that would make it true.
1.9Reproduce, extend, and create patterns and sequences of numbers using a variety of materials.
1.DAData Analysis
1.GGeometry
1.MMeasurement
1.OATOperations and Algebraic Thinking
1.ONBTOperations with Numbers: Base Ten

Grade 2

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2.1Use addition and subtraction within 100 to solve one- and two-step word problems by using drawings and equations with a symbol for the unknown number to represent the problem.
2.10Fluently add and subtract within 100, using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
2.11Use a variety of strategies to add up to four two-digit numbers.
2.12 Add and subtract within 1000 using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method.
2.12.aExplain that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
2.13Mentally add and subtract 10 or 100 to a given number between 100 and 900.
2.14Explain why addition and subtraction strategies work, using place value and the properties of operations.
2.15 Measure lengths of several objects to the nearest whole unit.
2.15.aCreate a line plot where the horizontal scale is marked off in whole-number units to show the lengths of several measured objects.
2.16 Create a picture graph and bar graph to represent data with up to four categories.
2.16.aUsing information presented in a bar graph, solve simple “put-together,” “take-apart,” and “compare” problems.
2.16.bUsing Venn diagrams, pictographs, and "yes-no" charts, analyze data to predict an outcome.
2.17Measure the length of an object by selecting and using standard units of measurement shown on rulers, yardsticks, meter sticks, or measuring tapes.
2.18Measure objects with two different units, and describe how the two measurements relate to each other and the size of the unit chosen.
2.19Estimate lengths using the following standard units of measurement: inches, feet, centimeters, and meters.
2.2 Fluently add and subtract within 20 using mental strategies such as counting on, making ten, decomposing a number leading to ten, using the relationship between addition and subtraction, and creating equivalent but easier or known sums.
2.2.aState automatically all sums of two one-digit numbers.
2.20Measure to determine how much longer one object is than another, expressing the length difference of the two objects using standard units of length.
2.21Use addition and subtraction within 100 to solve word problems involving same units of length, representing the problem with drawings (such as drawings of rulers) and/or equations with a symbol for the unknown number.
2.22Create a number line diagram using whole numbers and use it to represent whole-number sums and differences within 100.
2.23 Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
2.23.aExpress an understanding of common terms such as, but not limited to, quarter past, half past, and quarter to.
2.24 Solve problems with money.
2.24.aIdentify nickels and quarters by name and value.
2.24.bFind the value of a collection of quarters, dimes, nickels, and pennies.
2.24.cSolve word problems by adding and subtracting within one dollar, using the $ and ¢ symbols appropriately (not including decimal notation).
2.25 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
2.25.aRecognize and draw shapes having specified attributes.
2.26Partition a rectangle into rows and columns of same-size squares, and count to find the total number of squares.
2.27 Partition circles and rectangles into two, three, or four equal shares. Describe the shares using such terms as halves, thirds, half of, or a third of, and describe the whole as two halves, three thirds, or four fourths.
2.27.aExplain that equal shares of identical wholes need not have the same shape.
2.3 Use concrete objects to determine whether a group of up to 20 objects is even or odd.
2.3.aWrite an equation to express an even number as a sum of two equal addends.
2.4 Using concrete and pictorial representations and repeated addition, determine the total number of objects in a rectangular array with up to 5 rows and up to 5 columns.
2.4.aWrite an equation to express the total number of objects in a rectangular array with up to 5 rows and up to 5 columns as a sum of equal addends.
2.5Reproduce, extend, create, and describe patterns and sequences using a variety of materials.
2.6 Explain that the three digits of a three-digit number represent amounts of hundreds, tens, and ones.
2.6.aExplain the following three-digit numbers as special cases: 100 can be thought of as a bundle of ten tens, called a “hundred,” and the numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
2.7Count within 1000 by ones, fives, tens, and hundreds.
2.8Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
2.9Compare two three-digit numbers based on the value of the hundreds, tens, and ones digits, recording the results of comparisons with the symbols >, =, and < and orally with the words “is greater than,” “is equal to,” and “is less than.”
2.DAData Analysis
2.GGeometry
2.MMeasurement
2.OATOperations and Algebraic Thinking
2.ONBTOperations with Numbers: Base Ten

Grade 3

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3.1Illustrate the product of two whole numbers as equal groups by identifying the number of groups and the number in each group and represent as a written expression.
3.10Identify the nearest 10 or 100 when rounding whole numbers, using place value understanding.
3.11Use various strategies to add and subtract fluently within 1000.
3.12Use concrete materials and pictorial models based on place value and properties of operations to find the product of a one-digit whole number by a multiple of ten (from 10 to 90).
3.13Demonstrate that a unit fraction represents one part of an area model or length model of a whole that has been equally partitioned; explain that a numerator greater than one indicates the number of unit pieces represented by the fraction.
3.14 Interpret a fraction as a number on the number line; locate or represent fractions on a number line diagram.
3.14.aRepresent a unit fraction (1/b) on a number line by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts as specified by the denominator.
3.14.bRepresent a fraction (a/b) on a number line by marking off a lengths of size (1/b) from zero.
3.15 Explain equivalence and compare fractions by reasoning about their size using visual fraction models and number lines.
3.15.aExpress whole numbers as fractions and recognize fractions that are equivalent to whole numbers.
3.15.bCompare two fractions with the same numerator or with the same denominator by reasoning about their size (recognizing that fractions must refer to the same whole for the comparison to be valid). Record comparisons using < , >, or = and justify conclusions.
3.16 For a given or collected set of data, create a scaled (one-to-many) picture graph and scaled bar graph to represent a data set with several categories.
3.16.aDetermine a simple probability from a context that includes a picture.
3.16.bSolve one- and two-step “how many more” and “how many less” problems using information presented in scaled graphs.
3.17Measure lengths using rulers marked with halves and fourths of an inch to generate data and create a line plot marked off in appropriate units to display the data.
3.18 Tell and write time to the nearest minute; measure time intervals in minutes (within 90 minutes.)
3.18.aSolve real-world problems involving addition and subtraction of time intervals in minutes by representing the problem on a number line diagram.
3.19 Estimate and measure liquid volumes and masses of objects using liters (l), grams (g), and kilograms (kg).
3.19.aUse the four operations to solve one-step word problems involving masses or volumes given in the same metric units.
3.2Illustrate and interpret the quotient of two whole numbers as the number of objects in each group or the number of groups when the whole is partitioned into equal shares.
3.20Find the area of a rectangle with whole number side lengths by tiling without gaps or overlays and counting unit squares.
3.21Count unit squares (square cm, square m, square in, square ft, and improvised or non-standard units) to determine area.
3.22Relate area to the operations of multiplication using real-world problems, concrete materials, mathematical reasoning, and the distributive property.
3.23Decompose rectilinear figures into smaller rectangles to find the area, using concrete materials.
3.24Construct rectangles with the same perimeter and different areas or the same area and different perimeters.
3.25Solve real-world problems involving perimeters of polygons, including finding the perimeter given the side lengths and finding an unknown side length of rectangles.
3.26 Recognize and describe polygons (up to 8 sides), triangles, and quadrilaterals (rhombuses, rectangles, and squares) based on the number of sides and the presence or absence of square corners.
3.26.aDraw examples of quadrilaterals that are and are not rhombuses, rectangles, and squares.
3.3Solve word situations using multiplication and division within 100 involving equal groups, arrays, and measurement quantities; represent the situation using models, drawings, and equations with a symbol for the unknown number.
3.4Determine the unknown whole number in a multiplication or division equation relating three whole numbers.
3.5Develop and apply properties of operations as strategies to multiply and divide.
3.6Use the relationship between multiplication and division to represent division as an equation with an unknown factor.
3.7 Use strategies based on properties and patterns of multiplication to demonstrate fluency with multiplication and division within 100.
3.7.aFluently determine all products obtained by multiplying two one-digit numbers.
3.7.bState automatically all products of two one-digit numbers by the end of third grade.
3.8Determine and justify solutions for two-step word problems using the four operations and write an equation with a letter standing for the unknown quantity. Determine reasonableness of answers using number sense, context, mental computation, and estimation strategies including rounding.
3.9Recognize and explain arithmetic patterns using properties of operations.
3.DAData Analysis
3.GGeometry
3.MMeasurement
3.OATOperations and Algebraic Thinking
3.ONBTOperations with Numbers: Base Ten
3.ONFOperations with Numbers: Fractions

Grade 4

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4.1Interpret and write equations for multiplicative comparisons.
4.10Use place value strategies to fluently add and subtract multi-digit whole numbers and connect strategies to the standard algorithm.
4.11 Find the product of two factors (up to four digits by a one-digit number and two two-digit numbers), using strategies based on place value and the properties of operations.
4.11.aIllustrate and explain the product of two factors using equations, rectangular arrays, and area models.
4.12 Use strategies based on place value, properties of operations, and/or the relationship between multiplication and division to find whole-number quotients and remainders with one-digit divisors and up to four-digit dividends.
4.12.aIllustrate and/or explain quotients using equations, rectangular arrays, and/or area models.
4.13 Using area and length fraction models, explain why one fraction is equivalent to another, taking into account that the number and size of the parts differ even though the two fractions themselves are the same size.
4.13.aApply principles of fraction equivalence to recognize and generate equivalent fractions.
4.14 Compare two fractions with different numerators and different denominators using concrete models, benchmarks (0, ½, 1), common denominators, and/or common numerators, recording the comparisons with symbols >, =, or <, and justifying the conclusions.
4.14.aExplain that comparison of two fractions is valid only when the two fractions refer to the same whole.
4.15 Model and justify decompositions of fractions and explain addition and subtraction of fractions as joining or separating parts referring to the same whole.
4.15.aDecompose a fraction as a sum of unit fractions and as a sum of fractions with the same denominator in more than one way using area models, length models, and equations.
4.15.bAdd and subtract fractions and mixed numbers with like denominators using fraction equivalence, properties of operations, and the relationship between addition and subtraction.
4.15.cSolve word problems involving addition and subtraction of fractions and mixed numbers having like denominators, using drawings, visual fraction models, and equations to represent the problem.
4.16 Apply and extend previous understandings of multiplication to multiply a whole number times a fraction.
4.16.aModel and explain how a non-unit fraction can be represented by a whole number times the unit fraction.
4.16.bExtend previous understanding of multiplication to multiply a whole number times any fraction less than one.
4.16.cSolve word problems involving multiplying a whole number times a fraction using visual fraction models and equations to represent the problem.
4.17 Express, model, and explain the equivalence between fractions with denominators of 10 and 100.
4.17.aUse fraction equivalency to add two fractions with denominators of 10 and 100.
4.18Use models and decimal notation to represent fractions with denominators of 10 and 100.
4.19Use visual models and reasoning to compare two decimals to hundredths (referring to the same whole), recording comparisons using symbols >, =, or <, and justifying the conclusions.
4.2Solve word problems involving multiplicative comparison using drawings and write equations to represent the problem, using a symbol for the unknown number.
4.20 Interpret data in graphs (picture, bar, and line plots) to solve problems using numbers and operations.
4.20.aCreate a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8).
4.20.bSolve problems involving addition and subtraction of fractions using information presented in line plots.
4.21 Select and use an appropriate unit of measurement for a given attribute (length, mass, liquid volume, time) within one system of units: metric - km, m, cm; kg, g, l, ml; customary - lb, oz; time - hr, min, sec.
4.21.aWithin one system of units, express measurements of a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.
4.22 Use the four operations to solve measurement word problems with distance, intervals of time, liquid volume, mass of objects, and money.
4.22.aSolve measurement problems involving simple fractions or decimals.
4.22.bSolve measurement problems that require expressing measurements given in a larger unit in terms of a smaller unit.
4.22.cRepresent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
4.23Apply area and perimeter formulas for rectangles in real-world and mathematical situations.
4.24Identify an angle as a geometric shape formed wherever two rays share a common endpoint.
4.25Use a protractor to measure angles in whole-number degrees and sketch angles of specified measure.
4.26 Decompose an angle into non-overlapping parts to demonstrate that the angle measure of the whole is the sum of the angle measures of the parts.
4.26.aSolve addition and subtraction problems on a diagram to find unknown angles in real-world or mathematical problems.
4.27Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines, and identify these in two-dimensional figures.
4.28 Identify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size.
4.28.aDescribe right triangles as a category, and identify right triangles.
4.29 Define 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.
4.29.aIdentify line-symmetric figures and draw lines of symmetry.
4.3 Determine and justify solutions for multi-step word problems, including problems where remainders must be interpreted.
4.3.aWrite equations to show solutions for multi-step word problems with a letter standing for the unknown quantity.
4.3.bDetermine reasonableness of answers for multi-step word problems, using mental computation and estimation strategies including rounding.
4.4 For whole numbers in the range 1 to 100, find all factor pairs, identifying a number as a multiple of each of its factors.
4.4.aDetermine whether a whole number in the range 1 to 100 is a multiple of a given one-digit number.
4.4.bDetermine whether a whole number in the range 1 to 100 is prime or composite.
4.5Generate and analyze a number or shape pattern that follows a given rule.
4.6Using models and quantitative reasoning, explain that in a multi-digit whole number, a digit in any place represents ten times what it represents in the place to its right.
4.7Read and write multi-digit whole numbers using standard form, word form, and expanded form.
4.8Use place value understanding to compare two multi-digit numbers using >, =, and < symbols.
4.9Round multi-digit whole numbers to any place using place value understanding.

Grade 5

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5.1Write, explain, and evaluate simple numerical expressions involving the four operations to solve up to two-step problems. Include expressions involving parentheses, brackets, or braces, using commutative, associative, and distributive properties.
5.10Add and subtract fractions and mixed numbers with unlike denominators, using fraction equivalence to calculate a sum or difference of fractions or mixed numbers with like denominators.
5.11 Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.
5.11.aModel and interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b)
5.11.bUse visual fraction models, drawings, or equations to represent word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers
5.12 Apply and extend previous understandings of multiplication to find the product of a fraction times a whole number or a fraction times a fraction.
5.12.aUse a visual fraction model (area model, set model, or linear model) to show (a/b) x q and create a story context for this equation to interpret the product as a parts of a partition of q into b equal parts.
5.12.bUse a visual fraction model (area model, set model, or linear model) to show (a/b) x (c/d) and create a story context for this equation to interpret the product.
5.12.cMultiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
5.12.dFind the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths to show that the area is the same as would be found by multiplying the side lengths.
5.13 Interpret multiplication as scaling (resizing).
5.13.aCompare the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
5.13.bExplain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number and relate the principle of fraction equivalence.
5.13.cExplain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number and relate the principle of fraction equivalence.
5.14Model and solve real-world problems involving multiplication of fractions and mixed numbers using visual fraction models, drawings, or equations to represent the problem.
5.15 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
5.15.aSolve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions and illustrate using visual fraction models, drawings, and equations to represent the problem.
5.15.bCreate a story context for a unit fraction divided by a whole number, and use a visual fraction model to show the quotient.
5.15.cCreate a story context for a whole number divided by a unit fraction, and use a visual fraction model to show the quotient.
5.16 Make a line plot to display a data set of measurements in fractions of a unit (12, 14, 18).
5.16.aAdd, subtract, multiply, and divide fractions to solve problems involving information presented in line plots.
5.17Convert among different-sized standard measurement units within a given measurement system and use these conversions in solving multi-step, real-world problems.
5.18 Identify volume as an attribute of solid figures, and measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised (non-standard) units.
5.18.aPack a solid figure without gaps or overlaps using n unit cubes to demonstrate volume as n cubic units.
5.19 Relate volume to the operations of multiplication and addition, and solve real-world and mathematical problems involving volume.
5.19.aUse the associative property of multiplication to find the volume of a right rectangular prism and relate it to packing the prism with unit cubes. Show that the volume can be determined by multiplying the three edge lengths or by multiplying the height by the area of the base.
5.19.bApply the formulas V = l × w × h and V = B × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems.
5.19.cFind volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the two parts, applying this technique to solve real-world problems.
5.2 Generate two numerical patterns using two given rules and complete an input/output table for the data.
5.2.aUse data from an input/output table to identify apparent relationships between corresponding terms.
5.2.bForm ordered pairs from values in an input/output table.
5.2.cGraph ordered pairs from an input/output table on a coordinate plane.
5.20Graph points in the first quadrant of the coordinate plane, and interpret coordinate values of points to represent real-world and mathematical problems.
5.21Classify triangles according to side length (isosceles, equilateral, scalene) and angle measure (acute, obtuse, right, equiangular).
5.22Classify quadrilaterals in a hierarchy based on properties.
5.23Explain that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
5.3 Using models and quantitative reasoning, explain that in a multi-digit number, including decimals, a digit in any place represents ten times what it represents in the place to its right and 1/10 of what it represents in the place to its left.
5.3.aExplain patterns in the number of zeros of the product when multiplying a number by powers of 10, using whole-number exponents to denote powers of 10.
5.3.bExplain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10, using whole-number exponents to denote powers of 10.
5.4 Read, write, and compare decimals to thousandths.
5.4.aRead and write decimals to thousandths using base-ten numerals, number names, and expanded form.
5.4.bCompare two decimals to thousandths based on the meaning of the digits in each place, using >, =, and < to record the results of comparisons.
5.5Use place value understanding to round decimals to thousandths.
5.6Fluently multiply multi-digit whole numbers using the standard algorithm.
5.7Use strategies based on place value, properties of operations, and/or the relationship between multiplication and division to find whole-number quotients and remainders with up to four-digit dividends and two-digit divisors. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
5.8 Add, subtract, multiply, and divide decimals to hundredths using strategies based on place value, properties of operations, and/or the relationships between addition/subtraction and multiplication/division; relate the strategy to a written method, and explain the reasoning used.
5.8.aUse concrete models and drawings to solve problems with decimals to hundredths.
5.8.bSolve problems in a real-world context with decimals to hundredths.
5.9Model and solve real-word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally, and assess the reasonableness of answers.

Grade 6

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6.1Use appropriate notations [a/b, a to b, a:b] to represent a proportional relationship between quantities and use ratio language to describe the relationship between quantities.
6.10 Locate integers and other rational numbers on a horizontal or vertical line diagram.
6.10.aDefine opposites as numbers located on opposite sides of 0 and the same distance from 0 on a number line.
6.10.bUse rational numbers in real-world and mathematical situations, explaining the meaning of 0 in each situation.
6.11 Find the position of pairs of integers and other rational numbers on the coordinate plane.
6.11.aIdentify quadrant locations of ordered pairs on the coordinate plane based on the signs of the x and y coordinates.
6.11.bIdentify (a,b) and (a,-b) as reflections across the x-axis.
6.11.cIdentify (a,b) and (-a,b) as reflections across the y-axis.
6.11.dSolve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane, including finding distances between points with the same first or second coordinate.
6.12Explain the meaning of absolute value and determine the absolute value of rational numbers in real-world contexts.
6.13Compare and order rational numbers and absolute value of rational numbers with and without a number line in order to solve real-world and mathematical problems.
6.14Write, evaluate, and compare expressions involving whole number exponents.
6.15 Write, read, and evaluate expressions in which letters represent numbers in real-world contexts.
6.15.aInterpret a variable as an unknown value for any number in a specified set, depending on the context.
6.15.bWrite expressions to represent verbal statements and real-world scenarios.
6.15.cIdentify parts of an expression using mathematical terms such as sum, term, product, factor, quotient, and coefficient.
6.15.dEvaluate expressions (which may include absolute value and whole number exponents) with respect to order of operations.
6.16Generate equivalent algebraic expressions using the properties of operations, including inverse, identity, commutative, associative, and distributive.
6.17Determine whether two expressions are equivalent and justify the reasoning.
6.18Determine whether a value is a solution to an equation or inequality by using substitution to conclude whether a given value makes the equation or inequality true.
6.19 Write and solve an equation in the form of x+p=q or px=q for cases in which p, q, and x are all non-negative rational numbers to solve real-world and mathematical problems.
6.19.aInterpret the solution of an equation in the context of the problem.
6.2Use unit rates to represent and describe ratio relationships.
6.20 Write and solve inequalities in the form of x>c, x<c, x≥ c, or x ≤ c to represent a constraint or condition in a real-world or mathematical problem.
6.20.aInterpret the solution of an inequality in the context of a problem.
6.20.bRepresent the solutions of inequalities on a number line and explain that the solution set may contain infinitely many solutions.
6.21 Identify, represent, and analyze two quantities that change in relationship to one another in real-world or mathematical situations.
6.21.aUse tables, graphs, and equations to represent the relationship between independent and dependent variables.
6.22Write examples and non-examples of statistical questions, explaining that a statistical question anticipates variability in the data related to the question.
6.23 Calculate, interpret, and compare measures of center (mean, median, mode) and variability (range and interquartile range) in real-world data sets.
6.23.aDetermine which measure of center best represents a real-world data set.
6.23.bInterpret the measures of center and variability in the context of a problem.
6.24 Represent numerical data graphically, using dot plots, line plots, histograms, stem and leaf plots, and box plots.
6.24.aAnalyze the graphical representation of data by describing the center, spread, shape (including approximately symmetric or skewed), and unusual features (including gaps, peaks, clusters, and extreme values).
6.24.bUse graphical representations of real-world data to describe the context from which they were collected.
6.25 Graph polygons in the coordinate plane given coordinates of the vertices to solve real-world and mathematical problems.
6.25.aDetermine missing vertices of a rectangle with the same x-coordinate or the same y-coordinate when graphed in the coordinate plane.
6.25.bUse coordinates to find the length of a side between points having the same x-coordinate or the same y-coordinate.
6.25.cCalculate perimeter and area of a polygon graphed in the coordinate plane (limiting to polygons in which consecutive vertices have the same x-coordinate or the same y-coordinate).
6.26 Calculate the area of triangles, special quadilaterals, and other polygons by composing and decomposing them into known shapes.
6.26.aApply the techniques of composing and decomposing polygons to find area in the context of solving real-world and mathematical problems.
6.27Determine the surface area of three-dimensional figures by representing them with nets composed of rectangles and triangles to solve real-world and mathematical problems.
6.28 Apply previous understanding of volume of right rectangular prisms to those with fractional edge lengths to solve real-world and mathematical problems.
6.28.aUse models (cubes or drawings) and the volume formulas (V = lwh and V = Bh) to find and compare volumes of right rectangular prisms.
6.3Use ratio and rate reasoning to solve mathematical and real-world problems (including but not limited to percent, measurement conversion, and equivalent ratios) using a variety of models, including tables of equivalent ratios, tape diagrams, double number lines, and equations.
6.4 Interpret and compute quotients of fractions using visual models and equations to represent problems.
6.4.aUse quotients of fractions to analyze and solve problems.
6.5Fluently divide multi-digit whole numbers using a standard algorithm to solve real-world and mathematical problems.
6.6Add, subtract, multiply, and divide decimals using a standard algorithm.
6.7Use the distributive property to express the sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor.
6.8 Find the greatest common factor (GCF) and least common multiple (LCM) of two or more whole numbers.
6.8.aUse factors and multiples to determine prime factorization.
6.9Use signed numbers to describe quantities that have opposite directions or values and to represent quantities in real-world contexts.

Grade 7

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7.1Calculate unit rates of length, area, and other quantities measured in like or different units that include ratios or fractions.
7.10 Examine a sample of a population to generalize information about the population.
7.10.aDifferentiate between a sample and a population.
7.10.bCompare sampling techniques to determine whether a sample is random and thus representative of a population, explaining that random sampling tends to produce representative samples and support valid inferences.
7.10.cDetermine whether conclusions and generalizations can be made about a population based on a sample.
7.10.dUse data from a random sample to draw inferences about a population with an unknown characteristic of interest, generating multiple samples to gauge variation and making predictions or conclusions about the population.
7.10.eInformally explain situations in which statistical bias may exist.
7.11Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.
7.12Make informal comparative inferences about two populations using measures of center and variability and/or mean absolute deviation in context.
7.13Use a number from 0 to 1 to represent the probability of a chance event occurring, explaining that larger numbers indicate greater likelihood of the event occurring, while a number near zero indicates an unlikely event.
7.14 Define and develop a probability model, including models that may or may not be uniform, where uniform models assign equal probability to all outcomes and non-uniform models involve events that are not equally likely.
7.14.aCollect and use data to predict probabilities of events.
7.14.bCompare probabilities from a model to observed frequencies, explaining possible sources of discrepancy.
7.15 Approximate the probability of an event using data generated by a simulation (experimental probability) and compare it to the theoretical probability.
7.15.aObserve the relative frequency of an event over the long run, using simulation or technology, and use those results to predict approximate relative frequency.
7.16 Find probabilities of simple and compound events through experimentation or simulation and by analyzing the sample space, representing the probabilities as percents, decimals, or fractions.
7.16.aRepresent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams, and determine the probability of an event by finding the fraction of outcomes in the sample space for which the compound event occurred.
7.16.bDesign and use a simulation to generate frequencies for compound events.
7.16.cRepresent events described in everyday language in terms of outcomes in the sample space which composed the event.
7.17Solve problems involving scale drawings of geometric figures, including computation of actual lengths and areas from a scale drawing and reproduction of a scale drawing at a different scale.
7.18Construct geometric shapes (freehand, using a ruler and a protractor, and using technology), given a written description or measurement constraints with an emphasis on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
7.19Describe the two-dimensional figures created by slicing three-dimensional figures into plane sections.
7.2 Represent a relationship between two quantities and determine whether the two quantities are related proportionally.
7.2.aUse equivalent ratios displayed in a table or in a graph of the relationship in the coordinate plane to determine whether a relationship between two quantities is proportional.
7.2.bIdentify the constant of proportionality (unit rate) and express the proportional relationship using multiple representations including tables, graphs, equations, diagrams, and verbal descriptions.
7.2.cExplain in context the meaning of a point (x,y) on the graph of a proportional relationship, with special attention to the points (0,0) and (1, r) where r is the unit rate.
7.20 Explain the relationships among circumference, diameter, area, and radius of a circle to demonstrate understanding of formulas for the area and circumference of a circle.
7.20.aInformally derive the formula for area of a circle.
7.20.bSolve area and circumference problems in real-world and mathematical situations involving circles.
7.21Use facts about supplementary, complementary, vertical, and adjacent angles in multi-step problems to write and solve simple equations for an unknown angle in a figure.
7.22Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right rectangular prisms.
7.3Solve multi-step percent problems in context using proportional reasoning, including simple interest, tax, gratuities, commissions, fees, markups and markdowns, percent increase, and percent decrease.
7.4 Apply and extend knowledge of operations of whole numbers, fractions, and decimals to add, subtract, multiply, and divide rational numbers including integers, signed fractions, and decimals.
7.4.aIdentify and explain situations where the sum of opposite quantities is 0 and opposite quantities are defined as additive inverses.
7.4.bInterpret the sum of two or more rational numbers, by using a number line and in real-world contexts.
7.4.cExplain subtraction of rational numbers as addition of additive inverses.
7.4.dUse a number line to demonstrate that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
7.4.eExtend strategies of multiplication to rational numbers to develop rules for multiplying signed numbers, showing that the properties of the operations are preserved.
7.4.fDivide integers and explain that division by zero is undefined. Interpret the quotient of integers (with a non-zero divisor) as a rational number.
7.4.gConvert a rational number to a decimal using long division, explaining that the decimal form of a rational number terminates or eventually repeats.
7.5Solve real-world and mathematical problems involving the four operations of rational numbers, including complex fractions. Apply properties of operations as strategies where applicable.
7.6Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
7.7Generate expressions in equivalent forms based on context and explain how the quantities are related.
7.8Solve multi-step real-world and mathematical problems involving rational numbers (integers, signed fractions and decimals), converting between forms as needed. Assess the reasonableness of answers using mental computation and estimation strategies.
7.9 Use variables to represent quantities in real-world or mathematical problems and construct algebraic expressions, equations, and inequalities to solve problems by reasoning about the quantities.
7.9.aSolve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.
7.9.bSolve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality, and interpret it in the context of the problem.

Grade 8

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8.1 Define the real number system as composed of rational and irrational numbers.
8.1.aExplain that every number has a decimal expansion; for rational numbers, the decimal expansion repeats or terminates.
8.1.bConvert a decimal expansion that repeats into a rational number.
8.10Compare proportional and non-proportional linear relationships represented in different ways (algebraically, graphically, numerically in tables, or by verbal descriptions) to solve real-world problems.
8.11 Solve multi-step linear equations in one variable, including rational number coefficients, and equations that require using the distributive property and combining like terms.
8.11.aDetermine whether linear equations in one variable have one solution, no solution, or infinitely many solutions of the form x = a, a = a, or a = b (where a and b are different numbers).
8.11.bRepresent and solve real-world and mathematical problems with equations and interpret each solution in the context of the problem.
8.12 Solve systems of two linear equations in two variables by graphing and substitution.
8.12.aExplain that the solution(s) of systems of two linear equations in two variables corresponds to points of intersection on their graphs because points of intersection satisfy both equations simultaneously.
8.12.bInterpret and justify the results of systems of two linear equations in two variables (one solution, no solution, or infinitely many solutions) when applied to real-world and mathematical problems.
8.13Determine whether a relation is a function, defining a function as a rule that assigns to each input (independent value) exactly one output (dependent value), and given a graph, table, mapping, or set of ordered pairs.
8.14Evaluate functions defined by a rule or an equation, given values for the independent variable.
8.15 Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.
8.15.aDistinguish between linear and non-linear functions.
8.16 Construct a function to model a linear relationship between two variables.
8.16.aInterpret the rate of change (slope) and initial value of the linear function from a description of a relationship or from two points in a table or graph.
8.17Analyze the relationship (increasing or decreasing, linear or non-linear) between two quantities represented in a graph.
8.18Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities, describing patterns in terms of positive, negative, or no association, linear and non-linear association, clustering, and outliers.
8.19Given a scatter plot that suggests a linear association, informally draw a line to fit the data, and assess the model fit by judging the closeness of the data points to the line.
8.2Locate rational approximations of irrational numbers on a number line, compare their sizes, and estimate the values of the irrational numbers.
8.20 Use a linear model of a real-world situation to solve problems and make predictions.
8.20.aDescribe the rate of change and y-intercept in the context of a problem using a linear model of a real-world situation.
8.21Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects, using relative frequencies calculated for rows or columns to describe possible associations between the two variables.
8.22 Verify experimentally the properties of rigid motions (rotations, reflections, and translations): lines are taken to lines, and line segments are taken to line segments of the same length; angles are taken to angles of the same measure; and parallel lines are taken to parallel lines.
8.22.aGiven a pair of two-dimensional figures, determine if a series of rigid motions maps one figure onto the other, recognizing that if such a sequence exists the figures are congruent; describe the transformation sequence that verifies a congruence relationship.
8.23Use coordinates to describe the effect of transformations (dilations, translations, rotations, and reflections) on two-dimensional figures.
8.24Given a pair of two-dimensional figures, determine if a series of dilations and rigid motions maps one figure onto the other, recognizing that if such a sequence exists the figures are similar; describe the transformation sequence that exhibits the similarity between them.
8.25 Analyze and apply properties of parallel lines cut by a transversal to determine missing angle measures.
8.25.aUse informal arguments to establish that the sum of the interior angles of a triangle is 180 degrees.
8.26Informally justify the Pythagorean Theorem and its converse.
8.27Apply the Pythagorean Theorem to find the distance between two points in a coordinate plane.
8.28Apply the Pythagorean Theorem to determine unknown side lengths of right triangles, including real-world applications.
8.29Informally derive the formulas for the volume of cones and spheres by experimentally comparing the volumes of cones and spheres with the same radius and height to a cylinder with the same dimensions.
8.3Develop and apply properties of integer exponents to generate equivalent numerical and algebraic expressions.
8.30Use formulas to calculate the volumes of three-dimensional figures (cylinders, cones, and spheres) to solve real-world problems.
8.4 Use square root and cube root symbols to represent solutions to equations.
8.4.aEvaluate square roots of perfect squares (less than or equal to 225) and cube roots of perfect cubes (less than or equal to 1000).
8.4.bExplain that the square root of a non-perfect square is irrational.
8.5Estimate and compare very large or very small numbers in scientific notation.
8.6 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.
8.6.aUse scientific notation and choose units of appropriate size for measurements of very large or very small quantities.
8.6.bInterpret scientific notation that has been generated by technology.
8.7Determine whether a relationship between two variables is proportional or non-proportional.
8.8 Graph proportional relationships.
8.8.aInterpret the unit rate of a proportional relationship, describing the constant of proportionality as the slope of the graph which goes through the origin and has the equation y = mx where m is the slope.
8.9 Interpret y = mx + b as defining a linear equation whose graph is a line with m as the slope and b as the y-intercept.
8.9.aUse similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in a coordinate plane.
8.9.bGiven two distinct points in a coordinate plane, find the slope of the line containing the two points and explain why it will be the same for any two distinct points on the line.
8.9.cGraph linear relationships, interpreting the slope as the rate of change of the graph and the y-intercept as the initial value.
8.9.dGiven that the slopes for two different sets of points are equal, demonstrate that the linear equations that include those two sets of points may have different y-intercepts.

Grade 7 Accelerated

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7A.1Calculate unit rates of length, area, and other quantities measured in like or different units that include ratios or fractions. [Grade 7, 1]
7A.10 Define the real number system as composed of rational and irrational numbers.
7A.10.aExplain that every number has a decimal expansion; for rational numbers, the decimal expansion repeats in a pattern or terminates.
7A.10.bConvert a decimal expansion that repeats in a pattern into a rational number. [Grade 8, 1]
7A.11Locate rational approximations of irrational numbers on a number line, compare their sizes, and estimate the values of irrational numbers. [Grade 8, 2]
7A.12Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. [Grade 7, 6]
7A.13Generate expressions in equivalent forms based on context and explain how the quantities are related. [Grade 7, 7]
7A.14Develop and apply properties of integer exponents to generate equivalent numerical and algebraic expressions. [Grade 8, 3]
7A.15 Use square root and cube root symbols to represent solutions to equations.
7A.15.aEvaluate square roots of perfect squares (less than or equal to 225) and cube roots of perfect cubes (less than or equal to 1000).
7A.15.bExplain that the square root of a non-perfect square is irrational. [Grade 8, 4]
7A.16 Express and compare very large or very small numbers in scientific notation. [Grade 8, 5]
7A.16.aPerform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used, expressing answers in scientific notation. [Grade 8, 6]
7A.16.bUse scientific notation and choose units of appropriate size for measurements of very large or very small quantities. [Grade 8, 6a]
7A.16.cInterpret scientific notation that has been generated by technology. [Grade 8, 6b]
7A.17Solve multi-step real-world and mathematical problems involving rational numbers (integers, signed fractions, and decimals), converting between forms as needed. Assess the reasonableness of answers using mental computation and estimation strategies. [Grade 7, 8]
7A.18 Use variables to represent quantities in a real-world or mathematical problem and construct algebraic expressions, equations, and inequalities to solve problems by reasoning about the quantities.
7A.18.aSolve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.
7A.18.bSolve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. [Grade 7, 9, and linear portion of Algebra I with Probability, 11]
7A.19Create equations in two variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear functions. [Algebra I with Probability, 12 partial]
7A.2 Represent a relationship between two quantities and determine whether the two quantities are related proportionally.
7A.2.aUse equivalent ratios displayed in a table or in a graph of the relationship in the coordinate plane to determine whether a relationship between two quantities is proportional.
7A.2.bIdentify the constant of proportionality (unit rate) and express the proportional relationship using multiple representations including tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
7A.2.cExplain in context the meaning of a point (x,y) on the graph of a proportional relationship, with special attention to the points (0,0) and (1, r) where r is the unit rate. [Grade 7, 2]
7A.20Represent constraints by equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Limit to contexts arising from linear. [Algebra I with Probability, 13 partial]
7A.21 Solve multi-step linear equations in one variable, including rational number coefficients, and equations that require using the distributive property and combining like terms.
7A.21.aDetermine whether linear equations in one variable have one solution, no solution, or infinitely many solutions of the form x = a, a = a, or a = b (where a and b are different numbers).
7A.21.bRepresent and solve real-world and mathematical problems with equations and interpret each solution in the context of the problem. [Grade 8, 11]
7A.22Identify the effect on the graph of replacing f (x) by f (x) + k, k · f(x), f (kx), and f (x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and explain the effects on the graph using technology, where appropriate. Limit to linear functions. [Algebra I with Probability, 23]
7A.23 Construct a function to model the linear relationship between two variables.
7A.23.aInterpret the rate of change (slope) and initial value of the linear function from a description of a relationship from two points in a table or graph. [Grade 8, 16]
7A.24Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x). Limit to linear equations. [Algebra I with Probability, 19]
7A.25Find approximate solutions by graphing the functions, making tables of values, or finding successive approximations, using technology where appropriate. Note: Include cases where f(x) is linear and g(x) is constant or linear. [Algebra I with Probability, 19 edited]
7A.26 Examine a sample of a population to generalize information about the population.
7A.26.aDifferentiate between a sample and a population.
7A.26.bCompare sampling techniques to determine whether a sample is random and thus representative of a population, explaining that random sampling tends to produce representative samples and support valid inferences.
7A.26.cDetermine whether conclusions and generalizations can be made about a population based on a sample.
7A.26.dUse data from a random sample to draw inferences about a population with an unknown characteristic of interest, generating multiple samples to gauge variation and make predictions or conclusions about the population.
7A.26.eInformally explain situations in which statistical bias may exist. [Grade 7, 10]
7A.27Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. [Grade 7, 11]
7A.28Make informal comparative inferences about two populations using measures of center and variability and/or mean absolute deviation in context. [Grade 7, 12]
7A.29Use a number between 0 and 1 to represent the probability of a chance event occurring, explaining that larger numbers indicate greater likelihood of the event occurring, while a number near zero indicates an unlikely event. [Grade 7, 13]
7A.3Solve multi-step percent problems in context using proportional reasoning, including simple interest, tax, gratuities, commissions, fees, markups and markdowns, percent increase, and percent decrease. [Grade 7, 3]
7A.30 Define and develop a probability model, including models that may or may not be uniform, where uniform models assign equal probability to all outcomes and non-uniform models involve events that are not equally likely.
7A.30.aCollect and use data to predict probabilities of events.
7A.30.bCompare probabilities from a model to observe frequencies, explaining possible sources of discrepancy. [Grade 7, 14]
7A.31 Approximate the probability of an event by using data generated by a simulation (experimental probability) and compare it to theoretical probability.
7A.31.aObserve the relative frequency of an event over the long run, using simulation or technology, and use those results to predict approximate relative frequency. [Grade 7, 15]
7A.32 Find probabilities of simple and compound events through experimentation or simulation and by analyzing the sample space, representing the probabilities as percents, decimals, or fractions.
7A.32.aRepresent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams, and determine the probability of an event by finding the fraction of outcomes in the sample space for which the compound event occurred.
7A.32.bDesign and use a simulation to generate frequencies for compound events.
7A.32.cRepresent events described in everyday language in terms of outcomes in the sample space which composed the event. [Grade 7, 16]
7A.33Solve problems involving scale drawings of geometric figures including computation of actual lengths and areas from a scale drawing and reproduction of a scale drawing at a different scale. [Grade 7, 17]
7A.34Construct geometric shapes (freehand, using a ruler and a protractor, and using technology) given measurement constraints with an emphasis on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. [Grade 7, 18]
7A.35Describe the two-dimensional figures created by slicing three-dimensional figures into plane sections. [Grade 7, 19]
7A.36 Explain the relationships among circumference, diameter, area, and radius of a circle to demonstrate understanding of formulas for the area and circumference of a circle.
7A.36.aInformally derive the formula for area of a circle.
7A.36.bSolve area and circumference problems in real-world and mathematical situations involving circles. [Grade 7, 20]
7A.37Use facts about supplementary, complementary, vertical, and adjacent angles in multi-step problems to write and solve simple equations for an unknown angle in a figure. [Grade 7, 21]
7A.38 Analyze and apply properties of parallel lines cut by a transversal to determine missing angle measures.
7A.38.aUse informal arguments to establish that the sum of the interior angles of a triangle is 180 degrees. [Grade 8, 25]
7A.39Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right rectangular prisms. [Grade 7, 22]
7A.4Determine whether a relationship between two variables is proportional or non-proportional. [Grade 8, 7]
7A.40Informally derive the formulas for the volume of cones and spheres by experimentally comparing the volumes of cones and spheres with the same radius and height to a cylinder with the same dimensions. [Grade 8, 29]
7A.41Use formulas to calculate the volumes of three-dimensional figures to solve real-world problems. [Grade 8, 30]
7A.42 Verify experimentally the properties of rigid motions (rotations, reflections, and translations): lines are taken to lines, and line segments are taken to line segments of the same length; angles are taken to angles of the same measure; and parallel lines are taken to parallel lines.
7A.42.aGiven a pair of two-dimensional figures, determine if a series of rigid motions maps one figure onto the other, recognizing that if such a sequence exists the figures are congruent; describe the transformation sequence that verifies a congruence relationship. [Grade 8, 22]
7A.43Use coordinates to describe the effect of transformations (dilations, translations, rotations, and reflections) on two-dimensional figures. [Grade 8, 23]
7A.44Given a pair of two-dimensional figures, determine if a series of dilations and rigid motions maps one figure onto the other, recognizing that if such a sequence exists the figures are similar; describe the transformation sequence that exhibits the similarity between them. [Grade 8, 24]
7A.5 Graph proportional relationships.
7A.5.aInterpret the unit rate of a proportional relationship, describing the constant of proportionality as the slope of the graph which goes through the origin and has the equation y = mx where m is the slope. [Grade 8, 8]
7A.6 Interpret y = mx + b as defining a linear equation whose graph is a line with m as the slope and b as the y-intercept.
7A.6.aUse similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in a coordinate plane.
7A.6.bGiven two distinct points in a coordinate plane, find the slope of the line containing the two points and explain why it will be the same for any two distinct points on the line.
7A.6.cGraph linear relationships, interpreting the slope as the rate of change of the graph and the y-intercept as the initial value.
7A.6.dGiven that the slopes for two different sets of points are equal, demonstrate that the linear equations that include those two sets of points may have different y-intercepts. [Grade 8, 9]
7A.7Compare proportional and non-proportional linear relationships represented in different ways (algebraically, graphically, numerically in tables, or by verbal descriptions) to solve real-world problems. [Grade 8, 10]
7A.8 Apply and extend knowledge of operations of whole numbers, fractions, and decimals to add, subtract, multiply, and divide rational numbers including integers, signed fractions, and decimals.
7A.8.aIdentify and explain situations where the sum of opposite quantities is 0 and opposite quantities are defined as additive inverses.
7A.8.bInterpret the sum of two or more rational numbers, by using a number line and in real-world contexts.
7A.8.cExplain subtraction of rational numbers as addition of additive inverses.
7A.8.dUse a number line to demonstrate that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
7A.8.eExtend strategies of multiplication to rational numbers to develop rules for multiplying signed numbers, showing that the properties of the operations are preserved.
7A.8.fDivide integers and explain that division by zero is undefined. Interpret the quotient of integers (with a non-zero divisor) as a rational number.
7A.8.gConvert a rational number to a decimal using long division, explaining that the decimal form of a rational number terminates or eventually repeats. [Grade 7, 4]
7A.9Solve real-world and mathematical problems involving the four operations of rational numbers, including complex fractions. Apply properties of operations as strategies where applicable. [Grade 7, 5]

Grade 8 Accelerated

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8A.1Explain how the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for an additional notation for radicals in terms of rational exponents. [Algebra I with Probability, 1]
8A.10Explain why extraneous solutions to an equation involving absolute values may arise and how to check to be sure that a candidate solution satisfies an equation. [Algebra I with Probability, 8]
8A.11 Select an appropriate method to solve a quadratic equation in one variable.
8A.11.aUse the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)^2 = q that has the same solutions. Explain how the quadratic formula is derived from this form.
8A.11.bSolve quadratic equations by inspection (such as x^2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real. [Algebra I with Probability, 9]
8A.12 Select an appropriate method to solve a system of two linear equations in two variables.
8A.12.aSolve a system of two equations in two variables by using linear combinations; contrast situations in which use of linear combinations is more efficient with those in which substitution is more efficient.
8A.12.bContrast solutions to a system of two linear equations in two variables produced by algebraic methods with graphical and tabular methods. [Algebra I with Probability, 10]
8A.13Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately. Extend from contexts arising from linear functions to those involving quadratic, exponential, and absolute value functions. [Algebra I with Probability, 11]
8A.14Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions. [Algebra I with Probability, 12]
8A.15Represent constraints by equations and/or inequalities, and solve systems of equations and/or inequalities, interpreting solutions as viable or nonviable options in a modeling context. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions. [Algebra I with Probability, 13]
8A.16 Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range. [Grade 8, 13, edited for added content]
8A.16.aUse function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. [Grade 8, 14, edited for added content]
8A.16.bRelate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Limit to linear, quadratic, exponential, and absolute value functions. [Algebra I with Probability, 15]
8A.17Given a relation defined by an equation in two variables, identify the graph of the relation as the set of all its solutions plotted in the coordinate plane. [Algebra I with Probability, 14]
8A.18Compare and contrast relations and functions represented by equations, graphs, or tables that show related values; determine whether a relation is a function. Identify that a function f is a special kind of relation defined by the equation y = f(x). [Algebra I with Probability, 16]
8A.19 Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.
8A.19.aUse arithmetic operations to combine different types of standard functions to write and evaluate functions.
8A.19.bUse function composition to combine different types of standard functions to write and evaluate functions. [Algebra I with Probability, 17]
8A.2Rewrite expressions involving radicals and rational exponents using the properties of exponents. [Algebra I with Probability, 2]
8A.20 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x).
8A.20.aFind the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate. [Algebra I with Probability, 19] Note: Include cases where f(x) is linear, quadratic, exponential, or absolute value functions and g(x) is constant or linear.
8A.21Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes, using technology where appropriate. [Algebra I with Probability, 20]
8A.22Solve systems consisting of linear and/or quadratic equations in two variables graphically, using technology where appropriate. [Algebra I with Probability, 18]
8A.23 Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Include linear, quadratic, exponential, absolute value, and linear piecewise. [Algebra I with Probability, 21, edited]
8A.23.aDistinguish between linear and non-linear functions. [Grade 8, 15a]
8A.24 Define sequences as functions, including recursive definitions, whose domain is a subset of the integers.
8A.24.aWrite explicit and recursive formulas for arithmetic and geometric sequences and connect them to linear and exponential functions. [Algebra I with Probability, 22]
8A.25Identify the effect on the graph of replacing f (x) by f (x) + k, k·f (x), f (kx), and f (x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and explain the effects on the graph, using technology as appropriate. Extend from linear to quadratic, exponential, absolute value, and linear piecewise functions. [Algebra I with Probability, 23, edited]
8A.26 Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.
8A.26.aShow that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.
8A.26.bDefine linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.
8A.26.cDefine exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. [Algebra I with Probability, 24]
8A.27Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). [Algebra I with Probability, 25]
8A.28Use graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically. [Algebra I with Probability, 26]
8A.29Interpret the parameters of functions in terms of a context. Extend from linear functions, written in the form mx + b, to exponential functions, written in the form ab^x. [Algebra I with Probability, 27]
8A.3Define the imaginary number i such that i^2 = -1. [Algebra I with Probability, 3]
8A.30For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Note: Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries; and end behavior. Extend from relationships that can be represented by linear functions to quadratic, exponential, absolute value, and general piecewise functions. [Algebra I with Probability, 28]
8A.31Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Limit to linear, quadratic, exponential, and absolute value functions. [Algebra I with Probability, 29]
8A.32 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
8A.32.aGraph linear and quadratic functions and show intercepts, maxima, and minima.
8A.32.bGraph piecewise-defined functions, including step functions and absolute value functions.
8A.32.cGraph exponential functions, showing intercepts and end behavior. [Algebra I with Probability, 30]
8A.33Use the mathematical modeling cycle to solve real-world problems involving linear, quadratic, exponential, absolute value, and linear piecewise functions. [Algebra I with Probability, 31]
8A.34Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities, describing patterns in terms of positive, negative, or no association, linear and non-linear association, clustering, and outliers. [Grade 8, 18]
8A.35Given a scatter plot that suggests a linear association, informally draw a line to fit the data, and assess the model fit by judging the closeness of the data points to the line. [Grade 8, 19]
8A.36 Use a linear model of a real-world situation to solve problems and make predictions.
8A.36.aDescribe the rate of change and y-intercept in the context of a problem using a linear model of a real-world situation. [Grade 8, 20]
8A.37Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects, using relative frequencies calculated for rows or columns to describe possible associations between the two variables. [Grade 8, 21]
8A.38Distinguish between quantitative and categorical data and between the techniques that may be used for analyzing data of these two types. [Algebra I with Probability, 34]
8A.39 Analyze the possible association between two categorical variables.
8A.39.aSummarize categorical data for two categories in two-way frequency tables and represent using segmented bar graphs.
8A.39.bInterpret relative frequencies in the context of categorical data (including joint, marginal, and conditional relative frequencies).
8A.39.cIdentify possible associations and trends in categorical data. [Algebra I with Probability, 35]
8A.4Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity. [Algebra I with Probability, 4]
8A.40 Generate a two-way categorical table in order to find and evaluate solutions to real-world problems.
8A.40.aAggregate data from several groups to find an overall association between two categorical variables.
8A.40.bRecognize and explore situations where the association between two categorical variables is reversed when a third variable is considered (Simpson’s Paradox). [Algebra I with Probability, 36]
8A.41Use mathematical and statistical reasoning with bivariate categorical data in order to draw conclusions and assess risk. [Algebra I with Probability, 32]
8A.42Design and carry out an investigation to determine whether there appears to be an association between two categorical variables, and write a persuasive argument based on the results of the investigation. [Algebra I with Probability, 33]
8A.43Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not"). [Algebra I with Probability, 37]
8A.44Explain whether two events, A and B, are independent, using two-way tables or tree diagrams. [Algebra I with Probability, 38]
8A.45Compute the conditional probability of event A given event B, using two-way tables or tree diagrams. [Algebra I with Probability, 39]
8A.46Recognize and describe the concepts of conditional probability and independence in everyday situations and explain them using everyday language. [Algebra I with Probability, 40]
8A.47Explain why the conditional probability of A given B is the fraction of B's outcomes that also belong to A, and interpret the answer in context. [Algebra I with Probability, 41]
8A.48Informally justify the Pythagorean Theorem and its converse. [Grade 8, 26]
8A.49Apply the Pythagorean Theorem to find the distance between two points in a coordinate plane. [Grade 8, 27]
8A.5Use the structure of an expression to identify ways to rewrite it. [Algebra I with Probability, 5]
8A.50Apply the Pythagorean Theorem to determine unknown side lengths of right triangles, including real-world applications. [Grade 8, 28]
8A.6 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
8A.6.aFactor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.
8A.6.bUse the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.
8A.6.cUse the properties of exponents to transform expressions for exponential functions. [Algebra I with Probability, 6]
8A.7Add, subtract, and multiply polynomials, showing that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication. [Algebra I with Probability, 7]
8A.8Analyze the relationship (increasing or decreasing, linear or non-linear) between two quantities represented in a graph. [Grade 8, 17]
8A.9 Solve systems of two linear equations in two variables by graphing and substitution.
8A.9.aExplain that the solution(s) of systems of two linear equations in two variables corresponds to points of intersection on their graphs because points of intersection satisfy both equations simultaneously.
8A.9.bInterpret and justify the results of systems of two linear equations in two variables (one solution, no solution, or infinitely many solutions) when applied to real-world and mathematical problems. [Grade 8, 12]

Algebra I with Probability

Print
A1.1Explain how the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for an additional notation for radicals using rational exponents.
A1.10 Select an appropriate method to solve a system of two linear equations in two variables.
A1.10.aSolve a system of two equations in two variables by using linear combinations; contrast situations in which use of linear combinations is more efficient with those in which substitution is more efficient.
A1.10.bContrast solutions to a system of two linear equations in two variables produced by algebraic methods with graphical and tabular methods.
A1.11Create equations and inequalities in one variable and use them to solve problems in context, either exactly or approximately. Extend from contexts arising from linear functions to those involving quadratic, exponential, and absolute value functions.
A1.12Create equations in two or more variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.
A1.13Represent constraints by equations and/or inequalities, and solve systems of equations and/or inequalities, interpreting solutions as viable or nonviable options in a modeling context. Limit to contexts arising from linear, quadratic, exponential, absolute value, and linear piecewise functions.
A1.14Given a relation defined by an equation in two variables, identify the graph of the relation as the set of all its solutions plotted in the coordinate plane. Note: The graph of a relation often forms a curve (which could be a line).
A1.15 Define a function as a mapping from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range.
A1.15.aUse function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Note: If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x.
A1.15.bRelate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Limit to linear, quadratic, exponential, and absolute value functions.
A1.16Compare and contrast relations and functions represented by equations, graphs, or tables that show related values; determine whether a relation is a function. Explain that a function f is a special kind of relation defined by the equation y = f(x).
A1.17 Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.
A1.17.aUse arithmetic operations to combine different types of standard functions to write and evaluate functions.
A1.17.bUse function composition to combine different types of standard functions to write and evaluate functions.
A1.18Solve systems consisting of linear and/or quadratic equations in two variables graphically, using technology where appropriate.
A1.19 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x).
A1.19.aFind the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate. Note: Include cases where f(x) is a linear, quadratic, exponential, or absolute value function and g(x) is constant or linear.
A1.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.
A1.20Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes, using technology where appropriate.
A1.21Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Extend from linear to quadratic, exponential, absolute value, and general piecewise.
A1.22 Define sequences as functions, including recursive definitions, whose domain is a subset of the integers.
A1.22.aWrite explicit and recursive formulas for arithmetic and geometric sequences and connect them to linear and exponential functions.
A1.23Identify the effect on the graph of replacing f(x) by f(x) + k, k ∙ f(x), f(k ∙ x), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and explain the effects on the graph, using technology as appropriate. Limit to linear, quadratic, exponential, absolute value, and linear piecewise functions.
A1.24 Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.
A1.24.aShow that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.
A1.24.bDefine linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.
A1.24.cDefine exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
A1.25Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
A1.26Use graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.
A1.27Interpret the parameters of functions in terms of a context. Extend from linear functions, written in the form mx + b, to exponential functions, written in the form ab^x.
A1.28For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Note: Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries; and end behavior. Extend from relationships that can be represented by linear functions to quadratic, exponential, absolute value, and linear piecewise functions.
A1.29Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Limit to linear, quadratic, exponential, and absolute value functions.
A1.3Define the imaginary number i such that i^2 = -1.
A1.30 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
A1.30.aGraph linear and quadratic functions and show intercepts, maxima, and minima.
A1.30.bGraph piecewise-defined functions, including step functions and absolute value functions.
A1.30.cGraph exponential functions, showing intercepts and end behavior.
A1.31Use the mathematical modeling cycle to solve real-world problems involving linear, quadratic, exponential, absolute value, and linear piecewise functions.
A1.32Use mathematical and statistical reasoning with bivariate categorical data in order to draw conclusions and assess risk.
A1.33Design and carry out an investigation to determine whether there appears to be an association between two categorical variables, and write a persuasive argument based on the results of the investigation.
A1.34Distinguish between quantitative and categorical data and between the techniques that may be used for analyzing data of these two types.
A1.35 Analyze the possible association between two categorical variables.
A1.35.aSummarize categorical data for two categories in two-way frequency tables and represent using segmented bar graphs.
A1.35.bInterpret relative frequencies in the context of categorical data (including joint, marginal, and conditional relative frequencies).
A1.35.cIdentify possible associations and trends in categorical data.
A1.36 Generate a two-way categorical table in order to find and evaluate solutions to real-world problems.
A1.36.aAggregate data from several groups to find an overall association between two categorical variables.
A1.36.bRecognize and explore situations where the association between two categorical variables is reversed when a third variable is considered (Simpson's Paradox).
A1.37Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
A1.38Explain whether two events, A and B, are independent, using two-way tables or tree diagrams.
A1.39Compute the conditional probability of event A given event B, using two-way tables or tree diagrams.
A1.4Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.
A1.40Recognize and describe the concepts of conditional probability and independence in everyday situations and explain them using everyday language.
A1.41Explain why the conditional probability of A given B is the fraction of B's outcomes that also belong to A, and interpret the answer in context.
A1.5Use the structure of an expression to identify ways to rewrite it.
A1.6 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
A1.6.aFactor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.
A1.6.bUse the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.
A1.6.cUse the properties of exponents to transform expressions for exponential functions.
A1.7Add, subtract, and multiply polynomials, showing that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.
A1.8Explain why extraneous solutions to an equation involving absolute values may arise and how to check to be sure that a candidate solution satisfies an equation.
A1.9 Select an appropriate method to solve a quadratic equation in one variable.
A1.9.aUse the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)^2 = q that has the same solutions. Explain how the quadratic formula is derived from this form.
A1.9.bSolve quadratic equations by inspection (such as x^2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.

Algebra II with Statistics

Print
A2.1 Identify numbers written in the form a + bi, where a and b are real numbers and i^2 = –1, as complex numbers.
A2.1.aAdd, subtract, and multiply complex numbers using the commutative, associative, and distributive properties.
A2.10Create equations and inequalities in one variable and use them to solve problems. Extend to equations arising from polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions.
A2.11Solve quadratic equations with real coefficients that have complex solutions.
A2.12Solve simple equations involving exponential, radical, logarithmic, and trigonometric functions using inverse functions.
A2.13Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales and use them to make predictions. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
A2.14 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x).
A2.14.aFind the approximate solutions of an equation graphically, using tables of values, or finding successive approximations, using technology where appropriate. Extend to cases where f(x) and/or g(x) are polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions.
A2.15Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Extend to polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions.
A2.16Identify the effect on the graph of replacing f(x) by f(x) + k, k ∙ f(x), f(k ∙ x), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
A2.17For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; and periodicity. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
A2.18Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
A2.19Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
A2.2Use matrices to represent and manipulate data.
A2.20 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.
A2.20.aGraph polynomial functions expressed symbolically, identifying zeros when suitable factorizations are available, and showing end behavior.
A2.20.bGraph sine and cosine functions expressed symbolically, showing period, midline, and amplitude.
A2.20.cGraph logarithmic functions expressed symbolically, showing intercepts and end behavior.
A2.20.dGraph reciprocal functions expressed symbolically, identifying horizontal and vertical asymptotes.
A2.20.eGraph square root and cube root functions expressed symbolically.
A2.20.fCompare the graphs of inverse functions and the relationships between their key features, including but not limited to quadratic, square root, exponential, and logarithmic functions.
A2.21Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle, building on work with non-right triangle trigonometry.
A2.22Use the mathematical modeling cycle to solve real-world problems involving polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions, from the simplification of the problem through the solving of the simplified problem, the interpretation of its solution, and the checking of the solution’s feasibility.
A2.23Use mathematical and statistical reasoning about normal distributions to draw conclusions and assess risk; limit to informal arguments.
A2.24Design and carry out an experiment or survey to answer a question of interest, and write an informal persuasive argument based on the results.
A2.25 From a normal distribution, use technology to find the mean and standard deviation and estimate population percentages by applying the empirical rule.
A2.25.aUse technology to determine if a given set of data is normal by applying the empirical rule.
A2.25.bEstimate areas under a normal curve to solve problems in context, using calculators, spreadsheets, and tables as appropriate.
A2.26Describe the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
A2.27Distinguish between a statistic and a parameter and use statistical processes to make inferences about population parameters based on statistics from random samples from that population.
A2.28Describe differences between randomly selecting samples and randomly assigning subjects to experimental treatment groups in terms of inferences drawn regarding a population versus regarding cause and effect.
A2.29Explain the consequences, due to uncontrolled variables, of non-randomized assignment of subjects to groups in experiments.
A2.3Multiply matrices by scalars to produce new matrices.
A2.30Evaluate where bias, including sampling, response, or nonresponse bias, may occur in surveys, and whether results are representative of the population of interest.
A2.31 Evaluate the effect of sample size on the expected variability in the sampling distribution of a sample statistic.
A2.31.aSimulate a sampling distribution of sample means from a population with a known distribution, observing the effect of the sample size on the variability.
A2.31.bDemonstrate that the standard deviation of each simulated sampling distribution is the known standard deviation of the population divided by the square root of the sample size.
A2.32 Produce a sampling distribution by repeatedly selecting samples of the same size from a given population or from a population simulated by bootstrapping (resampling with replacement from an observed sample). Do initial examples by hand, then use technology to generate a large number of samples.
A2.32.aVerify that a sampling distribution is centered at the population mean and approximately normal if the sample size is large enough.
A2.32.bVerify that 95% of sample means are within two standard deviations of the sampling distribution from the population mean.
A2.32.cCreate and interpret a 95% confidence interval based on an observed mean from a sampling distribution.
A2.33Use data from a randomized experiment to compare two treatments; limit to informal use of simulations to decide if an observed difference in the responses of the two treatment groups is unlikely to have occurred due to randomization alone, thus implying that the difference between the treatment groups is meaningful.
A2.34Define the radian measure of an angle as the constant of proportionality of the length of an arc it intercepts to the radius of the circle; in particular, it is the length of the arc intercepted on the unit circle.
A2.35Choose trigonometric functions (sine and cosine) to model periodic phenomena with specified amplitude, frequency, and midline.
A2.36Prove the Pythagorean identity sin^2 (θ) + cos^2 (θ) = 1 and use it to calculate trigonometric ratios.
A2.37Derive and apply the formula A = ½·ab·sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side, extending the domain of sine to include right and obtuse angles.
A2.38Derive and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. Extend the domain of sine and cosine to include right and obtuse angles.
A2.4Add, subtract, and multiply matrices of appropriate dimensions.
A2.5 Describe the roles that zero and identity matrices play in matrix addition and multiplication, recognizing that they are similar to the roles of 0 and 1 in the real numbers.
A2.5.aFind the additive and multiplicative inverses of square matrices, using technology as appropriate.
A2.5.bExplain the role of the determinant in determining if a square matrix has a multiplicative inverse.
A2.6Factor polynomials using common factoring techniques, and use the factored form of a polynomial to reveal the zeros of the function it defines.
A2.7Prove polynomial identities and use them to describe numerical relationships.
A2.8Explain why extraneous solutions to an equation may arise and how to check to be sure that a candidate solution satisfies an equation. Extend to radical equations.
A2.9For exponential models, express as a logarithm the solution to ab^ct = d, where a, c, and d are real numbers and the base b is 2 or 10; evaluate the logarithm using technology to solve an exponential equation.

Applications of Finite Mathematics

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AFM.1Represent logic statements in words, with symbols, and in truth tables, including conditional, biconditional, converse, inverse, contrapositive, and quantified statements.
AFM.10Use the Pigeonhole Principle to solve counting problems.
AFM.11Find patterns in application problems involving series and sequences, and develop recursive and explicit formulas as models to understand and describe sequential change.
AFM.12Determine characteristics of sequences, including the Fibonacci Sequence, the triangular numbers, and pentagonal numbers.
AFM.13Use the recursive process and difference equations to create fractals, population growth models, sequences, and series.
AFM.14Use mathematical induction to prove statements involving the positive integers.
AFM.15Develop and apply connections between Pascal’s Triangle and combinations.
AFM.16 Use vertex and edge graphs to model mathematical situations involving networks.
AFM.16.aIdentify properties of simple graphs, complete graphs, bipartite graphs, complete bipartite graphs, and trees.
AFM.17 Solve problems involving networks through investigation and application of existence and nonexistence of Euler paths, Euler circuits, Hamilton paths, and Hamilton circuits. Note: Real-world contexts modeled by graphs may include roads or communication networks.
AFM.17.aDevelop optimal solutions of application-based problems using existing and student-created algorithms.
AFM.17.bGive an argument for graph properties.
AFM.18 Apply algorithms relating to minimum weight spanning trees, networks, flows, and Steiner trees.
AFM.18.aUse shortest path techniques to find optimal shipping routes.
AFM.18.bShow that every connected graph has a minimal spanning tree.
AFM.18.cUse Kruskal’s Algorithm and Prim’s Algorithm to determine the minimal spanning tree of a weighted graph.
AFM.19Use vertex-coloring, edge-coloring, and matching techniques to solve application-based problems involving conflict.
AFM.2Represent logic operations such as and, or, not, nor, and x or (exclusive or) in words, with symbols, and in truth tables.
AFM.20Determine the minimum time to complete a project using algorithms to schedule tasks in order, including critical path analysis, the list-processing algorithm, and student-created algorithms.
AFM.21Use the adjacency matrix of a graph to determine the number of walks of length n in a graph.
AFM.22 Analyze advantages and disadvantages of different types of ballot voting systems.
AFM.22.aIdentify impacts of using a preferential ballot voting system and compare it to single candidate voting and other voting systems.
AFM.22.bAnalyze the impact of legal and cultural features of political systems on the mathematical aspects of elections.
AFM.23Apply a variety of methods for determining a winner using a preferential ballot voting system, including plurality, majority, run-off with majority, sequential run-off with majority, Borda count, pairwise comparison, Condorcet, and approval voting.
AFM.24Identify issues of fairness for different methods of determining a winner using a preferential voting ballot and other voting systems and identify paradoxes that can result.
AFM.25 Use methods of weighted voting and identify issues of fairness related to weighted voting.
AFM.25.aDistinguish between weight and power in voting.
AFM.26Explain and apply mathematical aspects of fair division, with respect to classic problems of apportionment, cake cutting, and estate division. Include applications in other contexts and modern situations.
AFM.27Identify and apply historic methods of apportionment for voting districts including Hamilton, Jefferson, Adams, Webster, and Huntington-Hill. Identify issues of fairness and paradoxes that may result from methods.
AFM.28Use spreadsheets to examine apportionment methods in large problems.
AFM.29Critically analyze issues related to information processing including accuracy, efficiency, and security.
AFM.3 Use truth tables to solve application-based logic problems and determine the truth value of simple and compound statements including negations and implications.
AFM.3.aDetermine whether statements are equivalent and construct equivalent statements.
AFM.30 Apply ciphers (encryption and decryption algorithms) and cryptosystems for encrypting and decrypting including symmetric-key or public-key systems.
AFM.30.aUse modular arithmetic to apply RSA (Rivest-Shamir-Adleman) public-key cryptosystems.
AFM.30.bUse matrices and their inverses to encode and decode messages.
AFM.31Apply error-detecting codes and error-correcting codes to determine accuracy of information processing.
AFM.32Apply methods of data compression.
AFM.4 Determine whether a logical argument is valid or invalid, using laws of logic such as the law of syllogism and the law of detachment.
AFM.4.aDetermine whether a logical argument is a tautology or a contradiction.
AFM.5Prove a statement indirectly by proving the contrapositive of the statement.
AFM.6Use multiple representations and methods for counting objects and developing more efficient counting techniques. Note: Representations and methods may include tree diagrams, lists, manipulatives, overcounting methods, recursive patterns, and explicit formulas.
AFM.7 Develop and use the Fundamental Counting Principle for counting independent and dependent events.
AFM.7.aUse various counting models (including tree diagrams and lists) to identify the distinguishing factors of a context in which the Fundamental Counting Principle can be applied.
AFM.8 Using application-based problems, develop formulas for permutations, combinations, and combinations with repetition and compare student-derived formulas to standard representations of the formulas.
AFM.8.aIdentify differences between applications of combinations and permutations.
AFM.8.bUsing application-based problems, calculate the number of permutations of a set with n elements. Calculate the number of permutations of r elements taken from a set of n elements.
AFM.8.cUsing application-based problems, calculate the number of subsets of size r that can be chosen from a set of n elements, explaining this number as the number of combinations “n choose r.”
AFM.8.dUsing application-based problems, calculate the number of combinations with repetitions of r elements from a set of n elements as “(n + r – 1) choose r.”
AFM.9Use various counting techniques to determine probabilities of events.

Geometry with Data Analysis

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G.1Extend understanding of irrational and rational numbers by rewriting expressions involving radicals, including addition, subtraction, multiplication, and division, in order to recognize geometric patterns.
G.10 Use statistics appropriate to the shape of the data distribution to compare and contrast two or more data sets, utilizing the mean and median for center and the interquartile range and standard deviation for variability.
G.10.aExplain how standard deviation develops from mean absolute deviation.
G.10.bCalculate the standard deviation for a data set, using technology where appropriate.
G.11Interpret differences in shape, center, and spread in the context of data sets, accounting for possible effects of extreme data points (outliers) on mean and standard deviation.
G.12 Represent data of two quantitative variables on a scatter plot, and describe how the variables are related.
G.12.aFind a linear function for a scatter plot that suggests a linear association and informally assess its fit by plotting and analyzing residuals, including the squares of the residuals, in order to improve its fit.
G.12.bUse technology to find the least-squares line of best fit for two quantitative variables.
G.13Compute (using technology) and interpret the correlation coefficient of a linear relationship.
G.14Distinguish between correlation and causation.
G.15 Evaluate possible solutions to real-life problems by developing linear models of contextual situations and using them to predict unknown values.
G.15.aUse the linear model to solve problems in the context of the given data.
G.15.bInterpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the given data.
G.16Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
G.17 Model and solve problems using surface area and volume of solids, including composite solids and solids with portions removed.
G.17.aGive an informal argument for the formulas for the surface area and volume of a sphere, cylinder, pyramid, and cone using dissection arguments, Cavalieri's Principle, and informal limit arguments.
G.17.bApply geometric concepts to find missing dimensions to solve surface area or volume problems.
G.18Given the coordinates of the vertices of a polygon, compute its perimeter and area using a variety of methods, including the distance formula and dynamic geometry software, and evaluate the accuracy of the results.
G.19Derive and apply the relationships between the lengths, perimeters, areas, and volumes of similar figures in relation to their scale factor.
G.2 Use units as a way to understand problems and to guide the solution of multi-step problems.
G.2.aChoose and interpret units consistently in formulas.
G.2.bChoose and interpret the scale and the origin in graphs and data displays.
G.2.cDefine appropriate quantities for the purpose of descriptive modeling.
G.2.dChoose a level of accuracy appropriate to limitations of measurements when reporting quantities.
G.20Derive and apply the formula for the length of an arc and the formula for the area of a sector.
G.21 Represent transformations and compositions of transformations in the plane (coordinate and otherwise) using tools such as tracing paper and geometry software.
G.21.aDescribe transformations and compositions of transformations as functions that take points in the plane as inputs and give other points as outputs, using informal and formal notation.
G.21.bCompare transformations which preserve distance and angle measure to those that do not.
G.22 Explore rotations, reflections, and translations using graph paper, tracing paper, and geometry software.
G.22.aGiven a geometric figure and a rotation, reflection, or translation, draw the image of the transformed figure using graph paper, tracing paper, or geometry software.
G.22.bSpecify a sequence of rotations, reflections, or translations that will carry a given figure onto another.
G.22.cDraw figures with different types of symmetries and describe their attributes.
G.23Develop definitions of rotation, reflection, and translation in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G.24Define congruence of two figures in terms of rigid motions (a sequence of translations, rotations, and reflections); show that two figures are congruent by finding a sequence of rigid motions that maps one figure to the other.
G.25 Verify criteria for showing triangles are congruent using a sequence of rigid motions that map one triangle to another.
G.25.aVerify that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G.25.bVerify that two triangles are congruent if (but not only if) the following groups of corresponding parts are congruent: angle-side-angle (ASA), side-angle-side (SAS), side-side-side (SSS), and angle-angle-side (AAS).
G.26 Verify experimentally the properties of dilations given by a center and a scale factor.
G.26.aVerify that a 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.
G.26.bVerify that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G.27Given two figures, determine whether they are similar by identifying a similarity transformation (sequence of rigid motions and dilations) that maps one figure to the other.
G.28 Verify criteria for showing triangles are similar using a similarity transformation (sequence of rigid motions and dilations) that maps one triangle to another.
G.28.aVerify that two triangles are similar if and only if corresponding pairs of sides are proportional and corresponding pairs of angles are congruent.
G.28.bVerify that two triangles are similar if (but not only if) two pairs of corresponding angles are congruent (AA), the corresponding sides are proportional (SSS), or two pairs of corresponding sides are proportional and the pair of included angles is congruent (SAS).
G.29 Find patterns and relationships in figures including lines, triangles, quadrilaterals, and circles, using technology and other tools.
G.29.aConstruct figures, using technology and other tools, in order to make and test conjectures about their properties.
G.29.bIdentify different sets of properties necessary to define and construct figures.
G.3Find the coordinates of the vertices of a polygon determined by a set of lines, given their equations, by setting their function rules equal and solving, or by using their graphs.
G.30Develop and use precise definitions of figures such as angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G.31 Justify whether conjectures are true or false in order to prove theorems and then apply those theorems in solving problems, communicating proofs in a variety of ways, including flow chart, two-column, and paragraph formats.
G.31.aInvestigate, prove, and apply theorems about lines and angles, including but not limited to: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; the points on the perpendicular bisector of a line segment are those equidistant from the segment's endpoints.
G.31.bInvestigate, prove, and apply theorems about triangles, including but not limited to: the sum of the measures of the interior angles of a triangle is 180˚; the base angles of isosceles triangles are congruent; the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length; a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem using triangle similarity.
G.31.cInvestigate, prove, and apply theorems about parallelograms and other quadrilaterals, including but not limited to both necessary and sufficient conditions for parallelograms and other quadrilaterals, as well as relationships among kinds of quadrilaterals.
G.32Use coordinates to prove simple geometric theorems algebraically.
G.33Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.
G.34Use congruence and similarity criteria for triangles to solve problems in real-world contexts.
G.35 Discover and apply relationships in similar right triangles.
G.35.aDerive and apply the constant ratios of the sides in special right triangles (45˚-45˚-90˚ and 30˚-60˚-90˚).
G.35.bUse similarity to explore and define basic trigonometric ratios, including sine ratio, cosine ratio, and tangent ratio.
G.35.cExplain and use the relationship between the sine and cosine of complementary angles.
G.35.dDemonstrate the converse of the Pythagorean Theorem.
G.35.eUse trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems, including finding areas of regular polygons.
G.36Use geometric shapes, their measures, and their properties to model objects and use those models to solve problems.
G.37Investigate and apply relationships among inscribed angles, radii, and chords, including but not limited to: the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
G.38Use the mathematical modeling cycle involving geometric methods to solve design problems.
G.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
G.5Verify that the graph of a linear equation in two variables is the set of all its solutions plotted in the coordinate plane, which forms a line.
G.6 Derive the equation of a circle of given center and radius using the Pythagorean Theorem.
G.6.aGiven the endpoints of the diameter of a circle, use the midpoint formula to find its center and then use the Pythagorean Theorem to find its equation.
G.6.bDerive the distance formula from the Pythagorean Theorem.
G.7Use mathematical and statistical reasoning with quantitative data, both univariate data (set of values) and bivariate data (set of pairs of values) that suggest a linear association, in order to draw conclusions and assess risk.
G.8Use technology to organize data, including very large data sets, into a useful and manageable structure.
G.9Represent the distribution of univariate quantitative data with plots on the real number line, choosing a format (dot plot, histogram, or box plot) most appropriate to the data set, and represent the distribution of bivariate quantitative data with a scatter plot. Extend from simple cases by hand to more complex cases involving large data sets using technology.

Mathematical Modeling

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MM.1Use the full Mathematical Modeling Cycle or Statistical Problem-Solving Cycle to answer a real-world problem of particular student interest, incorporating standards from across the course.
MM.10 Construct a two-dimensional visual representation of a three-dimensional object or structure.
MM.10.aDetermine the level of precision and the appropriate tools for taking the measurements in constructing a two-dimensional visual representation of a three-dimensional object or structure.
MM.10.bCreate an elevation drawing to represent a given solid structure, using technology where appropriate.
MM.10.cDetermine which measurements cannot be taken directly and must be calculated based on other measurements when constructing a two-dimensional visual representation of a three-dimensional object or structure.
MM.10.dDetermine an appropriate means to visually represent an object or structure, such as drawings on paper or graphics on computer screens.
MM.11 Plot coordinates on a three-dimensional Cartesian coordinate system and use relationships between coordinates to solve design problems.
MM.11.aDescribe the features of a three-dimensional Cartesian coordinate system and use them to graph points.
MM.11.bGraph a point in space as the vertex of a right prism drawn in the appropriate octant with edges along the x, y, and z axes.
MM.11.cFind the distance between two objects in space given the coordinates of each.
MM.11.dFind the midpoint between two objects in space given the coordinates of each.
MM.12Use technology and other tools to explore the results of simple transformations using three-dimensional coordinates, including translations in the x, y, and/or z directions; rotations of 90º, 180º, or 270º about the x, y, and z axes; reflections over the xy, yz, and xy planes; and dilations from the origin.
MM.13Create a scale model of a complex three-dimensional structure based on observed measurements and indirect measurements, using translations, reflections, rotations, and dilations of its components.
MM.14Use elements of the Mathematical Modeling Cycle to make predictions based on measurements that change over time, including motion, growth, decay, and cycling.
MM.15 Use regression with statistical graphing technology to determine an equation that best fits a set of bivariate data, including nonlinear patterns.
MM.15.aCreate a scatter plot with a sufficient number of data points to predict a pattern.
MM.15.bDescribe the overall relationship between two quantitative variables (increase, decrease, linearity, concavity, extrema, inflection) or pattern of change.
MM.15.cMake a prediction based upon patterns.
MM.16Create a linear representation of non-linear data and interpret solutions, using technology and the process of linearization with logarithms.
MM.17Use the Statistical Problem Solving Cycle to answer real-world questions.
MM.18 Construct a probability distribution based on empirical observations of a variable.
MM.18.aEstimate the probability of each value for a random variable based on empirical observations or simulations, using technology.
MM.18.bRepresent a probability distribution by a relative frequency histogram and/or a cumulative relative frequency graph.
MM.18.cFind the mean, standard deviation, median, and interquartile range of a probability distribution and make long-term predictions about future possibilities. Determine which measures are most appropriate based upon the shape of the distribution.
MM.19 Construct a sampling distribution for a random event or random sample.
MM.19.aUse the binomial theorem to construct the sampling distribution for the number of successes in a binary event or the number of positive responses to a yes/no question in a random sample.
MM.19.bUse the normal approximation of a proportion from a random event or sample when conditions are met.
MM.19.cUse the central limit theorem to construct a normal sampling distribution for the sample mean when conditions are met.
MM.19.dFind the long-term probability of a given range of outcomes from a random event or random sample.
MM.2Use elements of the Mathematical Modeling Cycle to solve real-world problems involving finances.
MM.20 Perform inference procedures based on the results of samples and experiments.
MM.20.aUse a point estimator and margin of error to construct a confidence interval for a proportion or mean.
MM.20.bInterpret a confidence interval in context and use it to make strategic decisions.
MM.20.cPerform a significance test for null and alternative hypotheses.
MM.20.dInterpret the significance level of a test in the context of error probabilities, and use the results to make strategic decisions.
MM.21Critique the validity of reported conclusions from statistical studies in terms of bias and random error probabilities.
MM.22Conduct a randomized study on a topic of student interest (sample or experiment) and draw conclusions based upon the results.
MM.3Organize and display financial information using arithmetic sequences to represent simple interest and straight-line depreciation.
MM.4 Organize and display financial information using geometric sequences to represent compound interest and proportional depreciation, including periodic (yearly, monthly, weekly) and continuous compounding.
MM.4.aExplain the relationship between annual percentage yield (APY) and annual percentage rate (APR) as values for r in the formulas A=P(1+r)^t and A=Pe^rt.
MM.5Compare simple and compound interest, and straight-line and proportional depreciation.
MM.6Investigate growth and reduction of credit card debt using spreadsheets, including variables such as beginning balance, payment structures, credits, interest rates, new purchases, finance charges, and fees.
MM.7 Compare and contrast housing finance options including renting, leasing to purchase, purchasing with a mortgage, and purchasing with cash.
MM.7.aResearch and evaluate various mortgage products available to consumers.
MM.7.bCompare monthly mortgage payments for different terms, interest rates, and down payments.
MM.7.cAnalyze the financial consequence of buying a home (mortgage payments vs. potentially increasing resale value) versus investing the money saved when renting, assuming that renting is the less expensive option.
MM.8Investigate the advantages and disadvantages of various means of paying for an automobile, including leasing, purchasing by cash, and purchasing by loan.
MM.9Use the Mathematical Modeling Cycle to solve real-world problems involving the design of three-dimensional objects.

Precalculus

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PC.1 Define the constant e in a variety of contexts.
PC.1.aExplore the behavior of the function y=ex and its applications.
PC.1.bExplore the behavior of ln(x), the logarithmic function with base e, and its applications.
PC.10Solve problems involving velocity and other quantities that can be represented by vectors.
PC.11Find the scalar (dot) product of two vectors as the sum of the products of corresponding components and explain its relationship to the cosine of the angle formed by two vectors.
PC.12 Add and subtract vectors.
PC.12.aAdd vectors end-to-end, component-wise, and by the parallelogram rule, understanding that the magnitude of a sum of two vectors is not always the sum of the magnitudes.
PC.12.bGiven two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
PC.12.cExplain vector subtraction, v – w, as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
PC.13 Multiply a vector by a scalar.
PC.13.aRepresent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise.
PC.13.bCompute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
PC.14Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
PC.15Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems, extending to infinite geometric series.
PC.16Derive and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
PC.17Know and apply the Binomial Theorem for the expansion of (x + y)^n in powers of x and y for a positive integer, n, where x and y are any numbers.
PC.18Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated cases, a computer algebra system.
PC.19 Add, subtract, multiply, and divide rational expressions.
PC.19.aExplain why rational expressions form a system analogous to the rational numbers, which is closed under addition, subtraction, multiplication, and division by a non-zero rational expression.
PC.2Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
PC.20Explain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a clear-cut solution. Construct a viable argument to justify a solution method. Include equations that may involve linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, and trigonometric functions, and their inverses.
PC.21Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise.
PC.22Represent a system of linear equations as a single matrix equation in a vector variable.
PC.23Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).
PC.24Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features. Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.
PC.25 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Extend from polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions.
PC.25.aFind the difference quotient f(x+Δx)-f(x)/Δx of a function and use it to evaluate the average rate of change at a point.
PC.25.bExplore how the average rate of change of a function over an interval (presented symbolically or as a table) can be used to approximate the instantaneous rate of change at a point as the interval decreases.
PC.26 Graph functions expressed symbolically and show key features of the graph, by hand and using technology. Use the equation of functions to identify key features in order to generate a graph.
PC.26.aGraph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
PC.26.bGraph trigonometric functions and their inverses, showing period, midline, amplitude, and phase shift.
PC.27Compose functions. Extend to polynomial, trigonometric, radical, and rational functions.
PC.28 Find inverse functions.
PC.28.aGiven that a function has an inverse, write an expression for the inverse of the function.
PC.28.bVerify by composition that one function is the inverse of another.
PC.28.cRead values of an inverse function from a graph or a table, given that the function has an inverse.
PC.28.dProduce an invertible function from a non-invertible function by restricting the domain.
PC.29Use the inverse relationship between exponents and logarithms to solve problems involving logarithms and exponents. Extend from logarithms with base 2 and 10 to a base of e.
PC.3Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
PC.30Identify the effect on the graph of replacing f(x) by f(x) + k, k ∙ f(x), f(k ∙ x), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Extend the analysis to include all trigonometric, rational, and general piecewise-defined functions with and without technology.
PC.31 Graph conic sections from second-degree equations, extending from circles and parabolas to ellipses and hyperbolas, using technology to discover patterns.
PC.31.aGraph conic sections given their standard form.
PC.31.bIdentify the conic section that will be formed, given its equation in general form.
PC.32 Solve application-based problems involving parametric and polar equations.
PC.32.aGraph parametric and polar equations.
PC.32.bConvert parametric and polar equations to rectangular form.
PC.33Use special triangles to determine geometrically the values of sine, cosine, and 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.34Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
PC.35Demonstrate that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
PC.36Use 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.37 Use trigonometric identities to solve problems.
PC.37.aUse the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 to derive the other forms of the identity.
PC.37.bUse the angle sum formulas for sine, cosine, and tangent to derive the double angle formulas.
PC.37.cUse the Pythagorean and double angle identities to prove other simple identities.
PC.4Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
PC.5Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
PC.6Analyze possible zeros for a polynomial function over the complex numbers by applying the Fundamental Theorem of Algebra, using a graph of the function, or factoring with algebraic identities.
PC.7 Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.
PC.7.aApply limits of functions at specific values and at infinity in problems involving convergence and divergence.
PC.8Explain that vector quantities have both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes.
PC.9Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
PC.A-APREAlgebra - Arithmetic With Polynomials and Rational Expressions
PC.A-RWEIAlgebra - Reasoning With Equations and Inequalities
PC.A-SSEAlgebra - Seeing Structure in Expressions
PC.BFBuilding Functions
PC.FFunctions - Interpreting Functions
PC.NQ-CNSNumber and Quantity - The Complex Number System
PC.NQ-LNumber and Quantity - Limits
PC.NQ-VMQNumber and Quantity - Vector and Matrix Quantities
PC.TFTrigonometric Functions

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.