Massachusetts Math Standards - Massachusetts Curriculum Framework for Mathematics

1148 standards, Kindergarten through AP.

Massachusetts Curriculum Frameworks for Mathematics build on Common Core with state specific additions. Model courses in Algebra I/II and Geometry sit alongside integrated math pathways, giving districts flexibility in how they sequence courses.

Kindergarten

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K.CC Counting and Cardinality
K.CC.A Know number names and the count sequence.
K.CC.A.1Count to 100 by ones and by tens.
K.CC.A.2Count forward beginning from a given number within the known sequence (instead of having to begin at one).
K.CC.A.3Write numbers from 0 to 20. Represent a number of objects with a written numeral 0–20 (with 0 representing a count of no objects).
K.CC.B Count to tell the number of objects.
K.CC.B.4 Understand the relationship between numbers and quantities; connect counting to cardinality.
K.CC.B.4.aWhen counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.
K.CC.B.4.bUnderstand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.
K.CC.B.4.cUnderstand that each successive number name refers to a quantity that is one larger. Recognize the one more pattern of counting using objects.
K.CC.B.5Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects.
K.CC.C Compare numbers.
K.CC.C.6Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group for groups with up to 10 objects, e.g., by using matching and counting strategies.
K.CC.C.7Compare two numbers between 1 and 10 presented as written numerals.
K.G Geometry
K.G.A Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
K.G.A.1Describe 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.G.A.2Correctly name shapes regardless of their orientation or overall size.
K.G.A.3Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).
K.G.B Analyze, compare, create, and compose shapes.
K.G.B.4Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal length).
K.G.B.5Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
K.G.B.6Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?”
K.MD Measurement and Data
K.MD.A Describe and compare measurable attributes.
K.MD.A.1Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.
K.MD.A.2Directly compare two objects with a measurable attribute in common, to see which object has “more of”/“less of” the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.
K.MD.B Classify objects and count the number of objects in each category.
K.MD.B.3Classify objects into given categories; count the numbers of objects in each category (up to and including 10) and sort the categories by count.
K.NBT Number and Operations in Base Ten
K.NBT.A Work with numbers 11–19 to gain foundations for place value.
K.NBT.A.1Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.
K.OA Operations and Algebraic Thinking
K.OA.A Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
K.OA.A.1Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
K.OA.A.2Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
K.OA.A.3Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
K.OA.A.4For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
K.OA.A.5Fluently add and subtract within 5, including zero.

Grade 1

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1.G Geometry
1.G.A Reason with shapes and their attributes.
1.G.A.1Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes that possess defining attributes.
1.G.A.2Compose 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.G.A.3Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
1.MD Measurement and Data
1.MD.A Measure lengths indirectly and by iterating length units.
1.MD.A.1Order three objects by length; compare the lengths of two objects indirectly by using a third object.
1.MD.A.2Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.
1.MD.B Tell and write time.
1.MD.B.3Tell and write time in hours and half-hours using analog and digital clocks.
1.MD.C Represent and interpret data.
1.MD.C.4Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.
1.MD.D Work with money.
1.MD.D.5Identify the values of all U.S. coins and know their comparative values (e.g., a dime is of greater value than a nickel). Find equivalent values (e.g., a nickel is equivalent to five pennies). Use appropriate notation (e.g., 69¢). Use the values of coins in the solutions of problems (up to 100¢).
1.NBT Number and Operations in Base Ten
1.NBT.A Extend the counting sequence.
1.NBT.A.1Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.
1.NBT.B Understand place value.
1.NBT.B.2 Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
1.NBT.B.2.a10 can be thought of as a bundle of ten ones—called a “ten.”
1.NBT.B.2.bThe numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
1.NBT.B.2.cThe numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
1.NBT.B.3Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
1.NBT.C Use place value understanding and properties of operations to add and subtract.
1.NBT.C.4Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, 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. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.
1.NBT.C.5Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. Identify arithmetic patterns of 10 more and 10 less than using strategies based on place value.
1.NBT.C.6Subtract multiples of 10 in the range 10–90 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.OA Operations and Algebraic Thinking
1.OA.A Represent and solve problems involving addition and subtraction.
1.OA.A.1Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations (number sentences) with a symbol for the unknown number to represent the problem.
1.OA.A.2Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
1.OA.B Understand and apply properties of operations and the relationship between addition and subtraction.
1.OA.B.3Apply properties of operations to add. For example, when adding numbers order does not matter. If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known (Commutative property of addition). To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12 (Associative property of addition). When adding zero to a number, the result is the same number (Identity property of zero for addition).
1.OA.B.4Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.
1.OA.C Add and subtract within 20.
1.OA.C.5Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
1.OA.C.6Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use mental strategies such as counting on; making 10 (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a 10 (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
1.OA.D Work with addition and subtraction equations.
1.OA.D.7Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
1.OA.D.8Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = □ – 3, 6 + 6 = □.

Grade 2

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2.G Geometry
2.G.A Reason with shapes and their attributes.
2.G.A.1Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, squares, rectangles, rhombuses, trapezoids, pentagons, hexagons, and cubes.
2.G.A.2Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
2.G.A.3Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
2.MD Measurement and Data
2.MD.A Measure and estimate lengths in standard units.
2.MD.A.1Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.
2.MD.A.2Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.
2.MD.A.3Estimate lengths using units of inches, feet, centimeters, and meters.
2.MD.A.4Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.
2.MD.B Relate addition and subtraction to length.
2.MD.B.5Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.
2.MD.B.6Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.
2.MD.C Work with time and money.
2.MD.C.7 Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
2.MD.C.7.aKnow the relationships of time, including seconds in a minute, minutes in an hour, hours in a day, days in a week; days in a month and a year and approximate number of weeks in a month and weeks in a year.
2.MD.C.8Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies (up to $10), using $ and ¢ symbols appropriately and whole dollar amounts. For example, if you have 2 dimes and 3 pennies, how many cents do you have? If you have $3 and 4 quarters, how many dollars or cents do you have? (Students are not expected to use decimal notation.)
2.MD.D Represent and interpret data.
2.MD.D.10Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems, using information presented in a bar graph.
2.MD.D.9Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Organize and record the data on a line plot (dot plot) where the horizontal scale is marked off in whole-number units.
2.NBT Number and Operations in Base Ten
2.NBT.A Understand place value.
2.NBT.A.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
2.NBT.A.1.a100 can be thought of as a bundle of ten tens—called a “hundred.”
2.NBT.A.1.bThe 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.NBT.A.2Count within 1,000; skip-count by 5s, 10s, and 100s. Identify patterns in skip counting starting at any number.
2.NBT.A.3Read and write numbers to 1,000 using base-ten numerals, number names, and expanded form.
2.NBT.A.4Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
2.NBT.B Use place value understanding and properties of operations to add and subtract.
2.NBT.B.5Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
2.NBT.B.6Add up to four two-digit numbers using strategies based on place value and properties of operations.
2.NBT.B.7Add and subtract within 1,000, 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. Understand 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.NBT.B.8Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.
2.NBT.B.9Explain why addition and subtraction strategies work, using place value and the properties of operations.
2.OA Operations and Algebraic Thinking
2.OA.A Represent and solve problems involving addition and subtraction.
2.OA.A.1Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.=
2.OA.B Add and subtract within 20.
2.OA.B.2Fluently add and subtract within 20 using mental strategies. By end of grade 2, know from memory all sums of two single-digit numbers and related differences. For example, the sum 6 + 5 = 11 has related differences of 11 – 5 = 6 and 11 – 6 = 5.
2.OA.C Work with equal groups of objects to gain foundations for multiplication.
2.OA.C.3Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.
2.OA.C.4Use addition to find the total number of objects arranged in rectangular arrays with up to five rows and up to five columns; write an equation to express the total as a sum of equal addends.

Grade 3

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3.G Geometry
3.G.A Reason with shapes and their attributes.
3.G.A.1Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Compare and classify shapes by their sides and angles (right angle/non-right angle). Recognize rhombuses, rectangles, squares, and trapezoids as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.
3.G.A.2Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into four parts with equal areas and describe the area of each part as 1/4 of the area of the shape.
3.MD Measurement and Data
3.MD.A Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
3.MD.A.1Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.
3.MD.A.2Measure and estimate liquid volumes and masses of objects using standard metric units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same metric units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.
3.MD.B Represent and interpret data.
3.MD.B.3Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent five pets.
3.MD.B.4Generate measurement data by measuring lengths of objects using rulers marked with halves and fourths of an inch. Record and show the data by making a line plot (dot plot), where the horizontal scale is marked off in appropriate units—whole numbers, halves, or fourths.
3.MD.C Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
3.MD.C.5 Recognize area as an attribute of plane figures and understand concepts of area measurement.
3.MD.C.5.aA square with side length of one unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.
3.MD.C.5.bA plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
3.MD.C.6Measure areas by counting unit squares (square cm, square m, square in., square ft., and non-standard units).
3.MD.C.7 Relate area to the operations of multiplication and addition.
3.MD.C.7.aFind the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
3.MD.C.7.bMultiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real-world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
3.MD.C.7.cUse tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a x b and a x c. Use area models to represent the distributive property in mathematical reasoning.
3.MD.C.7.dRecognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real-world problems.
3.MD.D Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
3.MD.D.8Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
3.NBT Number and Operations in Base Ten
3.NBT.A Use place value understanding and properties of operations to perform multi-digit arithmetic.
3.NBT.A.1Use place value understanding to round whole numbers to the nearest 10 or 100.
3.NBT.A.2Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
3.NBT.A.3Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations.
3.NF Number and Operations—Fractions
3.NF.A Develop understanding of fractions as numbers for fractions with denominators 2, 3, 4, 6, and 8.
3.NF.A.1Understand a fraction 1∕b as the quantity formed by 1 part when a whole (a single unit) is partitioned into b equal parts; understand a fraction a∕b as the quantity formed by a parts of size 1∕b.
3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
3.NF.A.2.aRepresent a unit fraction, 1∕b, on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1∕b and that the fraction 1∕b is located 1∕b of a whole unit from 0 on the number line.
3.NF.A.2.bRepresent a fraction a∕b on a number line diagram by marking off a lengths 1∕b from 0. Recognize that the resulting interval has size a∕b and that its endpoint locates the number a∕b on the number line.
3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.A.3.aUnderstand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
3.NF.A.3.bRecognize and generate simple equivalent fractions, e.g., 1∕2 = 2∕4, 4∕6 = 2∕3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
3.NF.A.3.cExpress whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. For example, express 3 in the form 3 = 3∕1; recognize that 6∕1 = 6; locate 4∕4 and 1 at the same point of a number line diagram.
3.NF.A.3.dCompare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
3.OA Operations and Algebraic Thinking
3.OA.A Represent and solve problems involving multiplication and division.
3.OA.A.1Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in five groups of seven objects each. For example, describe a context in which a total number of objects can be expressed as 5 x 7.
3.OA.A.2Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
3.OA.A.3Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
3.OA.A.4Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 x ? = 48, 5 = □ ÷ 3, 6 x 6 = ?.
3.OA.B Understand properties of multiplication and the relationship between multiplication and division.
3.OA.B.5Apply properties of operations to multiply. For example: When multiplying numbers order does not matter. If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known (Commutative property of multiplication); The product 3 x 5 x 2 can be found by 3 x 5 = 15 then 15 x 2 = 30, or by 5 x 2 = 10 then 3 x 10 = 30 (Associative property of multiplication); When multiplying two numbers either number can be decomposed and multiplied; one can find 8 x 7 by knowing that 7 = 5 + 2 and that 8 x 5 = 40 and 8 x 2 = 16, resulting in 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56 (Distributive property); When a number is multiplied by 1 the result is the same number (Identity property of 1 for multiplication).
3.OA.B.6Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
3.OA.C Multiply and divide within 100.
3.OA.C.7Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of grade 3, know from memory all products of two single-digit numbers and related division facts. For example, the product 4 x 7 = 28 has related division facts 28 ÷ 7 = 4 and 28 ÷ 4 = 7.
3.OA.D Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3.OA.D.8Solve two-step word problems using the four operations for problems posed with whole numbers and having whole number answers. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies, including rounding.
3.OA.D.9Identify arithmetic patterns (including patterns in the addition table or multiplication table) and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Grade 4

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4.G Geometry
4.G.A Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
4.G.A.1Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
4.G.A.2Classify 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. Recognize right triangles as a category, and identify right triangles.
4.G.A.3Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.
4.MD Measurement and Data
4.MD.A Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
4.MD.A.1Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft. is 12 times as long as 1 in. Express the length of a 4 ft. snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), …
4.MD.A.2Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
4.MD.A.3Apply the area and perimeter formulas for rectangles in real-world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. (Note: When finding areas of rectangular regions answers will be in square units. For example, the area of a 1 cm x 1 cm rectangular region will be 1 square centimeter (1 cm^2, students are not expected to use this notation.) When finding the perimeter of a rectangular region answers will be in linear units. For example, the perimeter of the region is: 1cm + 1cm + 1cm +1cm = 4 cm or 2(1cm) + 2(1cm) = 4 cm).
4.MD.B Represent and interpret data.
4.MD.B.4Make a line plot (dot plot) representation to display a data set of measurements in fractions of a unit (1∕2, 1∕4, 1∕8). Solve problems involving addition and subtraction of fractions by using information presented in line plots (dot plots). For example, from a line plot (dot plot) find and interpret the difference in length between the longest and shortest specimens in an insect collection.
4.MD.C Geometric measurement: Understand concepts of angle and measure angles.
4.MD.C.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
4.MD.C.5.aAn angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1∕360 of a circle is called a “one-degree angle,” and can be used to measure angles.
4.MD.C.5.bAn angle that turns through n one-degree angles is said to have an angle measure of n degrees.
4.MD.C.6Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
4.MD.C.7Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
4.NBT Number and Operations in Base Ten
4.NBT.A Generalize place value understanding for multi-digit whole numbers less than or equal to 1,000,000.
4.NBT.A.1Recognize that in a multi-digit whole number, a digit in any place represents 10 times as much as it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
4.NBT.A.2Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
4.NBT.A.3Use place value understanding to round multi-digit whole numbers to any place.
4.NBT.B Use place value understanding and properties of operations to perform multi-digit arithmetic on whole numbers less than or equal to 1,000,000.
4.NBT.B.4Fluently add and subtract multi-digit whole numbers using the standard algorithm.
4.NBT.B.5Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
4.NF Number and Operations—Fractions
4.NF.A Extend understanding of fraction equivalence and ordering for fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
4.NF.A.1Explain why a fraction a∕b is equivalent to a fraction (n x a)∕(n x b) by using visual fraction models, with attention to how the numbers and sizes of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions, including fractions greater than 1.
4.NF.A.2Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1∕2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
4.NF.B Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers for fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
4.NF.B.3 Understand a fraction a∕b with a > 1 as a sum of fractions 1∕b.
4.NF.B.3.aUnderstand addition and subtraction of fractions as joining and separating parts referring to the same whole. (The whole can be a set of objects.)
4.NF.B.3.bDecompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using drawings or visual fraction models. Examples: 3∕8 = 1∕8 + 1∕8 + 1∕8; 3∕8 = 1∕8 + 2∕8; 2 1∕8 = 1 + 1 + 1∕8 = 8∕8 + 8∕8 + 1∕8.
4.NF.B.3.cAdd and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
4.NF.B.3.dSolve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using drawings or visual fraction models and equations to represent the problem.
4.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
4.NF.B.4.aUnderstand a fraction a∕b as a multiple of 1∕b. For example, use a visual fraction model to represent 5∕4 as the product 5 x (1∕4), recording the conclusion by the equation 5∕4 = 5 x (1∕4).
4.NF.B.4.bUnderstand a multiple of a∕b as a multiple of 1∕b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x (2∕5) as 6 x (1∕5), recognizing this product as 6∕5. (In general, n x (a∕b) = (n x a)∕b.)
4.NF.B.4.cSolve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3∕8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
4.NF.C Understand decimal notation for fractions, and compare decimal fractions.
4.NF.C.5Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3∕10 as 30∕100, and add 3∕10 + 4∕100 = 34∕100.
4.NF.C.6Use decimal notation to represent fractions with denominators 10 or 100. For example, rewrite 0.62 as 62∕100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
4.NF.C.7Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
4.OA Operations and Algebraic Thinking
4.OA.A Use the four operations with whole numbers to solve problems.
4.OA.A.1Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
4.OA.A.2Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
4.OA.A.3 Solve multi-step word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
4.OA.A.3.aKnow multiplication facts and related division facts through 12 x 12.
4.OA.B Gain familiarity with factors and multiples.
4.OA.B.4Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.
4.OA.C Generate and analyze patterns.
4.OA.C.5Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

Grade 5

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5.G Geometry
5.G.A Graph points on the coordinate plane to solve real-world and mathematical problems.
5.G.A.1Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the zero on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
5.G.A.2Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
5.G.B Classify two-dimensional figures into categories based on their properties.
5.G.B.3Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
5.G.B.4Classify two-dimensional figures in a hierarchy based on properties. For example, all rectangles are parallelograms because they are all quadrilaterals with two pairs of opposite sides parallel.
5.MD Measurement and Data
5.MD.A Convert like measurement units within a given measurement system.
5.MD.A.1Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real-world problems.
5.MD.B Represent and interpret data.
5.MD.B.2Make a line plot (dot plot) to display a data set of measurements in fractions of a unit. Use operations on fractions for this grade to solve problems involving information presented in line plot (dot plot). For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.
5.MD.C Geometric measurement: Understand concepts of volume and relate volume to multiplication and to addition.
5.MD.C.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
5.MD.C.3.aA cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.
5.MD.C.3.bA solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
5.MD.C.4Measure volumes by counting unit cubes, using cubic cm, cubic in., cubic ft., and non-standard units.
5.MD.C.5 Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume.
5.MD.C.5.aFind the volume of a right rectangular prism with whole-number edge lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
5.MD.C.5.bApply the formula V = l x w x h and V = B x h (where B stands for the area of the base) for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real-world and mathematical problems.
5.MD.C.5.cRecognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real-world problems.
5.NBT Number and Operations in Base Ten
5.NBT.A Understand the place value system.
5.NBT.A.1Recognize that in a multi-digit number, including decimals, a digit in any place represents 10 times as much as it represents in the place to its right and 1∕10 of what it represents in the place to its left.
5.NBT.A.2Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
5.NBT.A.3 Read, write, and compare decimals to thousandths.
5.NBT.A.3.aRead and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 x 100 + 4 x 10 + 7 x 1 + 3 x (1∕10) + 9 x (1∕100) + 2 x (1∕1000).
5.NBT.A.3.bCompare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
5.NBT.A.4Use place value understanding to round decimals to any place.
5.NBT.B Perform operations with multi-digit whole numbers and with decimals to hundredths.
5.NBT.B.5Fluently multiply multi-digit whole numbers. (Include two-digit x four-digit numbers and, three-digit x three-digit numbers) using the standard algorithm.
5.NBT.B.6Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
5.NBT.B.7Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction and between multiplication and division; relate the strategy to a written method and explain the reasoning used.
5.NF Number and Operations—Fractions
5.NF.A Use equivalent fractions as a strategy to add and subtract fractions.
5.NF.A.1Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2∕3 + 5∕4 = 8∕12 + 15∕12 = 23∕12. (In general, a∕b + c∕d = (ad + bc)∕bd.)
5.NF.A.2Solve word problems involving addition and subtraction of fractions referring to the same whole (the whole can be a set of objects), including cases of unlike denominators, e.g., by 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. For example, recognize an incorrect result 2∕5 + 1∕2 = 3∕7, by observing that 3∕7 < 1∕2.
5.NF.B Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.B.3Interpret a fraction as division of the numerator by the denominator (a∕b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when three wholes are shared equally among four people each person has a share of size 3/4. If nine people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.B.4.aInterpret the product (a∕b) x q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x q ÷ b. For example, use a visual fraction model and/or area model to show (2∕3) x 4 = 8∕3, and create a story context for this equation. Do the same with (2∕3) x (4∕5) = 8∕15. (In general, (a∕b) x (c∕d) = ac∕bd.)
5.NF.B.4.bFind the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
5.NF.B.5 Interpret multiplication as scaling (resizing), by:
5.NF.B.5.aComparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. For example, without multiplying tell which number is greater: 25 or 3/4 x 25; 11∕50 or 3∕2 x 11∕50?
5.NF.B.5.bExplaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n x a)/(n x b) to the effect of multiplying a∕b by 1.
5.NF.B.6Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
5.NF.B.7.aInterpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1∕3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1∕3) ÷ 4 = 1∕12 because (1∕12) × 4 = 1∕3.
5.NF.B.7.bInterpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1∕5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1∕5) = 20 because 20 ÷ (1∕5) = 4.
5.NF.B.7.cSolve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if three people share 1/2 lb. of chocolate equally? How many 1∕3-cup servings are in two cups of raisins?
5.OA Operations and Algebraic Thinking
5.OA.A Write and interpret numerical expressions.
5.OA.A.1Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols, e.g.,(6 x 30) + (6 x 1∕2).
5.OA.A.2Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “Add 8 and 7, then multiply by 2” as 2 x (8 + 7). Recognize that 3 x (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
5.OA.B Analyze patterns and relationships.
5.OA.B.3Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Grade 6

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6.EE Expressions and Equations
6.EE.A Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.
6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.A.2.aWrite expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y.
6.EE.A.2.bIdentify parts of an expression using mathematical terms (sum, term, product, factor, quotient, and coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
6.EE.A.2.cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s^3 and A = 6s^2 to find the volume and surface area of a cube with sides of length s = 1/2.
6.EE.A.3Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
6.EE.A.4Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
6.EE.B Reason about and solve one-variable equations and inequalities.
6.EE.B.5Understand solving an equation or inequality as a process of answering a question: Which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.EE.B.6Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.B.7Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers.
6.EE.B.8Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
6.EE.C Represent and analyze quantitative relationships between dependent and independent variables.
6.EE.C.9Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
6.G Geometry
6.G.A Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.A.1Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
6.G.A.2Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
6.G.A.3Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
6.G.A.4Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface areas of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
6.NS The Number System
6.NS.A Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
6.NS.A.1Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2∕3) ÷ (3∕4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2∕3) ÷ (3∕4) = 8∕9 because 3∕4 of 8∕9 is 2∕3. In general, (a∕b) ÷ (c∕d) = ad∕bc. How much chocolate will each person get if three people share 1∕2 lb. of chocolate equally? How many 3∕4-cup servings are in 2∕3 of a cup of yogurt? How wide is a rectangular strip of land with length 3∕4 mile and area 1∕2 square mile?
6.NS.B Compute fluently with multi-digit numbers and find common factors and multiples.
6.NS.B.2Fluently divide multi-digit numbers using the standard algorithm.
6.NS.B.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
6.NS.B.4Use prime factorization to find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two relatively prime numbers. For example, express 36 + 8 as 4(9 + 2).
6.NS.C Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.C.5Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, and positive/negative electric charge). Use positive and negative numbers (whole numbers, fractions, and decimals) to represent quantities in real-world contexts, explaining the meaning of zero in each situation.
6.NS.C.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
6.NS.C.6.aRecognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that zero is its own opposite.
6.NS.C.6.bUnderstand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
6.NS.C.6.cFind and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
6.NS.C.7 Understand ordering and absolute value of rational numbers.
6.NS.C.7.aInterpret statements of inequality as statements about the relative positions of two numbers on a number line diagram. For example, interpret –3 &gt; –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
6.NS.C.7.bWrite, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3°C &gt; –7°C to express the fact that –3°C is warmer than –7°C.
6.NS.C.7.cUnderstand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
6.NS.C.7.dDistinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
6.NS.C.8Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
6.RP Ratios and Proportional Relationships
6.RP.A Understand ratio and rate concepts and use ratio and rate reasoning to solve problems.
6.RP.A.1Understand the concept of a ratio including the distinctions between part:part and part:whole and the value of a ratio; part/part and part/whole. Use ratio language to describe a ratio relationship between two quantities. For example: The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every two wings there was one beak; For every vote candidate A received, candidate C received nearly three votes, meaning that candidate C received three out of every four votes or 3/4 of all votes.
6.RP.A.2Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship, including the use of units. For example: This recipe has a ratio of three cups of flour to four cups of sugar, so there is 3/4 cup of flour for each cup of sugar; We paid $75 for 15 hamburgers, which is a rate of five dollars per hamburger.
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
6.RP.A.3.aMake tables of equivalent ratios relating quantities with whole-number measurements. Find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
6.RP.A.3.bSolve unit rate problems, including those involving unit pricing, and constant speed. For example, if it took seven hours to mow four lawns, then, at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
6.RP.A.3.cFind a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30∕100 times the quantity); solve problems involving finding the whole, given a part and the percent.
6.RP.A.3.dUse ratio reasoning to convert measurement units within and between measurement systems; manipulate and transform units appropriately when multiplying or dividing quantities. For example, Malik is making a recipe, but he cannot find his measuring cups! He has, however, found a tablespoon. His cookbook says that 1 cup = 16 tablespoons. Explain how he could use the tablespoon to measure out the following ingredients: two cups of flour, 1/2 cup sunflower seed, and 1 1/4 cup of oatmeal.
6.RP.A.3.eSolve problems that relate the mass of an object to its volume.
6.SP Statistics and Probability
6.SP.A Develop understanding of statistical variability.
6.SP.A.1Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
6.SP.A.2Understand that a set of data collected to answer a statistical question has a distribution, which can be described by its center (median, mean, and/or mode), spread (range, interquartile range), and overall shape.
6.SP.A.3Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
6.SP.B Summarize and describe distributions.
6.SP.B.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
6.SP.B.4.aRead and interpret circle graphs.
6.SP.B.5 Summarize numerical data sets in relation to their context, such as by:
6.SP.B.5.aReporting the number of observations.
6.SP.B.5.bDescribing the nature of the attribute under investigation, including how it was measured and its units of measurement.
6.SP.B.5.cGiving quantitative measures of center (median, and/or mean) and variability (range and/or interquartile range), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
6.SP.B.5.dRelating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

Grade 7

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7.EE Expressions and Equations
7.EE.A Use properties of operations to generate equivalent expressions.
7.EE.A.1Apply properties of operations to add, subtract, factor, and expand linear expressions with rational coefficients. For example, 4x + 2 = 2(2x +1) and -3(x – 5∕3) = -3x + 5.
7.EE.A.2Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” A shirt at a clothing store is on sale for 20% off the regular price, “p”. The discount can be expressed as 0.2p. The new price for the shirt can be expressed as p – 0.2p or 0.8p.
7.EE.B Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
7.EE.B.3Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example, if a woman making $25 an hour gets a 10% raise, she will make an additional 1∕10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; This estimate can be used as a check on the exact computation.
7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
7.EE.B.4.aSolve word problems leading to equations of the form px + q = r and p(x ÷ q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
7.EE.B.4.bSolve word problems leading to inequalities of the form px + q &gt; r or px + q &lt; 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. For example, as a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
7.EE.B.4.cExtend analysis of patterns to include analyzing, extending, and determining an expression for simple arithmetic and geometric sequences (e.g., compounding, increasing area), using tables, graphs, words, and expressions.
7.G Geometry
7.G.A Draw, construct, and describe geometrical figures and describe the relationships between them.
7.G.A.1Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7.G.A.2Draw (freehand, with ruler and protractor, and with technology) two-dimensional geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
7.G.A.3Describe the shape of the two-dimensional face of the figure that results from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
7.G.B Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
7.G.B.4 Circles and measurement:
7.G.B.4.aKnow that a circle is a two-dimensional shape created by connecting all of the points equidistant from a fixed point called the center of the circle.
7.G.B.4.bUnderstand and describe the relationships among the radius, diameter, circumference and circumference of a circle.
7.G.B.4.cUnderstand and describe the relationship among the radius, diameter, and area of a circle.
7.G.B.4.dKnow the formulas for the area and circumference of a circle and use them to solve problems.
7.G.B.4.eGive an informal derivation of the relationship between the circumference and area of a circle.
7.G.B.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write simple equations and use them to solve for an unknown angle in a figure.
7.G.B.6Solve 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 prisms.
7.NS The Number System
7.NS.A Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract integers and other rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
7.NS.A.1.aDescribe situations in which opposite quantities combine to make zero. For example: A hydrogen atom has zero charge because its two constituents are oppositely charged; If you open a new bank account with a deposit of $30 and then withdraw $30, you are left with a $0 balance.
7.NS.A.1.bUnderstand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
7.NS.A.1.cUnderstand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
7.NS.A.1.dApply properties of operations as strategies to add and subtract rational numbers.
7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide integers and other rational numbers.
7.NS.A.2.aUnderstand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
7.NS.A.2.bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p∕q) = (–p)∕q = p∕(–q). Interpret quotients of rational numbers by describing real-world contexts.
7.NS.A.2.cApply properties of operations as strategies to multiply and divide rational numbers.
7.NS.A.2.dConvert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
7.NS.A.3Solve real-world and mathematical problems involving the four operations with integers and other rational numbers.
7.RP Ratios and Proportional Relationships
7.RP.A Analyze proportional relationships and use them to solve real-world and mathematical problems.
7.RP.A.1Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction (1/2)/(1/4) miles per hour, equivalently 2 miles per hour.
7.RP.A.2 Recognize and represent proportional relationships between quantities.
7.RP.A.2.aDecide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table, or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
7.RP.A.2.bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
7.RP.A.2.cRepresent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
7.RP.A.2.dExplain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
7.RP.A.3Use proportional relationships to solve multi-step ratio, rate, and percent problems. For example: simple interest, tax, price increases and discounts, gratuities and commissions, fees, percent increase and decrease, percent error.
7.SP Statistics and Probability
7.SP.A Use random sampling to draw inferences about a population.
7.SP.A.1Understand that statistics can be used to gain information about a population by examining a sample of the population; Generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
7.SP.A.2Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
7.SP.B Draw informal comparative inferences about two populations.
7.SP.B.3Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team and both distributions have similar variability (mean absolute deviation) of about 5 cm. The difference between the mean heights of the two teams (10 cm) is about twice the variability (5 cm) on either team. On a dot plot, the separation between the two distributions of heights is noticeable.
7.SP.B.4Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
7.SP.C Investigate chance processes and develop, use, and evaluate probability models.
7.SP.C.5Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
7.SP.C.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
7.SP.C.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
7.SP.C.7.aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
7.SP.C.7.bDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
7.SP.C.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
7.SP.C.8.aUnderstand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
7.SP.C.8.bRepresent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.
7.SP.C.8.cDesign and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least four donors to find one with type A blood?

Grade 8

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8.EE Expressions and Equations
8.EE.A Work with radicals and integer exponents.
8.EE.A.1Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^2 x 3^−5 = 3^−3 = (1∕3)^3 = 1∕27.
8.EE.A.2Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
8.EE.A.3Use numbers expressed in the form of a single digit multiplied by an integer power of 10 to estimate very large or very small quantities, and express how many times as much one is than the other. For example, estimate the population of the United States as 3 x 10^8 and the population of the world as 7 x 10^9, and determine that the world population is more than 20 times larger.
8.EE.A.4Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
8.EE.B Understand the connections between proportional relationships, lines, and linear equations.
8.EE.B.5Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
8.EE.B.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
8.EE.C Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.7 Solve linear equations in one variable.
8.EE.C.7.aGive examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
8.EE.C.7.bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
8.EE.C.8 Analyze and solve pairs of simultaneous linear equations.
8.EE.C.8.aUnderstand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
8.EE.C.8.bSolve systems of two linear equations in two variables algebraically (using substitution and elimination strategies), and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
8.EE.C.8.cSolve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
8.F Functions
8.F.A Define, evaluate, and compare functions.
8.F.A.1Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
8.F.A.2Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
8.F.A.3Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s^2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1, 1), (2, 4) and (3, 9), which are not on a straight line.
8.F.B Use functions to model relationships between quantities.
8.F.B.4Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
8.F.B.5Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
8.G Geometry
8.G.A Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations:
8.G.A.1.aLines are transformed to lines, and line segments to line segments of the same length.
8.G.A.1.bAngles are transformed to angles of the same measure.
8.G.A.1.cParallel lines are transformed to parallel lines.
8.G.A.2Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. Given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.A.3Describe the effects of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.A.4Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations. Given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
8.G.A.5Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
8.G.B Understand and apply the Pythagorean Theorem.
8.G.B.6aUnderstand the relationship among the sides of a right triangle.
8.G.B.6bAnalyze and justify the Pythagorean Theorem and its converse using pictures, diagrams, narratives, or models.
8.G.B.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.B.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
8.G.C Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
8.G.C.9Know the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems.
8.NS The Number System
8.NS.A Know that there are numbers that are not rational, and approximate them by rational numbers.
8.NS.A.1Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion. For rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
8.NS.A.2Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π^2). For example, by truncating the decimal expansion of √2 show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
8.SP Statistics and Probability
8.SP.A Investigate patterns of association in bivariate data.
8.SP.A.1Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
8.SP.A.2Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line and informally assess the model fit by judging the closeness of the data points to the line.
8.SP.A.3Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
8.SP.A.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

Algebra - Arithmetic with Polynomials and Rational Expressions

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A-APR.A Perform arithmetic operations on polynomials.
A-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under certain operations.
A-APR.A.1.aPerform operations on polynomial expressions (addition, subtraction, multiplication, division) and compare the system of polynomials to the system of integers when performing operations.
A-APR.A.1.bFactor and/or expand polynomial expressions, identify and combine like terms, and apply the Distributive property.
A-APR.B Understand the relationship between zeros and factors of polynomials.
A-APR.B.2Know 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).
A-APR.B.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
A-APR.C Use polynomial identities to solve problems.
A-APR.C.4Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2 + y^2)^2 = (x^2 – y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples.
A-APR.C.5(+) Know 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, with coefficients determined for example by Pascal’s Triangle.
A-APR.D Rewrite rational expressions.
A-APR.D.6Rewrite simple rational expressions in different forms; write a(x)∕b(x) in the form q(x) + r(x)∕b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
A-APR.D.7(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Algebra - Creating Equations

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A-CED.A Create equations that describe numbers or relationships.
A-CED.A.1Create equations and inequalities in one variable and use them to solve problems. (Include equations arising from linear and quadratic functions, and simple root and rational functions and exponential functions.)
A-CED.A.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A-CED.A.3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
A-CED.A.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance, R.

Algebra - Reasoning with Equations and Inequalities

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A-REI.A Understand solving equations as a process of reasoning and explain the reasoning.
A-REI.A.1Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify or refute a solution method.
A-REI.A.2Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
A-REI.B Solve equations and inequalities in one variable.
A-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
A-REI.B.3.aSolve linear equations and inequalities in one variable involving absolute value.
A-REI.B.4 Solve quadratic equations in one variable.
A-REI.B.4.aUse the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)^2 = q that has the same solutions. Derive the quadratic formula from this form.
A-REI.B.4.bSolve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
A-REI.C Solve systems of equations.
A-REI.C.5Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
A-REI.C.6Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
A-REI.C.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x^2 + y^2 = 3.
A-REI.C.8(+) Represent a system of linear equations as a single matrix equation in a vector variable.
A-REI.C.9(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).
A-REI.D Represent and solve equations and inequalities graphically.
A-REI.D.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Show that any point on the graph of an equation in two variables is a solution to the equation.
A-REI.D.11Explain 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); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
A-REI.D.12Graph the solutions of 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 of a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Algebra - Seeing Structure in Expressions

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A-SSE.A Interpret the structure of exponential, polynomial, and rational expressions.
A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
A-SSE.A.1.aInterpret parts of an expression, such as terms, factors, and coefficients.
A-SSE.A.1.bInterpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)^n as the product of P and a factor not depending on P.
A-SSE.A.2Use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 – (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 – y^2)(x^2 + y^2).
A-SSE.B Write expressions in equivalent forms to solve problems.
A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
A-SSE.B.3.aFactor a quadratic expression to reveal the zeros of the function it defines.
A-SSE.B.3.bComplete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
A-SSE.B.3.cUse the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15^t can be rewritten as (1.15^(1/12))^12t ≈ 1.012^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
A-SSE.B.4Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.

Model Algebra I [AI]

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AI.A-APR Algebra - Arithmetic with Polynomials and Rational Expressions
AI.A-APR.A Perform arithmetic operations on polynomials.
AI.A-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under certain operations.
AI.A-APR.A.1.aPerform operations on polynomial expressions (addition, subtraction, multiplication), and compare the system of polynomials to the system of integers when performing operations.
AI.A-APR.A.1.bFactor and/or expand polynomial expressions, identify and combine like terms, and apply the Distributive property.
AI.A-CED Algebra - Creating Equations
AI.A-CED.A Create equations that describe numbers or relationships.
AI.A-CED.A.1Create equations and inequalities in one variable and use them to solve problems. (Include equations arising from linear, quadratic, and exponential functions with integer exponents.)
AI.A-CED.A.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
AI.A-CED.A.3Represent constraints by linear equations or inequalities, and by systems of linear equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
AI.A-CED.A.4Rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations (Properties of equality). For example, rearrange Ohm’s law R = V^2/P to solve for voltage, V. Manipulate variables in formulas used in financial contexts such as for simple interest, I=Prt.
AI.A-REI Algebra - Reasoning with Equations and Inequalities
AI.A-REI.A Understand solving equations as a process of reasoning and explain the reasoning.
AI.A-REI.A.1Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify or refute a solution method.
AI.A-REI.B Solve equations and inequalities in one variable.
AI.A-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
AI.A-REI.B.3.aSolve linear equations and inequalities in one variable involving absolute value.
AI.A-REI.B.4 Solve quadratic equations in one variable.
AI.A-REI.B.4.aUse the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)^2 = q that has the same solutions. Derive the quadratic formula from this form.
AI.A-REI.B.4.bSolve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the solutions of a quadratic equation results in non-real solutions and write them as a ± bi for real numbers a and b.
AI.A-REI.C Solve systems of equations.
AI.A-REI.C.5Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
AI.A-REI.C.6Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
AI.A-REI.C.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x^2 + y^2 = 3.
AI.A-REI.D Represent and solve equations and inequalities graphically.
AI.A-REI.D.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Show that any point on the graph of an equation in two variables is a solution to the equation.
AI.A-REI.D.11Explain 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); find the solutions approximately, e.g., using technology to graph the functions and make tables of values. Include cases where f(x) and/or g(x) are linear and exponential functions.
AI.A-REI.D.12Graph the solutions of 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 of a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
AI.A-SSE Algebra - Seeing Structure in Expressions
AI.A-SSE.A Interpret the structure of linear, quadratic, and exponential expressions with integer exponents.
AI.A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
AI.A-SSE.A.1.aInterpret parts of an expression, such as terms, factors, and coefficients.
AI.A-SSE.A.1.bInterpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)^t as the product of P and a factor not depending on P.
AI.A-SSE.A.2Use the structure of an expression to identify ways to rewrite it. For example, see (x + 2)^2 – 9 as a difference of squares that can be factored as ((x + 2) + 3)((x + 2 ) – 3).
AI.A-SSE.B Write expressions in equivalent forms to solve problems.
AI.A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
AI.A-SSE.B.3.aFactor a quadratic expression to reveal the zeros of the function it defines.
AI.A-SSE.B.3.bComplete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
AI.A-SSE.B.3.cUse the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15^t can be rewritten as (1.15^(1/12))^12t ≈ 1.012^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
AI.F-BF Functions - Building Functions
AI.F-BF.A Build a function that models a relationship between two quantities.
AI.F-BF.A.1 Write linear, quadratic, and exponential functions that describe a relationship between two quantities.
AI.F-BF.A.1.aDetermine an explicit expression, a recursive process, or steps for calculation from a context.
AI.F-BF.A.1.bCombine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
AI.F-BF.A.2Write arithmetic and geometric sequences both recursively and with an explicit formula them to model situations, and translate between the two forms.
AI.F-BF.B Build new functions from existing functions.
AI.F-BF.B.3Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f (x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Include linear, quadratic, exponential, and absolute value functions. Utilize technology to experiment with cases and illustrate an explanation of the effects on the graph.
AI.F-BF.B.4 Find inverse functions algebraically and graphically.
AI.F-BF.B.4.aSolve an equation of the form f(x) = c for a linear function f that has an inverse and write an expression for the inverse.
AI.F-IF Functions - Interpreting Functions
AI.F-IF.A Understand the concept of a function and use function notation.
AI.F-IF.A.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output (range) of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
AI.F-IF.A.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. For example, given a function representing a car loan, determine the balance of the loan at different points in time.
AI.F-IF.A.3Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n - 1) for n ≥ 1.
AI.F-IF.B Interpret linear, quadratic, and exponential functions with integer exponents that arise in applications in terms of the context.
AI.F-IF.B.4For 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior.
AI.F-IF.B.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
AI.F-IF.B.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
AI.F-IF.C Analyze functions using different representations.
AI.F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
AI.F-IF.C.7.aGraph linear and quadratic functions and show intercepts, maxima, and minima.
AI.F-IF.C.7.bGraph piecewise-defined functions, including step functions and absolute value functions.
AI.F-IF.C.7.eGraph exponential functions showing intercepts and end behavior.
AI.F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
AI.F-IF.C.8.aUse the process of factoring and completing the square in a quadratic function to show zeros, maximum/minimum values, and symmetry of the graph, and interpret these in terms of a context.
AI.F-IF.C.8.bUse the properties of exponents to interpret expressions for exponential functions. Apply to financial situations such as identifying appreciation and depreciation rate for the value of a house or car some time after its initial purchase: V_n = P(1+r)^n. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^12t, and y = (1.2)^(t∕10), and classify them as representing exponential growth or decay.
AI.F-IF.C.9Translate among different representations of functions (algebraically, graphically, numerically in tables, or by verbal descriptions). Compare properties of two functions each represented in a different way. For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
AI.F-LE Functions - Linear, Quadratic, and Exponential Models
AI.F-LE.A Construct and compare linear, quadratic, and exponential models and solve problems.
AI.F-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
AI.F-LE.A.1.aProve that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
AI.F-LE.A.1.bRecognize situations in which one quantity changes at a constant rate per unit interval relative to another.
AI.F-LE.A.1.cRecognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
AI.F-LE.A.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (including reading these from a table).
AI.F-LE.A.3Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
AI.F-LE.B Interpret expressions for functions in terms of the situation they model.
AI.F-LE.B.5Interpret the parameters in a linear or exponential function (of the form f(x) = b^x + k) in terms of a context.
AI.N-Q Number and Quantity - Quantities
AI.N-Q.A Reason quantitatively and use units to solve problems.
AI.N-Q.A.1Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
AI.N-Q.A.2Define appropriate quantities for the purpose of descriptive modeling.
AI.N-Q.A.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
AI.N-RN Number and Quantity - The Real Number System
AI.N-RN.A Extend the properties of exponents to rational exponents.
AI.N-RN.A.1Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^1/3 to be the cube root of 5 because we want (5^1/3)^3 = 5^((1/3)3) to hold, so (5^1/3)^3 must equal 5.
AI.N-RN.A.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.
AI.N-RN.B Use properties of rational and irrational numbers.
AI.N-RN.B.3Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
AI.S-ID Statistics and Probability - Interpreting Categorical and Quantitative Data
AI.S-ID.A Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.
AI.S-ID.A.1Represent data with plots on the real number line (dot plots, histograms, and box plots).
AI.S-ID.A.2Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
AI.S-ID.A.3Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
AI.S-ID.B Summarize, represent, and interpret data on two categorical and quantitative variables.
AI.S-ID.B.5Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
AI.S-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
AI.S-ID.B.6.aFit a linear function to the data and use the fitted function to solve problems in the context of the data. Use functions fitted to data or choose a function suggested by the context (emphasize linear and exponential models).
AI.S-ID.B.6.bInformally assess the fit of a function by plotting and analyzing residuals.
AI.S-ID.B.6.cFit a linear function for a scatter plot that suggests a linear association.
AI.S-ID.C Interpret linear models.
AI.S-ID.C.7Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
AI.S-ID.C.8Compute (using technology) and interpret the correlation coefficient of a linear fit.
AI.S-ID.C.9Distinguish between correlation and causation.

Model Algebra II [AII]

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AII.A-APR Algebra - Arithmetic with Polynomials and Rational Expressions
AII.A-APR.A Perform arithmetic operations on polynomials.
AII.A-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under certain operations.
AII.A-APR.A.1.aPerform operations on polynomial expressions (addition, subtraction, multiplication, and division), and compare the system of polynomials to the system of integers when performing operations.
AII.A-APR.B Understand the relationship between zeros and factors of polynomials.
AII.A-APR.B.2Know 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).
AII.A-APR.B.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
AII.A-APR.C Use polynomial identities to solve problems.
AII.A-APR.C.4Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2 + y^2)^2 = (x^2 – y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples.
AII.A-APR.C.5(+) Know 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, with coefficients determined for example by Pascal’s Triangle.
AII.A-APR.D Rewrite rational expressions.
AII.A-APR.D.6Rewrite simple rational expressions in different forms; write a(x)∕b(x) in the form q(x) + r(x)∕b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
AII.A-APR.D.7(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
AII.A-CED Algebra - Creating Equations
AII.A-CED.A Create equations that describe numbers or relationships.
AII.A-CED.A.1Create equations and inequalities in one variable and use them to solve problems. Include equations arising from simple root and rational functions and exponential functions.
AII.A-CED.A.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
AII.A-CED.A.3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent equations describing satellites orbiting Earth and constraints on Earth’s size and atmosphere.
AII.A-REI Algebra - Reasoning with Equations and Inequalities
AII.A-REI.A Understand solving equations as a process of reasoning and explain the reasoning.
AII.A-REI.A.2Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
AII.A-REI.D Represent and solve equations and inequalities graphically.
AII.A-REI.D.11Explain 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); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are polynomial, rational, and logarithmic functions.
AII.A-SSE Algebra - Seeing Structure in Expressions
AII.A-SSE.A Interpret the structure of exponential, polynomial, and rational expressions.
AII.A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
AII.A-SSE.A.1.aInterpret parts of an expression, such as terms, factors, and coefficients.
AII.A-SSE.A.1.bInterpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)^n as the product of P and a factor not depending on P.
AII.A-SSE.A.2Use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 – (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 – y^2)(x^2 + y^2)and further factored (x-y)(x+y)(x-yi)(x+yi).
AII.A-SSE.B Write expressions in equivalent forms to solve problems.
AII.A-SSE.B.4Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.
AII.F-BF Functions - Building Functions
AII.F-BF.A Build a function that models a relationship between two quantities.
AII.F-BF.A.1 Write a function (simple rational, radical, logarithmic, and trigonometric functions) that describes a relationship between two quantities.
AII.F-BF.A.1.bCombine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
AII.F-BF.B Build new functions from existing functions.
AII.F-BF.B.3Identify 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. Include simple rational, radical, logarithmic, and trigonometric functions. Utilize technology to experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
AII.F-BF.B.4 Find inverse functions algebraically and graphically.
AII.F-BF.B.4.aSolve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x)=2x^3 or f(x)=(x + 1)∕(x - 1) for x ≠ 1.
AII.F-IF Functions - Interpreting Functions
AII.F-IF.B Interpret functions that arise in applications in terms of the context (polynomial, rational, square root and cube root, trigonometric, and logarithmic functions).
AII.F-IF.B.4For 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
AII.F-IF.B.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
AII.F-IF.B.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
AII.F-IF.C Analyze functions using different representations.
AII.F-IF.C.10Given algebraic, numeric and/or graphical representations of functions, recognize the function as polynomial, rational, logarithmic, exponential, or trigonometric.
AII.F-IF.C.7 Graph functions expressed symbolically and show key features of the graph; by hand in simple cases and using technology for more complicated cases.
AII.F-IF.C.7.bGraph square root and cube root functions.
AII.F-IF.C.7.cGraph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
AII.F-IF.C.7.eGraph exponential and logarithmic functions, showing intercepts and end behavior; and trigonometric functions, showing period, midline, and amplitude.
AII.F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
AII.F-IF.C.8.aUse the process of factoring in a polynomial to function to show zeros, extreme values, and symmetry of the graph and interpret these in terms of a context.
AII.F-IF.C.9Translate among different representations of functions (algebraically, graphically, numerically in tables, or by verbal descriptions). Compare properties of two functions each represented in a different way. For example, given a graph of one polynomial function and an algebraic expression for another, say which has the larger relative maximum and/or smaller relative minimum.
AII.F-LE Functions - Linear, Quadratic, and Exponential Models
AII.F-LE.A Construct and compare linear, quadratic, and exponential models and solve problems.
AII.F-LE.A.4For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
AII.F-TF Functions - Trigonometric Functions
AII.F-TF.A Extend the domain of trigonometric functions using the unit circle.
AII.F-TF.A.1Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
AII.F-TF.A.2Explain 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.
AII.F-TF.B Model periodic phenomena with trigonometric functions.
AII.F-TF.B.5Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
AII.F-TF.C Prove and apply trigonometric identities.
AII.F-TF.C.8Prove the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant.
AII.N-CN Number and Quantity - The Complex Number System
AII.N-CN.A Perform arithmetic operations with complex numbers.
AII.N-CN.A.1Know there is a complex number i such that i^2 = −1, and every complex number has the form a + bi with x-a and b real.
AII.N-CN.A.2Use the relation i^2 = –1 and the Commutative, Associative, and Distributive properties to add, subtract, and multiply complex numbers.
AII.N-CN.C Use complex numbers in polynomial identities and equations.
AII.N-CN.C.7Solve quadratic equations with real coefficients that have complex solutions.
AII.N-CN.C.8(+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x – 2i).
AII.N-CN.C.9(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
AII.N-VM Number and Quantity - Vector and Matrix Quantities
AII.N-VM.A Represent and model with vector quantities.
AII.N-VM.A.1(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
AII.N-VM.A.3(+) Solve problems involving velocity and other quantities that can be represented by vectors.
AII.N-VM.C Perform operations on matrices and use matrices in applications.
AII.N-VM.C.12(+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
AII.N-VM.C.6(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
AII.N-VM.C.8(+) Add, subtract, and multiply matrices of appropriate dimensions.
AII.S-IC Statistics and Probability - Making Inferences and Justifying Conclusions
AII.S-IC.A Understand and evaluate random processes underlying statistical experiments.
AII.S-IC.A.1Understand statistics as a process for making inferences to be made about population parameters based on a random sample from that population.
AII.S-IC.A.2Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of five tails in a row cause you to question the model?
AII.S-IC.B Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
AII.S-IC.B.3Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
AII.S-IC.B.4Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
AII.S-IC.B.5Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
AII.S-IC.B.6Evaluate reports based on data.
AII.S-ID Statistics and Probability - Interpreting Categorical and Quantitative Data
AII.S-ID.A Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.
AII.S-ID.A.4Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
AII.S-MD Statistics and Probability - Using Probability to Make Decisions
AII.S-MD.B Use probability to evaluate outcomes of decisions.
AII.S-MD.B.6(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
AII.S-MD.B.7(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game and replacing the goalie with an extra skater).

Model Advanced Quantitative Reasoning [AQR]

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AQR.A-APR Algebra - Arithmetic with Polynomials and Rational Expressions
AQR.A-APR.C Use polynomial identities to solve problems.
AQR.A-APR.C.5(+) Know 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, with coefficients determined for example by Pascal’s Triangle.
AQR.A-REI Algebra - Reasoning with Equations and Inequalities
AQR.A-REI.C Solve systems of equations.
AQR.A-REI.C.8(+) Represent a system of linear equations as a single matrix equation in a vector variable.
AQR.A-REI.C.9(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).
AQR.F-TF Functions - Trigonometric Functions
AQR.F-TF.A Extend the domain of trigonometric functions using the unit circle.
AQR.F-TF.A.3(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π∕3, π∕4 and π∕6, and use the unit circle to express the values of sine, cosine, and tangent for π - x, π + x, and 2π - x in terms of their values for x, where x is any real number.
AQR.F-TF.A.4(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
AQR.F-TF.B Model periodic phenomena with trigonometric functions.
AQR.F-TF.B.7(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
AQR.F-TF.C Prove and apply trigonometric identities.
AQR.F-TF.C.9(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
AQR.G-C Geometry - Circles
AQR.G-C.A Understand and apply theorems about circles.
AQR.G-C.A.4(+) Construct a tangent line from a point outside a given circle to the circle.
AQR.G-GMD Geometry - Geometric Measurement and Dimension
AQR.G-GMD.A Explain volume formulas and use them to solve problems.
AQR.G-GMD.A.2(+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
AQR.G-GPE Geometry - Expressing Geometric Properties with Equations
AQR.G-GPE.A Translate between the geometric description and the equation for a conic section.
AQR.G-GPE.A.3 (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
AQR.G-GPE.A.3.a(+) Use equations and graphs of conic sections to model real-world problems.
AQR.G-SRT Geometry - Similarity, Right Triangles, and Trigonometry
AQR.G-SRT.D Apply trigonometry to general triangles.
AQR.G-SRT.D.11(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
AQR.N-VM Number and Quantity - Vector and Matrix Quantities
AQR.N-VM.A Represent and model with vector quantities.
AQR.N-VM.A.1(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
AQR.N-VM.A.2(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
AQR.N-VM.A.3(+) Solve problems involving velocity and other quantities that can be represented by vectors.
AQR.N-VM.C Perform operations on matrices and use matrices in applications.
AQR.N-VM.C.10(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
AQR.N-VM.C.11(+) Multiply 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.
AQR.N-VM.C.12(+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
AQR.N-VM.C.6(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
AQR.N-VM.C.7(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
AQR.N-VM.C.8(+) Add, subtract, and multiply matrices of appropriate dimensions.
AQR.N-VM.C.9(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a Commutative operation, but still satisfies the Associative and Distributive properties.
AQR.S-CP Statistics and Probability - Conditional Probability and the Rules of Probability
AQR.S-CP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model.
AQR.S-CP.B.8(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
AQR.S-CP.B.9(+) Use permutations and combinations to compute probabilities of compound events and solve problems.
AQR.S-MD Statistics and Probability - Using Probability to Make Decisions
AQR.S-MD.A Calculate expected values and use them to solve problems.
AQR.S-MD.A.1(+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
AQR.S-MD.A.2(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
AQR.S-MD.A.3(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
AQR.S-MD.A.4(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?
AQR.S-MD.B Use probability to evaluate outcomes of decisions.
AQR.S-MD.B.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
AQR.S-MD.B.5.a(+) Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.
AQR.S-MD.B.5.b(+) Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
AQR.S-MD.B.6(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
AQR.S-MD.B.7(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game and replacing the goalie with an extra skater).

High School Content Standards by Conceptual Categories

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Functions - Building Functions

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F-BF.A Build a function that models a relationship between two quantities.
F-BF.A.1 Write a function (linear, quadratic, exponential, simple rational, radical, logarithmic, and trigonometric) that describes a relationship between two quantities.
F-BF.A.1.aDetermine an explicit expression, a recursive process, or steps for calculation from a context.
F-BF.A.1.bCombine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
F-BF.A.1.c(+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
F-BF.A.2Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
F-BF.B Build new functions from existing functions.
F-BF.B.3Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x, f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. (Include linear, quadratic, exponential, absolute value, simple rational and radical, logarithmic and trigonometric functions.) Utilize technology to experiment with cases and illustrate an explanation of the effects on the graph. (Include recognizing even and odd functions from their graphs and algebraic expressions for them.)
F-BF.B.4 Find inverse functions algebraically and graphically.
F-BF.B.4.aSolve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. (Include linear and simple polynomial, rational, and exponential functions.) For example, f(x) =2x^3 or f(x) = (x + 1)∕(x - 1) for x ≠ 1.
F-BF.B.4.b(+) Verify by composition that one function is the inverse of another.
F-BF.B.4.c(+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
F-BF.B.4.d(+) Produce an invertible function from a non-invertible function by restricting the domain.
F-BF.B.5(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Functions - Interpreting Functions

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F-IF.A Understand the concept of a function and use function notation.
F-IF.A.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
F-IF.A.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. For example, given a function representing a car loan, determine the balance of the loan at different points in time.
F-IF.A.3Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n - 1) for n ≥ 1.
F-IF.B Interpret functions that arise in applications in terms of the context (linear, quadratic, exponential, rational, polynomial, square root, cube root, trigonometric, logarithmic).
F-IF.B.4For 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
F-IF.B.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
F-IF.B.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
F-IF.C Analyze functions using different representations.
F-IF.C.10Given algebraic, numeric and/or graphical representations of functions, recognize the function as polynomial, rational, logarithmic, exponential, or trigonometric.
F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.C.7.aGraph linear and quadratic functions and show intercepts, maxima, and minima.
F-IF.C.7.bGraph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
F-IF.C.7.cGraph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
F-IF.C.7.d(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
F-IF.C.7.eGraph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F-IF.C.8.aUse the process of factoring and/or completing the square in quadratic and polynomial functions, where appropriate, to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
F-IF.C.8.bUse the properties of exponents to interpret expressions for exponential functions. Apply to financial situations such as identifying appreciation and depreciation rate for the value of a house or car some time after its initial purchase. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^12t, and y = (1.2)^(t∕10), and classify them as representing exponential growth or decay.
F-IF.C.9Translate among different representations of functions (algebraically, graphically, numerically in tables, or by verbal descriptions). Compare properties of two functions each represented in a different way. For example, given a graph of one polynomial function (including quadratic functions) and an algebraic expression for another, say which has the larger/smaller relative maximum and/or minimum.

Functions - Linear, Quadratic, and Exponential Models

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F-LE.A Construct and compare linear, quadratic, and exponential models and solve problems.
F-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
F-LE.A.1.aProve that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
F-LE.A.1.bRecognize situations in which one quantity changes at a constant rate per unit interval relative to another.
F-LE.A.1.cRecognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
F-LE.A.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (including reading these from a table).
F-LE.A.3Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
F-LE.A.4For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
F-LE.B Interpret expressions for functions in terms of the situation they model.
F-LE.B.5Interpret the parameters in a linear or exponential function (of the form f(x) = b^x + k) in terms of a context.

Functions - Trigonometric Functions

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F-TF.A Extend the domain of trigonometric functions using the unit circle.
F-TF.A.1Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
F-TF.A.2Explain 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.
F-TF.A.3(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π - x, π + x, and 2π - x in terms of their values for x, where x is any real number.
F-TF.A.4(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
F-TF.B Model periodic phenomena with trigonometric functions.
F-TF.B.5Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
F-TF.B.6(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
F-TF.B.7(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
F-TF.C Prove and apply trigonometric identities.
F-TF.C.8Prove the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant.
F-TF.C.9(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Geometry - Circles

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G-C.A Understand and apply theorems about circles.
G-C.A.1Prove that all circles are similar.
G-C.A.2Identify and describe relationships among inscribed angles, radii, and chords. Include 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-C.A.3Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral and other polygons inscribed in a circle.
G-C.A.4(+) Construct a tangent line from a point outside a given circle to the circle.
G-C.B Find arc lengths and areas of sectors of circles.
G-C.B.5Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Geometry - Congruence

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G-CO.A Experiment with transformations in the plane.
G-CO.A.1Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G-CO.A.2Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
G-CO.A.3Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
G-CO.A.4Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G-CO.A.5Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
G-CO.B Understand congruence in terms of rigid motions.
G-CO.B.6Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
G-CO.B.7Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G-CO.B.8Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G-CO.C Prove geometric theorems and, when appropriate, the converse of theorems.
G-CO.C.10Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent, and conversely prove a triangle is isosceles; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; and the medians of a triangle meet at a point.
G-CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
G-CO.C.11.aProve theorems about polygons. Theorems include the measures of interior and exterior angles. Apply properties of polygons to the solutions of mathematical and contextual problems.
G-CO.C.9Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent, and conversely prove lines are parallel; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
G-CO.D Make geometric constructions.
G-CO.D.12Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Constructions include: copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
G-CO.D.13Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Geometry - Geometric Measurement and Dimension

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Geometry - Expressing Geometric Properties with Equations

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G-GPE.A Translate between the geometric description and the equation for a conic section.
G-GPE.A.1Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
G-GPE.A.2Derive the equation of a parabola given a focus and directrix.
G-GPE.A.3 (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
G-GPE.A.3.a(+) Use equations and graphs of conic sections to model real-world problems.
G-GPE.B Use coordinates to prove simple geometric theorems algebraically.
G-GPE.B.4Use coordinates to prove simple geometric theorems algebraically including the distance formula and its relationship to the Pythagorean Theorem. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1,√3) lies on the circle centered at the origin and containing the point (0, 2).
G-GPE.B.5Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
G-GPE.B.6Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G-GPE.B.7Use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g., using the distance formula).

Geometry - Modeling with Geometry

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Geometry - Similarity, Right Triangles, and Trigonometry

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G-SRT.A Understand similarity in terms of similarity transformations.
G-SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor:
G-SRT.A.1.aA 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-SRT.A.1.bThe dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G-SRT.A.2Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G-SRT.A.3Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar.
G-SRT.B Prove theorems involving similarity.
G-SRT.B.4Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
G-SRT.B.5Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
G-SRT.C Define trigonometric ratios and solve problems involving right triangles.
G-SRT.C.6Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G-SRT.C.7Explain and use the relationship between the sine and cosine of complementary angles.
G-SRT.C.8Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G-SRT.D Apply trigonometry to general triangles.
G-SRT.D.10(+) Prove the Laws of Sines and Cosines and use them to solve problems.
G-SRT.D.11(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
G-SRT.D.9(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Model Geometry [GEO]

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GEO.G-C Geometry - Circles
GEO.G-C.A Understand and apply theorems about circles.
GEO.G-C.A.1Prove that all circles are similar.
GEO.G-C.A.2Identify and describe relationships among inscribed angles, radii, and chords. Include 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.
GEO.G-C.A.3Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral and other polygons inscribed in a circle.
GEO.G-C.A.4(+) Construct a tangent line from a point outside a given circle to the circle.
GEO.G-C.B Find arc lengths and areas of sectors of circles.
GEO.G-C.B.5Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
GEO.G-CO Geometry - Congruence
GEO.G-CO.A Experiment with transformations in the plane.
GEO.G-CO.A.1Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
GEO.G-CO.A.2Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
GEO.G-CO.A.3Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
GEO.G-CO.A.4Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
GEO.G-CO.A.5Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
GEO.G-CO.B Understand congruence in terms of rigid motions.
GEO.G-CO.B.6Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
GEO.G-CO.B.7Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
GEO.G-CO.B.8Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
GEO.G-CO.C Prove geometric theorems and, when appropriate, the converse of theorems.
GEO.G-CO.C.10Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent, and conversely prove a triangle is isosceles; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
GEO.G-CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
GEO.G-CO.C.11.aProve theorems about polygons. Theorems include the measures of interior and exterior angles. Apply properties of polygons to the solutions of mathematical and contextual problems.
GEO.G-CO.C.9Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent, and conversely prove lines are parallel; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
GEO.G-CO.D Make geometric constructions.
GEO.G-CO.D.12Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Constructions include: copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
GEO.G-CO.D.13Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
GEO.G-GMD Geometry - Geometric Measurement and Dimension
GEO.G-GMD.A Explain volume formulas and use them to solve problems.
GEO.G-GMD.A.1Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
GEO.G-GMD.A.2(+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
GEO.G-GMD.A.3Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
GEO.G-GMD.B Visualize relationships between two-dimensional and three-dimensional objects.
GEO.G-GMD.B.4Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
GEO.G-GPE Geometry - Expressing Geometric Properties with Equations
GEO.G-GPE.A Translate between the geometric description and the equation for a conic section.
GEO.G-GPE.A.1Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
GEO.G-GPE.A.2Derive the equation of a parabola given a focus and directrix.
GEO.G-GPE.B Use coordinates to prove simple geometric theorems algebraically.
GEO.G-GPE.B.4Use coordinates to prove simple geometric theorems algebraically, including the distance formula and its relationship to the Pythagorean Theorem. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1,√3) lies on the circle centered at the origin and containing the point (0,2).
GEO.G-GPE.B.5Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
GEO.G-GPE.B.6Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
GEO.G-GPE.B.7Use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g., using the distance formula).
GEO.G-MG Geometry - Modeling with Geometry
GEO.G-MG.A Apply geometric concepts in modeling situations.
GEO.G-MG.A.1Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
GEO.G-MG.A.2Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
GEO.G-MG.A.3Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
GEO.G-MG.A.4Use dimensional analysis for unit conversions to confirm that expressions and equations make sense.
GEO.G-SRT Geometry - Similarity, Right Triangles, and Trigonometry
GEO.G-SRT.A Understand similarity in terms of similarity transformations.
GEO.G-SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor:
GEO.G-SRT.A.1.aA 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.
GEO.G-SRT.A.1.bThe dilation of a line segment is longer or shorter in the ratio given by the scale factor.
GEO.G-SRT.A.2Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
GEO.G-SRT.A.3Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar.
GEO.G-SRT.B Prove theorems involving similarity.
GEO.G-SRT.B.4Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
GEO.G-SRT.B.5Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
GEO.G-SRT.C Define trigonometric ratios and solve problems involving right triangles.
GEO.G-SRT.C.6Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
GEO.G-SRT.C.7Explain and use the relationship between the sine and cosine of complementary angles.
GEO.G-SRT.C.8Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
GEO.G-SRT.D Apply trigonometry to general triangles.
GEO.G-SRT.D.10(+) Prove the Laws of Sines and Cosines and use them to solve problems.
GEO.G-SRT.D.11(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
GEO.G-SRT.D.9(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
GEO.N-Q Number and Quantity - Quantities
GEO.N-Q.A Reason quantitatively and use units to solve problems.
GEO.N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
GEO.N-Q.A.3.aDescribe the effects of approximate error in measurement and rounding on measurements and on computed values from measurements. Identify significant figures in recorded measures and computed values based on the context given and the precision of the tools used to measure.
GEO.S-CP Statistics and Probability - Conditional Probability and the Rules of Probability
GEO.S-CP.A Understand independence and conditional probability and use them to interpret data from simulations or experiments.
GEO.S-CP.A.1Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
GEO.S-CP.A.2Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
GEO.S-CP.A.3Understand the conditional probability of A given B as P(A and B)∕P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
GEO.S-CP.A.4Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
GEO.S-CP.A.5Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
GEO.S-CP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model.
GEO.S-CP.B.6Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
GEO.S-CP.B.7Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
GEO.S-CP.B.8(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
GEO.S-CP.B.9(+) Use permutations and combinations to compute probabilities of compound events and solve problems.

Model Mathematics I [MI]

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MI.A-CED Algebra - Creating Equations
MI.A-CED.A Create equations that describe numbers or relationships.
MI.A-CED.A.1Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and exponential functions with integer exponents.
MI.A-CED.A.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
MI.A-CED.A.3Represent constraints by linear equations or inequalities, and by systems of linear equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
MI.A-CED.A.4Rearrange formulas to highlight a quantity of interest, using the same reasoning (Properties of equality) as in solving equations. For example, rearrange Ohm’s law, V = IR, to solve for resistance, R. Manipulate variables in formulas used in financial contexts such as for simple interest, I=Prt.
MI.A-REI Algebra - Reasoning with Equations and Inequalities
MI.A-REI.A Understand solving equations as a process of reasoning and explain the reasoning.
MI.A-REI.A.1Explain each step in solving a simple linear equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify or refute a solution method.
MI.A-REI.B Solve equations and inequalities in one variable.
MI.A-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
MI.A-REI.B.3.aSolve linear equations and inequalities in one variable involving absolute value.
MI.A-REI.C Solve systems of equations.
MI.A-REI.C.5Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
MI.A-REI.C.6Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
MI.A-REI.D Represent and solve equations and inequalities graphically.
MI.A-REI.D.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Show that any point on the graph of an equation in two variables is a solution to the equation.
MI.A-REI.D.11Explain 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); find the solutions approximately, e.g., using technology to graph the functions and/or make tables of values. Include cases where f(x) and/or g(x) are linear and exponential functions.
MI.A-REI.D.12Graph the solutions of 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 of a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
MI.A-SSE Algebra - Seeing Structure in Expressions
MI.A-SSE.A Interpret the structure of linear and exponential expressions with integer exponents.
MI.A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
MI.A-SSE.A.1.aInterpret parts of an expression, such as terms, factors, and coefficients.
MI.A-SSE.A.1.bInterpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)^n as the product of P and a factor not depending on P.
MI.F-BF Functions - Building Functions
MI.F-BF.A Build a function that models a relationship between two quantities.
MI.F-BF.A.1 Write linear and exponential functions that describe a relationship between two quantities.
MI.F-BF.A.1.aDetermine an explicit expression, a recursive process, or steps for calculation from a context.
MI.F-BF.A.1.bCombine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
MI.F-BF.A.2Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
MI.F-BF.B Build new functions from existing functions.
MI.F-BF.B.3Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Include linear and exponential models. (Focus on vertical translations for exponential functions). Utilize technology to experiment with cases and illustrate an explanation of the effects on the graph.
MI.F-IF Functions - Interpreting Functions
MI.F-IF.A Understand the concept of a function and use function notation.
MI.F-IF.A.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
MI.F-IF.A.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. For example, given a function representing a car loan, determine the balance of the loan at different points in time.
MI.F-IF.A.3Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n - 1) for n ≥ 1.
MI.F-IF.B Interpret linear and exponential functions having integer exponents that arise in applications in terms of the context.
MI.F-IF.B.4For 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior.
MI.F-IF.B.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
MI.F-IF.B.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
MI.F-IF.C Analyze functions using different representations.
MI.F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
MI.F-IF.C.7.aGraph linear functions and show intercepts.
MI.F-IF.C.7.eGraph exponential functions, showing intercepts and end behavior.
MI.F-IF.C.9Translate among different representations of functions: (algebraically, graphically, numerically in tables, or by verbal descriptions). Compare properties of two functions each represented in a different way. For example, given a graph of one exponential function and an algebraic expression for another, say which has the larger y-intercept.
MI.F-LE Functions - Linear, Quadratic, and Exponential Models
MI.F-LE.A Construct and compare linear and exponential models and solve problems.
MI.F-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
MI.F-LE.A.1.aProve that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
MI.F-LE.A.1.bRecognize situations in which one quantity changes at a constant rate per unit interval relative to another.
MI.F-LE.A.1.cRecognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
MI.F-LE.A.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (including reading these from a table).
MI.F-LE.A.3Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly.
MI.F-LE.B Interpret expressions for functions in terms of the situation they model.
MI.F-LE.B.5Interpret the parameters in a linear function or exponential function (of the form f(x) = bx + k) in terms of a context.
MI.G-CO Geometry - Congruence
MI.G-CO.A Experiment with transformations in the plane.
MI.G-CO.A.1Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
MI.G-CO.A.2Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
MI.G-CO.A.3Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
MI.G-CO.A.4Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
MI.G-CO.A.5Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
MI.G-CO.B Understand congruence in terms of rigid motions.
MI.G-CO.B.6Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
MI.G-CO.B.7Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
MI.G-CO.B.8Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
MI.G-CO.D Make geometric constructions.
MI.G-CO.D.12Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
MI.G-CO.D.13Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
MI.G-GPE Geometry - Expressing Geometric Properties with Equations
MI.G-GPE.B Use coordinates to prove simple geometric theorems algebraically.
MI.G-GPE.B.5Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
MI.G-GPE.B.7Use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g., using the distance formula).
MI.N-Q Number and Quantity - Quantities
MI.N-Q.A Reason quantitatively and use units to solve problems.
MI.N-Q.A.1Use units as a way to understand problems; and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
MI.N-Q.A.2Define appropriate quantities for the purpose of descriptive modeling.
MI.N-Q.A.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
MI.S-ID Statistics and Probability - Interpreting Categorical and Quantitative Data
MI.S-ID.A Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.
MI.S-ID.A.1Represent data with plots on the real number line (dot plots, histograms, and box plots).
MI.S-ID.A.2Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
MI.S-ID.A.3Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
MI.S-ID.B Summarize, represent, and interpret data on two categorical and quantitative variables.
MI.S-ID.B.5Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
MI.S-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
MI.S-ID.B.6.aFit a linear function to the data and use the fitted function to solve problems in the context of the data. Use given functions fitted to data or choose a function suggested by the context. Emphasize linear and exponential models.
MI.S-ID.B.6.bInformally assess the fit of a function by plotting and analyzing residuals.
MI.S-ID.B.6.cFit a linear function for a scatter plot that suggests a linear association.
MI.S-ID.C Interpret linear models.
MI.S-ID.C.7Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
MI.S-ID.C.8Compute (using technology) and interpret the correlation coefficient of a linear fit.
MI.S-ID.C.9Distinguish between correlation and causation.

Model Mathematics II [MII]

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MII.A-APR Algebra - Arithmetic with Polynomials and Rational Expressions
MII.A-APR.A Perform arithmetic operations on polynomials.
MII.A-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under certain operations.
MII.A-APR.A.1.aPerform operations on polynomial expressions (addition, subtraction, multiplication), and compare the system of polynomials to the system of integers when performing operations.
MII.A-APR.A.1.bFactor and/or expand polynomial expressions; identify and combine like terms; and apply the Distributive property.
MII.A-CED Algebra - Creating Equations
MII.A-CED.A Create equations that describe numbers or relationships.
MII.A-CED.A.1Create equations and inequalities in one variable and use them to solve problems. Include equations arising from quadratic and exponential functions.
MII.A-CED.A.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
MII.A-CED.A.4Rearrange formulas, including formulas with quadratic terms, to highlight a quantity of interest using the same reasoning as in solving equations (Properties of equality). For example, rearrange Ohm’s law R = V^2/p to solve for voltage, V.
MII.A-REI Algebra - Reasoning with Equations and Inequalities
MII.A-REI.B Solve equations and inequalities in one variable.
MII.A-REI.B.4 Solve quadratic equations in one variable.
MII.A-REI.B.4.aUse the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)^2 = q that has the same solutions. Derive the quadratic formula from this form.
MII.A-REI.B.4.bSolve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
MII.A-REI.C Solve systems of equations.
MII.A-REI.C.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x^2 + y^2 = 3.
MII.A-SSE Algebra - Seeing Structure in Expressions
MII.A-SSE.A Interpret the structure of quadratic and exponential expressions.
MII.A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
MII.A-SSE.A.1.aInterpret parts of an expression, such as terms, factors, and coefficients.
MII.A-SSE.A.1.bInterpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)^t as the product of P and a factor not depending on P.
MII.A-SSE.A.2Use the structure of an expression to identify ways to rewrite it. For example, see (x + 2)^2 – 9 as a difference of squares that can be factored as ((x + 2) + 3)((x + 2) – 3).
MII.A-SSE.B Write quadratic and exponential expressions in equivalent forms to solve problems.
MII.A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
MII.A-SSE.B.3.aFactor a quadratic expression to reveal the zeros of the function it defines.
MII.A-SSE.B.3.bComplete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
MII.A-SSE.B.3.cUse the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15^t can be rewritten as (1.15^(1/12))^12t ≈ 1.012^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
MII.F-BF Functions - Building Functions
MII.F-BF.A Build a function that models a relationship between two quantities.
MII.F-BF.A.1 Write linear, quadratic, and exponential functions that describe relationships between two quantities.
MII.F-BF.A.1.aDetermine an explicit expression, a recursive process, or steps for calculation from a context.
MII.F-BF.A.1.bCombine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
MII.F-BF.B Build new functions from existing functions.
MII.F-BF.B.3Identify 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. Include exponential, quadratic, and absolute value functions. Utilize technology to experiment with cases and illustrate an explanation of the effects on the graph.
MII.F-BF.B.4 Find inverse functions algebraically and graphically.
MII.F-BF.B.4.aSolve an equation of the form f(x) = c for a linear function f that has an inverse and write an expression for the inverse.
MII.F-IF Functions - Interpreting Functions
MII.F-IF.B Interpret quadratic and exponential functions with integer exponents that arise in applications in terms of the context.
MII.F-IF.B.4For 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior.
MII.F-IF.B.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
MII.F-IF.B.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
MII.F-IF.C Analyze functions using different representations.
MII.F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
MII.F-IF.C.7.aGraph quadratic functions and show intercepts, maxima, and minima.
MII.F-IF.C.7.bGraph piecewise-defined functions, including step functions and absolute value functions.
MII.F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
MII.F-IF.C.8.aUse the process of factoring and completing the square in a quadratic function to show zeros, minimum/maximum values, and symmetry of the graph and interpret these in terms of a context.
MII.F-IF.C.8.bUse the properties of exponents to interpret expressions for exponential functions. Apply to financial situations such as Identifying appreciation/depreciation rate for the value of a house or car some time after its initial purchase:. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^12t, and y = (1.2)^t∕10, and classify them as representing exponential growth or decay.
MII.F-IF.C.9Translate among different representations of functions (algebraically, graphically, numerically in tables, or by verbal descriptions). Compare properties of two functions each represented in a different way. For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
MII.F-LE Functions - Linear, Quadratic, and Exponential Models
MII.F-LE.A Construct and compare linear, quadratic, and exponential models and solve problems.
MII.F-LE.A.3Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
MII.G-C Geometry - Circles
MII.G-C.A Understand and apply theorems about circles.
MII.G-C.A.1Prove that all circles are similar.
MII.G-C.A.2Identify and describe relationships among inscribed angles, radii, and chords. Include 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.
MII.G-C.A.3Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral and other polygons inscribed in a circle.
MII.G-C.A.4(+) Construct a tangent line from a point outside a given circle to the circle.
MII.G-C.B Find arc lengths and areas of sectors of circles.
MII.G-C.B.5Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
MII.G-CO Geometry - Congruence
MII.G-CO.C Prove geometric theorems and, when appropriate, the converse of theorems.
MII.G-CO.C.10Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent, and conversely prove a triangle is isosceles; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
MII.G-CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
MII.G-CO.C.11.aProve theorems about polygons. Theorems include the measures of interior and exterior angles. Apply properties of polygons to the solutions of mathematical and contextual problems.
MII.G-CO.C.9Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent, and conversely prove lines are parallel; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
MII.G-GMD Geometry - Geometric Measurement and Dimension
MII.G-GMD.A Explain volume formulas and use them to solve problems.
MII.G-GMD.A.1Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
MII.G-GMD.A.2(+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
MII.G-GMD.A.3Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
MII.G-GPE Geometry - Expressing Geometric Properties with Equations
MII.G-GPE.A Translate between the geometric description and the equation for a conic section.
MII.G-GPE.A.1Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
MII.G-GPE.A.2Derive the equation of a parabola given a focus and directrix.
MII.G-GPE.B Use coordinates to prove simple geometric theorems algebraically.
MII.G-GPE.B.4Use coordinates to prove simple geometric theorems algebraically including the distance formula and its relationship to the Pythagorean Theorem. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1,√3) lies on the circle centered at the origin and containing the point (0, 2).
MII.G-GPE.B.6Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
MII.G-SRT Geometry - Similarity, Right Triangles, and Trigonometry
MII.G-SRT.A Understand similarity in terms of similarity transformations.
MII.G-SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor:
MII.G-SRT.A.1.aA 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.
MII.G-SRT.A.1.bThe dilation of a line segment is longer or shorter in the ratio given by the scale factor.
MII.G-SRT.A.2Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
MII.G-SRT.A.3Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar.
MII.G-SRT.B Prove theorems involving similarity using a variety of ways of writing proofs, showing validity of underlying reasoning.
MII.G-SRT.B.4Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
MII.G-SRT.B.5Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
MII.G-SRT.C Define trigonometric ratios and solve problems involving right triangles.
MII.G-SRT.C.6Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
MII.G-SRT.C.7Explain and use the relationship between the sine and cosine of complementary angles.
MII.G-SRT.C.8Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
MII.N-CN Number and Quantity - The Complex Number System
MII.N-CN.A Perform arithmetic operations with complex numbers.
MII.N-CN.A.1Know there is a complex number i such that i^2 = −1, and every complex number has the form a + bi with a and b real.
MII.N-CN.A.2Use the relation i^2 = –1 and the Commutative, Associative, and Distributive properties to add, subtract, and multiply complex numbers.
MII.N-CN.C Use complex numbers in polynomial identities and equations.
MII.N-CN.C.7Solve quadratic equations with real coefficients that have complex solutions.
MII.N-Q Number and Quantity - Quantities
MII.N-Q.A Reason quantitatively and use units to solve problems.
MII.N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
MII.N-Q.A.3.aDescribe the effects of approximate error in measurement and rounding on measurements and on computed values from measurements. Identify significant figures in recorded measures and computed values based on the context given and the precision of the tools used to measure
MII.N-RN Number and Quantity - The Real Number System
MII.N-RN.A Extend the properties of exponents to rational exponents.
MII.N-RN.A.1Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))3 = 5^((1/3)3) to hold, so (5^(1/3))^3 must equal 5.
MII.N-RN.A.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.
MII.N-RN.B Use properties of rational and irrational numbers.
MII.N-RN.B.3Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
MII.S-CP Statistics and Probability - Conditional Probability and the Rules of Probability
MII.S-CP.A Understand independence and conditional probability and use them to interpret data from simulations or experiments.
MII.S-CP.A.1Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
MII.S-CP.A.2Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
MII.S-CP.A.3Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
MII.S-CP.A.4Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
MII.S-CP.A.5Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
MII.S-CP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model.
MII.S-CP.B.6Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
MII.S-CP.B.7Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
MII.S-CP.B.8(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
MII.S-CP.B.9(+) Use permutations and combinations to compute probabilities of compound events and solve problems.

Model Mathematics III [MIII]

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MIII.A-APR Algebra - Arithmetic with Polynomials and Rational Expressions
MIII.A-APR.A Perform arithmetic operations on polynomials.
MIII.A-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under certain operations.
MIII.A-APR.A.1.aPerform operations on polynomial expressions (addition, subtraction, multiplication, and division), and compare the system of polynomials to the system of integers when performing operations.
MIII.A-APR.B Understand the relationship between zeros and factors of polynomials.
MIII.A-APR.B.2Know 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).
MIII.A-APR.B.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
MIII.A-APR.C Use polynomial identities to solve problems.
MIII.A-APR.C.4Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2 + y^2)^2 = (x^2 – y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples.
MIII.A-APR.C.5(+) Know 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, with coefficients determined for example by Pascal’s Triangle.
MIII.A-APR.D Rewrite rational expressions.
MIII.A-APR.D.6Rewrite simple rational expressions in different forms; write a(x)∕b(x) in the form q(x) + r(x)∕b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
MIII.A-APR.D.7(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
MIII.A-CED Algebra - Creating Equations
MIII.A-CED.A Create equations that describe numbers or relationships.
MIII.A-CED.A.1Create equations and inequalities in one variable and use them to solve problems. (Include equations arising from simple root and rational functions.)
MIII.A-CED.A.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
MIII.A-CED.A.3Represent constraints by equations or inequalities and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent equations describing satellites orbiting earth and constraints on earth’s size and atmosphere.
MIII.A-REI Algebra - Reasoning with Equations and Inequalities
MIII.A-REI.A Understand solving equations as a process of reasoning and explain the reasoning.
MIII.A-REI.A.2Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
MIII.A-REI.D Represent and solve equations and inequalities graphically.
MIII.A-REI.D.11Explain 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); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are polynomial, rational, and logarithmic functions.
MIII.A-SSE Algebra - Seeing Structure in Expressions
MIII.A-SSE.A Interpret the structure of polynomial and rational expressions.
MIII.A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
MIII.A-SSE.A.1.aInterpret parts of an expression, such as terms, factors, and coefficients.
MIII.A-SSE.A.1.bInterpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.
MIII.A-SSE.A.2Use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 – (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 – y^2)(x^2 + y^2) and as (x-y)(x+y)(x-yi)(x+yi).
MIII.A-SSE.B Write expressions in equivalent forms to solve problems.
MIII.A-SSE.B.4Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.
MIII.F-BF Functions - Building Functions
MIII.F-BF.A Build a function that models a relationship between two quantities.
MIII.F-BF.A.1 Write simple rational and radical functions, logarithmic, and trigonometric functions that describes a relationship between two quantities.
MIII.F-BF.A.1.bCombine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model (include simple rational and radical functions, logarithmic, and trigonometric functions).
MIII.F-BF.B Build new functions from existing functions.
MIII.F-BF.B.3Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Include simple rational, radical, logarithmic, and trigonometric functions. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
MIII.F-BF.B.4 Find inverse functions algebraically and graphically.
MIII.F-BF.B.4.aSolve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x)=2x^3 or f(x) = (x + 1)∕(x - 1) for x ≠ 1.
MIII.F-IF Functions - Interpreting Functions
MIII.F-IF.B Interpret functions that arise in applications in terms of the context (rational, polynomial, square root, cube root, trigonometric, logarithmic).
MIII.F-IF.B.4For 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
MIII.F-IF.B.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
MIII.F-IF.B.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
MIII.F-IF.C Analyze functions using different representations.
MIII.F-IF.C.10Given algebraic, numeric and/or graphical representations of functions, recognize the function as polynomial, rational, logarithmic, exponential, or trigonometric.
MIII.F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
MIII.F-IF.C.7.bGraph square root and cube root functions.
MIII.F-IF.C.7.cGraph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
MIII.F-IF.C.7.eGraph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
MIII.F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
MIII.F-IF.C.8.aUse the process of factoring in a polynomial function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
MIII.F-IF.C.9Translate among different representations of functions: (algebraically, graphically, numerically in tables, or by verbal descriptions). Compare properties of two functions each represented in a different way. For example, given a graph of one polynomial function and an algebraic expression for another, say which has the larger relative maximum and/or smaller relative minimum.
MIII.F-LE Functions - Linear, Quadratic, and Exponential Models
MIII.F-LE.A Construct and compare linear, quadratic, and exponential models and solve problems.
MIII.F-LE.A.4For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
MIII.F-TF Functions - Trigonometric Functions
MIII.F-TF.A Extend the domain of trigonometric functions using the unit circle.
MIII.F-TF.A.1Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
MIII.F-TF.A.2Explain 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.
MIII.F-TF.B Model periodic phenomena with trigonometric functions.
MIII.F-TF.B.5Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
MIII.F-TF.C Prove and apply trigonometric identities.
MIII.F-TF.C.8Prove the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant.
MIII.G-GMD Geometry - Geometric Measurement and Dimension
MIII.G-GMD.B Visualize relationships between two-dimensional and three-dimensional objects.
MIII.G-GMD.B.4Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
MIII.G-MG Geometry - Modeling with Geometry
MIII.G-MG.A Apply geometric concepts in modeling situations.
MIII.G-MG.A.1Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
MIII.G-MG.A.2Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
MIII.G-MG.A.3Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
MIII.G-MG.A.4Use dimensional analysis for unit conversions to confirm that expressions and equations make sense.
MIII.G-SRT Geometry - Similarity, Right Triangles, and Trigonometry
MIII.G-SRT.D Apply trigonometry to general triangles.
MIII.G-SRT.D.10(+) Prove the Laws of Sines and Cosines and use them to solve problems.
MIII.G-SRT.D.11(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
MIII.G-SRT.D.9(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
MIII.N-CN Number and Quantity - The Complex Number System
MIII.N-CN.C Use complex numbers in polynomial identities and equations.
MIII.N-CN.C.8(+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x – 2i).
MIII.N-CN.C.9(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
MIII.N-VM Number and Quantity - Vector and Matrix Quantities
MIII.N-VM.A Represent and model with vector quantities.
MIII.N-VM.A.1(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
MIII.N-VM.A.3(+) Solve problems involving velocity and other quantities that can be represented by vectors.
MIII.N-VM.C Perform operations on matrices and use matrices in applications.
MIII.N-VM.C.12(+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
MIII.N-VM.C.6(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
MIII.N-VM.C.8(+) Add, subtract, and multiply matrices of appropriate dimensions.
MIII.S-IC Statistics and Probability - Making Inferences and Justifying Conclusions
MIII.S-IC.A Understand and evaluate random processes underlying statistical experiments.
MIII.S-IC.A.1Understand statistics as a process for making inferences to be made about population parameters based on a random sample from that population.
MIII.S-IC.A.2Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of five tails in a row cause you to question the model?
MIII.S-IC.B Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
MIII.S-IC.B.3Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
MIII.S-IC.B.4Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
MIII.S-IC.B.5Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
MIII.S-IC.B.6Evaluate reports based on data.
MIII.S-ID Statistics and Probability - Interpreting Categorical and Quantitative Data
MIII.S-ID.A Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.
MIII.S-ID.A.4Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
MIII.S-MD Statistics and Probability - Using Probability to Make Decisions
MIII.S-MD.B Use probability to evaluate outcomes of decisions.
MIII.S-MD.B.6(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
MIII.S-MD.B.7(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game and replacing the goalie with an extra skater).

Number and Quantity - The Complex Number System

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N-CN.A Perform arithmetic operations with complex numbers.
N-CN.A.1Know there is a complex number i such that i^2 = -1, and every complex number has the form a + bi with a and b real.
N-CN.A.2Use the relation i^2 = -1 and the Commutative, Associative, and Distributive properties to add, subtract, and multiply complex numbers.
N-CN.A.3(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
N-CN.B Represent complex numbers and their operations on the complex plane.
N-CN.B.4(+) Represent 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.
N-CN.B.5(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1+√3i)^3=8 because ((-1+√3i) has modulus 2 and argument 120°.
N-CN.B.6(+) Calculate 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.
N-CN.C Use complex numbers in polynomial identities and equations.
N-CN.C.7Solve quadratic equations with real coefficients that have complex solutions.
N-CN.C.8(+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x – 2i).
N-CN.C.9(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Number and Quantity - Quantities

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Number and Quantity - The Real Number System

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N-RN.A Extend the properties of exponents to rational exponents.
N-RN.A.1Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^3 = 5^((1/3)3) to hold, so (5^(1/3))^3 must equal 5.
N-RN.A.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.
N-RN.B Use properties of rational and irrational numbers.
N-RN.B.3Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Number and Quantity - Vector and Matrix Quantities

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N-VM.A Represent and model with vector quantities.
N-VM.A.1(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, || v ||, v).
N-VM.A.2(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
N-VM.A.3(+) Solve problems involving velocity and other quantities that can be represented by vectors.
N-VM.B Perform operations on vectors.
N-VM.B.4 (+) Add and subtract vectors.
N-VM.B.4.a(+) Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that (+) the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
N-VM.B.4.b(+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
N-VM.B.4.c(+) Understand 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.
N-VM.B.5 (+) Multiply a vector by a scalar.
N-VM.B.5.a(+) Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_x , v_y) = (cv_x, cv_y).
N-VM.B.5.b(+) Compute 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 &gt; 0) or against v (for c &lt; 0).
N-VM.C Perform operations on matrices and use matrices in applications.
N-VM.C.10(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
N-VM.C.11(+) Multiply 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.
N-VM.C.12(+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
N-VM.C.6(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
N-VM.C.7(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
N-VM.C.8(+) Add, subtract, and multiply matrices of appropriate dimensions.
N-VM.C.9(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a Commutative operation, but still satisfies the Associative and Distributive properties.

Model Precalculus [PC]

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PC.A-APR Algebra - Arithmetic with Polynomials and Rational Expressions
PC.A-APR.C Use polynomial identities to solve problems.
PC.A-APR.C.5(+) Know 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, with coefficients determined for example by Pascal’s Triangle.
PC.A-APR.D Rewrite rational expressions.
PC.A-APR.D.7(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
PC.A-REI Algebra - Reasoning with Equations and Inequalities
PC.A-REI.C Solve systems of equations.
PC.A-REI.C.8(+) Represent a system of linear equations as a single matrix equation in a vector variable.
PC.A-REI.C.9(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).
PC.F-BF Functions - Building Functions
PC.F-BF.A Build a function that models a relationship between two quantities.
PC.F-BF.A.1 Write a function that describes a relationship between two quantities.
PC.F-BF.A.1.c(+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
PC.F-BF.B Build new functions from existing functions.
PC.F-BF.B.4 Find inverse functions.
PC.F-BF.B.4.b(+) Verify by composition that one function is the inverse of another.
PC.F-BF.B.4.c(+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
PC.F-BF.B.4.d(+) Produce an invertible function from a non-invertible function by restricting the domain.
PC.F-BF.B.5(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
PC.F-IF Functions - Interpreting Functions
PC.F-IF.C Analyze functions using different representations.
PC.F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
PC.F-IF.C.7.d(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
PC.F-TF Functions - Trigonometric Functions
PC.F-TF.A Extend the domain of trigonometric functions using the unit circle.
PC.F-TF.A.3(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π∕3, π∕4 and π∕6, and use the unit circle to express the values of sine, cosine, and tangent for π = x, π + x, and 2π - x in terms of their values for x, where x is any real number.
PC.F-TF.A.4(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
PC.F-TF.B Model periodic phenomena with trigonometric functions.
PC.F-TF.B.6(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
PC.F-TF.B.7(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
PC.F-TF.C Prove and apply trigonometric identities.
PC.F-TF.C.9(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
PC.G-C Geometry - Circles
PC.G-C.A Understand and apply theorems about circles.
PC.G-C.A.4(+) Construct a tangent line from a point outside a given circle to the circle.
PC.G-GMD Geometry - Geometric Measurement and Dimension
PC.G-GMD.A Explain volume formulas and use them to solve problems.
PC.G-GMD.A.2(+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
PC.G-GPE Geometry - Expressing Geometric Properties with Equations
PC.G-GPE.A Translate between the geometric description and the equation for a conic section.
PC.G-GPE.A.3 (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
PC.G-GPE.A.3.a(+) Use equations and graphs of conic sections to model real-world problems.
PC.G-SRT Geometry - Similarity, Right Triangles, and Trigonometry
PC.G-SRT.D Apply trigonometry to general triangles.
PC.G-SRT.D.10(+) Prove the Laws of Sines and Cosines and use them to solve problems.
PC.G-SRT.D.11(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
PC.G-SRT.D.9(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
PC.N-CN Number and Quantity - The Complex Number System
PC.N-CN.A Perform arithmetic operations with complex numbers.
PC.N-CN.A.3(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
PC.N-CN.B Represent complex numbers and their operations on the complex plane.
PC.N-CN.B.4(+) Represent 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.N-CN.B.5(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example: (-1+√(3i))^3=8 because (-1+√(3i)) has modulus 2 and argument 120°.
PC.N-CN.B.6(+) Calculate 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.N-CN.C Use complex numbers in polynomial identities and equations.
PC.N-CN.C.8(+) Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x – 2i).
PC.N-CN.C.9(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
PC.N-VM Number and Quantity - Vector and Matrix Quantities
PC.N-VM.A Represent and model with vector quantities.
PC.N-VM.A.1(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
PC.N-VM.A.2(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
PC.N-VM.A.3(+) Solve problems involving velocity and other quantities that can be represented by vectors.
PC.N-VM.B Perform operations on vectors.
PC.N-VM.B.4 (+) Add and subtract vectors.
PC.N-VM.B.4.a(+) Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
PC.N-VM.B.4.b(+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
PC.N-VM.B.4.c(+) Understand 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.N-VM.B.5 (+) Multiply a vector by a scalar.
PC.N-VM.B.5.a(+) Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_x, v_y) = (cv_x, cv_y).
PC.N-VM.B.5.b(+) Compute 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 &gt; 0) or against v (for c &lt; 0).
PC.N-VM.C Perform operations on matrices and use matrices in applications.
PC.N-VM.C.10(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
PC.N-VM.C.11(+) Multiply 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.N-VM.C.12(+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
PC.N-VM.C.6(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
PC.N-VM.C.7(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
PC.N-VM.C.8(+) Add, subtract, and multiply matrices of appropriate dimensions.
PC.N-VM.C.9(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a Commutative operation, but still satisfies the Associative and Distributive properties.

Statistics and Probability - Conditional Probability and the Rules of Probability

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S-CP.A Understand independence and conditional probability and use them to interpret data from simulations or experiments.
S-CP.A.1Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
S-CP.A.2Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
S-CP.A.3Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
S-CP.A.4Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
S-CP.A.5Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
S-CP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model.
S-CP.B.6Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
S-CP.B.7Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
S-CP.B.8(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
S-CP.B.9(+) Use permutations and combinations to compute probabilities of compound events and solve problems.

Statistics and Probability - Making Inferences and Justifying Conclusions

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S-IC.A Understand and evaluate random processes underlying statistical experiments. Use calculators, spreadsheets, and other technology as appropriate.
S-IC.A.1Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
S-IC.A.2Decide if a specified model is consistent with results from a given data-generating process (e.g., using simulation). For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of five tails in a row cause you to question the model?
S-IC.B Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
S-IC.B.3Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
S-IC.B.4Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
S-IC.B.5Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
S-IC.B.6Evaluate reports based on data.

Statistics and Probability - Interpreting Categorical and Quantitative Data

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S-ID.A Summarize, represent, and interpret data on a single count or measurement variable. Use calculators, spreadsheets, and other technology as appropriate.
S-ID.A.1Represent data with plots on the real number line (dot plots, histograms, and box plots).
S-ID.A.2Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
S-ID.A.3Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
S-ID.A.4Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
S-ID.B Summarize, represent, and interpret data on two categorical and quantitative variables.
S-ID.B.5Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
S-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
S-ID.B.6.aFit a linear function to the data and use the fitted function to solve problems in the context of the data. Use functions fitted to data or choose a function suggested by the context. Emphasize linear and exponential models.
S-ID.B.6.bInformally assess the fit of a function by plotting and analyzing residuals.
S-ID.B.6.cFit a linear function for a scatter plot that suggests a linear association.
S-ID.C Interpret linear models.
S-ID.C.7Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
S-ID.C.8Compute (using technology) and interpret the correlation coefficient of a linear fit.
S-ID.C.9Distinguish between correlation and causation.

Statistics and Probability - Using Probability to Make Decisions

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S-MD.A Calculate expected values and use them to solve problems.
S-MD.A.1(+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
S-MD.A.2(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
S-MD.A.3(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
S-MD.A.4(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?
S-MD.B Use probability to evaluate outcomes of decisions.
S-MD.B.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
S-MD.B.5.a(+) Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.
S-MD.B.5.b(+) Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
S-MD.B.6(+) Use probabilities to make fair decisions (e.g., drawing by lots or using a random number generator).
S-MD.B.7(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, or pulling a hockey goalie at the end of a game and replacing the goalie with an extra skater).

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.