Kurs: Math Placement Test
Kurskompetenzen
D2867743.xml D2867743.xml Pfad:Aktivitäten
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Standards for Mathematical Practice S2896416 Standards for Mathematical Practice Aktivitäten Nutzerlernpläne
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MP.1 MP.1 Make sense of problems and persevere in solving them. Aktivitäten
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MP.2 MP.2 Reason abstractly and quantitatively. Aktivitäten
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MP.3 MP.3 Construct viable arguments and critique the reasoning of others. Aktivitäten
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MP.4 MP.4 Model with mathematics Aktivitäten
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MP.5 MP.5 Use appropriate tools strategically Aktivitäten
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MP.6 MP.6 Attend to precision. Aktivitäten
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MP.7 MP.7 Look for and make use of structure. Aktivitäten
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MP.8 MP.8 Look for and express regularity in repeated reasoning. Aktivitäten
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Counting and Cardinality S2896425 Counting and Cardinality Aktivitäten Nutzerlernpläne
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Know number names and the count sequence. S2896426 Know number names and the count sequence. Aktivitäten Nutzerlernpläne
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K.CC.1 K.CC.1 Count to 100 by ones and by tens and identify as a growth pattern. Aktivitäten Nutzerlernpläne
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K.CC.2 K.CC.2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1). Aktivitäten Nutzerlernpläne
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K.CC.3 K.CC.3 Read and write numerals from 0 to 20. Aktivitäten Nutzerlernpläne
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Count to tell the number of objects. S2896430 Count to tell the number of objects. Aktivitäten Nutzerlernpläne
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K.CC.4 K.CC.4 Understand the relationship between numbers and quantities; connect counting to cardinality. Aktivitäten Nutzerlernpläne
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K.CC.4.a K.CC.4.a When counting objects, say each number's name in sequential order, pairing each object with one and only one number name and each number name with one and only one object. Aktivitäten Nutzerlernpläne
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K.CC.4.b K.CC.4.b 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. Aktivitäten Nutzerlernpläne
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K.CC.4.c K.CC.4.c Understand that each successive number name refers to a quantity that is one larger. Aktivitäten Nutzerlernpläne
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K.CC.4.d K.CC.4.d Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects). Aktivitäten Nutzerlernpläne
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K.CC.5 K.CC.5 Count to answer "how many?" up to20 concrete or pictorial objects arranged in a line, a rectangular array, or a circle, or as many as 10 objects in a scattered configuration (subitizing); given a number from 1 to 20, count out that many objects. Aktivitäten Nutzerlernpläne
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Compare numbers. S2896437 Compare numbers. Aktivitäten Nutzerlernpläne
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K.CC.6 K.CC.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.) Include groups with up to ten objects. Aktivitäten Nutzerlernpläne
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K.CC.7 K.CC.7 Compare two numbers between 1 and 10 presented as written numerals. Aktivitäten Nutzerlernpläne
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Operations and Algebraic Thinking S2896440 Operations and Algebraic Thinking Aktivitäten
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S2896441 S2896441 Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. Aktivitäten
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K.OA.1 K.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g. claps), acting out situations, verbal explanations, expressions, or equations. Aktivitäten
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K.OA.2 K.OA.2 Solve addition and subtraction word problems, and add and subtract within 10, (e.g. by using objects or drawings to represent the problem.) Aktivitäten
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K.OA.3 K.OA.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 )). Aktivitäten
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K.OA.4 K.OA.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.). Aktivitäten
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K.OA.5 K.OA.5 Fluently (efficiently, accurately, and flexibly) add and subtract within 5. Aktivitäten
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Represent and solve problems involving addition and subtraction. S2896447 Represent and solve problems involving addition and subtraction. Aktivitäten
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1.OA.1 1.OA.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 situation equations and/or solution equations with a symbol for the unknown number to represent the problem.) Aktivitäten
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1.OA.2 1.OA.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.) Aktivitäten
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2.OA.1 2.OA.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 situation equations and/or solution equations with a symbol for the unknown number to represent the problem.) Aktivitäten
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S2896451 S2896451 Understand and apply properties of operations and the relationship between addition and subtraction. Aktivitäten
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1.OA.3 1.OA.3 Apply (not necessary to name) properties of operations as strategies to add and subtract. Aktivitäten
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1.OA.4 1.OA.4 Understand subtraction as an unknown-addend problem. For example, subtract 10-8 by finding the number that makes 10 when added to 8. Aktivitäten
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Add and subtract within 20. S2896454 Add and subtract within 20. Aktivitäten
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1.OA.5 1.OA.5 Relate counting to addition and subtraction (e.g. by counting on 2 to add 2, counting back 1 to subtract 1). Aktivitäten
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1.OA.6 1.OA.6 Add and subtract within 20, demonstrating fluency (efficiently, accurately, and flexibly) for addition and subtraction within 10. Use mental 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). Aktivitäten
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2.OA.2 2.OA.2 Fluently (efficiently, accurately, and flexibly) add and subtract within 20 using mental strategies (counting on, making a ten, decomposing a number, creating an equivalent but easier and known sum, and using the relationship between addition and subtraction). Aktivitäten
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Work with addition and subtraction equations. S2896458 Work with addition and subtraction equations. Aktivitäten
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1.OA.7 1.OA.7 Understand the meaning of the equal sign (the value is the same on both sides of the equal sign), and determine if equations involving addition and subtraction are true or false. Aktivitäten
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1.OA.8 1.OA.8 Using related equations, Determine the unknown whole number in an addition or subtraction equation. Aktivitäten
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Work with equal groups of objects to gain foundations for multiplication. S2896461 Work with equal groups of objects to gain foundations for multiplication. Aktivitäten
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2.OA.3 2.OA.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. Aktivitäten
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2.OA.4 2.OA.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. Aktivitäten
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Represent and solve problems involving multiplication and division. S2896464 Represent and solve problems involving multiplication and division. Aktivitäten
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3.OA.1 3.OA.1 Interpret products of whole numbers, (e.g. interpret 5·7 as the total number of objects in 5 groups of 7 objects each.) Aktivitäten
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3.OA.2 3.OA.2 Interpret whole-number quotients of whole numbers, (e.g. interpret 56÷8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.) Aktivitäten
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3.OA.3 3.OA.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.) Aktivitäten
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3.OA.4 3.OA.4 Determine the unknown whole number in a multiplication or division equation by using related equations. Aktivitäten
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S2896469 S2896469 Understand properties of multiplication and the relationship between multiplication and division. Aktivitäten
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3.OA.5 3.OA.5 Apply properties of operations as strategies to multiply and divide. Aktivitäten
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3.OA.6 3.OA.6 Understand division as an unknown-factor problem. Aktivitäten
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Multiply and divide within 100 (basic facts up to 10 x 10). S2896472 Multiply and divide within 100 (basic facts up to 10 x 10). Aktivitäten
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3.OA.7 3.OA.7 Fluently (efficiently, accurately, and flexibly) multiply and divide with single digit multiplications and related divisions using strategies (e.g. relationship between multiplication and division, doubles, double and double again, half and then double, etc.) or properties of operations. Aktivitäten
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S2896474 S2896474 Solve problems involving the four operations, and identify and explain patterns in arithmetic. Aktivitäten
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3.OA.8 3.OA.8 Solve two-step word problems using any of the four operations. Represent these problems using both situation equations and/or solution equations with a letter or symbol standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. This standard is limited to problems posed with whole numbers and having whole-number answers. Aktivitäten
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3.OA.9 3.OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. Aktivitäten
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Use the four operations with whole numbers to solve problems. S2896477 Use the four operations with whole numbers to solve problems. Aktivitäten
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4.OA.1 4.OA.1 Interpret a multiplication equation as a comparison, (e.g. interpret 35 = 5·7 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. Aktivitäten
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4.OA.2 4.OA.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.) Aktivitäten
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4.OA.3 4.OA.3 Solve multi-step word problem 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 situation equations and/or solution equations with a letter or symbol standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Aktivitäten
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Gain familiarity with factors and multiples. S2896481 Gain familiarity with factors and multiples. Aktivitäten
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4.OA.4 4.OA.4 Find all factor pairs for a whole number in the range 1 to 100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1 to 100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1 to 100 is prime or composite. Aktivitäten
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Generate and analyze patterns. S2896483 Generate and analyze patterns. Aktivitäten
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4.OA.5 4.OA.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. 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. Aktivitäten
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Write and interpret numerical expressions. S2896485 Write and interpret numerical expressions. Aktivitäten
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5.OA.1 5.OA.1 Use parentheses in numerical expressions and evaluate expressions with these symbols. Aktivitäten
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5.OA.2 5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. Aktivitäten
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Number and Operations in Base Ten S2896488 Number and Operations in Base Ten Aktivitäten
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Work with numbers 11–19 to gain foundations for place value. S2896489 Work with numbers 11–19 to gain foundations for place value. Aktivitäten
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K.NBT.1 K.NBT.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 (e.g. 10 + 8 = 18 and 19 = 10 + 9); ); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones. Aktivitäten
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Extend the counting sequence. S2896491 Extend the counting sequence. Aktivitäten
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1.NBT.1 1.NBT.1 Count to 120 (recognizing growth and repeating patterns), starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. Aktivitäten
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Understand place value. S2896493 Understand place value. Aktivitäten
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1.NBT.2 1.NBT.2 Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: Aktivitäten
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1.NBT.2.a 1.NBT.2.a 10 can be thought of as a grouping of ten ones—called a "ten." Aktivitäten
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1.NBT.2.b 1.NBT.2.b The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones. Aktivitäten
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1.NBT.2.c 1.NBT.2.c 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). Aktivitäten
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1.NBT.2.d 1.NBT.2.d Show flexibility in composing and decomposing tens and ones (e.g. 20 can be composed from 2 tens or 1 ten and 10 ones, or 20 ones.) Aktivitäten
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1.NBT.3 1.NBT.3 Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the relational symbols >, Aktivitäten
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2.NBT.1 2.NBT.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: Aktivitäten
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2.NBT.1.a 2.NBT.1.a 100 can be thought of as a bundle of ten tens—called a "hundred." Aktivitäten
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2.NBT.1.b 2.NBT.1.b The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds. Aktivitäten
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2.NBT.1.c 2.NBT.1.c Show flexibility in composing and decomposing hundreds, tens and ones (e.g. 207 can be composed from 2 hundreds 7 ones OR 20 tens 7 ones OR 207 ones OR 1 hundred 10 tens 7 ones OR 1 hundred 9 tens 17 ones, etc.) Aktivitäten
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2.NBT.2 2.NBT.2 Count within 1000; skip-count by 2s, 5s, 10s, and 100s; explain and generalize the patterns. Aktivitäten
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2.NBT.3 2.NBT.3 Read and write numbers within 1000 using base-ten numerals, number names, expanded form, and unit form. Aktivitäten
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2.NBT.4 2.NBT.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, Aktivitäten
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Use place value understanding and properties of operations to add and subtract. S2896507 Use place value understanding and properties of operations to add and subtract. Aktivitäten
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1.NBT.4 1.NBT.4 Add within 100 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 including: Aktivitäten
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1.NBT.4.a 1.NBT.4.a Adding a two-digit number and a one-digit number Aktivitäten
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1.NBT.4.b 1.NBT.4.b Adding a two-digit number and a multiple of 10 Aktivitäten
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1.NBT.4.c 1.NBT.4.c Understanding that when adding two-digit numbers, combine like base-ten units such as tens and tens, ones and ones; and sometimes it is necessary to compose a ten. Aktivitäten
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1.NBT.5 1.NBT.5 Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. Aktivitäten
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1.NBT.6 1.NBT.6 Subtract multiples of 10 in the range 10 to 90 from multiples of 10 in the range 10 to 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. Aktivitäten
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2.NBT.5 2.NBT.5 Fluently (efficiently, accurately, and flexibly) add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction (e.g. composing/decomposing by like base-10 units, using friendly or benchmark numbers, using related equations, compensation, number line, etc.). Aktivitäten
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2.NBT.6 2.NBT.6 Add up to four two-digit numbers using strategies based on place value and properties of operations. Aktivitäten
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2.NBT.7 2.NBT.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, like base-ten units such as hundreds and hundreds, tens and tens, ones and ones are used; and sometimes it is necessary to compose or decompose tens or hundreds. Aktivitäten
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2.NBT.8 2.NBT.8 Mentally add 10 or 100 to a given number 100 – 900, and mentally subtract 10 or 100 from a given number 100 – 900. Aktivitäten
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2.NBT.9 2.NBT.9 Explain why addition and subtraction strategies work using place value and the properties of operations. The explanations given may be supported by drawings or objects. Aktivitäten
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S2896519 S2896519 Use place value understanding and properties of operations to perform multi-digit arithmetic. Aktivitäten
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3.NBT.1 3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100. Aktivitäten
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3.NBT.2 3.NBT.2 Fluently (efficiently, accurately, & flexibly) add and subtract within 1000 using strategies (e.g. composing/decomposing by like base-10 units, using friendly or benchmark numbers, using related equations, compensation, number line, etc.) and algorithms (including, but not limited to: traditional, partial-sums, etc.) based on place value, properties of operations, and/or the relationship between addition and subtraction. Aktivitäten
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3.NBT.3 3.NBT.3 Multiply one-digit whole numbers by multiples of 10 in the range 10 to 90 (e.g. 9⋅80, 5⋅60) using strategies based on place value and properties of operations. Aktivitäten
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4.NBT.4 4.NBT.4 Fluently (efficiently, accurately, and flexibly) add and subtract multi-digit whole numbers using an efficient algorithm (including, but not limited to: traditional, partial-sums, etc.), based on place value understanding and the properties of operations. Aktivitäten
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4.NBT.5 4.NBT.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. Aktivitäten
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4.NBT.6 4.NBT.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. Aktivitäten
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Generalize place value understanding for multi-digit whole numbers. S2896526 Generalize place value understanding for multi-digit whole numbers. Aktivitäten
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4.NBT.1 4.NBT.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. Aktivitäten
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4.NBT.2 4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals, number names, expanded form, and unit form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, Aktivitäten
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4.NBT.3 4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place. Aktivitäten
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Understand the place value system. S2896530 Understand the place value system. Aktivitäten
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5.NBT.1 5.NBT.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. Aktivitäten
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5.NBT.2 5.NBT.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. Aktivitäten
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5.NBT.3 5.NBT.3 Read, write, and compare decimals to thousandths. Aktivitäten
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5.NBT.3.a 5.NBT.3.a Read and write decimals to thousandths using base-ten numerals, number names, expanded form, and unit form. Aktivitäten
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5.NBT.3.b 5.NBT.3.b Compare two decimals to thousandths based on meanings of the digits in each place, using >, Aktivitäten
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5.NBT.4 5.NBT.4 Use place value understanding to round decimals to any place Aktivitäten
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S2896537 S2896537 Perform operations with multi-digit whole numbers and with decimals to hundredths. Aktivitäten
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5.NBT.5 5.NBT.5 Fluently (efficiently, accurately, and flexibly) multiply multi-digit whole numbers using an efficient algorithm (ex., traditional, partial products, etc.) based on place value understanding and the properties of operations. Aktivitäten
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5.NBT.6 5.NBT.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. Aktivitäten
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5.NBT.7 5.NBT.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. Aktivitäten
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Measurement and Data S2896541 Measurement and Data Aktivitäten
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Describe and compare measurable attributes. S2896542 Describe and compare measurable attributes. Aktivitäten
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K.MD.1 K.MD.1 Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object. Aktivitäten
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K.MD.2 K.MD.2 Directly compare two objects, with a measureable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference. Aktivitäten
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Classify objects and count the number of objects in each category. S2896545 Classify objects and count the number of objects in each category. Aktivitäten
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K.MD.3 K.MD.3 Classify objects into given categories; count the numbers of objects in each category and sort the categories by count (Limit category counts to be less than or equal to 10). Aktivitäten
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Measure lengths indirectly and by iterating length units. S2896547 Measure lengths indirectly and by iterating length units. Aktivitäten
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1.MD.1 1.MD.1 Order three objects by length; compare the lengths of two objects indirectly by using a third object. Aktivitäten
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1.MD.2 1.MD.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. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps. Aktivitäten
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Tell and write time. S2896550 Tell and write time. Aktivitäten
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1.MD.3 1.MD.3 Tell and write time in hours and half-hours using analog and digital clocks. Aktivitäten
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Represent and interpret data. S2896552 Represent and interpret data. Aktivitäten
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1.MD.4 1.MD.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. Aktivitäten
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2.MD.10 2.MD.10 Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object using different units. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units. Aktivitäten
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2.MD.11 2.MD.11 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 using information presented in a bar graph Aktivitäten
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3.MD.4 3.MD.4 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. Aktivitäten
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3.MD.5 3.MD.5 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—whole numbers, halves, or quarters. Aktivitäten
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4.MD.4 4.MD.4 Make a data display (line plot, bar graph, pictograph) to show a set of measurements in fractions of a unit (½, ¼, 1/8, 1/16). Solve problems involving addition and subtraction of fractions by using information presented in the data display. Aktivitäten
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Convert like measurement units within a given measurement system. S2896559 Convert like measurement units within a given measurement system. Aktivitäten
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5.MD.1 5.MD.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. Aktivitäten
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Represent and interpret data. S2896561 Represent and interpret data. Aktivitäten
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5.MD.2 5.MD.2 Make a data display (line plot, bar graph, pictograph) to show a data set of measurements in fractions of a unit (½, ¼, 1/8). Use operations (add, subtract, multiply) on fractions for this grade to solve problems involving information presented in the data display. Aktivitäten
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S2896563 S2896563 Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. Aktivitäten
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5.MD.3 5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement. Aktivitäten
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5.MD.3.a 5.MD.3.a 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. Aktivitäten
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5.MD.3.b 5.MD.3.b A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. Aktivitäten
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5.MD.4 5.MD.4 Measure volumes by counting unit cubes such as cubic cm, cubic in, cubic ft. or non-standard cubic units. Aktivitäten
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5.MD.5 5.MD.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. Aktivitäten
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5.MD.5.a 5.MD.5.a 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 three-dimensional whole-number products as volumes, (e.g. to represent the associative property of multiplication.) Aktivitäten
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5.MD.5.b 5.MD.5.b Apply the formulas V = l·w·h and V = B·h (B represents the area of the base) for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems. Aktivitäten
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5.MD.5.c 5.MD.5.c 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. Aktivitäten
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Measure and estimate lengths in standard units. S2896572 Measure and estimate lengths in standard units. Aktivitäten
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2.MD.1 2.MD.1 Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. Aktivitäten
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2.MD.2 2.MD.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. Aktivitäten
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2.MD.3 2.MD.3 Estimate lengths using whole units of inches, feet, centimeters, and meters. Aktivitäten
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2.MD.4 2.MD.4 Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit (inches, feet, centimeters, and meters). Aktivitäten
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Relate addition and subtraction to length. S2896577 Relate addition and subtraction to length. Aktivitäten
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2.MD.5 2.MD.5 Use addition and subtraction within 100 to solve one- and two-step 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. Aktivitäten
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2.MD.6 2.MD.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. Aktivitäten
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Work with time and money. S2896580 Work with time and money. Aktivitäten
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2.MD.7 2.MD.7 Tell and write time from analog and digital clocks to the nearest five minutes. Aktivitäten
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2.MD.8 2.MD.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately (Do not use decimal point, if showing 25 cents, use the word cents or ¢). Aktivitäten
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2.MD.9 2.MD.9 Identify coins and bills and their values. Aktivitäten
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S2896584 S2896584 Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. Aktivitäten
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3.MD.1 3.MD.1 Tell and write time to the nearest minute using a.m. and p.m. 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.) Aktivitäten
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3.MD.2 3.MD.2 Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l) (Excludes cubed units such as cm³ and finding the geometric volume of a container). Aktivitäten
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3.MD.3 3.MD.3 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.) Aktivitäten
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S2896588 S2896588 Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. Aktivitäten
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4.MD.1 4.MD.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. Aktivitäten
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4.MD.2 4.MD.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. Aktivitäten
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4.MD.3 4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems explaining and justifying the appropriate unit of measure. Aktivitäten
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Geometry S2896592 Geometry Aktivitäten
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S2896593 S2896593 Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres). Aktivitäten
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K.G.1 K.G.1 Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to. Aktivitäten
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K.G.2 K.G.2 Correctly gives most precise name of shapes regardless of their orientations (position and direction in space) or overall size. Aktivitäten
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K.G.3 K.G.3 Identify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional ("solid"). Aktivitäten
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Analyze, compare, create, and compose shapes. S2896597 Analyze, compare, create, and compose shapes. Aktivitäten
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K.G.4 K.G.4 Analyze and compare two- and three-dimensional shapes, in different sizes and orientations (position and direction in space), 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). Aktivitäten
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K.G.5 K.G.5 Model shapes in the world by building shapes from components (e.g. sticks and clay balls) and drawing shapes. Aktivitäten
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K.G.6 K.G.6 Compose simple shapes to form larger shapes. For example, "Can you join these two triangles with full sides touching to make a rectangle?" Aktivitäten
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Reason with shapes and their attributes. S2896601 Reason with shapes and their attributes. Aktivitäten
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1.G.1 1.G.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 that possess defining attributes. Aktivitäten
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1.G.2 1.G.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. Students do not need to learn formal names such as "right rectangular prism." Aktivitäten
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1.G.3 1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Aktivitäten
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2.G.1 2.G.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. Aktivitäten
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2.G.2 2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. Aktivitäten
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2.G.3 2.G.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. Note: fraction notation 1/2, 1/3, 1/4 is not expected at this grade level. Recognize that equal shares of identical wholes need not have the same shape. Aktivitäten
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3.G.1 3.G.1 Understand that shapes in different categories (e.g. rhombuses, rectangles, trapezoids, kites 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. Aktivitäten
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3.G.2 3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. Aktivitäten
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S2896610 S2896610 Draw and identify lines and angles, and classify shapes by properties of their lines and angles. Aktivitäten
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4.G.1 4.G.1 Draw points, lines, line segments, rays, angles (right, acute, obtuse, straight, reflex), and perpendicular and parallel lines. Identify these in two-dimensional figures. Aktivitäten
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4.G.2 4.G.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles (right, acute, obtuse, straight, reflex). Recognize and categorize triangles based on angles (acute, obtuse, equiangular, and right) and/or sides (scalene, isosceles, and equilateral). Aktivitäten
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4.G.3 4.G.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. Aktivitäten
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S2896614 S2896614 Geometric measurement: understand concepts of area and relate area to multiplication and to addition. Aktivitäten
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3.MD.6 3.MD.6 Recognize area as an attribute of plane figures and understand concepts of area measurement. Aktivitäten
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3.MD.6.a 3.MD.6.a 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 (does not require standard square units). Aktivitäten
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3.MD.6.b 3.MD.6.b A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units (does not require standard square units). Aktivitäten
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3.MD.7 3.MD.7 Measure areas by counting unit squares (square cm, square m, square in, square ft, and non-standard square units). Aktivitäten
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3.MD.8 3.MD.8 Relate area to the operations of multiplication and addition Aktivitäten
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3.MD.8.a 3.MD.8.a 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. Aktivitäten
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3.MD.8.b 3.MD.8.b 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. Aktivitäten
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3.MD.8.c 3.MD.8.c Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b+c is the sum of a·b and a·c. Use area models to represent the distributive property in mathematical reasoning. Aktivitäten
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3.MD.8.d 3.MD.8.d 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. Aktivitäten
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S2896624 S2896624 Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. Aktivitäten
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3.MD.9 3.MD.9 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. Aktivitäten
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S2896626 S2896626 Graph points on the coordinate plane to solve real-world and mathematical problems. Aktivitäten
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5.G.1 5.G.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. x-axis and x-coordinate, y-axis and y-coordinate). Aktivitäten
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5.G.2 5.G.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. (e.g. plotting the relationship between two positive quantities such as maps, coordinate grid games (such as Battleship), time/temperature, time/distance, cost/quantity, etc.). Aktivitäten
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Classify two-dimensional figures into categories based on their properties S2896629 Classify two-dimensional figures into categories based on their properties Aktivitäten
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5.G.3 5.G.3 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. Aktivitäten
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5.G.4 5.G.4 Classify two-dimensional figures in a hierarchy based on properties. Aktivitäten
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S2896632 S2896632 Solve real-world and mathematical problems involving area, surface area, and volume. Aktivitäten
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6.G.1 6.G.1 Find the area of all triangles, special quadrilaterals (including parallelograms, kites and trapezoids), and polygons whose edges meet at right angles (rectilinear figure 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. Aktivitäten
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6.G.2 6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by applying the formulas V=lwh and V=Bh (B is the area of the base and h is the height) to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. Aktivitäten
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6.G.3 6.G.3 Draw polygons whose edges meet at right angles (rectilinear figure 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. Aktivitäten
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6.G.4 6.G.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. Aktivitäten
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7.G.4 7.G.4 .Use the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between the circumference and area of a circle. Aktivitäten
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7.G.5 7.G.5 Investigate the relationship between three-dimensional geometric shapes; Aktivitäten
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7.G.5.a 7.G.5.a Generalize the volume formula for prisms and cylinders (V = Bh where B is the base and h is the height). Aktivitäten
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7.G.5.b 7.G.5.b Generalize the surface area formula for prisms and cylinders (SA = 2B + Ph where B is the area of the base, P is the perimeter of the base, and h is the height (in the case of a cylinder, perimeter is replaced by circumference)). Aktivitäten
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7.G.6 7.G.6 Solve real-world and mathematical problems involving area of two-dimensional objects and volume and surface area of three-dimensional objects including cylinders and right prisms. (Solutions should not require students to take square roots or cube roots. Aktivitäten
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S2896642 S2896642 Draw, construct, and describe geometrical figures and describe the relationships between them. Aktivitäten
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7.G.1 7.G.1 Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Aktivitäten
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7.G.2 7.G.2 Identify three-dimensional objects generated by rotating a two-dimensional (rectangular or triangular) object around one edge. Aktivitäten
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7.G.3 7.G.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right cylinder. Aktivitäten
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Geometric measurement: understand concepts of angle and measure angles. S2896646 Geometric measurement: understand concepts of angle and measure angles. Aktivitäten
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8.G.1 8.G.1 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: Aktivitäten
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8.G.1.a 8.G.1.a 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. Aktivitäten
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8.G.1.b 8.G.1.b An angle that turns through n one-degree angles is said to have an angle measure of n degrees. Aktivitäten
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8.G.2 8.G.2 Measure angles in whole-number degrees using a protractor. Draw angles of specified measure using a protractor and straight edge. Aktivitäten
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8.G.3 8.G.3 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. Aktivitäten
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8.G.4 8.G.4 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and use them to solve simple equations for an unknown angle in a figure. Aktivitäten
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8.G.5 8.G.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. Aktivitäten
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8.G.6 8.G.6 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on drawing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. Aktivitäten
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Understand and apply the Pythagorean Theorem. S2896655 Understand and apply the Pythagorean Theorem. Aktivitäten
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8.G.7 8.G.7 Explain a proof of the Pythagorean Theorem and its converse. Aktivitäten
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8.G.8 8.G.8 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Aktivitäten
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8.G.9 8.G.9 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. Aktivitäten
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Solve real-world and mathematical problems involving measurement. S2896659 Solve real-world and mathematical problems involving measurement. Aktivitäten
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8.G.10 8.G.10 Use the formulas or informal reasoning to find the arc length, areas of sectors, surface areas and volumes of pyramids, cones, and spheres. Aktivitäten
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8.G.11 8.G.11 Investigate the relationship between the formulas of three dimensional geometric shapes; Aktivitäten
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8.G.11.a 8.G.11.a Generalize the volume formula for pyramids and cones (V = 1/3Bh). Aktivitäten
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8.G.11.b 8.G.11.b Generalize surface area formula of pyramids and cones (SA = B + ½Pl). Aktivitäten
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8.G.12 8.G.12 Solve real-world and mathematical problems involving arc length, area of two-dimensional shapes including sectors, volume and surface area of three-dimensional objects including pyramids, cones and spheres. Aktivitäten
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Number and Operations—Fractions S2896665 Number and Operations—Fractions Aktivitäten
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S2896666 S2896666 Develop understanding of fractions as numbers(Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.) Aktivitäten
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3.NF.1 3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. Aktivitäten
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3.NF.2 3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. Aktivitäten
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3.NF.2.a 3.NF.2.a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. Aktivitäten
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3.NF.2.b 3.NF.2.b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line (a is the countable units of 1/b that determines the place on the number line). Aktivitäten
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3.NF.3 3.NF.3 Explain equivalence of fractions, and compare fractions by reasoning about their size (it is a mathematical convention that when comparing fractions, the whole is the same size). Aktivitäten
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3.NF.3.a 3.NF.3.a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. Aktivitäten
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3.NF.3.b 3.NF.3.b Recognize and generate simple equivalent fractions, (e.g. ½ = 2/4, 4/6 = 2/3.) Explain why the fractions are equivalent, e.g. by using a visual fraction model. Aktivitäten
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3.NF.3.c 3.NF.3.c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Aktivitäten
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3.NF.3.d 3.NF.3.d 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 relational symbols >, Aktivitäten
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Extend understanding of fraction equivalence and ordering. S2896676 Extend understanding of fraction equivalence and ordering. Aktivitäten
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4.NF.1 4.NF.1 Explain why a fraction a/b is equivalent to a fraction ((n·a))/((n·b) ) 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. Aktivitäten
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4.NF.2 4.NF.2 Compare two fractions with different numerators and different denominators, (e.g. by creating common numerators or denominators, or by comparing to a benchmark fraction such as ½.) Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with relational symbols >, Aktivitäten
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S2896679 S2896679 Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. Aktivitäten
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4.NF.3 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. Aktivitäten
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4.NF.3.a 4.NF.3.a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Aktivitäten
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4.NF.3.b 4.NF.3.b 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. Aktivitäten
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4.NF.3.c 4.NF.3.c Add and subtract mixed numbers with like denominators, e.g. by replacing each mixed number with an equivalent fraction (simplest form is not an expectation), and/or by using properties of operations and the relationship between addition and subtraction. Aktivitäten
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4.NF.3.d 4.NF.3.d 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. Aktivitäten
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4.NF.4 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. Aktivitäten
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4.NF.4.a 4.NF.4.a Understand a fraction a/b as a multiple of 1/b. Aktivitäten
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4.NF.4.b 4.NF.4.b Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. Aktivitäten
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4.NF.4.c 4.NF.4.c Solve word problems involving multiplication of a fraction by a whole number. Aktivitäten
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Understand decimal notation for fractions, and compare decimal fractions. S2896689 Understand decimal notation for fractions, and compare decimal fractions. Aktivitäten
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4.NF.5 4.NF.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. Aktivitäten
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4.NF.6 4.NF.6 Use decimal notation for fractions with denominators 10 or 100. Aktivitäten
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4.NF.7 4.NF.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 relational symbols >, < =, or ≠, and justify the conclusions, (e.g. by using a visual model.). Aktivitäten
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Use equivalent fractions as a strategy to add and subtract fractions. S2896693 Use equivalent fractions as a strategy to add and subtract fractions. Aktivitäten
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5.NF.1 5.NF.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. Aktivitäten
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5.NF.2 5.NF.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. Aktivitäten
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S2896696 S2896696 Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Aktivitäten
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5.NF.3 5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b = a÷b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g. by using visual fraction models or equations to represent the problem. Aktivitäten
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5.NF.4 5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. Aktivitäten
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5.NF.4.a 5.NF.4.a Interpret the product a/b·q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a·q÷b. Aktivitäten
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5.NF.4.b 5.NF.4.b 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. Aktivitäten
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5.NF.5 5.NF.5 Interpret multiplication as scaling (resizing), by: Aktivitäten
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5.NF.5.a 5.NF.5.a Comparing the size of a product to the size of one factor based on the size of the other factor, without performing the indicated multiplication (e.g. They see (½·3) as half the size of 3.). Aktivitäten
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5.NF.5.b 5.NF.5.b Explain 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); explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = na/nb to the effect of multiplying a/b by 1. Aktivitäten
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5.NF.6 5.NF.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). Aktivitäten
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5.NF.7 5.NF.7 Apply and extend previous understandings of division, to divide unit fractions by whole numbers and whole numbers by unit fractions. Division of a fraction by a fraction is not a requirement at this grade. Aktivitäten
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5.NF.7.a 5.NF.7.a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. Aktivitäten
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5.NF.7.b 5.NF.7.b Interpret division of a whole number by a unit fraction, and compute such quotients. Aktivitäten
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5.NF.7.c 5.NF.7.c 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. Aktivitäten
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Ratios and Proportional Relationships S2896709 Ratios and Proportional Relationships Aktivitäten
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Understand ratio concepts and use ratio reasoning to solve problems. S2896710 Understand ratio concepts and use ratio reasoning to solve problems. Aktivitäten
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6.RP.1 6.RP.1 Use ratio language to describe a relationship between two quantities. Distinguish between part-to-part and part-to-whole relationships. Aktivitäten
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6.RP.2 6.RP.2 Use unit rate language ("for each one", "for every one" and "per") and unit rate notation to demonstrate understanding the concept of a unit rate a/b associated with a ratio a:b with b≠0, Aktivitäten
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6.RP.3 6.RP.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 diagram, or using calculations.) Aktivitäten
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6.RP.3.a 6.RP.3.a Make tables of equivalent ratios relating quantities with whole-number measurements, find the missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. Solve unit rate problems including those involving unit pricing and constant speed. Aktivitäten
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6.RP.3.b 6.RP.3.b 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. Aktivitäten
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6.RP.3.c 6.RP.3.c Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Aktivitäten
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S2896717 S2896717 Analyze proportional relationships and use them to solve real-world and mathematical problems. Aktivitäten
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7.RP.1 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. Aktivitäten
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7.RP.2 7.RP.2 Recognize and represent proportional relationships between quantities: Aktivitäten
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7.RP.2.a 7.RP.2.a Determine 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. Aktivitäten
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7.RP.2.b 7.RP.2.b Analyze a table or graph and recognize that, in a proportional relationship, every pair of numbers has the same unit rate (referred to as the "m"). Aktivitäten
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7.RP.2.c 7.RP.2.c Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t=pn. Aktivitäten
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7.RP.2.d 7.RP.2.d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Aktivitäten
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7.RP.3 7.RP.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. Aktivitäten
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The Number System S2896725 The Number System Aktivitäten
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S2896726 S2896726 Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Aktivitäten
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6.NS.1 6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, requiring multiple exposures connecting various concrete and abstract models. Aktivitäten
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S2896728 S2896728 Compute fluently (efficiently, accurately, and flexibly) with multi-digit numbers and find common factors and multiples. Aktivitäten
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6.NS.2 6.NS.2 Fluently (efficiently, accurately, and flexibly) divide multi-digit numbers using an efficient algorithm. Aktivitäten
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6.NS.3 6.NS.3 Fluently (efficiently, accurately, and flexibly) add, subtract, multiply, and divide multi-digit decimals using an efficient algorithm for each operation. Aktivitäten
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6.NS.4 6.NS.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. Aktivitäten
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S2896732 S2896732 Apply and extend previous understandings of numbers to the system of rational numbers. Aktivitäten
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6.NS.5 6.NS.5 Understand positive and negative numbers 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); Aktivitäten
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6.NS.5.a 6.NS.5.a Use positive and negative numbers to represent quantities in real-world contexts, Aktivitäten
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6.NS.5.b 6.NS.5.b Explaining the meaning of 0 in each situation. Aktivitäten
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6.NS.6 6.NS.6 Understand a rational number as a point on the number line and a coordinate pair as a location on a coordinate plane. Aktivitäten
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6.NS.6.a 6.NS.6.a 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. Aktivitäten
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6.NS.6.b 6.NS.6.b Recognize signs of numbers in ordered pairs indicate 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. Aktivitäten
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6.NS.6.c 6.NS.6.c 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. Aktivitäten
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6.NS.7 6.NS.7 Understand ordering and absolute value of rational numbers. Aktivitäten
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6.NS.7.a 6.NS.7.a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. Aktivitäten
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6.NS.7.b 6.NS.7.b Write, interpret, and explain statements of order for rational numbers in real-world contexts. Aktivitäten
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6.NS.7.c 6.NS.7.c Explain 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. Aktivitäten
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6.NS.7.d 6.NS.7.d Distinguish comparisons of absolute value from statements about order. Aktivitäten
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6.NS.8 6.NS.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. Aktivitäten
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S2896746 S2896746 Apply and extend previous understandings of operations with positive rational numbers to add, subtract, multiply, and divide all rational numbers. Aktivitäten
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7.NS.1 7.NS.1 Represent addition and subtraction on a horizontal or vertical number line diagram. Aktivitäten
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7.NS.1.a 7.NS.1.a Describe situations in which opposite quantities combine to make 0. Show that a number and its opposite have a sum of 0 (are additive inverses). Aktivitäten
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7.NS.1.b 7.NS.1.b Show p+q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Aktivitäten
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7.NS.1.c 7.NS.1.c Model subtraction of rational numbers as adding the additive inverse, p-q=p+(-q). Aktivitäten
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7.NS.1.d 7.NS.1.d Model subtraction as the distance between two rational numbers on the number line where the distance is the absolute value of their difference. Aktivitäten
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7.NS.1.e 7.NS.1.e Apply properties of operations as strategies to add and subtract rational numbers. Aktivitäten
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7.NS.2 7.NS.2 Apply and extend previous understandings of multiplication and division of positive rational numbers to multiply and divide all rational numbers. Aktivitäten
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7.NS.2.a 7.NS.2.a Describe how multiplication is extended from positive rational numbers to all 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. Aktivitäten
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7.NS.2.b 7.NS.2.b Explain 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. Leading to situations such that if p and q are integers, then -(p/q)=(-p)/q=p/(-q). Aktivitäten
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7.NS.2.c 7.NS.2.c Apply properties of operations as strategies to multiply and divide rational numbers. Aktivitäten
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7.NS.2.d 7.NS.2.d Convert a rational number in the form of a fraction to its decimal equivalent using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. Aktivitäten
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7.NS.3 7.NS.3 Solve and interpret real-world and mathematical problems involving the four operations with rational numbers. Aktivitäten
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S2896759 S2896759 Know that there are numbers that are not rational, and approximate them by rational numbers. Aktivitäten
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8.NS.1 8.NS.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. Aktivitäten
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8.NS.2 8.NS.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. π²). Aktivitäten
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Expressions and Equations S2896762 Expressions and Equations Aktivitäten
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Apply and extend previous understandings of arithmetic to algebraic expressions. S2896763 Apply and extend previous understandings of arithmetic to algebraic expressions. Aktivitäten
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6.EE.1 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents. Aktivitäten
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6.EE.2 6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers. Aktivitäten
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6.EE.2.a 6.EE.2.a Write expressions that record operations with numbers and with letters standing for numbers. Aktivitäten
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6.EE.2.b 6.EE.2.b 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. Aktivitäten
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6.EE.2.c 6.EE.2.c 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). Aktivitäten
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6.EE.3 6.EE.3 Apply the properties of operations and combine like terms, with the conventions of algebraic notation, to identify and generate equivalent expressions. Aktivitäten
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Reason about and solve one-variable equations and inequalities. S2896770 Reason about and solve one-variable equations and inequalities. Aktivitäten
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6.EE.4 6.EE.4 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. Aktivitäten
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6.EE.5 6.EE.5 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. Aktivitäten
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6.EE.6 6.EE.6 Solve one-step equations involving non-negative rational numbers using addition, subtraction, multiplication and division. Aktivitäten
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6.EE.7 6.EE.7 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. Aktivitäten
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S2896775 S2896775 Represent and analyze quantitative relationships between dependent and independent variables. Aktivitäten
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6.EE.8 6.EE.8 Use variables to represent two quantities in a real-world problem that change in relationship to one another. Aktivitäten
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6.EE.8.a 6.EE.8.a Identify the independent and dependent variable. Aktivitäten
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6.EE.8.b 6.EE.8.b 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. Aktivitäten
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6.EE.8.c 6.EE.8.c Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. Aktivitäten
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Use properties of operations to generate equivalent expressions. S2896780 Use properties of operations to generate equivalent expressions. Aktivitäten
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7.EE.1 7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with integer coefficients. Aktivitäten
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7.EE.2 7.EE.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. Aktivitäten
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S2896783 S2896783 Solve real-life and mathematical problems using numerical and algebraic expressions and equations. Aktivitäten
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7.EE.3 7.EE.3 Solve multi-step real-life and mathematical problems with rational numbers. 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. Aktivitäten
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7.EE.4 7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct two-step equations and inequalities to solve problems by reasoning about the quantities. Aktivitäten
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7.EE.4.a 7.EE.4.a Solve word problems leading to equations of the form px+q=r, and p(x+q)=r where p, q, and r are specific rational numbers. Solve equations of these forms fluently (efficiently, accurately, and flexibly). Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. Aktivitäten
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7.EE.4.b 7.EE.4.b Solve word problems leading to inequalities of the form px+q > r or px+q < r where p, q, and r are specific rational numbers and p > 0. Graph the solution set of the inequality and interpret it in the context of the problem. Aktivitäten
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Work with radicals and integer exponents. S2896788 Work with radicals and integer exponents. Aktivitäten
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8.EE.1 8.EE.1 Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of whole number perfect squares with solutions between 0 and 15 and cube roots of whole number perfect cubes with solutions between 0 and 5. Know that √2 is irrational. Aktivitäten
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8.EE.2 8.EE.2 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. Aktivitäten
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8.EE.3 8.EE.3 Read and write 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. Aktivitäten
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S2896792 S2896792 Understand the connections between proportional relationships, lines, and linear equations. Aktivitäten
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8.EE.4 8.EE.4 Graph proportional relationships, interpreting its unit rate as the slope (m) of the graph. Compare two different proportional relationships represented in different ways. Aktivitäten
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8.EE.5 8.EE.5 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 and extend to include the use of the slope formula (m = y2 - y1/x2 - x1when given two coordinate points (x1, y1) and (x2, y2)). Generate the equation y = mx for a line through the origin (proportional) and the equation y = mx + b for a line with slope m intercepting the vertical axis at y-intercept b (not proportional when b ≠ 0). Aktivitäten
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8.EE.6 8.EE.6 Describe the relationship between the proportional relationship expressed in y = mx and the non-proportional linear relationship y = mx + b as a result of a vertical translation. Aktivitäten
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Analyze and solve linear equations and inequalities. S2896796 Analyze and solve linear equations and inequalities. Aktivitäten
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8.EE.7 8.EE.7 Fluently (efficiently, accurately, and flexibly) solve one-step, two-step, and multi-step linear equations and inequalities in one variable, including situations with the same variable appearing on both sides of the equal sign. Aktivitäten
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8.EE.7.a 8.EE.7.a Give examples of linear equations in one variable with one solution (x = a), infinitely many solutions (a = a), or no solutions (a = b). Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). Aktivitäten
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8.EE.7.b 8.EE.7.b Solve linear equations and inequalities with rational number coefficients, including equations/inequalities whose solutions require expanding and/or factoring expressions using the distributive property and collecting like terms. Aktivitäten
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Statistics and Probability S2896800 Statistics and Probability Aktivitäten
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S2896801 S2896801 Develop concepts of statistical measures of center and variability and an informal understanding of outlier. Aktivitäten
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6.SP.1 6.SP.1 Recognize and generate a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. Aktivitäten
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6.SP.2 6.SP.2 Analyze a set of data collected to answer a statistical question with a distribution which can be described by its center (mean, median and/or mode), spread (range and/or interquartile range), and overall shape (cluster, peak, gap, symmetry, skew (data) and/or outlier). Aktivitäten
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6.SP.3 6.SP.3 Recognize that a measure of center (mean, median and/or mode) for a numerical data set summarizes all of its values with a single number, while a measure of variation (range and/or interquartile range) describes how its values vary with a single number. Aktivitäten
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Summarize and describe distributions. S2896805 Summarize and describe distributions. Aktivitäten
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6.SP.4 6.SP.4 Display numerical data on dot plots, histograms, stem-and-leaf plots, and box plots. (6.SP.4) Aktivitäten
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6.SP.5 6.SP.5 Summarize numerical data sets in relation to their context, such as by: Aktivitäten
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6.SP.5.a 6.SP.5.a Reporting the number of observations. Aktivitäten
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6.SP.5.b 6.SP.5.b Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. Aktivitäten
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6.SP.5.c 6.SP.5.c Giving quantitative measures of center (mean, median and/or mode) and variability (range and/or interquartile range), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. Aktivitäten
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6.SP.5.d 6.SP.5.d Relating the choice of measures of center and variability to the distribution of the data. Aktivitäten
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Use random sampling to draw inferences about a population. S2896812 Use random sampling to draw inferences about a population. Aktivitäten
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7.SP.1 7.SP.1 Use statistics to gain information about a population by examining a sample of the population; Aktivitäten
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7.SP.1.a 7.SP.1.a Know that generalizations about a population from a sample are valid only if the sample is representative of that population and generate a valid representative sample of a population. Aktivitäten
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7.SP.1.b 7.SP.1.b Identify if a particular random sample would be representative of a population and justify your reasoning. Aktivitäten
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7.SP.2 7.SP.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 informally gauge the variation in estimates or predictions. Aktivitäten
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Draw informal comparative inferences about two populations. S2896817 Draw informal comparative inferences about two populations. Aktivitäten
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7.SP.3 7.SP.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 (requires introduction of mean absolute deviation). Aktivitäten
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7.SP.4 7.SP.4 Use measures of center (mean, median and/or mode) and measures of variability (range, interquartile range and/or mean absolute deviation) for numerical data from random samples to draw informal comparative inferences about two populations. Aktivitäten
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Investigate chance processes and develop, use, and evaluate probability models. S2896820 Investigate chance processes and develop, use, and evaluate probability models. Aktivitäten
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7.SP.5 7.SP.5 Express the probability of a chance event as a number between 0 and 1 that represents the likelihood of the event occurring. (Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.) Aktivitäten
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7.SP.6 7.SP.6 Collect data from a chance process (probability experiment). Approximate the probability by observing its long-run relative frequency. Recognize that as the number of trials increase, the experimental probability approaches the theoretical probability. Conversely, predict the approximate relative frequency given the probability. Aktivitäten
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7.SP.7 7.SP.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. Aktivitäten
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7.SP.7.a 7.SP.7.a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. Aktivitäten
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7.SP.7.b 7.SP.7.b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. Aktivitäten
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7.SP.8 7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. Aktivitäten
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7.SP.8.a 7.SP.8.a Know 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. Aktivitäten
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7.SP.8.b 7.SP.8.b 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. Aktivitäten
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7.SP.8.c 7.SP.8.c Design and use a simulation to generate frequencies for compound events. Aktivitäten
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Investigate patterns of association in bivariate data. S2896830 Investigate patterns of association in bivariate data. Aktivitäten
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8.SP.1 8.SP.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. Aktivitäten
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8.SP.2 8.SP.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. Aktivitäten
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8.SP.3 8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Aktivitäten
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Functions S2896834 Functions Aktivitäten
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Define, evaluate, and compare functions. S2896835 Define, evaluate, and compare functions. Aktivitäten
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8.F.1 8.F.1 Explain 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. Aktivitäten
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8.F.2 8.F.2 Compare properties of two linear functions represented in a variety of ways (algebraically, graphically, numerically in tables, or by verbal descriptions). Aktivitäten
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8.F.3 8.F.3 Interpret the equation y=mx+b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Aktivitäten
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Use functions to model relationships between quantities. S2896839 Use functions to model relationships between quantities. Aktivitäten
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8.F.4 8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Aktivitäten
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8.F.5 8.F.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. Aktivitäten
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Number and Quantity S2896842 Number and Quantity Aktivitäten
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The Real Number System S2896843 The Real Number System Aktivitäten
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Use properties of rational numbers and irrational numbers. S2896844 Use properties of rational numbers and irrational numbers. Aktivitäten
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N.RN.1 N.RN.1 Know and apply the properties of integer exponents to generate equivalent numerical and algebraic expressions. Aktivitäten
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N.RN.2 N.RN.2 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. Aktivitäten
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N.RN.3 N.RN.3 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Aktivitäten
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Quantities S2896848 Quantities Aktivitäten
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Reason quantitatively and use units to solve problems. S2896849 Reason quantitatively and use units to solve problems. Aktivitäten
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N.Q.1 N.Q.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. Aktivitäten
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N.Q.2 N.Q.2 Define appropriate quantities for the purpose of descriptive modeling. Aktivitäten
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N.Q.3 N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Aktivitäten
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The Complex Number System S2896853 The Complex Number System Aktivitäten
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Perform arithmetic operations with complex numbers. S2896854 Perform arithmetic operations with complex numbers. Aktivitäten
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N.CN.1 N.CN.1 Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real. Aktivitäten
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N.CN.2 N.CN.2 Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Aktivitäten
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N.CN.3 N.CN.3 Find the conjugate of a complex number. Aktivitäten
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N.CN.4 N.CN.4 Use conjugates to find moduli and quotients of complex numbers. Aktivitäten
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Represent complex numbers and their operations on the complex plane. S2896859 Represent complex numbers and their operations on the complex plane. Aktivitäten
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N.CN.5 N.CN.5 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. Aktivitäten
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N.CN.6 N.CN.6 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. Aktivitäten
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N.CN.7 N.CN.7 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. Aktivitäten
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Use complex numbers in polynomial identities and equations. S2896863 Use complex numbers in polynomial identities and equations. Aktivitäten
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N.CN.10 N.CN.10 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Aktivitäten
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N.CN.8 N.CN.8 Solve quadratic equations with real coefficients that have complex solutions. Aktivitäten
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N.CN.9 N.CN.9 Extend polynomial identities to the complex numbers. Aktivitäten
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Vector and Matrix Quantities S2896867 Vector and Matrix Quantities Aktivitäten
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Represent and model with vector quantities. S2896868 Represent and model with vector quantities. Aktivitäten
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N.VM.1 N.VM.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v). Aktivitäten
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N.VM.2 N.VM.2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. Aktivitäten
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N.VM.3 N.VM.3 Solve problems involving velocity and other quantities that can be represented by vectors. Aktivitäten
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Perform operations on vectors. S2896872 Perform operations on vectors. Aktivitäten
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N.VM.4 N.VM.4 Add and subtract vectors. Aktivitäten
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N.VM.4.a N.VM.4.a Add vectors end-to-end, component-wise, and by the parallelogram rule . Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. Aktivitäten
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N.VM.4.b N.VM.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. Aktivitäten
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N.VM.4.c N.VM.4.c Understand vector subtraction v - w as v + (-w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise. Aktivitäten
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N.VM.5 N.VM.5 Multiply a vector by a scalar. Aktivitäten
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N.VM.5.a N.VM.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, (e.g. as c(vx, vy) = (cvx, cvy).) Aktivitäten
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N.VM.5.b N.VM.5.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). Aktivitäten
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Perform operations on matrices and use matrices in applications. S2896880 Perform operations on matrices and use matrices in applications. Aktivitäten
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N.VM.10 N.VM.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. Aktivitäten
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N.VM.11 N.VM.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. Aktivitäten
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N.VM.6 N.VM.6 Use matrices to represent and manipulate data, (e.g. to represent payoffs or incidence relationships in a network.) Aktivitäten
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N.VM.7 N.VM.7 Multiply matrices by scalars to produce new matrices, (e.g. as when all of the payoffs in a game are doubled.) Aktivitäten
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N.VM.8 N.VM.8 Add, subtract, and multiply matrices of appropriate dimensions; find determinants of 2 × 2 matrices. Aktivitäten
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N.VM.9 N.VM.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. Aktivitäten
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Algebra S2896887 Algebra Aktivitäten
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Seeing Structure in Expressions S2896888 Seeing Structure in Expressions Aktivitäten
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Interpret the structure of expressions. S2896889 Interpret the structure of expressions. Aktivitäten
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A.SSE.1 A.SSE.1 Interpret expressions that represent a quantity in terms of its context. Aktivitäten
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A.SSE.1.a A.SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients. Aktivitäten
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A.SSE.1.b A.SSE.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity. Aktivitäten
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A.SSE.2 A.SSE.2 Use the structure of an expression to identify ways to rewrite it. Aktivitäten
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Write expressions in equivalent forms to solve problems. S2896894 Write expressions in equivalent forms to solve problems. Aktivitäten
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A.SS.3 A.SS.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Aktivitäten
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A.SS.3.a A.SS.3.a Factor a quadratic expression to reveal the zeros of the function it defines. Aktivitäten
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A.SS.3.b A.SS.3.b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Aktivitäten
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A.SS.3.c A.SS.3.c Use the properties of exponents to transform expressions for exponential functions. Aktivitäten
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Arithmetic with Polynomials and Rational Expressions S2896899 Arithmetic with Polynomials and Rational Expressions Aktivitäten
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Perform arithmetic operations on polynomials. S2896900 Perform arithmetic operations on polynomials. Aktivitäten
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A.APR.1 A.APR.1 Add, subtract, and multiply polynomials. Aktivitäten
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A.APR.2 A.APR.2 Factor higher degree polynomials; identifying that some polynomials are prime. Aktivitäten
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A.APR.3 A.APR.3 Know and apply the Remainder Theorem: For a polynomial p(x) and a number c, the remainder on division by (x - c) is p(c), so p(c) = 0 if and only if (x - c) is a factor of p(x). Aktivitäten
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Use polynomial identities to solve problems. S2896904 Use polynomial identities to solve problems. Aktivitäten
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A.APR.4 A.APR.4 Generate polynomial identities from a pattern. Aktivitäten
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A.APR.5 A.APR.5 Know and apply the Binomial Theorem for the expansion of (x + y)² in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle. The Binomial Theorem can be proven by mathematical induction or by a combinatorial argument. Aktivitäten
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S2896907 S2896907 Rewrite rational expressions. Aktivitäten
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A.APR.6 A.APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. Aktivitäten
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A.APR.7 A.APR.7 Add, subtract, multiply, and divide rational expressions. Aktivitäten
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Creating Equations S2896910 Creating Equations Aktivitäten
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Create equations that describe numbers or relationships. S2896911 Create equations that describe numbers or relationships. Aktivitäten
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A.CED.1 A.CED.1 Apply and extend previous understanding to create equations and inequalities in one variable and use them to solve problems. Aktivitäten
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A.CED.2 A.CED.2 Apply and extend previous understanding to create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Aktivitäten
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A.CED.3 A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Aktivitäten
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A.CED.4 A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Aktivitäten
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Reasoning with Equations and Inequalities S2896916 Reasoning with Equations and Inequalities Aktivitäten
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S2896917 S2896917 Understand solving equations as a process of reasoning and explain the reasoning. Aktivitäten
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A.REI.1 A.REI.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. Aktivitäten
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Solve equations and inequalities in one variable. S2896919 Solve equations and inequalities in one variable. Aktivitäten
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A.REI.2 A.REI.2 Apply and extend previous understanding to solve equations, inequalities, and compound inequalities in one variable, including literal equations and inequalities. (A.REI.3) Aktivitäten
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A.REI.3 A.REI.3 Solve equations in one variable and give examples showing how extraneous solutions may arise. Aktivitäten
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A.REI.3.a A.REI.3.a Solve rational, absolute value and square root equations. Limited to simple equations such as, 2√x-3 + 8 = 16, x + 3/2x - 1 = 5, x ≠ ½. Aktivitäten
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A.REI.3.b A.REI.3.b Solve exponential and logarithmic equations. Aktivitäten
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A.REI.4 A.REI.4 Solve radical and rational exponent equations and inequalities in one variable, and give examples showing how extraneous solutions may arise. (A.REI.2) Aktivitäten
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A.REI.5 A.REI.5 Solve quadratic equations and inequalities Aktivitäten
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A.REI.5.a A.REI.5.a Solve quadratic equations by inspection (e.g. for x² = 49), taking square roots, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives no real solutions. Aktivitäten
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A.REI.5.b A.REI.5.b Solve quadratic equations with complex solutions written in the form a ± bi for real numbers a and b. Aktivitäten
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A.REI.5.c A.REI.5.c Use the method of completing the square to transform and solve any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Aktivitäten
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A.REI.5.d A.REI.5.d Solve quadratic inequalities and identify the domain. Aktivitäten
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Solve systems of equations. S2896930 Solve systems of equations. Aktivitäten
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A.REI.6 A.REI.6 Analyze and solve pairs of simultaneous linear equations. Aktivitäten
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A.REI.6.a A.REI.6.a 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. Aktivitäten
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A.REI.6.b A.REI.6.b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Aktivitäten
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A.REI.6.c A.REI.6.c Solve real-world and mathematical problems leading to two linear equations in two variables. Aktivitäten
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A.REI.7 A.REI.7 Represent a system of linear equations as a single matrix equation and solve (incorporating technology) for matrices of dimension 3 × 3 or greater. Aktivitäten
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Represent and solve equations and inequalities graphically. S2896936 Represent and solve equations and inequalities graphically. Aktivitäten
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A.REI.10 A.REI.10 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. Aktivitäten
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A.REI.8 A.REI.8 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). Aktivitäten
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A.REI.9 A.REI.9 Solve an equation f(x) = g(x) by graphing y = f(x) and y = g(x) and finding the x-value of the intersection point. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Aktivitäten
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Functions S2896940 Functions Aktivitäten
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Interpreting Functions S2896941 Interpreting Functions Aktivitäten
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Understand the concept of a function and use function notation. S2896942 Understand the concept of a function and use function notation. Aktivitäten
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F.IF.1 F.IF.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 f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Aktivitäten
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F.IF.2 F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Aktivitäten
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F.IF.3 F.IF.3 Recognize patterns in order to write functions whose domain is a subset of the integers. Limited to linear and quadratic. Aktivitäten
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Interpret functions that arise in applications in terms of the context. S2896946 Interpret functions that arise in applications in terms of the context. Aktivitäten
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F.IF.4 F.IF.4 For a function that models a relationship between two quantities, interpret key features of expressions, graphs and tables in terms of the quantities, and sketch graphs showing key features given a description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Aktivitäten
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F.IF.5 F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Aktivitäten
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F.IF.6 F.IF.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. Aktivitäten
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Analyze functions using different representations. S2896950 Analyze functions using different representations. Aktivitäten
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F.IF.7 F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Aktivitäten
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F.IF.7.a F.IF.7.a Graph linear, quadratic and absolute value functions and show intercepts, maxima, minima and end behavior. Aktivitäten
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F.IF.7.b F.IF.7.b Graph square root, cube root, and exponential functions. Aktivitäten
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F.IF.7.c F.IF.7.c Graph logarithmic functions, emphasizing the inverse relationship with exponentials and showing intercepts and end behavior. Aktivitäten
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F.IF.7.d F.IF.7.d Graph piecewise-defined functions, including step functions. Aktivitäten
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F.IF.7.e F.IF.7.e Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Aktivitäten
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F.IF.7.f F.IF.7.f Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Aktivitäten
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F.IF.7.g F.IF.7.g Graph trigonometric functions, showing period, midline, and amplitude. Aktivitäten
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F.IF.8 F.IF.8 Write a function in different but equivalent forms to reveal and explain different properties of the function. Aktivitäten
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F.IF.8.a F.IF.8.a Use different forms of linear functions, such as slope-intercept, standard, and point-slope form to show rate of change and intercepts. Aktivitäten
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F.IF.8.b F.IF.8.b 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. Aktivitäten
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F.IF.8.c F.IF.8.c Use the properties of exponents to interpret expressions for exponential functions. Aktivitäten
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F.IF.9 F.IF.9 Compare properties of two functions using a variety of representations (algebraically, graphically, numerically in tables, or by verbal descriptions). Aktivitäten
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Building Functions S2896964 Building Functions Aktivitäten
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F.BF.5 F.BF.5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Aktivitäten
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Build a function that models a relationship between two quantities. S2896965 Build a function that models a relationship between two quantities. Aktivitäten
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F.BF.1 F.BF.1 Use functions to model real-world relationships. Aktivitäten
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F.BF.1.a F.BF.1.a Combine multiple functions to model complex relationships. Aktivitäten
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F.BF.1.b F.BF.1.b Determine an explicit expression , a recursive function, or steps for calculation from a context. Aktivitäten
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F.BF.1.c F.BF.1.c Compose functions. Aktivitäten
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F.BF.2 F.BF.2 Write arithmetic and geometric sequences and series both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Aktivitäten
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Build new functions from existing functions. S2896971 Build new functions from existing functions. Aktivitäten
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F.BF.3 F.BF.3 Transform parent functions (f(x)) by replacing f(x) with f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. 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. Aktivitäten
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F.BF.4 F.BF.4 Find inverse functions. Aktivitäten
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F.BF.4.a F.BF.4.a Write an expression for the inverse of a function. Aktivitäten
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F.BF.4.b F.BF.4.b Read values of an inverse function from a graph or a table, given that the function has an inverse. Aktivitäten
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F.BF.4.c F.BF.4.c Verify by composition that one function is the inverse of another. Aktivitäten
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F.BF.4.d F.BF.4.d Produce an invertible function from a non-invertible function by restricting the domain. Aktivitäten
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Linear, Quadratic, and Exponential Models S2896979 Linear, Quadratic, and Exponential Models Aktivitäten
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S2896980 S2896980 Construct and compare linear, quadratic, and exponential models and solve problems. Aktivitäten
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F.LQE.1 F.LQE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. Aktivitäten
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F.LQE.1.a F.LQE.1.a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Aktivitäten
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F.LQE.1.b F.LQE.1.b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Aktivitäten
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F.LQE.1.c F.LQE.1.c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. Aktivitäten
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F.LQE.2 F.LQE.2 Construct exponential functions, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Aktivitäten
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Trigonometric Functions S2896986 Trigonometric Functions Aktivitäten
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Extend the domain of trigonometric functions using the unit circle. S2896987 Extend the domain of trigonometric functions using the unit circle. Aktivitäten
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F.TF.1 F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. Aktivitäten
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F.TF.2 F.TF.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. Aktivitäten
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F.TF.3 F.TF.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4, and π/6, use the unit circle to express the values of sine, cosine, and tangent for π - x, π + x, and 2π - x in terms of their values for x, where x is any real number. Aktivitäten
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F.TF.4 F.TF.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Aktivitäten
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Model periodic phenomena with trigonometric functions. S2896992 Model periodic phenomena with trigonometric functions. Aktivitäten
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F.TF.5 F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Aktivitäten
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F.TF.6 F.TF.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. Aktivitäten
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F.TF.7 F.TF.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. Aktivitäten
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Prove and apply trigonometric identities. S2896996 Prove and apply trigonometric identities. Aktivitäten
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F.TF.8 F.TF.8 Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin (θ), cos(θ), or tan(θ) and the quadrant. Aktivitäten
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F.TF.9 F.TF.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Aktivitäten
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Geometry S2896999 Geometry Aktivitäten
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Congruence S2897000 Congruence Aktivitäten
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Experiment with transformations in the plane. S2897001 Experiment with transformations in the plane. Aktivitäten
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G.CO.1 G.CO.1 Verify experimentally (for example, using patty paper or geometry software) the properties of rotations, reflections, translations, and symmetry: Aktivitäten
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G.CO.1.a G.CO.1.a Lines are taken to lines, and line segments to line segments of the same length. Aktivitäten
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G.CO.1.b G.CO.1.b Angles are taken to angles of the same measure. Aktivitäten
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G.CO.1.c G.CO.1.c Parallel lines are taken to parallel lines. Aktivitäten
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G.CO.1.d G.CO.1.d Identify any line and/or rotational symmetry within a figure. Aktivitäten
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G.CO.2 G.CO.2 Recognize transformations as functions that take points in the plane as inputs and give other points as outputs and describe the effect of translations, rotations, and reflections on two-dimensional figures. Aktivitäten
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Understand congruence in terms of rigid motions. S2897008 Understand congruence in terms of rigid motions. Aktivitäten
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G.CO.3 G.CO.3 Given two congruent figures, describe a sequence of rigid motions that exhibits the congruence (isometry) between them using coordinates and the non-coordinate plane. Aktivitäten
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G.CO.4 G.CO.4 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. Aktivitäten
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G.CO.5 G.CO.5 Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Aktivitäten
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G.CO.6 G.CO.6 Demonstrate triangle congruence using rigid motion (ASA, SAS, and SSS). Aktivitäten
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S2897013 S2897013 Construct arguments about geometric theorems using rigid transformations and/or logic. Aktivitäten
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G.CO.10 G.CO.10 Construct arguments about parallelograms using theorems. 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. Aktivitäten
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G.CO.7 G.CO.7 Construct arguments about lines and angles using theorems. 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. Aktivitäten
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G.CO.8 G.CO.8 Construct arguments about the relationships within one triangle using theorems. Theorems include: measures of interior angles of a triangle sum to 180°; 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; angle sum and exterior angle of triangles. Aktivitäten
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G.CO.9 G.CO.9 Construct arguments about the relationships between two triangles using theorems. Theorems include: SSS, SAS, ASA, AAS, and HL. Aktivitäten
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Make geometric constructions. S2897018 Make geometric constructions. Aktivitäten
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G.CO.11 G.CO.11 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Aktivitäten
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G.CO.12 G.CO.12 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Aktivitäten
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Similarity, Right Triangles, and Trigonometry S2897021 Similarity, Right Triangles, and Trigonometry Aktivitäten
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Understand similarity in terms of similarity transformations. S2897022 Understand similarity in terms of similarity transformations. Aktivitäten
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G.SRT.1 G.SRT.1 Use geometric constructions to verify the properties of dilations given by a center and a scale factor: Aktivitäten
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G.SRT.1.a G.SRT.1.a 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. Aktivitäten
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G.SRT.1.b G.SRT.1.b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Aktivitäten
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G.SRT.2 G.SRT.2 Recognize transformations as functions that take points in the plane as inputs and give other points as outputs and describe the effect of dilations on two-dimensional figures. Aktivitäten
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G.SRT.3 G.SRT.3 Given two similar figures, describe a sequence of transformations that exhibits the similarity between them using coordinates and the non-coordinate plane. Aktivitäten
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G.SRT.4 G.SRT.4 Understand the meaning of similarity for two-dimensional figures as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Aktivitäten
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Construct arguments about theorems involving similarity. S2897029 Construct arguments about theorems involving similarity. Aktivitäten
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G.SRT.5 G.SRT.5 Construct arguments about triangles using theorems. 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, and AA. Aktivitäten
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G.SRT.6 G.SRT.6 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Aktivitäten
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Define trigonometric ratios and solve problems involving right triangles. S2897032 Define trigonometric ratios and solve problems involving right triangles. Aktivitäten
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G.SRT.7 G.SRT.7 Show 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. Aktivitäten
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G.SRT.8 G.SRT.8 Explain and use the relationship between the sine and cosine of complementary angles. Aktivitäten
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G.SRT.9 G.SRT.9 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Aktivitäten
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Apply trigonometry to general triangles S2897036 Apply trigonometry to general triangles Aktivitäten
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G.SRT.10 G.SRT.10 Derive the formula A = ½ ab sin C for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Aktivitäten
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G.SRT.11 G.SRT.11 Prove the Laws of Sines and Cosines and use them to solve problems. Aktivitäten
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G.SRT.12 G.SRT.12 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). Aktivitäten
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Circles S2897040 Circles Aktivitäten
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Understand and apply theorems about circles. S2897041 Understand and apply theorems about circles. Aktivitäten
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G.C.1 G.C.1 Construct arguments that all circles are similar. Aktivitäten
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G.C.2 G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. Aktivitäten
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G.C.3 G.C.3 Construct arguments using properties of polygons inscribed and circumscribed about circles. Aktivitäten
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G.C.4 G.C.4 Construct inscribed and circumscribed circles for triangles. Aktivitäten
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G.C.5 G.C.5 Construct inscribed and circumscribed circles for polygons and tangent lines from a point outside a given circle to the circle. Aktivitäten
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Find arc lengths and areas of sectors of circles. S2897047 Find arc lengths and areas of sectors of circles. Aktivitäten
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G.C.6 G.C.6 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. Aktivitäten
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Expressing Geometric Properties with Equations S2897049 Expressing Geometric Properties with Equations Aktivitäten
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S2897050 S2897050 Translate between the geometric description and the equation for a conic section. Aktivitäten
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G.GPE.1 G.GPE.1 Write the equation of a circle given the center and radius or a graph of the circle; use the center and radius to graph the circle in the coordinate plane. Aktivitäten
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G.GPE.2 G.GPE.2 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; graph the circle in the coordinate plane; Aktivitäten
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G.GPE.3 G.GPE.3 Complete the square to find the center and radius of a circle given by an equation. Aktivitäten
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G.GPE.4 G.GPE.4 Derive the equation of a parabola given a focus and directrix; graph the parabola in the coordinate plane. Aktivitäten
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G.GPE.5 G.GPE.5 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; graph the ellipse or hyperbola in the coordinate plane. Aktivitäten
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Use coordinates to prove simple geometric theorems algebraically. S2897056 Use coordinates to prove simple geometric theorems algebraically. Aktivitäten
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G.GPE.6 G.GPE.6 Use coordinates to prove simple geometric theorems algebraically, including the use of slope, distance, and midpoint formulas For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle. Aktivitäten
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G.GPE.7 G.GPE.7 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). Aktivitäten
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G.GPE.8 G.GPE.8 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, including the use of the distance and midpoint formulas. Aktivitäten
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Geometric Measurement and Dimension S2897060 Geometric Measurement and Dimension Aktivitäten
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Explain volume formulas and use them to solve problems. S2897061 Explain volume formulas and use them to solve problems. Aktivitäten
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G.GMD.1 G.GMD.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. Use dissection arguments and informal limit arguments. Aktivitäten
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G.GMD.2 G.GMD.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a solid figure. Aktivitäten
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Modeling with Geometry S2897064 Modeling with Geometry Aktivitäten
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Apply geometric concepts in modeling situations. S2897065 Apply geometric concepts in modeling situations. Aktivitäten
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G.MG.1 G.MG.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). Aktivitäten
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G.MG.2 G.MG.2 Apply concepts of density and displacement based on area and volume in modeling situations (e.g. persons per square mile, BTUs per cubic foot). Aktivitäten
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G.MG.3 G.MG.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). Aktivitäten
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Statistics & Probability S2897069 Statistics & Probability Aktivitäten
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Interpreting Categorical and Quantitative Data S2897070 Interpreting Categorical and Quantitative Data Aktivitäten
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S2897071 S2897071 Summarize, represent, and interpret data on a single count or measurement variable. Aktivitäten
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S.ID.1 S.ID.1 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. Aktivitäten
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S.ID.2 S.ID.2 Interpret differences in shape, center, and spread in the context of the data sets using dot plots, histograms, and box plots, accounting for possible effects of extreme data points (outliers). Aktivitäten
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S.ID.3 S.ID.3 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. Aktivitäten
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S2897075 S2897075 Summarize, represent, and interpret data on two categorical and quantitative variables. Aktivitäten
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S.ID.4 S.ID.4 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. Aktivitäten
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S.ID.5 S.ID.5 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Aktivitäten
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S.ID.5.a S.ID.5.a Use a given linear function to solve problems in the context of data. Aktivitäten
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S.ID.5.b S.ID.5.b Fit a linear function to data and use it to solve problems in the context of the data. Aktivitäten
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S.ID.5.c S.ID.5.c Assess the fit of a function by plotting and analyzing residuals. Aktivitäten
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S.ID.5.d S.ID.5.d Fit quadratic and exponential functions to the data. Use functions fitted to data to solve problems in the context of the data. Aktivitäten
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Interpret linear models. S2897082 Interpret linear models. Aktivitäten
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S.ID.6 S.ID.6 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Aktivitäten
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S.ID.7 S.ID.7 Compute (using technology) and interpret the correlation coefficient of a linear fit. Aktivitäten
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S.ID.8 S.ID.8 Distinguish between correlation and causation. Aktivitäten
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Making Inferences and Justifying Conclusions S2897086 Making Inferences and Justifying Conclusions Aktivitäten
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Understand and evaluate random processes underlying statistical experiments. S2897087 Understand and evaluate random processes underlying statistical experiments. Aktivitäten
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S.IC.1 S.IC.1 Understand statistics as a process for making inferences to be made about population parameters based on a random sample from that population. Aktivitäten
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S.IC.2 S.IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g. using simulation. Aktivitäten
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S2897090 S2897090 Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Aktivitäten
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S.IC.3 S.IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. Aktivitäten
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S.IC.4 S.IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error, (e.g. through the use of simulation models for random sampling.) Aktivitäten
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S.IC.5 S.IC.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. Aktivitäten
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S.IC.6 S.IC.6 Evaluate reports based on data. Aktivitäten
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Conditional Probability and the Rules of Probability S2897095 Conditional Probability and the Rules of Probability Aktivitäten
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S2897096 S2897096 Understand independent and conditional probability and use them to interpret data. Aktivitäten
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S.CP.1 S.CP.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"). Aktivitäten
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S.CP.2 S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. Aktivitäten
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S.CP.3 S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B) , and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Aktivitäten
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S.CP.4 S.CP.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. Aktivitäten
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S.CP.5 S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. Aktivitäten
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S2897102 S2897102 Use the rules of probability to compute probabilities of compound events in a uniform probability model. Aktivitäten
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S.CP.6 S.CP.6 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. Aktivitäten
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S.CP.7 S.CP.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. Aktivitäten
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S.CP.8 S.CP.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. Aktivitäten
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S.CP.9 S.CP.9 Use permutations and combinations to compute probabilities of compound events and solve problems. Aktivitäten
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Using Probability to Make Decisions S2897107 Using Probability to Make Decisions Aktivitäten
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Calculate expected values and use them to solve problems. S2897108 Calculate expected values and use them to solve problems. Aktivitäten
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S.MD.1 S.MD.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. Aktivitäten
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S.MD.2 S.MD.2 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. Aktivitäten
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S.MD.3 S.MD.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. Aktivitäten
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S.MD.4 S.MD.4 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. Aktivitäten
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Use probability to evaluate outcomes of decisions. S2897113 Use probability to evaluate outcomes of decisions. Aktivitäten
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S.MD.5 S.MD.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. Aktivitäten
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S.MD.5.a S.MD.5.a Find the expected payoff for a game of chance. Aktivitäten
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S.MD.5.b S.MD.5.b Evaluate and compare strategies on the basis of expected values. Aktivitäten
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S.MD.6 S.MD.6 Use probabilities to make fair decisions (e.g. drawing by lots, using a random number generator). Aktivitäten
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S.MD.7 S.MD.7 Analyze decisions and strategies using probability concepts (e.g. product testing, medical testing, pulling a hockey goalie at the end of a game). Aktivitäten
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