1.2 http://corestandards.org/Math/Content/K/CC/A/1/ 7 Standard CCSS.Math.Content.K.CC.A.1 Count to 100 by ones and by tens. K B62C1C106873438AA0126760075A65A3 4F4106218F834258BCDDB7EB39806880 http://corestandards.org/Math/Content/K/CC/A/2/ 7 Standard CCSS.Math.Content.K.CC.A.2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1). K B62C1C106873438AA0126760075A65A3 4F4106218F834258BCDDB7EB39806880 http://corestandards.org/Math/Content/K/CC/A/3/ 7 Standard CCSS.Math.Content.K.CC.A.3 Write 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 B62C1C106873438AA0126760075A65A3 4F4106218F834258BCDDB7EB39806880 http://corestandards.org/Math/Content/K/CC/B/4/ 7 Standard CCSS.Math.Content.K.CC.B.4 Understand the relationship between numbers and quantities; connect counting to cardinality. K B62C1C106873438AA0126760075A65A3 BB49D28AFFFE46FBB8F4ECA58C88B001 7B20214AA4AA445AA720062C6F1B5C58 3DEE205D86BC461FA4271EF4BD190A0C A863C4E937BE4307A241D126B8FBD342 http://corestandards.org/Math/Content/K/CC/B/4/a/ 8 Component CCSS.Math.Content.K.CC.B.4a When 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 B62C1C106873438AA0126760075A65A3 A0586B74C257473C98F38DD353D39BC5 http://corestandards.org/Math/Content/K/CC/B/4/b/ 8 Component CCSS.Math.Content.K.CC.B.4b Understand 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 B62C1C106873438AA0126760075A65A3 A0586B74C257473C98F38DD353D39BC5 http://corestandards.org/Math/Content/K/CC/B/4/c/ 8 Component CCSS.Math.Content.K.CC.B.4c Understand that each successive number name refers to a quantity that is one larger. K B62C1C106873438AA0126760075A65A3 A0586B74C257473C98F38DD353D39BC5 http://corestandards.org/Math/Content/K/CC/B/5/ 7 Standard CCSS.Math.Content.K.CC.B.5 Count 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 B62C1C106873438AA0126760075A65A3 BB49D28AFFFE46FBB8F4ECA58C88B001 http://corestandards.org/Math/Content/K/CC/C/6/ 7 Standard CCSS.Math.Content.K.CC.C.6 Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.<sup>1</sup> K B62C1C106873438AA0126760075A65A3 74695C233ACE4F458E47C26F8D9FFF86 http://corestandards.org/Math/Content/K/CC/C/7/ 7 Standard CCSS.Math.Content.K.CC.C.7 Compare two numbers between 1 and 10 presented as written numerals. K B62C1C106873438AA0126760075A65A3 74695C233ACE4F458E47C26F8D9FFF86 http://corestandards.org/Math/Content/K/G/A/1/ 7 Standard CCSS.Math.Content.K.G.A.1 Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as <i>above</i>, <i>below</i>, <i>beside</i>, <i>in front of</i>, <i>behind</i>, and <i>next to</i>. K B62C1C106873438AA0126760075A65A3 B1AC98EADE4145689E70EEEBD9B8CC18 http://corestandards.org/Math/Content/K/G/A/2/ 7 Standard CCSS.Math.Content.K.G.A.2 Correctly name shapes regardless of their orientations or overall size. K B62C1C106873438AA0126760075A65A3 B1AC98EADE4145689E70EEEBD9B8CC18 http://corestandards.org/Math/Content/K/G/A/3/ 7 Standard CCSS.Math.Content.K.G.A.3 Identify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional ("solid"). K B62C1C106873438AA0126760075A65A3 B1AC98EADE4145689E70EEEBD9B8CC18 http://corestandards.org/Math/Content/K/G/B/4/ 7 Standard CCSS.Math.Content.K.G.B.4 Analyze 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 B62C1C106873438AA0126760075A65A3 5F9DA7B7BFD04F33B5A925A4825B1DB1 http://corestandards.org/Math/Content/K/G/B/5/ 7 Standard CCSS.Math.Content.K.G.B.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes. K B62C1C106873438AA0126760075A65A3 5F9DA7B7BFD04F33B5A925A4825B1DB1 http://corestandards.org/Math/Content/K/G/B/6/ 7 Standard CCSS.Math.Content.K.G.B.6 Compose simple shapes to form larger shapes. <i>For example, "Can you join these two triangles with full sides touching to make a rectangle</i>?" K B62C1C106873438AA0126760075A65A3 5F9DA7B7BFD04F33B5A925A4825B1DB1 http://corestandards.org/Math/Content/K/MD/A/1/ 7 Standard CCSS.Math.Content.K.MD.A.1 Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object. K B62C1C106873438AA0126760075A65A3 48519EC7CCA74533BF4D4052B47BFB48 http://corestandards.org/Math/Content/K/MD/A/2/ 7 Standard CCSS.Math.Content.K.MD.A.2 Directly compare two objects with a measurable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference. <i>For example, directly compare the heights of two children and describe one child as taller/shorter</i>. K B62C1C106873438AA0126760075A65A3 48519EC7CCA74533BF4D4052B47BFB48 http://corestandards.org/Math/Content/K/MD/B/3/ 7 Standard CCSS.Math.Content.K.MD.B.3 Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.<sup>1</sup> K B62C1C106873438AA0126760075A65A3 B1AC98EADE4145689E70EEEBD9B8CC19 http://corestandards.org/Math/Content/K/NBT/A/1/ 7 Standard CCSS.Math.Content.K.NBT.A.1 Compose 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 (such as 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 B62C1C106873438AA0126760075A65A3 F2D60C80EBD94C939E2BA189CCD56860 http://corestandards.org/Math/Content/K/OA/A/1/ 7 Standard CCSS.Math.Content.K.OA.A.1 Represent addition and subtraction with objects, fingers, mental images, drawings<sup>1</sup>, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. K B62C1C106873438AA0126760075A65A3 E754542E6C144EEC8D7E332063FF4F67 http://corestandards.org/Math/Content/K/OA/A/2/ 7 Standard CCSS.Math.Content.K.OA.A.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. K B62C1C106873438AA0126760075A65A3 E754542E6C144EEC8D7E332063FF4F67 http://corestandards.org/Math/Content/K/OA/A/3/ 7 Standard CCSS.Math.Content.K.OA.A.3 Decompose 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 B62C1C106873438AA0126760075A65A3 E754542E6C144EEC8D7E332063FF4F67 http://corestandards.org/Math/Content/K/OA/A/4/ 7 Standard CCSS.Math.Content.K.OA.A.4 For 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 B62C1C106873438AA0126760075A65A3 E754542E6C144EEC8D7E332063FF4F67 http://corestandards.org/Math/Content/K/OA/A/5/ 7 Standard CCSS.Math.Content.K.OA.A.5 Fluently add and subtract within 5. K B62C1C106873438AA0126760075A65A3 E754542E6C144EEC8D7E332063FF4F67 http://corestandards.org/Math/Content/1/G/A/1/ 7 Standard CCSS.Math.Content.1.G.A.1 Distinguish 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 to possess defining attributes. 01 B62C1C106873438AA0126760075A65A3 17A1459188434C839096339B99C457B1 http://corestandards.org/Math/Content/1/G/A/2/ 7 Standard CCSS.Math.Content.1.G.A.2 Compose 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.<sup>1</sup> 01 B62C1C106873438AA0126760075A65A3 17A1459188434C839096339B99C457B1 http://corestandards.org/Math/Content/1/G/A/3/ 7 Standard CCSS.Math.Content.1.G.A.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words <i>halves</i>, <i>fourths</i>, and <i>quarters</i>, and use the phrases <i>half of</i>, <i>fourth of</i>, and <i>quarter of</i>. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. 01 B62C1C106873438AA0126760075A65A3 17A1459188434C839096339B99C457B1 http://corestandards.org/Math/Content/1/MD/A/1/ 7 Standard CCSS.Math.Content.1.MD.A.1 Order three objects by length; compare the lengths of two objects indirectly by using a third object. 01 B62C1C106873438AA0126760075A65A3 F50913CE7F82434D98350AEA2774BA42 http://corestandards.org/Math/Content/1/MD/A/2/ 7 Standard CCSS.Math.Content.1.MD.A.2 Express 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. <i>Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps</i>. 01 B62C1C106873438AA0126760075A65A3 F50913CE7F82434D98350AEA2774BA42 http://corestandards.org/Math/Content/1/MD/B/3/ 7 Standard CCSS.Math.Content.1.MD.B.3 Tell and write time in hours and half-hours using analog and digital clocks. 01 B62C1C106873438AA0126760075A65A3 A06A607DB379444391776C0C13B73A6E http://corestandards.org/Math/Content/1/MD/C/4/ 7 Standard CCSS.Math.Content.1.MD.C.4 Organize, 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. 01 B62C1C106873438AA0126760075A65A3 B9CAA38C0B114B969475DE86455B7B46 http://corestandards.org/Math/Content/1/NBT/A/1/ 7 Standard CCSS.Math.Content.1.NBT.A.1 Count 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. 01 B62C1C106873438AA0126760075A65A3 10EF7C4FC6974243A0B6BC075E49B1D4 http://corestandards.org/Math/Content/1/NBT/B/2/ 7 Standard CCSS.Math.Content.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: 01 B62C1C106873438AA0126760075A65A3 97A52B26167842E49CD8C78C04481304 537AAD09E07C43D0968D7C9D9E07FAC4 B9A7CBDAD2C94286837A25AE81B46920 1125E1AD13944E35A3BF5358FD157DF3 http://corestandards.org/Math/Content/1/NBT/B/2/a/ 8 Component CCSS.Math.Content.1.NBT.B.2a 10 can be thought of as a bundle of ten ones &mdash; called a "ten." 01 B62C1C106873438AA0126760075A65A3 0B8F8764427D4A1D9FE9EBA6D2EC0C95 http://corestandards.org/Math/Content/1/NBT/B/2/b/ 8 Component CCSS.Math.Content.1.NBT.B.2b The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones. 01 B62C1C106873438AA0126760075A65A3 0B8F8764427D4A1D9FE9EBA6D2EC0C95 http://corestandards.org/Math/Content/1/NBT/B/2/c/ 8 Component CCSS.Math.Content.1.NBT.B.2c The 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). 01 B62C1C106873438AA0126760075A65A3 0B8F8764427D4A1D9FE9EBA6D2EC0C95 http://corestandards.org/Math/Content/1/NBT/B/3/ 7 Standard CCSS.Math.Content.1.NBT.B.3 Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols &gt;, =, and &lt;. 01 B62C1C106873438AA0126760075A65A3 97A52B26167842E49CD8C78C04481304 http://corestandards.org/Math/Content/1/NBT/C/4/ 7 Standard CCSS.Math.Content.1.NBT.C.4 Add 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. 01 B62C1C106873438AA0126760075A65A3 68702F4D7C994AF1AEFF699CEB567C56 http://corestandards.org/Math/Content/1/NBT/C/5/ 7 Standard CCSS.Math.Content.1.NBT.C.5 Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. 01 B62C1C106873438AA0126760075A65A3 68702F4D7C994AF1AEFF699CEB567C56 http://corestandards.org/Math/Content/1/NBT/C/6/ 7 Standard CCSS.Math.Content.1.NBT.C.6 Subtract 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. 01 B62C1C106873438AA0126760075A65A3 68702F4D7C994AF1AEFF699CEB567C56 http://corestandards.org/Math/Content/1/OA/A/1/ 7 Standard CCSS.Math.Content.1.OA.A.1 Use 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 with a symbol for the unknown number to represent the problem.<sup>1</sup> 01 B62C1C106873438AA0126760075A65A3 1A7D11B497334220BAFD174AE988EE0C http://corestandards.org/Math/Content/1/OA/A/2/ 7 Standard CCSS.Math.Content.1.OA.A.2 Solve 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. 01 B62C1C106873438AA0126760075A65A3 1A7D11B497334220BAFD174AE988EE0C http://corestandards.org/Math/Content/1/OA/B/3/ 7 Standard CCSS.Math.Content.1.OA.B.3 Apply properties of operations as strategies to add and subtract.<sup>2</sup> <i>Examples: 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.) </i> 01 B62C1C106873438AA0126760075A65A3 FDFBABAA6BFA4FBDACA10CA3898A2D6E http://corestandards.org/Math/Content/1/OA/B/4/ 7 Standard CCSS.Math.Content.1.OA.B.4 Understand subtraction as an unknown-addend problem. <i>For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.</i> 01 B62C1C106873438AA0126760075A65A3 FDFBABAA6BFA4FBDACA10CA3898A2D6E http://corestandards.org/Math/Content/1/OA/C/5/ 7 Standard CCSS.Math.Content.1.OA.C.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 01 B62C1C106873438AA0126760075A65A3 BDB6E9CCB87F466786AAFABB611B726D http://corestandards.org/Math/Content/1/OA/C/6/ 7 Standard CCSS.Math.Content.1.OA.C.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (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). 01 B62C1C106873438AA0126760075A65A3 BDB6E9CCB87F466786AAFABB611B726D http://corestandards.org/Math/Content/1/OA/D/7/ 7 Standard CCSS.Math.Content.1.OA.D.7 Understand 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. 01 B62C1C106873438AA0126760075A65A3 23FDB68FBA7C4ACFA753306D02D0343F http://corestandards.org/Math/Content/1/OA/D/8/ 7 Standard CCSS.Math.Content.1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. <i>For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _</i>. 01 B62C1C106873438AA0126760075A65A3 23FDB68FBA7C4ACFA753306D02D0343F http://corestandards.org/Math/Content/2/G/A/1/ 7 Standard CCSS.Math.Content.2.G.A.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.<sup>1</sup> Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. 02 B62C1C106873438AA0126760075A65A3 E82ECFD919D748a7AD0F9AFDDF8EA50F http://corestandards.org/Math/Content/2/G/A/2/ 7 Standard CCSS.Math.Content.2.G.A.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. 02 B62C1C106873438AA0126760075A65A3 E82ECFD919D748a7AD0F9AFDDF8EA50F http://corestandards.org/Math/Content/2/G/A/3/ 7 Standard CCSS.Math.Content.2.G.A.3 Partition 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. 02 B62C1C106873438AA0126760075A65A3 E82ECFD919D748a7AD0F9AFDDF8EA50F http://corestandards.org/Math/Content/2/MD/A/1/ 7 Standard CCSS.Math.Content.2.MD.A.1 Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. 02 B62C1C106873438AA0126760075A65A3 CB91A45FEBF84FF2A442D51FF3660B82 http://corestandards.org/Math/Content/2/MD/A/2/ 7 Standard CCSS.Math.Content.2.MD.A.2 Measure 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. 02 B62C1C106873438AA0126760075A65A3 CB91A45FEBF84FF2A442D51FF3660B82 http://corestandards.org/Math/Content/2/MD/A/3/ 7 Standard CCSS.Math.Content.2.MD.A.3 Estimate lengths using units of inches, feet, centimeters, and meters. 02 B62C1C106873438AA0126760075A65A3 CB91A45FEBF84FF2A442D51FF3660B82 http://corestandards.org/Math/Content/2/MD/A/4/ 7 Standard CCSS.Math.Content.2.MD.A.4 Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit. 02 B62C1C106873438AA0126760075A65A3 CB91A45FEBF84FF2A442D51FF3660B82 http://corestandards.org/Math/Content/2/MD/B/5/ 7 Standard CCSS.Math.Content.2.MD.B.5 Use 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. 02 B62C1C106873438AA0126760075A65A3 8183F864EBEE4FA19252DF67462B3156 http://corestandards.org/Math/Content/2/MD/B/6/ 7 Standard CCSS.Math.Content.2.MD.B.6 Represent 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. 02 B62C1C106873438AA0126760075A65A3 8183F864EBEE4FA19252DF67462B3156 http://corestandards.org/Math/Content/2/MD/C/7/ 7 Standard CCSS.Math.Content.2.MD.C.7 Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. 02 B62C1C106873438AA0126760075A65A3 8EF9A4A45830489882546A7D3C10EF57 http://corestandards.org/Math/Content/2/MD/C/8/ 7 Standard CCSS.Math.Content.2.MD.C.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using \$ and &cent; symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have? 02 B62C1C106873438AA0126760075A65A3 8EF9A4A45830489882546A7D3C10EF57 http://corestandards.org/Math/Content/2/MD/D/9/ 7 Standard CCSS.Math.Content.2.MD.D.9 Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units. 02 B62C1C106873438AA0126760075A65A3 24A998ECFA5B4E7FAC36036671790ED7 http://corestandards.org/Math/Content/2/MD/D/10/ 7 Standard CCSS.Math.Content.2.MD.D.10 Draw 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<sup>1</sup> using information presented in a bar graph. 02 B62C1C106873438AA0126760075A65A3 24A998ECFA5B4E7FAC36036671790ED7 http://corestandards.org/Math/Content/2/NBT/A/1/ 7 Standard CCSS.Math.Content.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: 02 B62C1C106873438AA0126760075A65A3 D15326F9F775410681FF079A08EAAC14 C9623D25BEB947A38CA8A92AB009E699 0544B12300304F739F00C96785DCE7D2 http://corestandards.org/Math/Content/2/NBT/A/1/a/ 8 Component CCSS.Math.Content.2.NBT.A.1a 100 can be thought of as a bundle of ten tens &mdash; called a "hundred." 02 B62C1C106873438AA0126760075A65A3 3B25AF48C22D4668A6085998F847B56E http://corestandards.org/Math/Content/2/NBT/A/1/b/ 8 Component CCSS.Math.Content.2.NBT.A.1b The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones). 02 B62C1C106873438AA0126760075A65A3 3B25AF48C22D4668A6085998F847B56E http://corestandards.org/Math/Content/2/NBT/A/2/ 7 Standard CCSS.Math.Content.2.NBT.A.2 Count within 1000; skip-count by 5s, 10s, and 100s. 02 B62C1C106873438AA0126760075A65A3 D15326F9F775410681FF079A08EAAC14 http://corestandards.org/Math/Content/2/NBT/A/3/ 7 Standard CCSS.Math.Content.2.NBT.A.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form. 02 B62C1C106873438AA0126760075A65A3 D15326F9F775410681FF079A08EAAC14 http://corestandards.org/Math/Content/2/NBT/A/4/ 7 Standard CCSS.Math.Content.2.NBT.A.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using &gt;, =, and &lt; symbols to record the results of comparisons. 02 B62C1C106873438AA0126760075A65A3 D15326F9F775410681FF079A08EAAC14 http://corestandards.org/Math/Content/2/NBT/B/5/ 7 Standard CCSS.Math.Content.2.NBT.B.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. 02 B62C1C106873438AA0126760075A65A3 E0C9E87B224A43E8A0EFB8503905B720 http://corestandards.org/Math/Content/2/NBT/B/6/ 7 Standard CCSS.Math.Content.2.NBT.B.6 Add up to four two-digit numbers using strategies based on place value and properties of operations. 02 B62C1C106873438AA0126760075A65A3 E0C9E87B224A43E8A0EFB8503905B720 http://corestandards.org/Math/Content/2/NBT/B/7/ 7 Standard CCSS.Math.Content.2.NBT.B.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. 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. 02 B62C1C106873438AA0126760075A65A3 E0C9E87B224A43E8A0EFB8503905B720 http://corestandards.org/Math/Content/2/NBT/B/8/ 7 Standard CCSS.Math.Content.2.NBT.B.8 Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900. 02 B62C1C106873438AA0126760075A65A3 E0C9E87B224A43E8A0EFB8503905B720 http://corestandards.org/Math/Content/2/NBT/B/9/ 7 Standard CCSS.Math.Content.2.NBT.B.9 Explain why addition and subtraction strategies work, using place value and the properties of operations.<sup>1</sup> 02 B62C1C106873438AA0126760075A65A3 E0C9E87B224A43E8A0EFB8503905B720 http://corestandards.org/Math/Content/2/OA/A/1/ 7 Standard CCSS.Math.Content.2.OA.A.1 Use 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.<sup>1</sup> 02 B62C1C106873438AA0126760075A65A3 25F1A2E4A27442D886723ADD7B62C09E http://corestandards.org/Math/Content/2/OA/B/2/ 7 Standard CCSS.Math.Content.2.OA.B.2 Fluently add and subtract within 20 using mental strategies.<sup>2</sup> By end of Grade 2, know from memory all sums of two one-digit numbers. 02 B62C1C106873438AA0126760075A65A3 21FF72D85AF248E28B8AD028ABF94DDE http://corestandards.org/Math/Content/2/OA/C/3/ 7 Standard CCSS.Math.Content.2.OA.C.3 Determine 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. 02 B62C1C106873438AA0126760075A65A3 1E2E6DE1CD454AF99E15FA153F58DF3E http://corestandards.org/Math/Content/2/OA/C/4/ 7 Standard CCSS.Math.Content.2.OA.C.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. 02 B62C1C106873438AA0126760075A65A3 1E2E6DE1CD454AF99E15FA153F58DF3E http://corestandards.org/Math/Content/3/G/A/1/ 7 Standard CCSS.Math.Content.3.G.A.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. 03 B62C1C106873438AA0126760075A65A3 834B17E279C64263AA83F7625F5D2994 http://corestandards.org/Math/Content/3/G/A/2/ 7 Standard CCSS.Math.Content.3.G.A.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. <i>For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape</i>. 03 B62C1C106873438AA0126760075A65A3 834B17E279C64263AA83F7625F5D2994 http://corestandards.org/Math/Content/3/MD/A/1/ 7 Standard CCSS.Math.Content.3.MD.A.1 Tell 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. 03 B62C1C106873438AA0126760075A65A3 CDF685BAD71C449594666CFF41F7171E http://corestandards.org/Math/Content/3/MD/A/2/ 7 Standard CCSS.Math.Content.3.MD.A.2 Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).<sup>1</sup> Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.<sup>2</sup> 03 B62C1C106873438AA0126760075A65A3 CDF685BAD71C449594666CFF41F7171E http://corestandards.org/Math/Content/3/MD/B/3/ 7 Standard CCSS.Math.Content.3.MD.B.3 Draw 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. <i>For example, draw a bar graph in which each square in the bar graph might represent 5 pets</i>. 03 B62C1C106873438AA0126760075A65A3 8F925240A0754F1DAABA5E18D8D83BE5 http://corestandards.org/Math/Content/3/MD/B/4/ 7 Standard CCSS.Math.Content.3.MD.B.4 Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units&mdash; whole numbers, halves, or quarters. 03 B62C1C106873438AA0126760075A65A3 8F925240A0754F1DAABA5E18D8D83BE5 http://corestandards.org/Math/Content/3/MD/C/5/ 7 Standard CCSS.Math.Content.3.MD.C.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. 03 B62C1C106873438AA0126760075A65A3 834B17E279C64263AA83F7625F5D2993 F82EDE7D54704B7F97AB74D10A56D228 F0B097C190F4482BA735FEC560DA0A7F http://corestandards.org/Math/Content/3/MD/C/5/a/ 8 Component CCSS.Math.Content.3.MD.C.5a A square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area. 03 B62C1C106873438AA0126760075A65A3 C3AE47EBADAC4FD3806961236A97793F http://corestandards.org/Math/Content/3/MD/C/5/b/ 8 Component CCSS.Math.Content.3.MD.C.5b A plane figure which can be covered without gaps or overlaps by <i>n</i> unit squares is said to have an area of <i>n</i> square units. 03 B62C1C106873438AA0126760075A65A3 C3AE47EBADAC4FD3806961236A97793F http://corestandards.org/Math/Content/3/MD/C/6/ 7 Standard CCSS.Math.Content.3.MD.C.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). 03 B62C1C106873438AA0126760075A65A3 834B17E279C64263AA83F7625F5D2993 http://corestandards.org/Math/Content/3/MD/C/7/ 7 Standard CCSS.Math.Content.3.MD.C.7 Relate area to the operations of multiplication and addition. 03 B62C1C106873438AA0126760075A65A3 834B17E279C64263AA83F7625F5D2993 83167CA50D1442788B85C6140A20E39D C18643B92D4D47398EC8F283A2947631 4E5415C8BEA348EEA13B1A1C825A5036 B0DFFB3792F84CDAACFF99CE6F0381C6 http://corestandards.org/Math/Content/3/MD/C/7/a/ 8 Component CCSS.Math.Content.3.MD.C.7a Find 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. 03 B62C1C106873438AA0126760075A65A3 DB1670D9BA2E4F6BA040EEA5EE826670 http://corestandards.org/Math/Content/3/MD/C/7/b/ 8 Component CCSS.Math.Content.3.MD.C.7b Multiply 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. 03 B62C1C106873438AA0126760075A65A3 DB1670D9BA2E4F6BA040EEA5EE826670 http://corestandards.org/Math/Content/3/MD/C/7/c/ 8 Component CCSS.Math.Content.3.MD.C.7c Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths <i>a</i> and <i>b</i> + <i>c</i> is the sum of <i>a</i> &times; <i>b</i> and <i>a</i> &times; <i>c</i>. Use area models to represent the distributive property in mathematical reasoning. 03 B62C1C106873438AA0126760075A65A3 DB1670D9BA2E4F6BA040EEA5EE826670 http://corestandards.org/Math/Content/3/MD/C/7/d/ 8 Component CCSS.Math.Content.3.MD.C.7d Recognize 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. 03 B62C1C106873438AA0126760075A65A3 DB1670D9BA2E4F6BA040EEA5EE826670 http://corestandards.org/Math/Content/3/MD/D/8/ 7 Standard CCSS.Math.Content.3.MD.D.8 Solve 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. 03 B62C1C106873438AA0126760075A65A3 EE8FEE0FAA2F435496406A7EA6CCFC56 http://corestandards.org/Math/Content/3/NBT/A/1/ 7 Standard CCSS.Math.Content.3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100. 03 B62C1C106873438AA0126760075A65A3 600ED56CB17B45BCA84F742EF6CE451C http://corestandards.org/Math/Content/3/NBT/A/2/ 7 Standard CCSS.Math.Content.3.NBT.A.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. 03 B62C1C106873438AA0126760075A65A3 600ED56CB17B45BCA84F742EF6CE451C http://corestandards.org/Math/Content/3/NBT/A/3/ 7 Standard CCSS.Math.Content.3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 &times; 80, 5 &times; 60) using strategies based on place value and properties of operations. 03 B62C1C106873438AA0126760075A65A3 600ED56CB17B45BCA84F742EF6CE451C http://corestandards.org/Math/Content/3/NF/A/1/ 7 Standard CCSS.Math.Content.3.NF.A.1 Understand a fraction 1/<i>b</i> as the quantity formed by 1 part when a whole is partitioned into <i>b</i> equal parts; understand a fraction <i>a</i>/<i>b</i> as the quantity formed by <i>a</i> parts of size 1/<i>b</i>. 03 B62C1C106873438AA0126760075A65A3 FB0C3EF6E0254112BE3E1A3B3BBE691C http://corestandards.org/Math/Content/3/NF/A/2/ 7 Standard CCSS.Math.Content.3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. 03 B62C1C106873438AA0126760075A65A3 FB0C3EF6E0254112BE3E1A3B3BBE691C 74D0C6FACE7E4939BD8B01C5E729871C D31E57DF888541EAA539B9C1954C76E1 http://corestandards.org/Math/Content/3/NF/A/2/a/ 8 Component CCSS.Math.Content.3.NF.A.2a Represent a fraction 1/<i>b</i> on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into <i>b</i> equal parts. Recognize that each part has size 1/<i>b</i> and that the endpoint of the part based at 0 locates the number 1/<i>b</i> on the number line. 03 B62C1C106873438AA0126760075A65A3 8C551F4EF2224A61AED96E27F7BB8852 http://corestandards.org/Math/Content/3/NF/A/2/b/ 8 Component CCSS.Math.Content.3.NF.A.2b Represent a fraction <i>a</i>/<i>b</i> on a number line diagram by marking off a lengths 1/<i>b</i> from 0. Recognize that the resulting interval has size <i>a</i>/<i>b</i> and that its endpoint locates the number <i>a</i>/<i>b</i> on the number line. 03 B62C1C106873438AA0126760075A65A3 8C551F4EF2224A61AED96E27F7BB8852 http://corestandards.org/Math/Content/3/NF/A/3/ 7 Standard CCSS.Math.Content.3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. 03 B62C1C106873438AA0126760075A65A3 FB0C3EF6E0254112BE3E1A3B3BBE691C F80BD4BFFDB946DCB74AFA807DF07F0C 8A6F987095B24FD8825585A2510C4356 70C12332BEED45098AA0A0BBCD977376 43F9FB0B127B45738F173195B04CAF6B http://corestandards.org/Math/Content/3/NF/A/3/a/ 8 Component CCSS.Math.Content.3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. 03 B62C1C106873438AA0126760075A65A3 95A47E3EB7134F7FA65667B692D6F23E http://corestandards.org/Math/Content/3/NF/A/3/b/ 8 Component CCSS.Math.Content.3.NF.A.3b Recognize 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. 03 B62C1C106873438AA0126760075A65A3 95A47E3EB7134F7FA65667B692D6F23E http://corestandards.org/Math/Content/3/NF/A/3/c/ 8 Component CCSS.Math.Content.3.NF.A.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. <i>Examples: 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</i>. 03 B62C1C106873438AA0126760075A65A3 95A47E3EB7134F7FA65667B692D6F23E http://corestandards.org/Math/Content/3/NF/A/3/d/ 8 Component CCSS.Math.Content.3.NF.A.3d Compare 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 &gt;, =, or &lt;, and justify the conclusions, e.g., by using a visual fraction model. 03 B62C1C106873438AA0126760075A65A3 95A47E3EB7134F7FA65667B692D6F23E http://corestandards.org/Math/Content/3/OA/A/1/ 7 Standard CCSS.Math.Content.3.OA.A.1 Interpret products of whole numbers, e.g., interpret 5 &times; 7 as the total number of objects in 5 groups of 7 objects each. <i>For example, describe a context in which a total number of objects can be expressed as 5 &times; 7</i>. 03 B62C1C106873438AA0126760075A65A3 13E091EB7A6C41949CFF99799A39D3C2 http://corestandards.org/Math/Content/3/OA/A/2/ 7 Standard CCSS.Math.Content.3.OA.A.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 &divide; 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. <i>For example, describe a context in which a number of shares or a number of groups can be expressed as 56 &divide; 8</i>. 03 B62C1C106873438AA0126760075A65A3 13E091EB7A6C41949CFF99799A39D3C2 http://corestandards.org/Math/Content/3/OA/A/3/ 7 Standard CCSS.Math.Content.3.OA.A.3 Use 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.<sup>1</sup> 03 B62C1C106873438AA0126760075A65A3 13E091EB7A6C41949CFF99799A39D3C2 http://corestandards.org/Math/Content/3/OA/A/4/ 7 Standard CCSS.Math.Content.3.OA.A.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. <i>For example, determine the unknown number that makes the equation true in each of the equations 8 &times; ? = 48, 5 = _ &divide; 3, 6 &times; 6 = ?</i> 03 B62C1C106873438AA0126760075A65A3 13E091EB7A6C41949CFF99799A39D3C2 http://corestandards.org/Math/Content/3/OA/B/5/ 7 Standard CCSS.Math.Content.3.OA.B.5 Apply properties of operations as strategies to multiply and divide.<sup>2</sup> <i>Examples: If 6 &times; 4 = 24 is known, then 4 &times; 6 = 24 is also known. (Commutative property of multiplication.) 3 &times; 5 &times; 2 can be found by 3 &times; 5 = 15, then 15 &times; 2 = 30, or by 5 &times; 2 = 10, then 3 &times; 10 = 30. (Associative property of multiplication.) Knowing that 8 &times; 5 = 40 and 8 &times; 2 = 16, one can find 8 &times; 7 as 8 &times; (5 + 2) = (8 &times; 5) + (8 &times; 2) = 40 + 16 = 56. (Distributive property.)</i> 03 B62C1C106873438AA0126760075A65A3 2553905A2EE948F2924E4E48ECDC1274 http://corestandards.org/Math/Content/3/OA/B/6/ 7 Standard CCSS.Math.Content.3.OA.B.6 Understand division as an unknown-factor problem. <i>For example, find 32 &divide; 8 by finding the number that makes 32 when multiplied by 8</i>. 03 B62C1C106873438AA0126760075A65A3 2553905A2EE948F2924E4E48ECDC1274 http://corestandards.org/Math/Content/3/OA/C/7/ 7 Standard CCSS.Math.Content.3.OA.C.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 &times; 5 = 40, one knows 40 &divide; 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. 03 B62C1C106873438AA0126760075A65A3 FD7A8A1461E048A1BB220ABA40139E3E http://corestandards.org/Math/Content/3/OA/D/8/ 7 Standard CCSS.Math.Content.3.OA.D.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.<sup>3</sup> 03 B62C1C106873438AA0126760075A65A3 D99E14C6EA514E36A82F63FC3EEA56A0 http://corestandards.org/Math/Content/3/OA/D/9/ 7 Standard CCSS.Math.Content.3.OA.D.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. <i>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</i>. 03 B62C1C106873438AA0126760075A65A3 D99E14C6EA514E36A82F63FC3EEA56A0 http://corestandards.org/Math/Content/4/G/A/1/ 7 Standard CCSS.Math.Content.4.G.A.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. 04 B62C1C106873438AA0126760075A65A3 0B670B7BA80B457D82D11DD3D54DCE92 http://corestandards.org/Math/Content/4/G/A/2/ 7 Standard CCSS.Math.Content.4.G.A.2 Classify 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. 04 B62C1C106873438AA0126760075A65A3 0B670B7BA80B457D82D11DD3D54DCE92 http://corestandards.org/Math/Content/4/G/A/3/ 7 Standard CCSS.Math.Content.4.G.A.3 Recognize 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. 04 B62C1C106873438AA0126760075A65A3 0B670B7BA80B457D82D11DD3D54DCE92 http://corestandards.org/Math/Content/4/MD/A/1/ 7 Standard CCSS.Math.Content.4.MD.A.1 Know 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. <i>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), ...</i> 04 B62C1C106873438AA0126760075A65A3 618C4BCCE83C4945B4B5F627C87798FE http://corestandards.org/Math/Content/4/MD/A/2/ 7 Standard CCSS.Math.Content.4.MD.A.2 Use 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. 04 B62C1C106873438AA0126760075A65A3 618C4BCCE83C4945B4B5F627C87798FE http://corestandards.org/Math/Content/4/MD/A/3/ 7 Standard CCSS.Math.Content.4.MD.A.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. <i>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</i>. 04 B62C1C106873438AA0126760075A65A3 618C4BCCE83C4945B4B5F627C87798FE http://corestandards.org/Math/Content/4/MD/B/4/ 7 Standard CCSS.Math.Content.4.MD.B.4 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. <i>For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection</i>. 04 B62C1C106873438AA0126760075A65A3 AA7E8B827B5C48EB8BB337F72B78ABEA http://corestandards.org/Math/Content/4/MD/C/5/ 7 Standard CCSS.Math.Content.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: 04 B62C1C106873438AA0126760075A65A3 BBBDB104931D41A0B3436D9DFC5E36C0 56F1AA30380B4D998B334C2741E398A1 A5B4E57485C849E1B96A79E4F3CF9E75 http://corestandards.org/Math/Content/4/MD/C/5/a/ 8 Component CCSS.Math.Content.4.MD.C.5a An 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. 04 B62C1C106873438AA0126760075A65A3 3ECC888B18184139A496F870F884FCC0 http://corestandards.org/Math/Content/4/MD/C/5/b/ 8 Component CCSS.Math.Content.4.MD.C.5b An angle that turns through <i>n</i> one-degree angles is said to have an angle measure of <i>n</i> degrees. 04 B62C1C106873438AA0126760075A65A3 3ECC888B18184139A496F870F884FCC0 http://corestandards.org/Math/Content/4/MD/C/6/ 7 Standard CCSS.Math.Content.4.MD.C.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. 04 B62C1C106873438AA0126760075A65A3 BBBDB104931D41A0B3436D9DFC5E36C0 http://corestandards.org/Math/Content/4/MD/C/7/ 7 Standard CCSS.Math.Content.4.MD.C.7 Recognize 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. 04 B62C1C106873438AA0126760075A65A3 BBBDB104931D41A0B3436D9DFC5E36C0 http://corestandards.org/Math/Content/4/NBT/A/1/ 7 Standard CCSS.Math.Content.4.NBT.A.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.<i> For example, recognize that 700 &divide; 70 = 10 by applying concepts of place value and division</i>. 04 B62C1C106873438AA0126760075A65A3 8FD3F69556544361BDEA345B9D5DAC6B http://corestandards.org/Math/Content/4/NBT/A/2/ 7 Standard CCSS.Math.Content.4.NBT.A.2 Read 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 &gt;, =, and &lt; symbols to record the results of comparisons. 04 B62C1C106873438AA0126760075A65A3 8FD3F69556544361BDEA345B9D5DAC6B http://corestandards.org/Math/Content/4/NBT/A/3/ 7 Standard CCSS.Math.Content.4.NBT.A.3 Use place value understanding to round multi-digit whole numbers to any place. 04 B62C1C106873438AA0126760075A65A3 8FD3F69556544361BDEA345B9D5DAC6B http://corestandards.org/Math/Content/4/NBT/B/4/ 7 Standard CCSS.Math.Content.4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm. 04 B62C1C106873438AA0126760075A65A3 239876423FE74AE7AF262256E547698D http://corestandards.org/Math/Content/4/NBT/B/5/ 7 Standard CCSS.Math.Content.4.NBT.B.5 Multiply 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. 04 B62C1C106873438AA0126760075A65A3 239876423FE74AE7AF262256E547698D http://corestandards.org/Math/Content/4/NBT/B/6/ 7 Standard CCSS.Math.Content.4.NBT.B.6 Find 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. 04 B62C1C106873438AA0126760075A65A3 239876423FE74AE7AF262256E547698D http://corestandards.org/Math/Content/4/NF/A/1/ 7 Standard CCSS.Math.Content.4.NF.A.1 Explain why a fraction <i>a</i>/<i>b</i> is equivalent to a fraction (<i>n</i> &times; <i>a</i>)/(<i>n</i> &times; <i>b</i>) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 04 B62C1C106873438AA0126760075A65A3 5E9ED66B8DBE490EB71B397896148D60 http://corestandards.org/Math/Content/4/NF/A/2/ 7 Standard CCSS.Math.Content.4.NF.A.2 Compare 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 &gt;, =, or &lt;, and justify the conclusions, e.g., by using a visual fraction model. 04 B62C1C106873438AA0126760075A65A3 5E9ED66B8DBE490EB71B397896148D60 http://corestandards.org/Math/Content/4/NF/B/3/ 7 Standard CCSS.Math.Content.4.NF.B.3 Understand a fraction <i>a</i>/<i>b</i> with <i>a</i> &gt; 1 as a sum of fractions 1/<i>b</i>. 04 B62C1C106873438AA0126760075A65A3 CAF90B8CE96A4458829E7C7C376A7126 F6933013AE4F438BB52517F900140E51 82832D1795064E29BA1CD2CCE62FC28F 3E844B9ECC954ACAA0E0177CE452D4C1 863D0723B4434959A7B9449A41488937 http://corestandards.org/Math/Content/4/NF/B/3/a/ 8 Component CCSS.Math.Content.4.NF.B.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 04 B62C1C106873438AA0126760075A65A3 1195E43626BF4A6B8AA96AF62AFB4283 http://corestandards.org/Math/Content/4/NF/B/3/b/ 8 Component CCSS.Math.Content.4.NF.B.3b Decompose 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 a visual fraction model. <i>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</i>. 04 B62C1C106873438AA0126760075A65A3 1195E43626BF4A6B8AA96AF62AFB4283 http://corestandards.org/Math/Content/4/NF/B/3/c/ 8 Component CCSS.Math.Content.4.NF.B.3c Add 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. 04 B62C1C106873438AA0126760075A65A3 1195E43626BF4A6B8AA96AF62AFB4283 http://corestandards.org/Math/Content/4/NF/B/3/d/ 8 Component CCSS.Math.Content.4.NF.B.3d Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. 04 B62C1C106873438AA0126760075A65A3 1195E43626BF4A6B8AA96AF62AFB4283 http://corestandards.org/Math/Content/4/NF/B/4/ 7 Standard CCSS.Math.Content.4.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 04 B62C1C106873438AA0126760075A65A3 CAF90B8CE96A4458829E7C7C376A7126 5AA07FB2E9934DCAB59F4AE0D1624C8B AD8924A66D334EF59D20C304573728DC 697F85A12F954712923BD35862DEC2BB http://corestandards.org/Math/Content/4/NF/B/4/a/ 8 Component CCSS.Math.Content.4.NF.B.4a Understand a fraction <i>a</i>/<i>b</i> as a multiple of 1/<i>b</i>. <i>For example, use a visual fraction model to represent 5/4 as the product 5 &times; (1/4), recording the conclusion by the equation 5/4 = 5 &times; (1/4)</i>. 04 B62C1C106873438AA0126760075A65A3 D7B0A4C892974E05B3789666B34F8C50 http://corestandards.org/Math/Content/4/NF/B/4/b/ 8 Component CCSS.Math.Content.4.NF.B.4b Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. <i>For example, use a visual fraction model to express 3 &times; (2/5) as 6 &times; (1/5), recognizing this product as 6/5. (In general, n &times; (a/b) = (n &times; a)/b.)</i> 04 B62C1C106873438AA0126760075A65A3 D7B0A4C892974E05B3789666B34F8C50 http://corestandards.org/Math/Content/4/NF/B/4/c/ 8 Component CCSS.Math.Content.4.NF.B.4c Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. <i>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?</i> 04 B62C1C106873438AA0126760075A65A3 D7B0A4C892974E05B3789666B34F8C50 http://corestandards.org/Math/Content/4/NF/C/5/ 7 Standard CCSS.Math.Content.4.NF.C.5 Express 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.<sup>2</sup> <i>For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100</i>. 04 B62C1C106873438AA0126760075A65A3 30C671F2536E4F57B7A32A782A6E00B9 http://corestandards.org/Math/Content/4/NF/C/6/ 7 Standard CCSS.Math.Content.4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. <i>For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram</i>. 04 B62C1C106873438AA0126760075A65A3 30C671F2536E4F57B7A32A782A6E00B9 http://corestandards.org/Math/Content/4/NF/C/7/ 7 Standard CCSS.Math.Content.4.NF.C.7 Compare 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 &gt;, =, or &lt;, and justify the conclusions, e.g., by using a visual model. 04 B62C1C106873438AA0126760075A65A3 30C671F2536E4F57B7A32A782A6E00B9 http://corestandards.org/Math/Content/4/OA/A/1/ 7 Standard CCSS.Math.Content.4.OA.A.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 &times; 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. 04 B62C1C106873438AA0126760075A65A3 C23D0AFF59694FACA65ECEEBDE3F1698 http://corestandards.org/Math/Content/4/OA/A/2/ 7 Standard CCSS.Math.Content.4.OA.A.2 Multiply 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.<sup>1</sup> 04 B62C1C106873438AA0126760075A65A3 C23D0AFF59694FACA65ECEEBDE3F1698 http://corestandards.org/Math/Content/4/OA/A/3/ 7 Standard CCSS.Math.Content.4.OA.A.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 04 B62C1C106873438AA0126760075A65A3 C23D0AFF59694FACA65ECEEBDE3F1698 http://corestandards.org/Math/Content/4/OA/B/4/ 7 Standard CCSS.Math.Content.4.OA.B.4 Find 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. 04 B62C1C106873438AA0126760075A65A3 1FAE8D11BF744C22A5FC37D6A49587C5 http://corestandards.org/Math/Content/4/OA/C/5/ 7 Standard CCSS.Math.Content.4.OA.C.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. <i>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</i>. 04 B62C1C106873438AA0126760075A65A3 114C44E82C464238BC714BA5B546A7D9 http://corestandards.org/Math/Content/5/G/A/1/ 7 Standard CCSS.Math.Content.5.G.A.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., <i>x</i>-axis and <i>x</i>-coordinate, <i>y</i>-axis and<i> y</i>-coordinate). 05 B62C1C106873438AA0126760075A65A3 3363FCD84F3C4CB4B928C0BE1151DBDC http://corestandards.org/Math/Content/5/G/A/2/ 7 Standard CCSS.Math.Content.5.G.A.2 Represent 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. 05 B62C1C106873438AA0126760075A65A3 3363FCD84F3C4CB4B928C0BE1151DBDC http://corestandards.org/Math/Content/5/G/B/3/ 7 Standard CCSS.Math.Content.5.G.B.3 Understand 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. 05 B62C1C106873438AA0126760075A65A3 5C683021E8394B7597AE0222D0BBE59E http://corestandards.org/Math/Content/5/G/B/4/ 7 Standard CCSS.Math.Content.5.G.B.4 Classify two-dimensional figures in a hierarchy based on properties. 05 B62C1C106873438AA0126760075A65A3 5C683021E8394B7597AE0222D0BBE59E http://corestandards.org/Math/Content/5/MD/A/1/ 7 Standard CCSS.Math.Content.5.MD.A.1 Convert 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. 05 B62C1C106873438AA0126760075A65A3 DC3BDA42AAD245D6AD4025C0C1AA61F1 http://corestandards.org/Math/Content/5/MD/B/2/ 7 Standard CCSS.Math.Content.5.MD.B.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. <i>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</i>. 05 B62C1C106873438AA0126760075A65A3 3FFC6CAA5E994DA180B63210C3DA5142 http://corestandards.org/Math/Content/5/MD/C/3/ 7 Standard CCSS.Math.Content.5.MD.C.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement. 05 B62C1C106873438AA0126760075A65A3 295D70687EC34B7C87B27C0D3D9A39AA 164CCBB72823487A88079EC2593D9354 E92C6ADEAB084243BE335C4B561D1CEA http://corestandards.org/Math/Content/5/MD/C/3/a/ 8 Component CCSS.Math.Content.5.MD.C.3a A 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. 05 B62C1C106873438AA0126760075A65A3 36D5E16885004574B88BAF6886FA2911 http://corestandards.org/Math/Content/5/MD/C/3/b/ 8 Component CCSS.Math.Content.5.MD.C.3b A solid figure which can be packed without gaps or overlaps using <i>n</i> unit cubes is said to have a volume of <i>n</i> cubic units. 05 B62C1C106873438AA0126760075A65A3 36D5E16885004574B88BAF6886FA2911 http://corestandards.org/Math/Content/5/MD/C/4/ 7 Standard CCSS.Math.Content.5.MD.C.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 05 B62C1C106873438AA0126760075A65A3 295D70687EC34B7C87B27C0D3D9A39AA http://corestandards.org/Math/Content/5/MD/C/5/ 7 Standard CCSS.Math.Content.5.MD.C.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. 05 B62C1C106873438AA0126760075A65A3 295D70687EC34B7C87B27C0D3D9A39AA 90E0C13F4411422189D10B2BB526E609 C5BB4D4030BE40A4A51FD07282BBEF75 6B05F71ECDDA44CB9148AC09EB30CE9C http://corestandards.org/Math/Content/5/MD/C/5/a/ 8 Component CCSS.Math.Content.5.MD.C.5a Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, 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. 05 B62C1C106873438AA0126760075A65A3 B8EEE0176EDE4F9998AB63E1055E0068 http://corestandards.org/Math/Content/5/MD/C/5/b/ 8 Component CCSS.Math.Content.5.MD.C.5b Apply the formulas <i>V</i> = <i>l</i> &times; <i>w</i> &times; <i>h</i> and <i>V</i> = <i>b</i> &times; <i>h</i> 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. 05 B62C1C106873438AA0126760075A65A3 B8EEE0176EDE4F9998AB63E1055E0068 http://corestandards.org/Math/Content/5/MD/C/5/c/ 8 Component CCSS.Math.Content.5.MD.C.5c Recognize 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. 05 B62C1C106873438AA0126760075A65A3 B8EEE0176EDE4F9998AB63E1055E0068 http://corestandards.org/Math/Content/5/NBT/A/1/ 7 Standard CCSS.Math.Content.5.NBT.A.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 05 B62C1C106873438AA0126760075A65A3 7226510F74B34FD3ACAEE829293D7576 http://corestandards.org/Math/Content/5/NBT/A/2/ 7 Standard CCSS.Math.Content.5.NBT.A.2 Explain 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. 05 B62C1C106873438AA0126760075A65A3 7226510F74B34FD3ACAEE829293D7576 http://corestandards.org/Math/Content/5/NBT/A/3/ 7 Standard CCSS.Math.Content.5.NBT.A.3 Read, write, and compare decimals to thousandths. 05 B62C1C106873438AA0126760075A65A3 7226510F74B34FD3ACAEE829293D7576 959ACB4A9E1D4A6897A43FA910E55589 8353967DFA6844188C5A5B4E596BBD2D http://corestandards.org/Math/Content/5/NBT/A/3/a/ 8 Component CCSS.Math.Content.5.NBT.A.3a Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 &times; 100 + 4 &times; 10 + 7 &times; 1 + 3 &times; (1/10) + 9 &times; (1/100) + 2 &times; (1/1000). 05 B62C1C106873438AA0126760075A65A3 BE6257FC08DA4AA896C87FC4A19F6520 http://corestandards.org/Math/Content/5/NBT/A/3/b/ 8 Component CCSS.Math.Content.5.NBT.A.3b Compare two decimals to thousandths based on meanings of the digits in each place, using &gt;, =, and &lt; symbols to record the results of comparisons. 05 B62C1C106873438AA0126760075A65A3 BE6257FC08DA4AA896C87FC4A19F6520 http://corestandards.org/Math/Content/5/NBT/A/4/ 7 Standard CCSS.Math.Content.5.NBT.A.4 Use place value understanding to round decimals to any place. 05 B62C1C106873438AA0126760075A65A3 7226510F74B34FD3ACAEE829293D7576 http://corestandards.org/Math/Content/5/NBT/B/5/ 7 Standard CCSS.Math.Content.5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm. 05 B62C1C106873438AA0126760075A65A3 63C3F21F71684B79B0C994BDCF9AC481 http://corestandards.org/Math/Content/5/NBT/B/6/ 7 Standard CCSS.Math.Content.5.NBT.B.6 Find 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. 05 B62C1C106873438AA0126760075A65A3 63C3F21F71684B79B0C994BDCF9AC481 http://corestandards.org/Math/Content/5/NBT/B/7/ 7 Standard CCSS.Math.Content.5.NBT.B.7 Add, 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; relate the strategy to a written method and explain the reasoning used. 05 B62C1C106873438AA0126760075A65A3 63C3F21F71684B79B0C994BDCF9AC481 http://corestandards.org/Math/Content/5/NF/A/1/ 7 Standard CCSS.Math.Content.5.NF.A.1 Add 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. <i>For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)</i> 05 B62C1C106873438AA0126760075A65A3 89DB00AD36014824847EBE83D78E1399 http://corestandards.org/Math/Content/5/NF/A/2/ 7 Standard CCSS.Math.Content.5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, 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. <i>For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 &lt; 1/2</i>. 05 B62C1C106873438AA0126760075A65A3 89DB00AD36014824847EBE83D78E1399 http://corestandards.org/Math/Content/5/NF/B/3/ 7 Standard CCSS.Math.Content.5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (<i>a</i>/<i>b</i> = <i>a</i> &divide; <i>b</i>). 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. <i>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 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 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?</i> 05 B62C1C106873438AA0126760075A65A3 366753EC60DE4400AC95657B754A9F34 http://corestandards.org/Math/Content/5/NF/B/4/ 7 Standard CCSS.Math.Content.5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 05 B62C1C106873438AA0126760075A65A3 366753EC60DE4400AC95657B754A9F34 6B92A5720478456FAFF81454BA4CFDA2 7A2B7EB8894848DD84B9B4E869755FE1 http://corestandards.org/Math/Content/5/NF/B/4/a/ 8 Component CCSS.Math.Content.5.NF.B.4a Interpret the product (<i>a</i>/<i>b</i>) × <i>q</i> as <em>a</em> parts of a partition of <i>q</i> into <i>b</i> equal parts; equivalently, as the result of a sequence of operations<i> a</i> × <i>q</i> ÷ <i>b</i>. <i>For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = (ac)/(bd).</i> 05 B62C1C106873438AA0126760075A65A3 D270A624395246338200753501C0BF0E http://corestandards.org/Math/Content/5/NF/B/4/b/ 8 Component CCSS.Math.Content.5.NF.B.4b Find 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. 05 B62C1C106873438AA0126760075A65A3 D270A624395246338200753501C0BF0E http://corestandards.org/Math/Content/5/NF/B/5/ 7 Standard CCSS.Math.Content.5.NF.B.5 Interpret multiplication as scaling (resizing), by: 05 B62C1C106873438AA0126760075A65A3 366753EC60DE4400AC95657B754A9F34 036DA6C72EEB4DA38D456F6CD768A182 FD677276B89E4F55AEEC482260D345C2 http://corestandards.org/Math/Content/5/NF/B/5/a/ 8 Component CCSS.Math.Content.5.NF.B.5a Comparing 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. 05 B62C1C106873438AA0126760075A65A3 F81FE2BA03074CD7AFB0C48A37A9D51A http://corestandards.org/Math/Content/5/NF/B/5/b/ 8 Component CCSS.Math.Content.5.NF.B.5b Explaining 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 <i>a</i>/<i>b</i> = (<i>n</i> &times; <i>a</i>)/(<i>n</i> &times; <i>b</i>) to the effect of multiplying <i>a</i>/<i>b</i> by 1. 05 B62C1C106873438AA0126760075A65A3 F81FE2BA03074CD7AFB0C48A37A9D51A http://corestandards.org/Math/Content/5/NF/B/6/ 7 Standard CCSS.Math.Content.5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 05 B62C1C106873438AA0126760075A65A3 366753EC60DE4400AC95657B754A9F34 http://corestandards.org/Math/Content/5/NF/B/7/ 7 Standard CCSS.Math.Content.5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.<sup>1</sup> 05 B62C1C106873438AA0126760075A65A3 366753EC60DE4400AC95657B754A9F34 4D8482255802431492B2AB084EA082F0 0FD052F169B04888A6259C2A5D126892 34128D58FD89458C84AFB84FE54AB1C8 http://corestandards.org/Math/Content/5/NF/B/7/a/ 8 Component CCSS.Math.Content.5.NF.B.7a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. <i>For example, create a story context for (1/3) &divide; 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) &divide; 4 = 1/12 because (1/12) &times; 4 = 1/3</i>. 05 B62C1C106873438AA0126760075A65A3 6662A64E904C4480B0940FA7F7EAE5BC http://corestandards.org/Math/Content/5/NF/B/7/b/ 8 Component CCSS.Math.Content.5.NF.B.7b Interpret division of a whole number by a unit fraction, and compute such quotients. <i>For example, create a story context for 4 &divide; (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 &divide; (1/5) = 20 because 20 &times; (1/5) = 4</i>. 05 B62C1C106873438AA0126760075A65A3 6662A64E904C4480B0940FA7F7EAE5BC http://corestandards.org/Math/Content/5/NF/B/7/c/ 8 Component CCSS.Math.Content.5.NF.B.7c Solve 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. <i>For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?</i> 05 B62C1C106873438AA0126760075A65A3 6662A64E904C4480B0940FA7F7EAE5BC http://corestandards.org/Math/Content/5/OA/A/1/ 7 Standard CCSS.Math.Content.5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. 05 B62C1C106873438AA0126760075A65A3 BFE360826DBB414F917BF59E9D2BB5D1 http://corestandards.org/Math/Content/5/OA/A/2/ 7 Standard CCSS.Math.Content.5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. <i>For example, express the calculation "add 8 and 7, then multiply by 2" as 2 &times; (8 + 7). Recognize that 3 &times; (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product</i>. 05 B62C1C106873438AA0126760075A65A3 BFE360826DBB414F917BF59E9D2BB5D1 http://corestandards.org/Math/Content/5/OA/B/3/ 7 Standard CCSS.Math.Content.5.OA.B.3 Generate 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. <i>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</i>. 05 B62C1C106873438AA0126760075A65A3 E9FDE5E39856429F89106808E49664ED http://corestandards.org/Math/Content/6/EE/A/1/ 7 Standard CCSS.Math.Content.6.EE.A.1 Write and evaluate numerical expressions involving whole-number exponents. 06 B62C1C106873438AA0126760075A65A3 A7D3275BC52147618D6CFEE43FB1A47E http://corestandards.org/Math/Content/6/EE/A/2/ 7 Standard CCSS.Math.Content.6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers. 06 B62C1C106873438AA0126760075A65A3 A7D3275BC52147618D6CFEE43FB1A47E C1A47EC07F9F49C481DE52A8235E27B6 3639C586FB844ED3AD7D4232451C44F1 DC40EAF30F894820988517FBA44D9601 http://corestandards.org/Math/Content/6/EE/A/2/a/ 8 Component CCSS.Math.Content.6.EE.A.2a Write expressions that record operations with numbers and with letters standing for numbers. <i>For example, express the calculation "Subtract y from 5" as 5 - y</i>. 06 B62C1C106873438AA0126760075A65A3 F78ACD4FA2D74173998D2EA26FBA1E1B http://corestandards.org/Math/Content/6/EE/A/2/b/ 8 Component CCSS.Math.Content.6.EE.A.2b Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. <i>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</i>. 06 B62C1C106873438AA0126760075A65A3 F78ACD4FA2D74173998D2EA26FBA1E1B http://corestandards.org/Math/Content/6/EE/A/2/c/ 8 Component CCSS.Math.Content.6.EE.A.2c Evaluate 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). <i>For example, use the formulas V = s<sup>3</sup> and A = 6 s<sup>2</sup> to find the volume and surface area of a cube with sides of length s = 1/2</i>. 06 B62C1C106873438AA0126760075A65A3 F78ACD4FA2D74173998D2EA26FBA1E1B http://corestandards.org/Math/Content/6/EE/A/3/ 7 Standard CCSS.Math.Content.6.EE.A.3 Apply the properties of operations to generate equivalent expressions. <i>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</i>. 06 B62C1C106873438AA0126760075A65A3 A7D3275BC52147618D6CFEE43FB1A47E http://corestandards.org/Math/Content/6/EE/A/4/ 7 Standard CCSS.Math.Content.6.EE.A.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). <i>For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.</i>. 06 B62C1C106873438AA0126760075A65A3 A7D3275BC52147618D6CFEE43FB1A47E http://corestandards.org/Math/Content/6/EE/B/5/ 7 Standard CCSS.Math.Content.6.EE.B.5 Understand 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. 06 B62C1C106873438AA0126760075A65A3 E67594E323034EDFAD123E9BC4B258BE http://corestandards.org/Math/Content/6/EE/B/6/ 7 Standard CCSS.Math.Content.6.EE.B.6 Use 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. 06 B62C1C106873438AA0126760075A65A3 E67594E323034EDFAD123E9BC4B258BE http://corestandards.org/Math/Content/6/EE/B/7/ 7 Standard CCSS.Math.Content.6.EE.B.7 Solve real-world and mathematical problems by writing and solving equations of the form <i>x</i> + <i>p</i> = <i>q</i> and<i> px</i> = <i>q</i> for cases in which <i>p</i>, <i>q</i> and <i>x</i> are all nonnegative rational numbers. 06 B62C1C106873438AA0126760075A65A3 E67594E323034EDFAD123E9BC4B258BE http://corestandards.org/Math/Content/6/EE/B/8/ 7 Standard CCSS.Math.Content.6.EE.B.8 Write an inequality of the form <i>x</i> &gt; <i>c</i> or <i>x</i> &lt; <i>c</i> to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form <i>x</i> &gt; <i>c</i> or<i> x</i> &lt; c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. 06 B62C1C106873438AA0126760075A65A3 E67594E323034EDFAD123E9BC4B258BE http://corestandards.org/Math/Content/6/EE/C/9/ 7 Standard CCSS.Math.Content.6.EE.C.9 Use 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. 06 B62C1C106873438AA0126760075A65A3 7FCA1FBCCFE846C3A774502E1F65B093 http://corestandards.org/Math/Content/6/G/A/1/ 7 Standard CCSS.Math.Content.6.G.A.1 Find 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. 06 B62C1C106873438AA0126760075A65A3 4672773E1A8744EEA5318249CA2D8884 http://corestandards.org/Math/Content/6/G/A/2/ 7 Standard CCSS.Math.Content.6.G.A.2 Find 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 <i>V = l w h</i> and <i>V = b h</i> to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. 06 B62C1C106873438AA0126760075A65A3 4672773E1A8744EEA5318249CA2D8884 http://corestandards.org/Math/Content/6/G/A/3/ 7 Standard CCSS.Math.Content.6.G.A.3 Draw 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. 06 B62C1C106873438AA0126760075A65A3 4672773E1A8744EEA5318249CA2D8884 http://corestandards.org/Math/Content/6/G/A/4/ 7 Standard CCSS.Math.Content.6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. 06 B62C1C106873438AA0126760075A65A3 4672773E1A8744EEA5318249CA2D8884 http://corestandards.org/Math/Content/6/NS/A/1/ 7 Standard CCSS.Math.Content.6.NS.A.1 Interpret 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. <i>For example, create a story context for (2/3) &divide; (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) &divide; (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) &divide; (c/d) = ad/bc.) How much chocolate will each person get if 3 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 mi and area 1/2 square mi?</i>. 06 B62C1C106873438AA0126760075A65A3 91FABAB899814C55851003A0EE98F8FC http://corestandards.org/Math/Content/6/NS/B/2/ 7 Standard CCSS.Math.Content.6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm. 06 B62C1C106873438AA0126760075A65A3 E0D6F554561F4C4EB524449269E39665 http://corestandards.org/Math/Content/6/NS/B/3/ 7 Standard CCSS.Math.Content.6.NS.B.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. 06 B62C1C106873438AA0126760075A65A3 E0D6F554561F4C4EB524449269E39665 http://corestandards.org/Math/Content/6/NS/B/4/ 7 Standard CCSS.Math.Content.6.NS.B.4 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 whole numbers with no common factor. <i>For example, express 36 + 8 as 4 (9 + 2).</i>. 06 B62C1C106873438AA0126760075A65A3 E0D6F554561F4C4EB524449269E39665 http://corestandards.org/Math/Content/6/NS/C/5/ 7 Standard CCSS.Math.Content.6.NS.C.5 Understand 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, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. 06 B62C1C106873438AA0126760075A65A3 DED7E6EBA72E4243ACAD94A13CCC92E9 http://corestandards.org/Math/Content/6/NS/C/6/ 7 Standard CCSS.Math.Content.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. 06 B62C1C106873438AA0126760075A65A3 DED7E6EBA72E4243ACAD94A13CCC92E9 5EE6ABF39BA64476BBD43564FBF2B2F8 C81686A64B584D928944DF54E754B474 A5AF20605DE94E0CA3EDDE67F4EF3818 http://corestandards.org/Math/Content/6/NS/C/6/a/ 8 Component CCSS.Math.Content.6.NS.C.6a Recognize 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 0 is its own opposite. 06 B62C1C106873438AA0126760075A65A3 A476322A7B9545339733892F2F6BF97F http://corestandards.org/Math/Content/6/NS/C/6/b/ 8 Component CCSS.Math.Content.6.NS.C.6b Understand 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. 06 B62C1C106873438AA0126760075A65A3 A476322A7B9545339733892F2F6BF97F http://corestandards.org/Math/Content/6/NS/C/6/c/ 8 Component CCSS.Math.Content.6.NS.C.6c Find 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. 06 B62C1C106873438AA0126760075A65A3 A476322A7B9545339733892F2F6BF97F http://corestandards.org/Math/Content/6/NS/C/7/ 7 Standard CCSS.Math.Content.6.NS.C.7 Understand ordering and absolute value of rational numbers. 06 B62C1C106873438AA0126760075A65A3 DED7E6EBA72E4243ACAD94A13CCC92E9 4590DEFD95E44B6B9BF6A9425DE30D1A 3D676E16444546DAB6327396969B83D4 EDA8659B8F2F468C891C301FF54B210C 21FA1EE07B5641ED877C720AD65ECF60 http://corestandards.org/Math/Content/6/NS/C/7/a/ 8 Component CCSS.Math.Content.6.NS.C.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. <i>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</i>. 06 B62C1C106873438AA0126760075A65A3 71F96ADF450A4133A258FDACE2D3910D http://corestandards.org/Math/Content/6/NS/C/7/b/ 8 Component CCSS.Math.Content.6.NS.C.7b Write, interpret, and explain statements of order for rational numbers in real-world contexts. <i>For example, write -3 <sup>o</sup>C &gt; -7 <sup>o</sup>C to express the fact that -3 <sup>o</sup>C is warmer than -7 <sup>o</sup>C</i>. 06 B62C1C106873438AA0126760075A65A3 71F96ADF450A4133A258FDACE2D3910D http://corestandards.org/Math/Content/6/NS/C/7/c/ 8 Component CCSS.Math.Content.6.NS.C.7c Understand 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. <i>For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars</i>. 06 B62C1C106873438AA0126760075A65A3 71F96ADF450A4133A258FDACE2D3910D http://corestandards.org/Math/Content/6/NS/C/7/d/ 8 Component CCSS.Math.Content.6.NS.C.7d Distinguish comparisons of absolute value from statements about order. <i>For example, recognize that an account balance less than -30 dollars represents a debt greater than 30 dollars</i>. 06 B62C1C106873438AA0126760075A65A3 71F96ADF450A4133A258FDACE2D3910D http://corestandards.org/Math/Content/6/NS/C/8/ 7 Standard CCSS.Math.Content.6.NS.C.8 Solve 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. 06 B62C1C106873438AA0126760075A65A3 DED7E6EBA72E4243ACAD94A13CCC92E9 http://corestandards.org/Math/Content/6/RP/A/1/ 7 Standard CCSS.Math.Content.6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. <i>For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."</i> 06 B62C1C106873438AA0126760075A65A3 91FABAB899814C55851003A0EE98F8FB http://corestandards.org/Math/Content/6/RP/A/2/ 7 Standard CCSS.Math.Content.6.RP.A.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b &ne; 0, and use rate language in the context of a ratio relationship. <i>For example, "This recipe has a ratio of 3 cups of flour to 4 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 \$5 per hamburger."</i><sup>1</sup> 06 B62C1C106873438AA0126760075A65A3 91FABAB899814C55851003A0EE98F8FB http://corestandards.org/Math/Content/6/RP/A/3/ 7 Standard CCSS.Math.Content.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. 06 B62C1C106873438AA0126760075A65A3 91FABAB899814C55851003A0EE98F8FB 7C6AD49FC1EC4401A3AFA1ACA63BF4A9 D31433DF25FB4CD78EB603BAF929B294 A068CFC353A845239E73BF7580A2F924 FA34B81A7B23457B81D873613F7E7B75 http://corestandards.org/Math/Content/6/RP/A/3/a/ 8 Component CCSS.Math.Content.6.RP.A.3a Make 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. 06 B62C1C106873438AA0126760075A65A3 76D9105C412B45739450D4FAF3667D6B http://corestandards.org/Math/Content/6/RP/A/3/b/ 8 Component CCSS.Math.Content.6.RP.A.3b Solve unit rate problems including those involving unit pricing and constant speed. <i>For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?</i> 06 B62C1C106873438AA0126760075A65A3 76D9105C412B45739450D4FAF3667D6B http://corestandards.org/Math/Content/6/RP/A/3/c/ 8 Component CCSS.Math.Content.6.RP.A.3c Find 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. 06 B62C1C106873438AA0126760075A65A3 76D9105C412B45739450D4FAF3667D6B http://corestandards.org/Math/Content/6/RP/A/3/d/ 8 Component CCSS.Math.Content.6.RP.A.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. 06 B62C1C106873438AA0126760075A65A3 76D9105C412B45739450D4FAF3667D6B http://corestandards.org/Math/Content/6/SP/A/1/ 7 Standard CCSS.Math.Content.6.SP.A.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. <i>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</i>. 06 B62C1C106873438AA0126760075A65A3 A999DF3166834919A4C8EA1415BEF8A7 http://corestandards.org/Math/Content/6/SP/A/2/ 7 Standard CCSS.Math.Content.6.SP.A.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. 06 B62C1C106873438AA0126760075A65A3 A999DF3166834919A4C8EA1415BEF8A7 http://corestandards.org/Math/Content/6/SP/A/3/ 7 Standard CCSS.Math.Content.6.SP.A.3 Recognize 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. 06 B62C1C106873438AA0126760075A65A3 A999DF3166834919A4C8EA1415BEF8A7 http://corestandards.org/Math/Content/6/SP/B/4/ 7 Standard CCSS.Math.Content.6.SP.B.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 06 B62C1C106873438AA0126760075A65A3 D0BBF510F9F543828994AF283582A18F http://corestandards.org/Math/Content/6/SP/B/5/ 7 Standard CCSS.Math.Content.6.SP.B.5 Summarize numerical data sets in relation to their context, such as by: 06 B62C1C106873438AA0126760075A65A3 D0BBF510F9F543828994AF283582A18F F2D6B6FB63E14C8098655981124EBC79 61E5ED58B8D14F3DBEBF8C11108ADDD4 00314B63F7C044B48D1E6021D30FA77A EEB4EB9C540C46E79C61B867FF4B1BF9 http://corestandards.org/Math/Content/6/SP/B/5/a/ 8 Component CCSS.Math.Content.6.SP.B.5a Reporting the number of observations. 06 B62C1C106873438AA0126760075A65A3 25270B3E4531410DA4BECFDE1F058E6A http://corestandards.org/Math/Content/6/SP/B/5/b/ 8 Component CCSS.Math.Content.6.SP.B.5b Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. 06 B62C1C106873438AA0126760075A65A3 25270B3E4531410DA4BECFDE1F058E6A http://corestandards.org/Math/Content/6/SP/B/5/c/ 8 Component CCSS.Math.Content.6.SP.B.5c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. 06 B62C1C106873438AA0126760075A65A3 25270B3E4531410DA4BECFDE1F058E6A http://corestandards.org/Math/Content/6/SP/B/5/d/ 8 Component CCSS.Math.Content.6.SP.B.5d Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. 06 B62C1C106873438AA0126760075A65A3 25270B3E4531410DA4BECFDE1F058E6A http://corestandards.org/Math/Content/7/EE/A/1/ 7 Standard CCSS.Math.Content.7.EE.A.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 07 B62C1C106873438AA0126760075A65A3 B8B28A72AADF4D5C9D819A0AD29E830F http://corestandards.org/Math/Content/7/EE/A/2/ 7 Standard CCSS.Math.Content.7.EE.A.2 Understand 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. <i>For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05."</i> 07 B62C1C106873438AA0126760075A65A3 B8B28A72AADF4D5C9D819A0AD29E830F http://corestandards.org/Math/Content/7/EE/B/3/ 7 Standard CCSS.Math.Content.7.EE.B.3 Solve 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. <i>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</i>. 07 B62C1C106873438AA0126760075A65A3 28C6157CE6114983AE2E7334B88BB477 http://corestandards.org/Math/Content/7/EE/B/4/ 7 Standard CCSS.Math.Content.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. 07 B62C1C106873438AA0126760075A65A3 28C6157CE6114983AE2E7334B88BB477 2F1135F18E434B77BB694AC9D3EEE575 38D1A0732B9A4055BA1E28B6D44D2493 http://corestandards.org/Math/Content/7/EE/B/4/a/ 8 Component CCSS.Math.Content.7.EE.B.4a Solve word problems leading to equations of the form <i>px</i> + <i>q</i> = <i>r</i> and <i>p</i>(<i>x</i> + <i>q</i>) = <i>r</i>, where <i>p</i>, <i>q</i>, and <i>r</i> 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. <i>For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?</i> 07 B62C1C106873438AA0126760075A65A3 5C4A177A795D42178DD81C960C176EAC http://corestandards.org/Math/Content/7/EE/B/4/b/ 8 Component CCSS.Math.Content.7.EE.B.4b Solve word problems leading to inequalities of the form <i>px</i> + <i>q</i> &gt; <i>r</i> or <i>px</i> + <i>q</i> &lt; <i>r</i>, where <i>p</i>, <i>q</i>, and <i>r</i> are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. <i>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</i>. 07 B62C1C106873438AA0126760075A65A3 5C4A177A795D42178DD81C960C176EAC http://corestandards.org/Math/Content/7/G/A/1/ 7 Standard CCSS.Math.Content.7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. 07 B62C1C106873438AA0126760075A65A3 5995C336BE744A67B7D2D5FD8DC54086 http://corestandards.org/Math/Content/7/G/A/2/ 7 Standard CCSS.Math.Content.7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) 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. 07 B62C1C106873438AA0126760075A65A3 5995C336BE744A67B7D2D5FD8DC54086 http://corestandards.org/Math/Content/7/G/A/3/ 7 Standard CCSS.Math.Content.7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. 07 B62C1C106873438AA0126760075A65A3 5995C336BE744A67B7D2D5FD8DC54086 http://corestandards.org/Math/Content/7/G/B/4/ 7 Standard CCSS.Math.Content.7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. 07 B62C1C106873438AA0126760075A65A3 D8C53E0B5E7445D1B3493BEB30AAD017 http://corestandards.org/Math/Content/7/G/B/5/ 7 Standard CCSS.Math.Content.7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.</li> 07 B62C1C106873438AA0126760075A65A3 D8C53E0B5E7445D1B3493BEB30AAD017 http://corestandards.org/Math/Content/7/G/B/6/ 7 Standard CCSS.Math.Content.7.G.B.6 Solve 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. 07 B62C1C106873438AA0126760075A65A3 D8C53E0B5E7445D1B3493BEB30AAD017 http://corestandards.org/Math/Content/7/NS/A/1/ 7 Standard CCSS.Math.Content.7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 07 B62C1C106873438AA0126760075A65A3 3DDCF0B192B04D66AEE3E2C9EC8B7CAC AEFCAD7F6D18406aA4769EF34D8AEAC6 F020570FB095480193A4F969241B8029 9C1435F358A84468956E149747DF8D85 CECCADE8384141139CA663AEC03D37B5 http://corestandards.org/Math/Content/7/NS/A/1/a/ 8 Component CCSS.Math.Content.7.NS.A.1a Describe situations in which opposite quantities combine to make 0. <i>For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged</i>. 07 B62C1C106873438AA0126760075A65A3 3C0139D0DA344DBCAC06DBC4CBC41F9C http://corestandards.org/Math/Content/7/NS/A/1/b/ 8 Component CCSS.Math.Content.7.NS.A.1b Understand <i>p</i> + <i>q</i> as the number located a distance |<i>q</i>| from <i>p</i>, in the positive or negative direction depending on whether <i>q</i> 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. 07 B62C1C106873438AA0126760075A65A3 3C0139D0DA344DBCAC06DBC4CBC41F9C http://corestandards.org/Math/Content/7/NS/A/1/c/ 8 Component CCSS.Math.Content.7.NS.A.1c Understand subtraction of rational numbers as adding the additive inverse, <i>p</i> - <i>q</i> = <i>p</i> + (-<i>q</i>). 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. 07 B62C1C106873438AA0126760075A65A3 3C0139D0DA344DBCAC06DBC4CBC41F9C http://corestandards.org/Math/Content/7/NS/A/1/d/ 8 Component CCSS.Math.Content.7.NS.A.1d Apply properties of operations as strategies to add and subtract rational numbers. 07 B62C1C106873438AA0126760075A65A3 3C0139D0DA344DBCAC06DBC4CBC41F9C http://corestandards.org/Math/Content/7/NS/A/2/ 7 Standard CCSS.Math.Content.7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 07 B62C1C106873438AA0126760075A65A3 3DDCF0B192B04D66AEE3E2C9EC8B7CAC 90A64AED5CA141F2844E11A8A381F0F9 8E31252383F148FEA1E748BEB841036E 23541ADA26D346F09FCC989C5E04848E 23541ADA26D346F09FCC989C5E04848F http://corestandards.org/Math/Content/7/NS/A/2/a/ 8 Component CCSS.Math.Content.7.NS.A.2a Understand 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. 07 B62C1C106873438AA0126760075A65A3 416206BA82F74726987E836991FB89F8 http://corestandards.org/Math/Content/7/NS/A/2/b/ 8 Component CCSS.Math.Content.7.NS.A.2b Understand 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 <i>p</i> and <i>q</i> are integers, then -(<i>p</i>/<i>q</i>) = (-<i>p</i>)/<i>q</i> = <i>p</i>/(-<i>q</i>). Interpret quotients of rational numbers by describing real-world contexts. 07 B62C1C106873438AA0126760075A65A3 416206BA82F74726987E836991FB89F8 http://corestandards.org/Math/Content/7/NS/A/2/c/ 8 Component CCSS.Math.Content.7.NS.A.2c Apply properties of operations as strategies to multiply and divide rational numbers. 07 B62C1C106873438AA0126760075A65A3 416206BA82F74726987E836991FB89F8 http://corestandards.org/Math/Content/7/NS/A/2/d/ 8 Component CCSS.Math.Content.7.NS.A.2d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 07 B62C1C106873438AA0126760075A65A3 416206BA82F74726987E836991FB89F8 http://corestandards.org/Math/Content/7/NS/A/3/ 7 Standard CCSS.Math.Content.7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers.<sup>1</sup> 07 B62C1C106873438AA0126760075A65A3 3DDCF0B192B04D66AEE3E2C9EC8B7CAC http://corestandards.org/Math/Content/7/RP/A/1/ 7 Standard CCSS.Math.Content.7.RP.A.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. <i>For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction <sup>1/2</sup>/<sub>1/4</sub> miles per hour, equivalently 2 miles per hour</i>. 07 B62C1C106873438AA0126760075A65A3 71C7D669B568463CB1189D089CF4C723 http://corestandards.org/Math/Content/7/RP/A/2/ 7 Standard CCSS.Math.Content.7.RP.A.2 Recognize and represent proportional relationships between quantities. 07 B62C1C106873438AA0126760075A65A3 71C7D669B568463CB1189D089CF4C723 46FC84C961254E79BFBEE8399477632D 3ADAFFEF079D4FC3809513B650D91D9F A0D3F72097A44707B921C2A4AD288AAE C2918ADA67414c57AC6A37C2F3BB7815 http://corestandards.org/Math/Content/7/RP/A/2/a/ 8 Component CCSS.Math.Content.7.RP.A.2a Decide 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. 07 B62C1C106873438AA0126760075A65A3 A57D2EDF9830409CB97D9F8DBD6763DD http://corestandards.org/Math/Content/7/RP/A/2/b/ 8 Component CCSS.Math.Content.7.RP.A.2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 07 B62C1C106873438AA0126760075A65A3 A57D2EDF9830409CB97D9F8DBD6763DD http://corestandards.org/Math/Content/7/RP/A/2/c/ 8 Component CCSS.Math.Content.7.RP.A.2c Represent proportional relationships by equations. <i>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</i>. 07 B62C1C106873438AA0126760075A65A3 A57D2EDF9830409CB97D9F8DBD6763DD http://corestandards.org/Math/Content/7/RP/A/2/d/ 8 Component CCSS.Math.Content.7.RP.A.2d Explain what a point (<i>x</i>, <i>y</i>) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, <i>r</i>) where r is the unit rate. 07 B62C1C106873438AA0126760075A65A3 A57D2EDF9830409CB97D9F8DBD6763DD http://corestandards.org/Math/Content/7/RP/A/3/ 7 Standard CCSS.Math.Content.7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. 07 B62C1C106873438AA0126760075A65A3 71C7D669B568463CB1189D089CF4C723 http://corestandards.org/Math/Content/7/SP/A/1/ 7 Standard CCSS.Math.Content.7.SP.A.1 Understand 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. 07 B62C1C106873438AA0126760075A65A3 30690CE0AA584E25B4C8AA897DC93E34 http://corestandards.org/Math/Content/7/SP/A/2/ 7 Standard CCSS.Math.Content.7.SP.A.2 Use 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. <i>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</i>. 07 B62C1C106873438AA0126760075A65A3 30690CE0AA584E25B4C8AA897DC93E34 http://corestandards.org/Math/Content/7/SP/B/3/ 7 Standard CCSS.Math.Content.7.SP.B.3 Informally 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. <i>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, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable</i>. 07 B62C1C106873438AA0126760075A65A3 929EBEA1A93948A3B0B8B31E3CA106FC http://corestandards.org/Math/Content/7/SP/B/4/ 7 Standard CCSS.Math.Content.7.SP.B.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. <i>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</i>. 07 B62C1C106873438AA0126760075A65A3 929EBEA1A93948A3B0B8B31E3CA106FC http://corestandards.org/Math/Content/7/SP/C/5/ 7 Standard CCSS.Math.Content.7.SP.C.5 Understand 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.</li> 07 B62C1C106873438AA0126760075A65A3 C62D7937C1DD4C009128AE62E48A92C3 http://corestandards.org/Math/Content/7/SP/C/6/ 7 Standard CCSS.Math.Content.7.SP.C.6 Approximate 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. <i>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</i>. 07 B62C1C106873438AA0126760075A65A3 C62D7937C1DD4C009128AE62E48A92C3 http://corestandards.org/Math/Content/7/SP/C/7/ 7 Standard CCSS.Math.Content.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. 07 B62C1C106873438AA0126760075A65A3 C62D7937C1DD4C009128AE62E48A92C3 716CB223FD464557990DA2C256BDC030 09E56ADF76064716A2076F71820F95F3 http://corestandards.org/Math/Content/7/SP/C/7/a/ 8 Component CCSS.Math.Content.7.SP.C.7a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. <i>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</i>. 07 B62C1C106873438AA0126760075A65A3 472222394A204D26844BA834703EAC55 http://corestandards.org/Math/Content/7/SP/C/7/b/ 8 Component CCSS.Math.Content.7.SP.C.7b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. <i>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?</i> 07 B62C1C106873438AA0126760075A65A3 472222394A204D26844BA834703EAC55 http://corestandards.org/Math/Content/7/SP/C/8/ 7 Standard CCSS.Math.Content.7.SP.C.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. 07 B62C1C106873438AA0126760075A65A3 C62D7937C1DD4C009128AE62E48A92C3 F9897607C3E8445E8391D0F7284D37E0 0ED4E865656449238229475118C793D4 1159DE3EB8BF4B24AE2DB38F66D346CF http://corestandards.org/Math/Content/7/SP/C/8/a/ 8 Component CCSS.Math.Content.7.SP.C.8a Understand 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. 07 B62C1C106873438AA0126760075A65A3 1001223E3087486CBB73C6BAA21516DF http://corestandards.org/Math/Content/7/SP/C/8/b/ 8 Component CCSS.Math.Content.7.SP.C.8b Represent 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. 07 B62C1C106873438AA0126760075A65A3 1001223E3087486CBB73C6BAA21516DF http://corestandards.org/Math/Content/7/SP/C/8/c/ 8 Component CCSS.Math.Content.7.SP.C.8c Design and use a simulation to generate frequencies for compound events. <i>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 4 donors to find one with type A blood?</i> 07 B62C1C106873438AA0126760075A65A3 1001223E3087486CBB73C6BAA21516DF http://corestandards.org/Math/Content/8/EE/A/1/ 7 Standard CCSS.Math.Content.8.EE.A.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3<sup>2</sup>&nbsp;&times;&nbsp;3<sup>-5</sup> = 3<sup>-3</sup> = 1/3<sup>3</sup> = 1/27. 08 B62C1C106873438AA0126760075A65A3 42D43D973FEF4A028D2AEAC7151A01DB http://corestandards.org/Math/Content/8/EE/A/2/ 7 Standard CCSS.Math.Content.8.EE.A.2 Use square root and cube root symbols to represent solutions to equations of the form <i>x</i><sup>2</sup> = <i>p</i> and <i>x</i><sup>3</sup> = p, where <i>p</i> is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that &radic;2 is irrational. 08 B62C1C106873438AA0126760075A65A3 42D43D973FEF4A028D2AEAC7151A01DB http://corestandards.org/Math/Content/8/EE/A/3/ 7 Standard CCSS.Math.Content.8.EE.A.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. <em>For example, estimate the population of the United States as 3 times 10<sup>8</sup> and the population of the world as 7 times 10<sup>9</sup>, and determine that the world population is more than 20 times larger</em>. 08 B62C1C106873438AA0126760075A65A3 42D43D973FEF4A028D2AEAC7151A01DB http://corestandards.org/Math/Content/8/EE/A/4/ 7 Standard CCSS.Math.Content.8.EE.A.4 Perform 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 08 B62C1C106873438AA0126760075A65A3 42D43D973FEF4A028D2AEAC7151A01DB http://corestandards.org/Math/Content/8/EE/B/5/ 7 Standard CCSS.Math.Content.8.EE.B.5 Graph 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. 08 B62C1C106873438AA0126760075A65A3 D41D63A930114002AC63910B8A561DBD http://corestandards.org/Math/Content/8/EE/B/6/ 7 Standard CCSS.Math.Content.8.EE.B.6 Use 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 <i>y</i> = <i>mx</i> + <i>b</i> for a line intercepting the vertical axis at <i>b</i>. 08 B62C1C106873438AA0126760075A65A3 D41D63A930114002AC63910B8A561DBD http://corestandards.org/Math/Content/8/EE/C/7/ 7 Standard CCSS.Math.Content.8.EE.C.7 Solve linear equations in one variable. 08 B62C1C106873438AA0126760075A65A3 29419E561D0F4947929F5400799CB6ED 37D0AF8DE45246A1BE82DA6ACB8DC0D4 DC8116E17B134614B9C311DB843BD9FA http://corestandards.org/Math/Content/8/EE/C/7/a/ 8 Component CCSS.Math.Content.8.EE.C.7a Give 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 <i>x</i> = <i>a</i>, <i>a</i> = <i>a</i>, or <i>a</i> = <i>b</i> results (where <i>a</i> and <i>b</i> are different numbers). 08 B62C1C106873438AA0126760075A65A3 7276B02F0B004E889E6355C98B8C1F62 http://corestandards.org/Math/Content/8/EE/C/7/b/ 8 Component CCSS.Math.Content.8.EE.C.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 08 B62C1C106873438AA0126760075A65A3 7276B02F0B004E889E6355C98B8C1F62 http://corestandards.org/Math/Content/8/EE/C/8/ 7 Standard CCSS.Math.Content.8.EE.C.8 Analyze and solve pairs of simultaneous linear equations. 08 B62C1C106873438AA0126760075A65A3 29419E561D0F4947929F5400799CB6ED C8EFEC63BD1C4E8AB40AE467EF904347 C837C873D4C94DC9BB08D7525EA73A5C 6319C40BAA4B4B2F8D2B8BF2CA02D465 http://corestandards.org/Math/Content/8/EE/C/8/a/ 8 Component CCSS.Math.Content.8.EE.C.8a Understand 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. 08 B62C1C106873438AA0126760075A65A3 6FC2501BA58B4695979EA68AC7E2C987 http://corestandards.org/Math/Content/8/EE/C/8/b/ 8 Component CCSS.Math.Content.8.EE.C.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. <i>For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6</i>. 08 B62C1C106873438AA0126760075A65A3 6FC2501BA58B4695979EA68AC7E2C987 http://corestandards.org/Math/Content/8/EE/C/8/c/ 8 Component CCSS.Math.Content.8.EE.C.8c Solve real-world and mathematical problems leading to two linear equations in two variables. <i>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</i>. 08 B62C1C106873438AA0126760075A65A3 6FC2501BA58B4695979EA68AC7E2C987 http://corestandards.org/Math/Content/8/F/A/1/ 7 Standard CCSS.Math.Content.8.F.A.1 Understand 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.<sup>1</sup> 08 B62C1C106873438AA0126760075A65A3 B9013BEAF8364E749A2E6C67FDA59EE5 http://corestandards.org/Math/Content/8/F/A/2/ 7 Standard CCSS.Math.Content.8.F.A.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). <i>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</i>. 08 B62C1C106873438AA0126760075A65A3 B9013BEAF8364E749A2E6C67FDA59EE5 http://corestandards.org/Math/Content/8/F/A/3/ 7 Standard CCSS.Math.Content.8.F.A.3 Interpret the equation <i>y = mx + b</i> as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. <i>For example, the function A = s<sup>2</sup> 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</i>. 08 B62C1C106873438AA0126760075A65A3 B9013BEAF8364E749A2E6C67FDA59EE5 http://corestandards.org/Math/Content/8/F/B/4/ 7 Standard CCSS.Math.Content.8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change&nbsp; and initial value of the function from a description of a relationship or from two (<i>x, y</i>) 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. 08 B62C1C106873438AA0126760075A65A3 EA16A7D416D74957AD1F70155C0F9DC0 http://corestandards.org/Math/Content/8/F/B/5/ 7 Standard CCSS.Math.Content.8.F.B.5 Describe 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. 08 B62C1C106873438AA0126760075A65A3 EA16A7D416D74957AD1F70155C0F9DC0 http://corestandards.org/Math/Content/8/G/A/1/ 7 Standard CCSS.Math.Content.8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations: 08 B62C1C106873438AA0126760075A65A3 C0C985F7FFE7423D835C537871E191F5 C5B0127A97064227B311F52DC8A657C9 8423353534EF4EB9B529A48B5A96B2BD 8423353534EF4EB9B529A48B5A96B2BE http://corestandards.org/Math/Content/8/G/A/1/a/ 8 Component CCSS.Math.Content.8.G.A.1a Lines are taken to lines, and line segments to line segments of the same length. 08 B62C1C106873438AA0126760075A65A3 72594BE267C043CC819168C2580494B9 http://corestandards.org/Math/Content/8/G/A/1/b/ 8 Component CCSS.Math.Content.8.G.A.1b Angles are taken to angles of the same measure. 08 B62C1C106873438AA0126760075A65A3 72594BE267C043CC819168C2580494B9 http://corestandards.org/Math/Content/8/G/A/1/c/ 8 Component CCSS.Math.Content.8.G.A.1c Parallel lines are taken to parallel lines. 08 B62C1C106873438AA0126760075A65A3 72594BE267C043CC819168C2580494B9 http://corestandards.org/Math/Content/8/G/A/2/ 7 Standard CCSS.Math.Content.8.G.A.2 Understand 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. 08 B62C1C106873438AA0126760075A65A3 C0C985F7FFE7423D835C537871E191F5 http://corestandards.org/Math/Content/8/G/A/3/ 7 Standard CCSS.Math.Content.8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 08 B62C1C106873438AA0126760075A65A3 C0C985F7FFE7423D835C537871E191F5 http://corestandards.org/Math/Content/8/G/A/4/ 7 Standard CCSS.Math.Content.8.G.A.4 Understand 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. 08 B62C1C106873438AA0126760075A65A3 C0C985F7FFE7423D835C537871E191F5 http://corestandards.org/Math/Content/8/G/A/5/ 7 Standard CCSS.Math.Content.8.G.A.5 Use 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. <i>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</i>. 08 B62C1C106873438AA0126760075A65A3 C0C985F7FFE7423D835C537871E191F5 http://corestandards.org/Math/Content/8/G/B/6/ 7 Standard CCSS.Math.Content.8.G.B.6 Explain a proof of the Pythagorean Theorem and its converse. 08 B62C1C106873438AA0126760075A65A3 1843F8FBB5AE461199C9463C49D26B74 http://corestandards.org/Math/Content/8/G/B/7/ 7 Standard CCSS.Math.Content.8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 08 B62C1C106873438AA0126760075A65A3 1843F8FBB5AE461199C9463C49D26B74 http://corestandards.org/Math/Content/8/G/B/8/ 7 Standard CCSS.Math.Content.8.G.B.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. 08 B62C1C106873438AA0126760075A65A3 1843F8FBB5AE461199C9463C49D26B74 http://corestandards.org/Math/Content/8/G/C/9/ 7 Standard CCSS.Math.Content.8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. 08 B62C1C106873438AA0126760075A65A3 236B7B9A5B6142e4867DF7E103BE81FE http://corestandards.org/Math/Content/8/NS/A/1/ 7 Standard CCSS.Math.Content.8.NS.A.1 Know 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. 08 B62C1C106873438AA0126760075A65A3 73B661BC829045CA834CB4F5419EB83E http://corestandards.org/Math/Content/8/NS/A/2/ 7 Standard CCSS.Math.Content.8.NS.A.2 Use 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., &pi;<sup>2</sup>). <i>For example, by truncating the decimal expansion of &radic;2, show that &radic;2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations</i>. 08 B62C1C106873438AA0126760075A65A3 73B661BC829045CA834CB4F5419EB83E http://corestandards.org/Math/Content/8/SP/A/1/ 7 Standard CCSS.Math.Content.8.SP.A.1 Construct 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. 08 B62C1C106873438AA0126760075A65A3 55CAA84D9CDF446093D0196E65CCE796 http://corestandards.org/Math/Content/8/SP/A/2/ 7 Standard CCSS.Math.Content.8.SP.A.2 Know 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. 08 B62C1C106873438AA0126760075A65A3 55CAA84D9CDF446093D0196E65CCE796 http://corestandards.org/Math/Content/8/SP/A/3/ 7 Standard CCSS.Math.Content.8.SP.A.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. <i>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</i>. 08 B62C1C106873438AA0126760075A65A3 55CAA84D9CDF446093D0196E65CCE796 http://corestandards.org/Math/Content/8/SP/A/4/ 7 Standard CCSS.Math.Content.8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. 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. <i>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?</i> 08 B62C1C106873438AA0126760075A65A3 55CAA84D9CDF446093D0196E65CCE796 http://corestandards.org/Math/Content/HSA/APR/A/1/ 7 Standard CCSS.Math.Content.HSA-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. 09 10 11 12 B62C1C106873438AA0126760075A65A3 86029C6936A24246AC435CA696634800 http://corestandards.org/Math/Content/HSA/APR/B/2/ 7 Standard CCSS.Math.Content.HSA-APR.B.2 Know and apply the Remainder Theorem: For a polynomial <i>p</i>(<i>x</i>) and a number <i>a</i>, the remainder on division by <i>x - a</i> is <i>p</i>(<i>a</i>), so <i>p</i>(<i>a</i>) = 0 if and only if (<i>x - a</i>) is a factor of <i>p</i>(<i>x</i>). 09 10 11 12 B62C1C106873438AA0126760075A65A3 F807C53F9EDE43FDB34DF614CA072F87 http://corestandards.org/Math/Content/HSA/APR/B/3/ 7 Standard CCSS.Math.Content.HSA-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. 09 10 11 12 B62C1C106873438AA0126760075A65A3 F807C53F9EDE43FDB34DF614CA072F87 http://corestandards.org/Math/Content/HSA/APR/C/4/ 7 Standard CCSS.Math.Content.HSA-APR.C.4 Prove polynomial identities and use them to describe numerical relationships. <i>For example, the polynomial identity (x<sup>2</sup> + y<sup>2</sup>)<sup>2</sup> = (x<sup>2</sup> - y<sup>2</sup>)<sup>2</sup> + (2xy)<sup>2</sup> can be used to generate Pythagorean triples.</i></li> 09 10 11 12 B62C1C106873438AA0126760075A65A3 A2539A99653A47DBAFA1740D2F0BE124 http://corestandards.org/Math/Content/HSA/APR/C/5/ 7 Standard CCSS.Math.Content.HSA-APR.C.5 (+) Know and apply the Binomial Theorem for the expansion of (<i>x</i> + <i>y</i>)<sup><i>n</i></sup> in powers of <i>x</i> and <i>y</i> for a positive integer <i>n</i>, where <i>x</i> and <i>y</i> are any numbers, with coefficients determined for example by Pascal's Triangle.<sup>1</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 A2539A99653A47DBAFA1740D2F0BE124 http://corestandards.org/Math/Content/HSA/APR/D/6/ 7 Standard CCSS.Math.Content.HSA-APR.D.6 Rewrite simple rational expressions in different forms; write <sup><i>a</i>(<i>x</i>)</sup>/<sub><i>b</i>(<i>x</i>)</sub> in the form <i>q</i>(<i>x</i>) + <sup><i>r</i>(<i>x</i>)</sup>/<sub><i>b</i>(<i>x</i>)</sub>, where <i>a</i>(<i>x</i>), <i>b</i>(<i>x</i>), <i>q</i>(<i>x</i>), and <i>r</i>(<i>x</i>) are polynomials with the degree of <i>r</i>(<i>x</i>) less than the degree of <i>b</i>(<i>x</i>), using inspection, long division, or, for the more complicated examples, a computer algebra system. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0EC032FAF5F34E04B00A864C2E588A09 http://corestandards.org/Math/Content/HSA/APR/D/7/ 7 Standard CCSS.Math.Content.HSA-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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0EC032FAF5F34E04B00A864C2E588A09 http://corestandards.org/Math/Content/HSA/CED/A/1/ 7 Standard CCSS.Math.Content.HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. <i>Include equations arising from linear and quadratic functions, and simple rational and exponential functions</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 2B387BDF43CB43B58983E460B46BE605 http://corestandards.org/Math/Content/HSA/CED/A/2/ 7 Standard CCSS.Math.Content.HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 09 10 11 12 B62C1C106873438AA0126760075A65A3 2B387BDF43CB43B58983E460B46BE605 http://corestandards.org/Math/Content/HSA/CED/A/3/ 7 Standard CCSS.Math.Content.HSA-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. <i>For example, represent inequalities describing nutritional and cost constraints on combinations of different foods</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 2B387BDF43CB43B58983E460B46BE605 http://corestandards.org/Math/Content/HSA/CED/A/4/ 7 Standard CCSS.Math.Content.HSA-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <i>For example, rearrange Ohm's law V = IR to highlight resistance R</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 2B387BDF43CB43B58983E460B46BE605 http://corestandards.org/Math/Content/HSA/REI/A/1/ 7 Standard CCSS.Math.Content.HSA-REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 09 10 11 12 B62C1C106873438AA0126760075A65A3 76A52762F6C540829E34B961AC57ED32 http://corestandards.org/Math/Content/HSA/REI/A/2/ 7 Standard CCSS.Math.Content.HSA-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 09 10 11 12 B62C1C106873438AA0126760075A65A3 76A52762F6C540829E34B961AC57ED32 http://corestandards.org/Math/Content/HSA/REI/B/3/ 7 Standard CCSS.Math.Content.HSA-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 09 10 11 12 B62C1C106873438AA0126760075A65A3 7F7DAA3BD975420E9A21319845B45A2A http://corestandards.org/Math/Content/HSA/REI/B/4/ 7 Standard CCSS.Math.Content.HSA-REI.B.4 Solve quadratic equations in one variable. 09 10 11 12 B62C1C106873438AA0126760075A65A3 7F7DAA3BD975420E9A21319845B45A2A 5BD6F36D021A429E951EE42C678F1CF5 617091A725F44F42A823052465447328 http://corestandards.org/Math/Content/HSA/REI/B/4/a/ 8 Component CCSS.Math.Content.HSA-REI.B.4a Use the method of completing the square to transform any quadratic equation in <i>x</i> into an equation of the form (<i>x</i> - <i>p</i>)<sup>2</sup> = <i>q</i> that has the same solutions. Derive the quadratic formula from this form. 09 10 11 12 B62C1C106873438AA0126760075A65A3 87B0241DE22D4A18943F3BC6CBF5DA82 http://corestandards.org/Math/Content/HSA/REI/B/4/b/ 8 Component CCSS.Math.Content.HSA-REI.B.4b Solve quadratic equations by inspection (e.g., for <i>x</i><sup>2</sup> = 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 <i>a</i> &plusmn; <i>bi</i> for real numbers <i>a</i> and <i>b</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 87B0241DE22D4A18943F3BC6CBF5DA82 http://corestandards.org/Math/Content/HSA/REI/C/5/ 7 Standard CCSS.Math.Content.HSA-REI.C.5 Prove 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 DA79BD4E58434CAABB4C784D994D19E8 http://corestandards.org/Math/Content/HSA/REI/C/6/ 7 Standard CCSS.Math.Content.HSA-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 09 10 11 12 B62C1C106873438AA0126760075A65A3 DA79BD4E58434CAABB4C784D994D19E8 http://corestandards.org/Math/Content/HSA/REI/C/7/ 7 Standard CCSS.Math.Content.HSA-REI.C.7 Solve 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 <i>y</i> = -3<i>x</i> and the circle <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 3. 09 10 11 12 B62C1C106873438AA0126760075A65A3 DA79BD4E58434CAABB4C784D994D19E8 http://corestandards.org/Math/Content/HSA/REI/C/8/ 7 Standard CCSS.Math.Content.HSA-REI.C.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable. 09 10 11 12 B62C1C106873438AA0126760075A65A3 DA79BD4E58434CAABB4C784D994D19E8 http://corestandards.org/Math/Content/HSA/REI/C/9/ 7 Standard CCSS.Math.Content.HSA-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 &times; 3 or greater). 09 10 11 12 B62C1C106873438AA0126760075A65A3 DA79BD4E58434CAABB4C784D994D19E8 http://corestandards.org/Math/Content/HSA/REI/D/10/ 7 Standard CCSS.Math.Content.HSA-REI.D.10 Understand 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). 09 10 11 12 B62C1C106873438AA0126760075A65A3 A65B3884A9784D21B409AC13B11DFD9B http://corestandards.org/Math/Content/HSA/REI/D/11/ 7 Standard CCSS.Math.Content.HSA-REI.D.11 Explain why the <i>x</i>-coordinates of the points where the graphs of the equations <i>y</i> = <i>f</i>(<i>x</i>) and <i>y</i> = <i>g</i>(<i>x</i>) intersect are the solutions of the equation <i>f</i>(<i>x</i>) = <i>g</i>(<i>x</i>); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where <i>f</i>(<i>x</i>) and/or <i>g</i>(<i>x</i>) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 A65B3884A9784D21B409AC13B11DFD9B http://corestandards.org/Math/Content/HSA/REI/D/12/ 7 Standard CCSS.Math.Content.HSA-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. 09 10 11 12 B62C1C106873438AA0126760075A65A3 A65B3884A9784D21B409AC13B11DFD9B http://corestandards.org/Math/Content/HSA/SSE/A/1/ 7 Standard CCSS.Math.Content.HSA-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 16247FEF153B4404A115600AF3C81E03 529B17211FDB499292FE9DCD0CAAA928 5155CED2475A42D88AD48B7A676C7943 http://corestandards.org/Math/Content/HSA/SSE/A/1/a/ 8 Component CCSS.Math.Content.HSA-SSE.A.1a Interpret parts of an expression, such as terms, factors, and coefficients. 09 10 11 12 B62C1C106873438AA0126760075A65A3 137C90D24DE949558F17713B84161B4D http://corestandards.org/Math/Content/HSA/SSE/A/1/b/ 8 Component CCSS.Math.Content.HSA-SSE.A.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. <i>For example, interpret P(1+r)<sup>n</sup> as the product of P and a factor not depending on P</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 137C90D24DE949558F17713B84161B4D http://corestandards.org/Math/Content/HSA/SSE/A/2/ 7 Standard CCSS.Math.Content.HSA-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. <i>For example, see x<sup>4</sup> - y<sup>4</sup> as (x<sup>2</sup>)<sup>2</sup> - (y<sup>2</sup>)<sup>2</sup>, thus recognizing it as a difference of squares that can be factored as (x<sup>2</sup> - y<sup>2</sup>)(x<sup>2</sup> + y<sup>2</sup>)</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 16247FEF153B4404A115600AF3C81E03 http://corestandards.org/Math/Content/HSA/SSE/B/3/ 7 Standard CCSS.Math.Content.HSA-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 BEA7E3EF2A4C4C8F85F12D9743C25155 194A6E971ED74A3695C6CE468E337562 C057EE31BA98447FB241F2F732B896BD D07C6A24D9F045EFACB20582472FC7EB http://corestandards.org/Math/Content/HSA/SSE/B/3/a/ 8 Component CCSS.Math.Content.HSA-SSE.B.3a Factor a quadratic expression to reveal the zeros of the function it defines. 09 10 11 12 B62C1C106873438AA0126760075A65A3 C705AD3CCF9641C9A3101BDB10737B11 http://corestandards.org/Math/Content/HSA/SSE/B/3/b/ 8 Component CCSS.Math.Content.HSA-SSE.B.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 09 10 11 12 B62C1C106873438AA0126760075A65A3 C705AD3CCF9641C9A3101BDB10737B11 http://corestandards.org/Math/Content/HSA/SSE/B/3/c/ 8 Component CCSS.Math.Content.HSA-SSE.B.3c Use the properties of exponents to transform expressions for exponential functions. <i>For example the expression 1.15<sup>t</sup> can be rewritten as (1.15<sup>1/12</sup>)<sup>12t</sup> &asymp; 1.012<sup>12t</sup> to reveal the approximate equivalent monthly interest rate if the annual rate is 15%</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 C705AD3CCF9641C9A3101BDB10737B11 http://corestandards.org/Math/Content/HSA/SSE/B/4/ 7 Standard CCSS.Math.Content.HSA-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. <i>For example, calculate mortgage payments.</i><sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 BEA7E3EF2A4C4C8F85F12D9743C25155 http://corestandards.org/Math/Content/HSF/BF/A/1/ 7 Standard CCSS.Math.Content.HSF-BF.A.1 Write a function that describes a relationship between two quantities.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 9B65633847F14c7eBE6901214D7CCE72 70C634815522412aA1C2601B0B0990FF 7C24A27D8C154d37A8F4C00AFA8EE4A1 EB54465D5EA44f868FCD2C73D4E782AC http://corestandards.org/Math/Content/HSF/BF/A/1/a/ 8 Component CCSS.Math.Content.HSF-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. 09 10 11 12 B62C1C106873438AA0126760075A65A3 148058CA1C12486fBED93EA48399BFB5 http://corestandards.org/Math/Content/HSF/BF/A/1/b/ 8 Component CCSS.Math.Content.HSF-BF.A.1b Combine standard function types using arithmetic operations. <i>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</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 148058CA1C12486fBED93EA48399BFB5 http://corestandards.org/Math/Content/HSF/BF/A/1/c/ 8 Component CCSS.Math.Content.HSF-BF.A.1c (+) Compose functions. <i>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</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 148058CA1C12486fBED93EA48399BFB5 http://corestandards.org/Math/Content/HSF/BF/A/2/ 7 Standard CCSS.Math.Content.HSF-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 9B65633847F14c7eBE6901214D7CCE72 http://corestandards.org/Math/Content/HSF/BF/B/3/ 7 Standard CCSS.Math.Content.HSF-BF.B.3 Identify the effect on the graph of replacing <i>f</i>(<i>x</i>) by <i>f</i>(<i>x</i>) + <i>k</i>,<i> k</i> <i>f</i>(<i>x</i>), <i>f</i>(<i>kx</i>), and <i>f</i>(<i>x</i> + <i>k</i>) for specific values of <i>k</i> (both positive and negative); find the value of <i>k</i> given the graphs. 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0D80A8E0EDC34cfeA85B47771F813955 http://corestandards.org/Math/Content/HSF/BF/B/4/ 7 Standard CCSS.Math.Content.HSF-BF.B.4 Find inverse functions. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0D80A8E0EDC34cfeA85B47771F813955 392F16871F5443c9BE065FD1A9DE39C2 F4B7CBBED04C4960A3F16E86069BE376 5A331E7758B240c389EA982FA8605DAE 2CE8D7B11DF246bbA2D5DC96DE5BE282 http://corestandards.org/Math/Content/HSF/BF/B/4/a/ 8 Component CCSS.Math.Content.HSF-BF.B.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. <i>For example, f(x) =2 x<sup>3</sup> or f(x) = (x+1)/(x-1) for x &ne; 1</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 D600962249FF4b1aADC5857575D228AE http://corestandards.org/Math/Content/HSF/BF/B/4/b/ 8 Component CCSS.Math.Content.HSF-BF.B.4b (+) Verify by composition that one function is the inverse of another. 09 10 11 12 B62C1C106873438AA0126760075A65A3 D600962249FF4b1aADC5857575D228AE http://corestandards.org/Math/Content/HSF/BF/B/4/c/ 8 Component CCSS.Math.Content.HSF-BF.B.4c (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. 09 10 11 12 B62C1C106873438AA0126760075A65A3 D600962249FF4b1aADC5857575D228AE http://corestandards.org/Math/Content/HSF/BF/B/4/d/ 8 Component CCSS.Math.Content.HSF-BF.B.4d (+) Produce an invertible function from a non-invertible function by restricting the domain. 09 10 11 12 B62C1C106873438AA0126760075A65A3 D600962249FF4b1aADC5857575D228AE http://corestandards.org/Math/Content/HSF/BF/B/5/ 7 Standard CCSS.Math.Content.HSF-BF.B.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0D80A8E0EDC34cfeA85B47771F813955 http://corestandards.org/Math/Content/HSF/IF/A/1/ 7 Standard CCSS.Math.Content.HSF-IF.A.1 Understand 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 <i>f</i> is a function and <i>x</i> is an element of its domain, then <i>f</i>(<i>x</i>) denotes the output of <i>f</i> corresponding to the input <i>x</i>. The graph of <i>f</i> is the graph of the equation <i>y</i> = <i>f</i>(<i>x</i>). 09 10 11 12 B62C1C106873438AA0126760075A65A3 59E4F5F26EF044869249ED3E7CC0AD95 http://corestandards.org/Math/Content/HSF/IF/A/2/ 7 Standard CCSS.Math.Content.HSF-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 09 10 11 12 B62C1C106873438AA0126760075A65A3 59E4F5F26EF044869249ED3E7CC0AD95 http://corestandards.org/Math/Content/HSF/IF/A/3/ 7 Standard CCSS.Math.Content.HSF-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. <i>For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n &ge; 1</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 59E4F5F26EF044869249ED3E7CC0AD95 http://corestandards.org/Math/Content/HSF/IF/B/4/ 7 Standard CCSS.Math.Content.HSF-IF.B.4 For 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. <i>Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity</i>.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 E9DDBF450BFB48a79C8EF29E1EB211CD http://corestandards.org/Math/Content/HSF/IF/B/5/ 7 Standard CCSS.Math.Content.HSF-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. <i>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.</i><sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 E9DDBF450BFB48a79C8EF29E1EB211CD http://corestandards.org/Math/Content/HSF/IF/B/6/ 7 Standard CCSS.Math.Content.HSF-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 E9DDBF450BFB48a79C8EF29E1EB211CD http://corestandards.org/Math/Content/HSF/IF/C/7/ 7 Standard CCSS.Math.Content.HSF-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.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 F99D9C6750C84f45A30719B919C80A8B BDAA9678DF8E48a7B07B6546CA928796 E7A923E2D1DA493e8B2B5146B298B677 45B93BCB8B1B4b388574F0CA327C48BD 890632566D8440189B4D825B5EE6BFD3 282642278ABD4b32911BDCD1EA2C6468 http://corestandards.org/Math/Content/HSF/IF/C/7/a/ 8 Component CCSS.Math.Content.HSF-IF.C.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. 09 10 11 12 B62C1C106873438AA0126760075A65A3 5E15A791B8614350AC4E7E11AACFC5B2 http://corestandards.org/Math/Content/HSF/IF/C/7/b/ 8 Component CCSS.Math.Content.HSF-IF.C.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 09 10 11 12 B62C1C106873438AA0126760075A65A3 5E15A791B8614350AC4E7E11AACFC5B2 http://corestandards.org/Math/Content/HSF/IF/C/7/c/ 8 Component CCSS.Math.Content.HSF-IF.C.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 09 10 11 12 B62C1C106873438AA0126760075A65A3 5E15A791B8614350AC4E7E11AACFC5B2 http://corestandards.org/Math/Content/HSF/IF/C/7/d/ 8 Component CCSS.Math.Content.HSF-IF.C.7d (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. 09 10 11 12 B62C1C106873438AA0126760075A65A3 5E15A791B8614350AC4E7E11AACFC5B2 http://corestandards.org/Math/Content/HSF/IF/C/7/e/ 8 Component CCSS.Math.Content.HSF-IF.C.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. 09 10 11 12 B62C1C106873438AA0126760075A65A3 5E15A791B8614350AC4E7E11AACFC5B2 http://corestandards.org/Math/Content/HSF/IF/C/8/ 7 Standard CCSS.Math.Content.HSF-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 09 10 11 12 B62C1C106873438AA0126760075A65A3 F99D9C6750C84f45A30719B919C80A8B E8C42265F5F341ea9C0284AA7BDC65AE E8C42265F5F341ea9C0284AA7BDC65AF http://corestandards.org/Math/Content/HSF/IF/C/8/a/ 8 Component CCSS.Math.Content.HSF-IF.C.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 09 10 11 12 B62C1C106873438AA0126760075A65A3 B6B625DA12C4423d955A66BFD1193175 http://corestandards.org/Math/Content/HSF/IF/C/8/b/ 8 Component CCSS.Math.Content.HSF-IF.C.8b Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)ᵗ, y = (0.97)ᵗ, y = (1.01)12ᵗ, y = (1.2)ᵗ/10, and classify them as representing exponential growth or decay. 09 10 11 12 B62C1C106873438AA0126760075A65A3 B6B625DA12C4423d955A66BFD1193175 http://corestandards.org/Math/Content/HSF/IF/C/9/ 7 Standard CCSS.Math.Content.HSF-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). <i>For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 F99D9C6750C84f45A30719B919C80A8B http://corestandards.org/Math/Content/HSF/LE/A/1/ 7 Standard CCSS.Math.Content.HSF-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. 09 10 11 12 B62C1C106873438AA0126760075A65A3 EA31B803149948c1AB6FCB71C7EE6E23 39F3AF9140744f049D8A7F16C1E1C11B 40FD278805FB4f0091985412D2591720 5E2567D4DEEA4e71971D8405AD83282D http://corestandards.org/Math/Content/HSF/LE/A/1/a/ 8 Component CCSS.Math.Content.HSF-LE.A.1a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. 09 10 11 12 B62C1C106873438AA0126760075A65A3 13898F5B9227486aBB749E5A8D5F0BA2 http://corestandards.org/Math/Content/HSF/LE/A/1/b/ 8 Component CCSS.Math.Content.HSF-LE.A.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 09 10 11 12 B62C1C106873438AA0126760075A65A3 13898F5B9227486aBB749E5A8D5F0BA2 http://corestandards.org/Math/Content/HSF/LE/A/1/c/ 8 Component CCSS.Math.Content.HSF-LE.A.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 09 10 11 12 B62C1C106873438AA0126760075A65A3 13898F5B9227486aBB749E5A8D5F0BA2 http://corestandards.org/Math/Content/HSF/LE/A/2/ 7 Standard CCSS.Math.Content.HSF-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). 09 10 11 12 B62C1C106873438AA0126760075A65A3 EA31B803149948c1AB6FCB71C7EE6E23 http://corestandards.org/Math/Content/HSF/LE/A/3/ 7 Standard CCSS.Math.Content.HSF-LE.A.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. 09 10 11 12 B62C1C106873438AA0126760075A65A3 EA31B803149948c1AB6FCB71C7EE6E23 http://corestandards.org/Math/Content/HSF/LE/A/4/ 7 Standard CCSS.Math.Content.HSF-LE.A.4 For exponential models, express as a logarithm the solution to <i>ab<sup>ct</sup></i> = <i>d</i> where <i>a</i>, <i>c</i>, and <i>d</i> are numbers and the base <i>b</i> is 2, 10, or <i>e</i>; evaluate the logarithm using technology. 09 10 11 12 B62C1C106873438AA0126760075A65A3 EA31B803149948c1AB6FCB71C7EE6E23 http://corestandards.org/Math/Content/HSF/LE/B/5/ 7 Standard CCSS.Math.Content.HSF-LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context. 09 10 11 12 B62C1C106873438AA0126760075A65A3 06327BD558CD4f66B40934E0E0CAD555 http://corestandards.org/Math/Content/HSF/TF/A/1/ 7 Standard CCSS.Math.Content.HSF-TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0B13FE26FA344dca998C7DEDEBC7E0BF http://corestandards.org/Math/Content/HSF/TF/A/2/ 7 Standard CCSS.Math.Content.HSF-TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0B13FE26FA344dca998C7DEDEBC7E0BF http://corestandards.org/Math/Content/HSF/TF/A/3/ 7 Standard CCSS.Math.Content.HSF-TF.A.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for &pi;/3, &pi;/4 and &pi;/6, and use the unit circle to express the values of sine, cosine, and tangent for <i>x</i>, &pi; + <i>x</i>, and 2&pi; - <i>x</i> in terms of their values for <i>x</i>, where <i>x</i> is any real number. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0B13FE26FA344dca998C7DEDEBC7E0BF http://corestandards.org/Math/Content/HSF/TF/A/4/ 7 Standard CCSS.Math.Content.HSF-TF.A.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0B13FE26FA344dca998C7DEDEBC7E0BF http://corestandards.org/Math/Content/HSF/TF/B/5/ 7 Standard CCSS.Math.Content.HSF-TF.B.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 CEBB90EF52E44ac196F126C321E1E446 http://corestandards.org/Math/Content/HSF/TF/B/6/ 7 Standard CCSS.Math.Content.HSF-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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 CEBB90EF52E44ac196F126C321E1E446 http://corestandards.org/Math/Content/HSF/TF/B/7/ 7 Standard CCSS.Math.Content.HSF-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.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 CEBB90EF52E44ac196F126C321E1E446 http://corestandards.org/Math/Content/HSF/TF/C/8/ 7 Standard CCSS.Math.Content.HSF-TF.C.8 Prove the Pythagorean identity sin<sup>2</sup>(&theta;) + cos<sup>2</sup>(&theta;) = 1 and use it to find sin(&theta;), cos(&theta;), or tan(&theta;) given sin(&theta;), cos(&theta;), or tan(&theta;) and the quadrant of the angle. 09 10 11 12 B62C1C106873438AA0126760075A65A3 601ED08534CF40a5977CEBAD49863725 http://corestandards.org/Math/Content/HSF/TF/C/9/ 7 Standard CCSS.Math.Content.HSF-TF.C.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. 09 10 11 12 B62C1C106873438AA0126760075A65A3 601ED08534CF40a5977CEBAD49863725 http://corestandards.org/Math/Content/HSG/C/A/1/ 7 Standard CCSS.Math.Content.HSG-C.A.1 Prove that all circles are similar. 09 10 11 12 B62C1C106873438AA0126760075A65A3 6FCFE0FCB1D94D9DB0CBE4E55A4C4689 http://corestandards.org/Math/Content/HSG/C/A/2/ 7 Standard CCSS.Math.Content.HSG-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. <i>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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 6FCFE0FCB1D94D9DB0CBE4E55A4C4689 http://corestandards.org/Math/Content/HSG/C/A/3/ 7 Standard CCSS.Math.Content.HSG-C.A.3 </i>Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.</i> 09 10 11 12 B62C1C106873438AA0126760075A65A3 6FCFE0FCB1D94D9DB0CBE4E55A4C4689 http://corestandards.org/Math/Content/HSG/C/A/4/ 7 Standard CCSS.Math.Content.HSG-C.A.4 (+) Construct a tangent line from a point outside a given circle to the circle. 09 10 11 12 B62C1C106873438AA0126760075A65A3 6FCFE0FCB1D94D9DB0CBE4E55A4C4689 http://corestandards.org/Math/Content/HSG/C/B/5/ 7 Standard CCSS.Math.Content.HSG-C.B.5 Derive 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 E85D80D849884C7E87DF3B6F703E08BF http://corestandards.org/Math/Content/HSG/CO/A/1/ 7 Standard CCSS.Math.Content.HSG-CO.A.1 Know 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0A4FB274306F48B8849D365982EEE811 http://corestandards.org/Math/Content/HSG/CO/A/2/ 7 Standard CCSS.Math.Content.HSG-CO.A.2 Represent 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). 09 10 11 12 B62C1C106873438AA0126760075A65A3 0A4FB274306F48B8849D365982EEE811 http://corestandards.org/Math/Content/HSG/CO/A/3/ 7 Standard CCSS.Math.Content.HSG-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0A4FB274306F48B8849D365982EEE811 http://corestandards.org/Math/Content/HSG/CO/A/4/ 7 Standard CCSS.Math.Content.HSG-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0A4FB274306F48B8849D365982EEE811 http://corestandards.org/Math/Content/HSG/CO/A/5/ 7 Standard CCSS.Math.Content.HSG-CO.A.5 Given 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0A4FB274306F48B8849D365982EEE811 http://corestandards.org/Math/Content/HSG/CO/B/6/ 7 Standard CCSS.Math.Content.HSG-CO.B.6 Use 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 2C230AEDFC714C45A4DE2FE3C0D07FC0 http://corestandards.org/Math/Content/HSG/CO/B/7/ 7 Standard CCSS.Math.Content.HSG-CO.B.7 Use 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 2C230AEDFC714C45A4DE2FE3C0D07FC0 http://corestandards.org/Math/Content/HSG/CO/B/8/ 7 Standard CCSS.Math.Content.HSG-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. 09 10 11 12 B62C1C106873438AA0126760075A65A3 2C230AEDFC714C45A4DE2FE3C0D07FC0 http://corestandards.org/Math/Content/HSG/CO/C/9/ 7 Standard CCSS.Math.Content.HSG-CO.C.9 Prove theorems about lines and angles. <i>Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 3950A96EBD5349E4ADFFA1F417DFAE3E http://corestandards.org/Math/Content/HSG/CO/C/10/ 7 Standard CCSS.Math.Content.HSG-CO.C.10 Prove theorems about triangles. <i>Theorems include: measures of interior angles of a triangle sum to 180&deg;; base angles of isosceles triangles are congruent; 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</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 3950A96EBD5349E4ADFFA1F417DFAE3E http://corestandards.org/Math/Content/HSG/CO/C/11/ 7 Standard CCSS.Math.Content.HSG-CO.C.11 Prove theorems about parallelograms. <i>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</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 3950A96EBD5349E4ADFFA1F417DFAE3E http://corestandards.org/Math/Content/HSG/CO/D/12/ 7 Standard CCSS.Math.Content.HSG-CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). <i>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</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 53D370CCC2F2442DA6B3B2FB0957F6B7 http://corestandards.org/Math/Content/HSG/CO/D/13/ 7 Standard CCSS.Math.Content.HSG-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 09 10 11 12 B62C1C106873438AA0126760075A65A3 53D370CCC2F2442DA6B3B2FB0957F6B7 http://corestandards.org/Math/Content/HSG/GMD/A/1/ 7 Standard CCSS.Math.Content.HSG-GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. <i>Use dissection arguments, Cavalieri's principle, and informal limit arguments</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 5B85430986994BB498573FE89639B3A7 http://corestandards.org/Math/Content/HSG/GMD/A/2/ 7 Standard CCSS.Math.Content.HSG-GMD.A.2 </i>(+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.</i> 09 10 11 12 B62C1C106873438AA0126760075A65A3 5B85430986994BB498573FE89639B3A7 http://corestandards.org/Math/Content/HSG/GMD/A/3/ 7 Standard CCSS.Math.Content.HSG-GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 5B85430986994BB498573FE89639B3A7 http://corestandards.org/Math/Content/HSG/GMD/B/4/ 7 Standard CCSS.Math.Content.HSG-GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. 09 10 11 12 B62C1C106873438AA0126760075A65A3 68FC0B7F46D44814802F51352CE0A108 http://corestandards.org/Math/Content/HSG/GPE/A/1/ 7 Standard CCSS.Math.Content.HSG-GPE.A.1 Derive 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 20163EF6BD26432AA11A2E0E37083DE9 http://corestandards.org/Math/Content/HSG/GPE/A/2/ 7 Standard CCSS.Math.Content.HSG-GPE.A.2 Derive the equation of a parabola given a focus and directrix. 09 10 11 12 B62C1C106873438AA0126760075A65A3 20163EF6BD26432AA11A2E0E37083DE9 http://corestandards.org/Math/Content/HSG/GPE/A/3/ 7 Standard CCSS.Math.Content.HSG-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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 20163EF6BD26432AA11A2E0E37083DE9 http://corestandards.org/Math/Content/HSG/GPE/B/4/ 7 Standard CCSS.Math.Content.HSG-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. <i>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, &radic;3) lies on the circle centered at the origin and containing the point (0, 2). 09 10 11 12 B62C1C106873438AA0126760075A65A3 10D0867BA9B248F79713088456252172 http://corestandards.org/Math/Content/HSG/GPE/B/5/ 7 Standard CCSS.Math.Content.HSG-GPE.B.5 </i>Prove 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).</i> 09 10 11 12 B62C1C106873438AA0126760075A65A3 10D0867BA9B248F79713088456252172 http://corestandards.org/Math/Content/HSG/GPE/B/6/ 7 Standard CCSS.Math.Content.HSG-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. 09 10 11 12 B62C1C106873438AA0126760075A65A3 10D0867BA9B248F79713088456252172 http://corestandards.org/Math/Content/HSG/GPE/B/7/ 7 Standard CCSS.Math.Content.HSG-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.<sup>* 09 10 11 12 B62C1C106873438AA0126760075A65A3 10D0867BA9B248F79713088456252172 http://corestandards.org/Math/Content/HSG/MG/A/1/ 7 Standard CCSS.Math.Content.HSG-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 0C8CD3F231AC4EA89C7247F43A7B5E0A http://corestandards.org/Math/Content/HSG/MG/A/2/ 7 Standard CCSS.Math.Content.HSG-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 0C8CD3F231AC4EA89C7247F43A7B5E0A http://corestandards.org/Math/Content/HSG/MG/A/3/ 7 Standard CCSS.Math.Content.HSG-MG.A.3 Apply 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).<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 0C8CD3F231AC4EA89C7247F43A7B5E0A http://corestandards.org/Math/Content/HSG/SRT/A/1/ 7 Standard CCSS.Math.Content.HSG-SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor: 09 10 11 12 B62C1C106873438AA0126760075A65A3 1D96FE69A8D94CCFAA827DBD53424C3E DDCD2BFAF9A041A4B8094EE33FD1F27C 01F21CA7909F4C52AC16B840DAF353A9 http://corestandards.org/Math/Content/HSG/SRT/A/1/a/ 8 Component CCSS.Math.Content.HSG-SRT.A.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. 09 10 11 12 B62C1C106873438AA0126760075A65A3 686678EDBF7A4F93A09BAAC0B4C2CBFC http://corestandards.org/Math/Content/HSG/SRT/A/1/b/ 8 Component CCSS.Math.Content.HSG-SRT.A.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. 09 10 11 12 B62C1C106873438AA0126760075A65A3 686678EDBF7A4F93A09BAAC0B4C2CBFC http://corestandards.org/Math/Content/HSG/SRT/A/2/ 7 Standard CCSS.Math.Content.HSG-SRT.A.2 Given 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 1D96FE69A8D94CCFAA827DBD53424C3E http://corestandards.org/Math/Content/HSG/SRT/A/3/ 7 Standard CCSS.Math.Content.HSG-SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 09 10 11 12 B62C1C106873438AA0126760075A65A3 1D96FE69A8D94CCFAA827DBD53424C3E http://corestandards.org/Math/Content/HSG/SRT/B/4/ 7 Standard CCSS.Math.Content.HSG-SRT.B.4 Prove theorems about triangles. <i>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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 4F170A4349664CC59D6A26D6CBC822E7 http://corestandards.org/Math/Content/HSG/SRT/B/5/ 7 Standard CCSS.Math.Content.HSG-SRT.B.5 </i>Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.</i> 09 10 11 12 B62C1C106873438AA0126760075A65A3 4F170A4349664CC59D6A26D6CBC822E7 http://corestandards.org/Math/Content/HSG/SRT/C/6/ 7 Standard CCSS.Math.Content.HSG-SRT.C.6 Understand 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 3759148F143D4274B532EF24C7D28D76 http://corestandards.org/Math/Content/HSG/SRT/C/7/ 7 Standard CCSS.Math.Content.HSG-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. 09 10 11 12 B62C1C106873438AA0126760075A65A3 3759148F143D4274B532EF24C7D28D76 http://corestandards.org/Math/Content/HSG/SRT/C/8/ 7 Standard CCSS.Math.Content.HSG-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 3759148F143D4274B532EF24C7D28D76 http://corestandards.org/Math/Content/HSG/SRT/D/9/ 7 Standard CCSS.Math.Content.HSG-SRT.D.9 (+) Derive the formula <i>A</i> = 1/2 <i>ab</i> sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 09 10 11 12 B62C1C106873438AA0126760075A65A3 F892626502934381A5369630B6D847AA http://corestandards.org/Math/Content/HSG/SRT/D/10/ 7 Standard CCSS.Math.Content.HSG-SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. 09 10 11 12 B62C1C106873438AA0126760075A65A3 F892626502934381A5369630B6D847AA http://corestandards.org/Math/Content/HSG/SRT/D/11/ 7 Standard CCSS.Math.Content.HSG-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). 09 10 11 12 B62C1C106873438AA0126760075A65A3 F892626502934381A5369630B6D847AA http://corestandards.org/Math/Content/HSN/CN/A/1/ 7 Standard CCSS.Math.Content.HSN-CN.A.1 Know there is a complex number <i>i</i> such that <i>i</i><sup>2</sup> = -1, and every complex number has the form <i>a + bi</i> with <i>a</i> and <i>b</i> real. 09 10 11 12 B62C1C106873438AA0126760075A65A3 34739C4FC4AD468B909C63CB696B4153 http://corestandards.org/Math/Content/HSN/CN/A/2/ 7 Standard CCSS.Math.Content.HSN-CN.A.2 Use the relation <i>i</i><sup>2</sup> = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 09 10 11 12 B62C1C106873438AA0126760075A65A3 34739C4FC4AD468B909C63CB696B4153 http://corestandards.org/Math/Content/HSN/CN/A/3/ 7 Standard CCSS.Math.Content.HSN-CN.A.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. 09 10 11 12 B62C1C106873438AA0126760075A65A3 34739C4FC4AD468B909C63CB696B4153 http://corestandards.org/Math/Content/HSN/CN/B/4/ 7 Standard CCSS.Math.Content.HSN-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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 B8FB763556FF41D9B95242FA4D2BBC70 http://corestandards.org/Math/Content/HSN/CN/B/5/ 7 Standard CCSS.Math.Content.HSN-CN.B.5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. <i>For example, (-1 + &radic;3 i)<sup>3</sup> = 8 because (-1 + &radic;3 i) has modulus 2 and argument 120&deg;.</i> 09 10 11 12 B62C1C106873438AA0126760075A65A3 B8FB763556FF41D9B95242FA4D2BBC70 http://corestandards.org/Math/Content/HSN/CN/B/6/ 7 Standard CCSS.Math.Content.HSN-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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 B8FB763556FF41D9B95242FA4D2BBC70 http://corestandards.org/Math/Content/HSN/CN/C/7/ 7 Standard CCSS.Math.Content.HSN-CN.C.7 Solve quadratic equations with real coefficients that have complex solutions. 09 10 11 12 B62C1C106873438AA0126760075A65A3 ADC66FBA84CC47C982E9AFFA1EEBBF15 http://corestandards.org/Math/Content/HSN/CN/C/8/ 7 Standard CCSS.Math.Content.HSN-CN.C.8 (+) Extend polynomial identities to the complex numbers. <i>For example, rewrite x<sup>2</sup> + 4 as (x + 2i)(x - 2i)</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 ADC66FBA84CC47C982E9AFFA1EEBBF15 http://corestandards.org/Math/Content/HSN/CN/C/9/ 7 Standard CCSS.Math.Content.HSN-CN.C.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. 09 10 11 12 B62C1C106873438AA0126760075A65A3 ADC66FBA84CC47C982E9AFFA1EEBBF15 http://corestandards.org/Math/Content/HSN/Q/A/1/ 7 Standard CCSS.Math.Content.HSN-Q.A.1 Use 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 E3814968AC9A45868EE75DAB6BDC3E56 http://corestandards.org/Math/Content/HSN/Q/A/2/ 7 Standard CCSS.Math.Content.HSN-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. 09 10 11 12 B62C1C106873438AA0126760075A65A3 E3814968AC9A45868EE75DAB6BDC3E56 http://corestandards.org/Math/Content/HSN/Q/A/3/ 7 Standard CCSS.Math.Content.HSN-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 09 10 11 12 B62C1C106873438AA0126760075A65A3 E3814968AC9A45868EE75DAB6BDC3E56 http://corestandards.org/Math/Content/HSN/RN/A/1/ 7 Standard CCSS.Math.Content.HSN-RN.A.1 Explain 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. <i>For example, we define 5<sup>1/3</sup> to be the cube root of 5 because we want (5<sup>1/3</sup>)<sup>3</sup> = 5<sup>(1/3)3</sup> to hold, so (5<sup>1/3</sup>)<sup>3</sup> must equal 5</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 DC59C069EEA449788E1D8B0F23EF0FDC http://corestandards.org/Math/Content/HSN/RN/A/2/ 7 Standard CCSS.Math.Content.HSN-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. 09 10 11 12 B62C1C106873438AA0126760075A65A3 DC59C069EEA449788E1D8B0F23EF0FDC http://corestandards.org/Math/Content/HSN/RN/B/3/ 7 Standard CCSS.Math.Content.HSN-RN.B.3 Explain 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 AA68661CDFE24E4FB1F3BBC792D85F4C http://corestandards.org/Math/Content/HSN/VM/A/1/ 7 Standard CCSS.Math.Content.HSN-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., <b><i>v</i></b>, |<b><i>v</i></b>|, ||<b><i>v</i></b>||, <i>v</i>). 09 10 11 12 B62C1C106873438AA0126760075A65A3 E509597287964EA2B70FDD2F10BF988E http://corestandards.org/Math/Content/HSN/VM/A/2/ 7 Standard CCSS.Math.Content.HSN-VM.A.2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. 09 10 11 12 B62C1C106873438AA0126760075A65A3 E509597287964EA2B70FDD2F10BF988E http://corestandards.org/Math/Content/HSN/VM/A/3/ 7 Standard CCSS.Math.Content.HSN-VM.A.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors. 09 10 11 12 B62C1C106873438AA0126760075A65A3 E509597287964EA2B70FDD2F10BF988E http://corestandards.org/Math/Content/HSN/VM/B/4/ 7 Standard CCSS.Math.Content.HSN-VM.B.4 (+) Add and subtract vectors. 09 10 11 12 B62C1C106873438AA0126760075A65A3 608A98D224DE4CA9B945897C7A58799C E71EAB46BC9E474C861A53C60CAB9D2C 984534D6C05B4CFB9E257E5BB29EF018 B5969B9BA727452CAB1EF05C4694974C http://corestandards.org/Math/Content/HSN/VM/B/4/a/ 8 Component CCSS.Math.Content.HSN-VM.B.4a 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 FA9C047FE1C547998A6CBB16C323FAD8 http://corestandards.org/Math/Content/HSN/VM/B/4/b/ 8 Component CCSS.Math.Content.HSN-VM.B.4b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. 09 10 11 12 B62C1C106873438AA0126760075A65A3 FA9C047FE1C547998A6CBB16C323FAD8 http://corestandards.org/Math/Content/HSN/VM/B/4/c/ 8 Component CCSS.Math.Content.HSN-VM.B.4c Understand vector subtraction <b><i>v</i></b> - <b><i>w</i></b> as <b><i>v</i></b> + (-<b><i>w</i></b>), where -<b><i>w</i></b> is the additive inverse of <b><i>w</i></b>, with the same magnitude as <b><i>w</i></b> and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise. 09 10 11 12 B62C1C106873438AA0126760075A65A3 FA9C047FE1C547998A6CBB16C323FAD8 http://corestandards.org/Math/Content/HSN/VM/B/5/ 7 Standard CCSS.Math.Content.HSN-VM.B.5 (+) Multiply a vector by a scalar. 09 10 11 12 B62C1C106873438AA0126760075A65A3 608A98D224DE4CA9B945897C7A58799C FA032D8A02564B37A0BAF505A0296FB5 FA032D8A02564B37A0BAF505A0296FB4 http://corestandards.org/Math/Content/HSN/VM/B/5/a/ 8 Component CCSS.Math.Content.HSN-VM.B.5a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as <i>c</i>(<i>v</i><sub><i>x</i></sub>, <i>v</i><sub><i>y</i></sub>) = (<i>cv</i><sub><i>x</i></sub>, <i>cv</i><sub><i>y</i></sub>). 09 10 11 12 B62C1C106873438AA0126760075A65A3 5B4228464AF24FC2A8B23CF733ABD5D8 http://corestandards.org/Math/Content/HSN/VM/B/5/b/ 8 Component CCSS.Math.Content.HSN-VM.B.5b Compute the magnitude of a scalar multiple <i>c</i><b><i>v</i></b> using ||<i>c</i><b><i>v</i></b>|| = |<i>c</i>|<b><i>v</i></b>. Compute the direction of <i>c</i><b><i>v</i></b> knowing that when |<i>c</i>|<b><i>v</i></b> &ne; 0, the direction of <i>c</i><b><i>v</i></b> is either along <b><i>v</i></b> (for <i>c</i> &gt; 0) or against <b><i>v</i></b> (for <i>c</i> &lt; 0). 09 10 11 12 B62C1C106873438AA0126760075A65A3 5B4228464AF24FC2A8B23CF733ABD5D8 http://corestandards.org/Math/Content/HSN/VM/C/6/ 7 Standard CCSS.Math.Content.HSN-VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. 09 10 11 12 B62C1C106873438AA0126760075A65A3 22F88FBD48CB4D73A3A0D0177BE00314 http://corestandards.org/Math/Content/HSN/VM/C/7/ 7 Standard CCSS.Math.Content.HSN-VM.C.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. 09 10 11 12 B62C1C106873438AA0126760075A65A3 22F88FBD48CB4D73A3A0D0177BE00314 http://corestandards.org/Math/Content/HSN/VM/C/8/ 7 Standard CCSS.Math.Content.HSN-VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions. 09 10 11 12 B62C1C106873438AA0126760075A65A3 22F88FBD48CB4D73A3A0D0177BE00314 http://corestandards.org/Math/Content/HSN/VM/C/9/ 7 Standard CCSS.Math.Content.HSN-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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 22F88FBD48CB4D73A3A0D0177BE00314 http://corestandards.org/Math/Content/HSN/VM/C/10/ 7 Standard CCSS.Math.Content.HSN-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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 22F88FBD48CB4D73A3A0D0177BE00314 http://corestandards.org/Math/Content/HSN/VM/C/11/ 7 Standard CCSS.Math.Content.HSN-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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 22F88FBD48CB4D73A3A0D0177BE00314 http://corestandards.org/Math/Content/HSN/VM/C/12/ 7 Standard CCSS.Math.Content.HSN-VM.C.12 (+) Work with 2 &times; 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area. 09 10 11 12 B62C1C106873438AA0126760075A65A3 22F88FBD48CB4D73A3A0D0177BE00314 http://corestandards.org/Math/Content/HSS/CP/A/1/ 7 Standard CCSS.Math.Content.HSS-CP.A.1 Describe 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"). 09 10 11 12 B62C1C106873438AA0126760075A65A3 EAD8FB0CC6514306B4018FF8AA698AFC http://corestandards.org/Math/Content/HSS/CP/A/2/ 7 Standard CCSS.Math.Content.HSS-CP.A.2 Understand that two events <i>A</i> and <i>B</i> are independent if the probability of <i>A</i> and <i>B</i> occurring together is the product of their probabilities, and use this characterization to determine if they are independent. 09 10 11 12 B62C1C106873438AA0126760075A65A3 EAD8FB0CC6514306B4018FF8AA698AFC http://corestandards.org/Math/Content/HSS/CP/A/3/ 7 Standard CCSS.Math.Content.HSS-CP.A.3 Understand the conditional probability of <i>A</i> given <i>B</i> as <i>P</i>(<i>A</i> and <i>B</i>)/<i>P</i>(<i>B</i>), and interpret independence of <i>A</i> and <i>B</i> as saying that the conditional probability of <i>A</i> given <i>B</i> is the same as the probability of <i>A</i>, and the conditional probability of <i>B</i> given <i>A</i> is the same as the probability of <i>B</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 EAD8FB0CC6514306B4018FF8AA698AFC http://corestandards.org/Math/Content/HSS/CP/A/4/ 7 Standard CCSS.Math.Content.HSS-CP.A.4 Construct 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. <i>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.</i> 09 10 11 12 B62C1C106873438AA0126760075A65A3 EAD8FB0CC6514306B4018FF8AA698AFC http://corestandards.org/Math/Content/HSS/CP/A/5/ 7 Standard CCSS.Math.Content.HSS-CP.A.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. <i>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.</i> 09 10 11 12 B62C1C106873438AA0126760075A65A3 EAD8FB0CC6514306B4018FF8AA698AFC http://corestandards.org/Math/Content/HSS/CP/B/6/ 7 Standard CCSS.Math.Content.HSS-CP.B.6 Find the conditional probability of <i>A</i> given <i>B</i> as the fraction of <i>B</i>'s outcomes that also belong to <i>A</i>, and interpret the answer in terms of the model. 09 10 11 12 B62C1C106873438AA0126760075A65A3 4174F5D4916B4D11BD2B5964660E814D http://corestandards.org/Math/Content/HSS/CP/B/7/ 7 Standard CCSS.Math.Content.HSS-CP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model. 09 10 11 12 B62C1C106873438AA0126760075A65A3 4174F5D4916B4D11BD2B5964660E814D http://corestandards.org/Math/Content/HSS/CP/B/8/ 7 Standard CCSS.Math.Content.HSS-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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 4174F5D4916B4D11BD2B5964660E814D http://corestandards.org/Math/Content/HSS/CP/B/9/ 7 Standard CCSS.Math.Content.HSS-CP.B.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems. 09 10 11 12 B62C1C106873438AA0126760075A65A3 4174F5D4916B4D11BD2B5964660E814D http://corestandards.org/Math/Content/HSS/IC/A/1/ 7 Standard CCSS.Math.Content.HSS-IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. 09 10 11 12 B62C1C106873438AA0126760075A65A3 666C59E34C6C458F80B403EDB8ADD7C1 http://corestandards.org/Math/Content/HSS/IC/A/2/ 7 Standard CCSS.Math.Content.HSS-IC.A.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. <i>For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model</i>? 09 10 11 12 B62C1C106873438AA0126760075A65A3 666C59E34C6C458F80B403EDB8ADD7C1 http://corestandards.org/Math/Content/HSS/IC/B/3/ 7 Standard CCSS.Math.Content.HSS-IC.B.3 </i>Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.</i> 09 10 11 12 B62C1C106873438AA0126760075A65A3 472C49B4DC9A494AA4804C46D08C4CA6 http://corestandards.org/Math/Content/HSS/IC/B/4/ 7 Standard CCSS.Math.Content.HSS-IC.B.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. 09 10 11 12 B62C1C106873438AA0126760075A65A3 472C49B4DC9A494AA4804C46D08C4CA6 http://corestandards.org/Math/Content/HSS/IC/B/5/ 7 Standard CCSS.Math.Content.HSS-IC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. 09 10 11 12 B62C1C106873438AA0126760075A65A3 472C49B4DC9A494AA4804C46D08C4CA6 http://corestandards.org/Math/Content/HSS/IC/B/6/ 7 Standard CCSS.Math.Content.HSS-IC.B.6 Evaluate reports based on data. 09 10 11 12 B62C1C106873438AA0126760075A65A3 472C49B4DC9A494AA4804C46D08C4CA6 http://corestandards.org/Math/Content/HSS/ID/A/1/ 7 Standard CCSS.Math.Content.HSS-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots). 09 10 11 12 B62C1C106873438AA0126760075A65A3 8EDD3006C95F458A936A9A103964FD3A http://corestandards.org/Math/Content/HSS/ID/A/2/ 7 Standard CCSS.Math.Content.HSS-ID.A.2 Use 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 8EDD3006C95F458A936A9A103964FD3A http://corestandards.org/Math/Content/HSS/ID/A/3/ 7 Standard CCSS.Math.Content.HSS-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 09 10 11 12 B62C1C106873438AA0126760075A65A3 8EDD3006C95F458A936A9A103964FD3A http://corestandards.org/Math/Content/HSS/ID/A/4/ 7 Standard CCSS.Math.Content.HSS-ID.A.4 Use 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 8EDD3006C95F458A936A9A103964FD3A http://corestandards.org/Math/Content/HSS/ID/B/5/ 7 Standard CCSS.Math.Content.HSS-ID.B.5 Summarize 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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 CD5D9C5E086B4FBC8570AAD25285B297 http://corestandards.org/Math/Content/HSS/ID/B/6/ 7 Standard CCSS.Math.Content.HSS-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. 09 10 11 12 B62C1C106873438AA0126760075A65A3 CD5D9C5E086B4FBC8570AAD25285B297 A5C66573CEE844968D5B13903D709F64 E83A0AEC69604F37962EDF492D955DDC FFC82EDD5128419B801F4B2E05DE7909 http://corestandards.org/Math/Content/HSS/ID/B/6/a/ 8 Component CCSS.Math.Content.HSS-ID.B.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. 09 10 11 12 B62C1C106873438AA0126760075A65A3 50A3FD15E91A495195429B64E628AC06 http://corestandards.org/Math/Content/HSS/ID/B/6/b/ 8 Component CCSS.Math.Content.HSS-ID.B.6b Informally assess the fit of a function by plotting and analyzing residuals. 09 10 11 12 B62C1C106873438AA0126760075A65A3 50A3FD15E91A495195429B64E628AC06 http://corestandards.org/Math/Content/HSS/ID/B/6/c/ 8 Component CCSS.Math.Content.HSS-ID.B.6c Fit a linear function for a scatter plot that suggests a linear association. 09 10 11 12 B62C1C106873438AA0126760075A65A3 50A3FD15E91A495195429B64E628AC06 http://corestandards.org/Math/Content/HSS/ID/C/7/ 7 Standard CCSS.Math.Content.HSS-ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. 09 10 11 12 B62C1C106873438AA0126760075A65A3 822E400507344744A3FACC6F93D4A8E0 http://corestandards.org/Math/Content/HSS/ID/C/8/ 7 Standard CCSS.Math.Content.HSS-ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit. 09 10 11 12 B62C1C106873438AA0126760075A65A3 822E400507344744A3FACC6F93D4A8E0 http://corestandards.org/Math/Content/HSS/ID/C/9/ 7 Standard CCSS.Math.Content.HSS-ID.C.9 Distinguish between correlation and causation. 09 10 11 12 B62C1C106873438AA0126760075A65A3 822E400507344744A3FACC6F93D4A8E0 http://corestandards.org/Math/Content/HSS/MD/A/1/ 7 Standard CCSS.Math.Content.HSS-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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 2ABAD306C29949F899683A13FFC985BE http://corestandards.org/Math/Content/HSS/MD/A/2/ 7 Standard CCSS.Math.Content.HSS-MD.A.2 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. 09 10 11 12 B62C1C106873438AA0126760075A65A3 2ABAD306C29949F899683A13FFC985BE http://corestandards.org/Math/Content/HSS/MD/A/3/ 7 Standard CCSS.Math.Content.HSS-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. <i>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.</i> 09 10 11 12 B62C1C106873438AA0126760075A65A3 2ABAD306C29949F899683A13FFC985BE http://corestandards.org/Math/Content/HSS/MD/A/4/ 7 Standard CCSS.Math.Content.HSS-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. <i>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?</i> 09 10 11 12 B62C1C106873438AA0126760075A65A3 2ABAD306C29949F899683A13FFC985BE http://corestandards.org/Math/Content/HSS/MD/B/5/ 7 Standard CCSS.Math.Content.HSS-MD.B.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. 09 10 11 12 B62C1C106873438AA0126760075A65A3 C36505788B144F25AB2E5D3693E257B6 9B5191DE265C4408985B7D86C943EC07 26F801DADEEB411EAF34A557F4C5D3C8 http://corestandards.org/Math/Content/HSS/MD/B/5/a/ 8 Component CCSS.Math.Content.HSS-MD.B.5a Find the expected payoff for a game of chance. <i>For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.</i> 09 10 11 12 B62C1C106873438AA0126760075A65A3 1724BB9BB4964D4BBFA0D6E4BFCEBBC7 http://corestandards.org/Math/Content/HSS/MD/B/5/b/ 8 Component CCSS.Math.Content.HSS-MD.B.5b Evaluate and compare strategies on the basis of expected values. <i>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.</i> 09 10 11 12 B62C1C106873438AA0126760075A65A3 1724BB9BB4964D4BBFA0D6E4BFCEBBC7 http://corestandards.org/Math/Content/HSS/MD/B/6/ 7 Standard CCSS.Math.Content.HSS-MD.B.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). 09 10 11 12 B62C1C106873438AA0126760075A65A3 C36505788B144F25AB2E5D3693E257B6 http://corestandards.org/Math/Content/HSS/MD/B/7/ 7 Standard CCSS.Math.Content.HSS-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). 09 10 11 12 B62C1C106873438AA0126760075A65A3 C36505788B144F25AB2E5D3693E257B6 http://corestandards.org/Math/Practice/MP1/ 7 Standard CCSS.Math.Practice.MP1 <p>Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.</p> K 01 02 03 04 05 06 07 08 09 10 11 12 B62C1C106873438AA0126760075A65A3 FBCBB7C696FE475695920CA622B1C854 http://corestandards.org/Math/Practice/MP2/ 7 Standard CCSS.Math.Practice.MP2 <p>Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to <i>decontextualize</i>&mdash;to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents&mdash;and the ability to <i>contextualize</i>, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.</p> K 01 02 03 04 05 06 07 08 09 10 11 12 B62C1C106873438AA0126760075A65A3 FBCBB7C696FE475695920CA622B1C854 http://corestandards.org/Math/Practice/MP3/ 7 Standard CCSS.Math.Practice.MP3 <p>Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and&mdash;if there is a flaw in an argument&mdash;explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.</p> K 01 02 03 04 05 06 07 08 09 10 11 12 B62C1C106873438AA0126760075A65A3 FBCBB7C696FE475695920CA622B1C854 http://corestandards.org/Math/Practice/MP4/ 7 Standard CCSS.Math.Practice.MP4 <p>Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.</p> K 01 02 03 04 05 06 07 08 09 10 11 12 B62C1C106873438AA0126760075A65A3 FBCBB7C696FE475695920CA622B1C854 http://corestandards.org/Math/Practice/MP5/ 7 Standard CCSS.Math.Practice.MP5 <p>Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.</p> K 01 02 03 04 05 06 07 08 09 10 11 12 B62C1C106873438AA0126760075A65A3 FBCBB7C696FE475695920CA622B1C854 http://corestandards.org/Math/Practice/MP6/ 7 Standard CCSS.Math.Practice.MP6 <p>Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.</p> K 01 02 03 04 05 06 07 08 09 10 11 12 B62C1C106873438AA0126760075A65A3 FBCBB7C696FE475695920CA622B1C854 http://corestandards.org/Math/Practice/MP7/ 7 Standard CCSS.Math.Practice.MP7 <p>Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 &times; 8 equals the well remembered 7 &times; 5 + 7 &times; 3, in preparation for learning about the distributive property. In the expression <i>x</i><sup>2</sup> + 9<i>x</i> + 14, older students can see the 14 as 2 &times; 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(<i>x</i> - <i>y</i>)<sup>2</sup> as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers <i>x</i> and <i>y</i>.</p> K 01 02 03 04 05 06 07 08 09 10 11 12 B62C1C106873438AA0126760075A65A3 FBCBB7C696FE475695920CA622B1C854 http://corestandards.org/Math/Practice/MP8/ 7 Standard CCSS.Math.Practice.MP8 <p>Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (<i>y</i> - 2)/(<i>x</i> - 1) = 3. Noticing the regularity in the way terms cancel when expanding (<i>x</i> - 1)(<i>x</i> + 1), (<i>x</i> - 1)(<i>x</i><sup>2</sup> + <i>x</i> + 1), and (<i>x</i> - 1)(<i>x</i><sup>3</sup> + <i>x</i>2 + <i>x</i> + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.</p> <h2>Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content</h2> <p>The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.</p> <p>The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word "understand" are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.</p> <p>In this respect, those content standards which set an expectation of understanding are potential "points of intersection" between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.</p> K 01 02 03 04 05 06 07 08 09 10 11 12 B62C1C106873438AA0126760075A65A3 FBCBB7C696FE475695920CA622B1C854