Patterns, Relations, and Functions 2
From NCSC Wiki
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{{BACK TO|[[Core Content Connectors]]}} | {{BACK TO|[[Core Content Connectors]]}} | ||
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| + | |width = "833"|'''(K-4) Elementary School Learning Targets''' | ||
| − | + | |width = "833"|'''(5-8) Middle School Learning Targets''' | |
| − | |width = " | + | |width = "834"|'''(9-12) High School Learning Targets''' |
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| + | ||'''''E.PRF-2 '''Give examples, interpret, and analyze repeating and growing patterns and functions involving the four basic operations.'' | ||
| − | | | + | ||'''''M.PRF-2 '''Give examples, interpret, and analyze a variety of mathematical patterns, relations, and explicit and recursive functions.'' |
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| − | | | + | ||'''''H.PRF-2 '''Use trends and analyze a variety of mathematical patterns, relations, and explicit and recursive functions.'' |
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| + | |colspan= "5" align= "center"|'''Arithmetic Patterns''' | ||
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| − | |width = " | + | |width = "500"|'''Grades K-2''' |
| − | |width = " | + | |width = "500"|'''Grades 3-4''' |
| − | |width = " | + | |width = "500"|'''Grades 5-6''' |
| − | |width = " | + | |width = "500"|'''Grades 7-8''' |
| − | |width = " | + | |width = "500"|'''HS''' |
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| + | ||'''K.PRF.2a1 Describe or select the repeating pattern using objects or pictures (AB or ABC)''' | ||
| − | | | + | ||'''3.PRF.2d1 Identify multiplication patterns in a real world setting''' |
| − | | | + | ||'''5.PRF.2a1 Generate a pattern that follows the provided rule''' |
| − | | | + | ||'''8.PRF.2c1 Given two graphs, describe the function as linear and not linear''' |
| − | | | + | ||'''H.PRF. 2c1 Make predictions based on a given model (for example, a weather model, data for athletes over years)''' |
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| + | ||'''K.PRF.2a2 Extend a repeating pattern using objects or pictures (AB or ABC)''' | ||
| − | | | + | ||'''3.PRF.2d2 Apply properties of operations as strategies to multiply and divide''' |
| − | | | + | ||'''5.PRF.2b1 Generate or select a comparison between two graphs from a similar situation''' |
| − | | | + | ||'''8.PRF.2e1 Distinguish between functions and non-functions, using equations, graphs or tables''' |
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| + | ||'''K.PRF.2a3 Extend a repeating numerical AB pattern''' | ||
| − | | | + | ||'''4.PRF.2d3 Generate a pattern when given a rule and word problem (I run 3 miles every day, how many miles have I run in 3 days)''' |
| − | | | + | ||'''6.PRF.2b2 Using provided table with numerical patterns, form ordered pairs''' |
| − | | | + | ||'''8.PRF.2e2 Identify the rate of change (slope) and initial value (y-intercept) from graphs''' |
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| + | ||'''K.PRF.2b1 Create a repeating pattern using objects, pictures, or numbers''' | ||
| − | | | + | ||'''4.PRF.2e1 Extend a numerical pattern when the rule is provided ''' |
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| − | | | + | ||'''8.PRF.2e3 Given a verbal description of a situation, create or identify a graph to model the situation''' |
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| − | | | + | ||'''8.PRF.2e4 Given a graph of a situation, generate a description of the situation''' |
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| + | ||'''1.PRF.2a4 Use a number line to extend the numerical patterns that grow at a constant rate (2,4,6,8) ''' | ||
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| − | + | ||'''1.PRF.2b2 ''' | |
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'''Create a growing pattern using numbers or objects''' | '''Create a growing pattern using numbers or objects''' | ||
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| + | ||'''1.PRF.2c1 Identify the rule of a given arithmetic pattern ''' | ||
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| − | | | + | ||'''2.PRF.2a6 Use a number line to extend the numerical patterns that grow at a constant rate (2,4,6,8)''' |
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| + | ||'''2.PRF.2b3 Use a number line to extend arithmetic patterns that are decreasing ''' | ||
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| + | ||'''2.PRF.2c2 Identify the rule of arithmetic patterns that are increasing ''' | ||
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| + | ||'''2.PRF.2c3 Identify the rule of arithmetic patterns that are decreasing ''' | ||
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| + | | colspan= "5" align= "center"|'''Expressions and Equations''' | ||
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| − | |width = " | + | |width = "500"|'''Grades K-2''' |
| − | |width = " | + | |width = "500"|'''Grades 3-4''' |
| − | |width = " | + | |width = "500"|'''Grades 5-6''' |
| − | |width = " | + | |width = "500"|'''Grades 7-8''' |
| − | |width = " | + | |width = "500"|'''HS''' |
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| − | | | + | ||'''6.PRF.2a2 Use variables to represent numbers and write expressions when solving real-world problems''' |
| − | | | + | ||'''7.PRF.2a5 Use variables to represent two quantities in a real-world problem that change in relationship to one another''' |
| − | | | + | ||'''H.PRF.2a1 Translate an algebraic expression into a word problem''' |
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| − | | | + | ||'''6.PRF.2a3''' |
| + | '''Use variables to represent two quantities in a real-world problem that change in relationship to one another''' | ||
| − | | | + | ||'''7.PRF.2d1 ''' |
| + | '''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''' | ||
| − | | | + | ||'''H.PRF.2b1 Translate a real-world problem into a one variable equation''' |
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| − | | | + | ||'''6.PRF.2a4''' |
| + | '''Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation''' | ||
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| − | | | + | ||'''H.PRF.2b2 ''' |
| + | '''Solve equations with one or two variables using equations or graphs''' | ||
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| − | | | + | ||'''6.PRF.2b3''' |
| + | '''Complete a statement that describes the ratio relationship between two quantities ''' | ||
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| − | | | + | ||'''6.PRF.2b4 ''' |
| + | '''Determine the unit rate in a variety of contextual situations''' | ||
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| − | | | + | ||'''6.PRF.2b5 Use ratios and reasoning to solve real-world mathematical problems (e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations)''' |
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=Grade Differentiation= | =Grade Differentiation= | ||
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==Elementary School Progress Indicators== | ==Elementary School Progress Indicators== | ||
| − | {| border=1 | + | {|border=1 |
| − | | | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: E.PRF.2a recognizing, describing, and extending simple repeating (ABAB) and growing (A+1, A+2, A+3) patterns (e.g., colors, sounds, words, shapes, numeric – counting, odd, even) ''' |
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| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''Core Content Connectors: K''' |
| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | |width = "834" style="background-color: | + | |width = "834" style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
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| − | | | + | ||'''K.PRF.2a1 Describe or select the repeating pattern using objects or pictures (AB or ABC)''' |
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| − | | | + | ||No CCSS linked |
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| − | | | + | ||'''K.PRF.2a2 Extend a repeating pattern using objects or pictures (AB or ABC)''' |
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| − | | | + | ||No CCSS linked |
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| − | | | + | ||'''K.PRF.2a3 Extend a repeating numerical AB pattern''' |
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| − | | | + | ||No CCSS linked |
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| − | | | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: E.PRF.2b creating and explaining repeating and growing patterns using objects or numbers ''' |
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| − | | | + | | style="background-color:#D9D9D9;"|'''Core Content Connectors: K''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
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| − | | | + | ||'''K.PRF.2b1 Create a repeating pattern using objects, pictures, or numbers''' |
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| − | | | + | ||No CCSS linked |
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| − | + | {|border=1 | |
| − | {| border=1 | + | |width = "33%" style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: E.PRF.2a recognizing, describing, and extending simple repeating (ABAB) and growing (A+1, A+2, A+3) patterns (e.g., colors, sounds, words, shapes, numeric – counting, odd, even)''' |
| − | |width = " | + | |
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| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''Core Content Connectors: 1''' |
| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | |width = "834" style="background-color: | + | |width = "834" style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
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| − | | | + | ||'''1.PRF.2a4 Use a number line to extend the numerical patterns that grow at a constant rate (2,4,6,8) ''' |
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| − | | | + | ||No CCSS linked |
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| − | | | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: E.PRF.2b creating and explaining repeating and growing patterns using objects or numbers ''' |
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| − | | | + | | style="background-color:#D9D9D9;"|'''Core Content Connectors: 1''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
|- | |- | ||
| − | | | + | ||'''1.PRF.2b2 Create a growing pattern using numbers or objects''' |
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| − | | | + | ||No CCSS linked |
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| − | | | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: E.PRF.2c extending and analyzing simple numeric patterns with rules that involve addition and subtraction ''' |
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| − | | | + | | style="background-color:#D9D9D9;"|'''Core Content Connectors: 1''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
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| − | | | + | ||'''1.PRF.2c1 Identify the rule of a given arithmetic pattern''' |
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| − | | | + | ||No CCSS linked |
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| − | + | {|border=1 | |
| − | {| border=1 | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: E.PRF.2a recognizing, describing, and extending simple repeating (ABAB) and growing (A+1, A+2, A+3) patterns (e.g., colors, sounds, words, shapes, numeric – counting, odd, even) ''' |
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| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''Core Content Connectors: 2''' |
| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | |width = "834" style="background-color: | + | |width = "834" style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
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| − | | | + | ||'''2.PRF.2a6 Use a number line to extend the numerical patterns that grow at a constant rate (2,4,6,8)''' |
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| − | | | + | ||No CCSS linked |
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| − | | | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: E.PRF.2b creating and explaining repeating and growing patterns using objects or numbers ''' |
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| − | | | + | | style="background-color:#D9D9D9;"|'''Core Content Connectors: 2''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
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| − | | | + | ||'''2.PRF.2b3 Use a number line to extend arithmetic patterns that are decreasing''' |
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| − | | | + | ||No CCSS linked |
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| − | | | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: E.PRF.2c extending and analyzing simple numeric patterns with rules that involve addition and subtraction ''' |
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| − | | | + | | style="background-color:#D9D9D9;"|'''Core Content Connectors: 2''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
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| − | | | + | ||'''2.PRF.2c2 Identify the rule of arithmetic patterns that are increasing ''' |
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| − | | | + | ||No CCSS linked |
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| − | | | + | ||'''2.PRF.2c3 Identify the rule of arithmetic patterns that are decreasing ''' |
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| − | | | + | ||No CCSS linked |
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| − | {| border=1 | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: E.PRF.2d representing and analyzing patterns and rules (e.g., doubling, adding 3) using words, tables, graphs, and models ''' |
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| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''Core Content Connectors: 3''' |
| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | |width = "834" style="background-color: | + | |width = "834" style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
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| − | | | + | ||'''3.PRF.2d1 Identify multiplication patterns in a real world setting''' |
| − | | | + | ||'''Operations and Algebraic Thinking''' |
3 OA Solve problems involving the four operations, and identify and explain patterns in arithmetic. | 3 OA Solve problems involving the four operations, and identify and explain patterns in arithmetic. | ||
| − | | | + | ||3.OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. ''For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.'' |
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| − | | | + | ||'''3.PRF.2d2 Apply properties of operations as strategies to multiply and divide''' |
| − | | | + | ||'''Operations and Algebraic Thinking''' |
3 OA Understand properties of multiplication and the relationship between multiplication and division. | 3 OA Understand properties of multiplication and the relationship between multiplication and division. | ||
| − | | | + | ||3.OA.5 Apply properties of operations as strategies to multiply and divide. ''Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)'' |
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| − | + | {|border=1 | |
| − | {| border=1 | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: E.PRF.2d representing and analyzing patterns and rules (e.g., doubling, adding 3) using words, tables, graphs, and models ''' |
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| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''Core Content Connectors: 4''' |
| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | |width = "834" style="background-color: | + | |width = "834" style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
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| − | | | + | ||'''4.PRF.2d3 Generate a pattern when given a rule and word problem (I run 3 miles every day, how many miles have I run in 3 days)''' |
| − | | | + | ||'''Operations and Algebraic Thinking''' |
4 OA Generate and analyze patterns. | 4 OA Generate and analyze patterns. | ||
| − | | | + | ||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.'' |
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| − | | | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: E.PRF.2e extending, translating, and analyzing numeric patterns and their rules using addition, subtraction, multiplication, and division ''' |
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| − | | | + | | style="background-color:#D9D9D9;"|'''Core Content Connectors: 4''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
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| − | | | + | ||'''4.PRF.2e1 Extend a numerical pattern when the rule is provided''' |
| − | | | + | ||'''Operations and Algebraic Thinking''' |
4 OA Generate and analyze patterns. | 4 OA Generate and analyze patterns. | ||
| − | | | + | ||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.'' |
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| − | {| border=1 | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: M.PRF.2a representing, analyzing, extending, and generalizing a variety of patterns using tables, graphs, words, and symbolic rules ''' |
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| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''Core Content Connectors: 5''' |
| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | |width = "834" style="background-color: | + | |width = "834" style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
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| − | | | + | ||'''5.PRF.2a1 Generate a pattern that follows the provided rule''' |
| − | | | + | ||'''Operations and Algebraic Thinking''' |
4 OA Generate and analyze patterns. | 4 OA Generate and analyze patterns. | ||
| − | | | + | ||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.'' |
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| − | | | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: M.PRF.2b relating and comparing different forms of representation and identifying functions as linear or nonlinear ''' |
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| − | | | + | | style="background-color:#D9D9D9;"|'''Core Content Connectors: 5''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
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| − | | | + | ||'''5.PRF.2b1 Generate or select a comparison between two graphs from a similar situation''' |
| − | | | + | ||'''Operations and Algebraic Thinking''' |
5 OA Analyze patterns and relationships. | 5 OA Analyze patterns and relationships. | ||
| − | | | + | ||5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. ''For example, given the rule "Add 3"'' ''and the starting number 0, and given the rule "Add 6" and the starting'' ''number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.'' |
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==Middle School Progress Indicators== | ==Middle School Progress Indicators== | ||
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| + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: M.PRF.2a representing, analyzing, extending, and generalizing a variety of patterns using tables, graphs, words, and symbolic rules ''' | ||
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| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''Core Content Connectors: 6''' |
| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | |width = "834" style="background-color: | + | |width = "834" style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
|- | |- | ||
| − | | | + | ||'''6.PRF.2a2 Use variables to represent numbers and write expressions when solving real-world problems''' |
| − | | | + | ||'''Expressions and Equations''' |
6 EE Reason about and solve one-variable equations and inequalities. | 6 EE Reason about and solve one-variable equations and inequalities. | ||
| − | | | + | ||6.EE.6 Use variables to represent numbers and write expressions when solving real-world or mathematical problems; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. |
|- | |- | ||
| − | | | + | ||'''6.PRF.2a3 Use variables to represent two quantities in a real-world problem that change in relationship to one another''' |
| − | | | + | ||'''Expressions and Equations''' |
6 EE Represent and analyze quantitative relationships between dependent and independent variables. | 6 EE Represent and analyze quantitative relationships between dependent and independent variables. | ||
| − | | | + | ||# 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. ''Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. '' |
|- | |- | ||
| − | | | + | ||'''6.PRF.2a4 Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation''' |
| − | | | + | ||'''Expressions and Equations''' |
6 EE Represent and analyze quantitative relationships between dependent and independent variables. | 6 EE Represent and analyze quantitative relationships between dependent and independent variables. | ||
| − | | | + | ||# 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. ''Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.'' |
|- | |- | ||
| − | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: M.PRF.2b relating and comparing different forms of representation and identifying functions as linear or nonlinear ''' | |
| − | | | + | |
|- | |- | ||
| − | | | + | | style="background-color:#D9D9D9;"|'''Core Content Connectors: 6''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
|- | |- | ||
| − | | | + | ||'''6.PRF.2b2 Using provided table with numerical patterns, form ordered pairs''' |
| − | | | + | ||'''Operations and Algebraic Thinking''' |
5 OA Analyze patterns and relationships. | 5 OA Analyze patterns and relationships. | ||
| − | | | + | ||5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. ''For example, given the rule "Add 3"'' ''and the starting number 0, and given the rule "Add 6" and the starting'' ''number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.'' |
|- | |- | ||
| − | | | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: M.PRF.2b relating and comparing different forms of representation and identifying functions as linear or nonlinear ''' |
|- | |- | ||
| − | | | + | | style="background-color:#D9D9D9;"|'''Core Content Connectors: 6''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
|- | |- | ||
| − | | | + | ||'''6.PRF.2b3 Complete a statement that describes the ratio relationship between two quantities ''' |
| − | | | + | ||'''Ratios and Proportional Relationships''' |
6 RP Understand ratio concepts and use ratio reasoning to solve problems. | 6 RP Understand ratio concepts and use ratio reasoning to solve problems. | ||
| − | | | + | ||6.RP.1''' '''Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. ''For example, "The ratio of wings to beaks in the'' ''bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every'' ''vote candidate A received, candidate C received nearly three votes." '' |
|- | |- | ||
| − | | | + | ||'''6.PRF.2b4 Determine the unit rate in a variety of contextual situations''' |
| − | | | + | ||'''Ratios and Proportional Relationships''' |
6 RP Understand ratio concepts and use ratio reasoning to solve problems. | 6 RP Understand ratio concepts and use ratio reasoning to solve problems. | ||
| − | | | + | ||6.RP.2 Understand the concept of a unit rate a/b associated with a ration a:b with b ≠ 0, and use rate language in the context of a ratio relationship. ''For example, "this recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is ¾ cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."'' |
|- | |- | ||
| − | | | + | ||'''6.PRF.2b5 Use ratios and reasoning to solve real-world mathematical problems (e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams,''' |
'''or equations)''' | '''or equations)''' | ||
| − | | | + | ||'''Ratios and Proportional Relationships''' |
6 RP Understand ratio concepts and use ratio reasoning to solve problems. | 6 RP Understand ratio concepts and use ratio reasoning to solve problems. | ||
| − | | | + | ||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 diagrams, or equations. |
| − | + | a) Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. | |
| − | + | b) Solve unit rate problems including those involving unit pricing and constant speed. ''For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?'' | |
| − | + | c) 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. | |
| − | + | d) Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. | |
|- | |- | ||
| Line 673: | Line 662: | ||
| − | {| border=1 | + | {|border=1 |
| − | | | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: M.PRF.2a representing, analyzing, extending, and generalizing a variety of patterns using tables, graphs, words, and symbolic rules ''' |
|- | |- | ||
| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''Core Content Connectors: 7''' |
| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | |width = "834" style="background-color: | + | |width = "834" style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
|- | |- | ||
| − | | | + | ||'''7.PRF.2a5 Use variables to represent two quantities in a real-world problem that change in relationship to one another''' |
| − | | | + | ||'''Expressions and Equations''' |
6 EE Represent and analyze quantitative relationships between dependent and independent variables. | 6 EE Represent and analyze quantitative relationships between dependent and independent variables. | ||
| − | | | + | ||# 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. ''Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. '' |
|- | |- | ||
| − | | | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: M.PRF.2d solving linear equations and formulating and explaining reasoning about expressions and equations ''' |
|- | |- | ||
| − | | | + | | style="background-color:#D9D9D9;"|'''Core Content Connectors: 7''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
|- | |- | ||
| − | | | + | ||'''7.PRF.2d1 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''' |
| − | | | + | ||'''Expressions and Equations''' |
7 EE Solve real-life and mathematical problems using numerical and algebraic expressions and equations. | 7 EE Solve real-life and mathematical problems using numerical and algebraic expressions and equations. | ||
| − | | | + | ||7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. |
| − | + | 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. Graph the solution set of the inequality and interpret it in the context of the problem. ''For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.'' | |
|- | |- | ||
| Line 716: | Line 705: | ||
| − | + | {|border=1 | |
| − | {| border=1 | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: M.PRF.2c relating and comparing different forms of representation and identifying functions as linear or nonlinear ''' |
| − | | | + | |
|- | |- | ||
| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''Core Content Connectors: 8''' |
| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | |width = "834" style="background-color: | + | |width = "834" style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
|- | |- | ||
| − | | | + | ||'''8.PRF.2c1 Given two graphs, describe the function as linear and not linear''' |
| − | | | + | ||'''Functions''' |
8 F Define, evaluate, and compare functions. | 8 F Define, evaluate, and compare functions. | ||
8 F Use functions to model relationships between quantities. | 8 F Use functions to model relationships between quantities. | ||
| − | | | + | ||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. ''For example, the function A = s<sup>2</sup> giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.'' |
| − | 8.F | + | 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. |
|- | |- | ||
| − | | | + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: M.PRF.2e using functions to describe quantitative relationships ''' |
|- | |- | ||
| − | | | + | | style="background-color:#D9D9D9;"|'''Core Content Connectors: 8''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | | | + | | style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
|- | |- | ||
| − | | | + | ||'''8.PRF.2e1 Distinguish between functions and non-functions, using equations, graphs or tables''' |
| − | | | + | ||'''Functions''' |
| + | 8 F Define, evaluate, and compare functions. | ||
| − | | | + | ||8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. |
|- | |- | ||
| − | | | + | ||'''8.PRF.2e2 Identify the rate of change (slope) and initial value (y-intercept) from graphs''' |
| − | | | + | ||'''Functions''' |
8 F Use functions to model relationships between quantities. | 8 F Use functions to model relationships between quantities. | ||
| − | | | + | ||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. |
|- | |- | ||
| − | | | + | ||'''8.PRF.2e3 Given a verbal description of a situation, create or identify a graph to model the situation''' |
| − | | | + | ||'''Functions''' |
8 F Use functions to model relationships between quantities. | 8 F Use functions to model relationships between quantities. | ||
| − | | | + | ||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. |
|- | |- | ||
| − | | | + | ||'''8.PRF.2e4 Given a graph of a situation, generate a description of the situation''' |
| − | | | + | ||'''Functions''' |
8 F Use functions to model relationships between quantities. | 8 F Use functions to model relationships between quantities. | ||
| − | | | + | ||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. |
| + | |||
| + | |- | ||
| + | ||'''8.PRF.2e5''' '''Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.''' | ||
| + | |||
| + | ||'''Functions''' | ||
| + | 8 F Define, evaluate, and compare functions. | ||
| + | |||
| + | |||
| + | ||8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. | ||
|- | |- | ||
|} | |} | ||
| + | |||
| + | |||
| + | |||
==High School Progress Indicators== | ==High School Progress Indicators== | ||
| − | |||
| − | |||
| + | {|border=1 | ||
| + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: H.PRF.2a interpreting and rewriting a variety of expressions or functions to solve problems''' | ||
|- | |- | ||
| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''Core Content Connectors: 9-12''' |
| − | |width = "833" style="background-color: | + | |width = "833" style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' |
| − | |width = " | + | |width = "833" style="background-color:#D9D9D9;"|'''Common Core State Standard''' |
|- | |- | ||
| − | | | + | ||'''H.PRF.2a1 Translate an algebraic expression into a word problem''' |
| − | | | + | ||'''Seeing Structure in Expressions''' |
A SSE Interpret the structure of expressions. | A SSE Interpret the structure of expressions. | ||
| − | | | + | ||A.SSE.1 Interpret expressions that represent a quantity in terms of its context. |
| − | + | ||
| − | + | a) Interpret parts of an expression, such as terms, factors, and coefficients. | |
| + | |||
| + | b) Interpret complicated expressions by viewing one or more of their parts as a single entity. ''For example, interpret P(1+r)n as the product of P and a factor not depending on P. '' | ||
|- | |- | ||
| − | | | + | ||'''H.PRF.2a2 Factor a quadratic expression.''' |
| + | |||
| + | ||'''Seeing Structure in Expressions''' | ||
| + | Write expressions in equivalent forms to solve problems | ||
| + | |||
| + | ||A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.\* | ||
| + | a) Factor a quadratic expression to reveal the zeros of the function it defines. | ||
|- | |- | ||
| − | | | + | ||'''H.PRF.2a3 Given a quadratic expression, explain the meaning of the zeros graphically. That is for an expression (x – a) (x – c), a and c correspond to the x-intercepts (if a and c are real).''' |
| − | | | + | ||'''Seeing Structure in Expressions''' |
| + | Write expressions in equivalent forms to solve problems | ||
| − | | | + | ||A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.\* |
| + | a) Factor a quadratic expression to reveal the zeros of the function it defines. | ||
|- | |- | ||
| − | | | + | ||'''H.PRF.2a4 Use the formula to solve real world problems such as calculating the height of a tree after n years given the initial height of the tree and the rate the tree grows each year.''' |
| − | |width = "833" style="background-color:# | + | ||'''Seeing Structure in Expressions''' |
| + | Write expressions in equivalent forms to solve problems | ||
| + | |||
| + | ||A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.\* | ||
| + | |||
| + | |- | ||
| + | ||'''H.PRF.2a5 Rewrite rational expressions, a(x)/b(x), in the form q(x) + r(x)/b(x) by using factoring, long division, or synthetic division.''' | ||
| + | |||
| + | ||'''Arithmetic with Polynomials and Rational Expressions''' | ||
| + | Rewrite rational expressions | ||
| + | |||
| + | ||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. | ||
| + | |||
| + | |- | ||
| + | ||'''H.PRF.2a6 Write and use a system of equations and/or inequalities to solve a real world problem.''' | ||
| + | |||
| + | ||'''Creating Equations''' | ||
| + | Create equations that describe numbers or relationships | ||
| + | |||
| + | ||A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. | ||
| + | |||
| + | |- | ||
| + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: H.PRF.2b creating equations and inequalities (in one or two variables) and use them to solve problems and graph solutions ''' | ||
| + | |||
| + | |- | ||
| + | |width = "833" style="background-color:#D9D9D9;"|'''Core Content Connectors: 9-12''' | ||
| + | |||
| + | |width = "833" style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' | ||
| + | |||
| + | |width = "834" style="background-color:#D9D9D9;"|'''Common Core State Standard''' | ||
| + | |||
| + | |- | ||
| + | ||'''H.PRF.2b1 Translate a real-world problem into a one variable equation''' | ||
| + | |||
| + | ||'''Creating Equations''' | ||
A CED Create equations that describe numbers or relationships. | A CED Create equations that describe numbers or relationships. | ||
| − | | | + | ||A.CED.1 Create equations and inequalities in one variable and use them to solve problems. ''Include equations arising from linear and quadratic functions, and simple rational and exponential functions.'' |
|- | |- | ||
| − | | | + | ||'''H.PRF.2b2 Solve equations with one or two variables using equations or graphs''' |
| − | | | + | ||'''Reasoning with Equations''' |
'''and Inequalities''' | '''and Inequalities''' | ||
A REI Understand solving equations as a process of reasoning and explain the reasoning. | A REI Understand solving equations as a process of reasoning and explain the reasoning. | ||
| Line 835: | Line 880: | ||
A CED Create equations that describe numbers and relationships. | A CED Create equations that describe numbers and relationships. | ||
| − | | | + | ||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. |
| − | + | A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. | |
| − | + | A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. | |
|- | |- | ||
| − | | | + | ||'''H.PRF.2b3 Transform a quadratic equation written in standard form to an equation in vertex form (x - p) = q 2 by completing the square.''' |
| + | |||
| + | ||'''Reasoning with Equations''' | ||
| + | '''and Inequalities''' | ||
| + | A REI Solve equations and inequalities in one variable. | ||
| + | |||
| + | |||
| + | ||A.REI.4 Solve quadratic equations in one variable. | ||
| + | a) Use the method of completing the square to transform and quadratic equation in ''x ''into an equation of the form (''x ''– ''p'')2 = q that has the same solutions. Derive the quadratic formula from this form. | ||
| + | |||
|- | |- | ||
| − | | | + | ||'''H.PRF.2b4 Derive the quadratic formula by completing the square on the standard form of a quadratic equation.''' |
| − | | | + | ||'''Reasoning with Equations''' |
| + | '''and Inequalities''' | ||
| + | A REI Solve equations and inequalities in one variable. | ||
| + | |||
| + | |||
| + | ||A.REI.4 Solve quadratic equations in one variable. | ||
| + | a) Use the method of completing the square to transform and quadratic equation in ''x ''into an equation of the form (''x ''– ''p'')2 = q that has the same solutions. Derive the quadratic formula from this form. | ||
| − | |||
|- | |- | ||
| − | | | + | ||'''H.PRF.2b5 Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and completing the square.''' |
| − | |width = "833" style="background-color:# | + | ||'''Reasoning with Equations''' |
| + | '''and Inequalities''' | ||
| + | A REI Solve equations and inequalities in one variable. | ||
| + | |||
| + | |||
| + | ||A.REI.4 Solve quadratic equations in one variable. | ||
| + | b) Solve quadratic equations by inspection (e.g., for ''x''2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as ''a ''± ''bi ''for real numbers ''a ''and ''''b''''. | ||
| + | |||
| + | |- | ||
| + | ||'''H.PRF.2b6 Solve systems of equations using the elimination method (sometimes called linear combinations).''' | ||
| + | |||
| + | ||'''Reasoning with Equations ''' | ||
| + | '''and Inequalities''' | ||
| + | Solve systems of equations | ||
| + | |||
| + | ||A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. | ||
| + | |||
| + | |- | ||
| + | ||'''H.PRF.2b7 Solve a system of equations by substitution (solving for one variable in the first equation and substitution it into the second equation).''' | ||
| + | |||
| + | ||'''Reasoning with Equations''' | ||
| + | '''and Inequalities''' | ||
| + | Solve systems of equations | ||
| + | |||
| + | ||A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. | ||
| + | |||
| + | |- | ||
| + | ||'''H.PRF.2b8 Solve systems of equations using graphs.''' | ||
| + | |||
| + | ||'''Reasoning with Equations''' | ||
| + | '''and Inequalities''' | ||
| + | Solve systems of equations | ||
| + | |||
| + | ||A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. | ||
| + | |||
| + | |- | ||
| + | ||'''H.PRF.2b9 Solve a system containing a linear equation and a quadratic equation in two variables graphically and symbolically.''' | ||
| + | |||
| + | ||'''Reasoning with Equations''' | ||
| + | '''and Inequalities''' | ||
| + | Solve systems of equations | ||
| + | |||
| + | ||A.REI.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. | ||
| + | |||
| + | |- | ||
| + | ||'''H.PRF.2b10 Understand that all solutions to an equation in two variables are contained on the graph of that equation.''' | ||
| + | |||
| + | ||'''Reasoning with Equations''' | ||
| + | '''and Inequalities''' | ||
| + | Represent and solve equations and inequalities graphically | ||
| + | |||
| + | ||A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). | ||
| + | |||
| + | |- | ||
| + | ||'''H.PRF.2b11 Graph the solutions to a linear inequality in two variables as a half-plane, excluding the boundary for non-inclusive inequalities.''' | ||
| + | |||
| + | ||'''Reasoning with Equations''' | ||
| + | '''and Inequalities''' | ||
| + | Represent and solve equations and inequalities graphically | ||
| + | |||
| + | ||A.REI.12 Graph the solutions to a linear inequality in two variables as a halfplane (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. | ||
| + | |||
| + | |- | ||
| + | ||'''H.PRF.2b12 Graph the solution set to a system of linear inequalities in two variables as the intersection of their corresponding half-planes. ''' | ||
| + | |||
| + | ||'''Reasoning with Equations''' | ||
| + | '''and Inequalities''' | ||
| + | Represent and solve equations and inequalities graphically | ||
| + | |||
| + | ||A.REI.12 Graph the solutions to a linear inequality in two variables as a halfplane (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. | ||
| + | |||
| + | |- | ||
| + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: H.PRF. 2c using trends that follow a pattern and are described mathematically to make generalizations or predictions ''' | ||
| + | |||
| + | |- | ||
| + | |width = "833" style="background-color:#D9D9D9;"|'''Core Content Connectors: 9-12''' | ||
| + | |||
| + | |width = "833" style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' | ||
| + | |||
| + | |width = "834" style="background-color:#D9D9D9;"|'''Common Core State Standard''' | ||
| + | |||
| + | |- | ||
| + | ||'''H.PRF. 2c1 Make predictions based on a given model (for example, a weather model, data for athletes over years)''' | ||
| + | |||
| + | ||'''Linear, Quadratic, and Exponential Models''' | ||
F LE Construct and compare linear, quadratic, and exponential models and solve problems. | F LE Construct and compare linear, quadratic, and exponential models and solve problems. | ||
| − | | | + | ||F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. |
| + | |||
| + | |- | ||
| + | | colspan=3|Explanations and clarifications: | ||
| + | |||
| + | |- | ||
| + | | style="background-color:#D9D9D9;" colspan=3|'''Progress Indicator: H.PRF. 2d:''' '''analyzing functions (using technology) by investigating significant characteristics (e.g. intercepts, asymptotes)''' | ||
| + | |||
| + | |- | ||
| + | |width = "833" style="background-color:#D9D9D9;"|'''Core Content Connectors: 9-12''' | ||
| + | |||
| + | |width = "833" style="background-color:#D9D9D9;"|'''CCSS Domain/Cluster''' | ||
| + | |||
| + | |width = "834" style="background-color:#D9D9D9;"|'''Common Core State Standard''' | ||
|- | |- | ||
| − | | | + | ||'''H.PRF. 2d1 Explain why the intersection of y = f(x) and y = g(x) is the solution of f(x) = g(x) for any combination of linear or exponential. Find the solution(s) by: Using technology to graph the equations and determine their point of intersection, Using tables of values, or Using successive approximations that become closer and closer to the actual value.''' |
| − | | | + | ||'''Reasoning with Equations''' |
| + | ''''''and Inequalities'''''' | ||
| + | Represent and solve equations and inequalities graphically | ||
| − | | | + | ||A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.\* |
|- | |- | ||
Revision as of 11:01, 2 July 2014
BACK TO Core Content Connectors
| (K-4) Elementary School Learning Targets | (5-8) Middle School Learning Targets | (9-12) High School Learning Targets |
| E.PRF-2 Give examples, interpret, and analyze repeating and growing patterns and functions involving the four basic operations. | M.PRF-2 Give examples, interpret, and analyze a variety of mathematical patterns, relations, and explicit and recursive functions. | H.PRF-2 Use trends and analyze a variety of mathematical patterns, relations, and explicit and recursive functions. |
| Arithmetic Patterns | ||||
| Grades K-2 | Grades 3-4 | Grades 5-6 | Grades 7-8 | HS |
| K.PRF.2a1 Describe or select the repeating pattern using objects or pictures (AB or ABC) | 3.PRF.2d1 Identify multiplication patterns in a real world setting | 5.PRF.2a1 Generate a pattern that follows the provided rule | 8.PRF.2c1 Given two graphs, describe the function as linear and not linear | H.PRF. 2c1 Make predictions based on a given model (for example, a weather model, data for athletes over years) |
| K.PRF.2a2 Extend a repeating pattern using objects or pictures (AB or ABC) | 3.PRF.2d2 Apply properties of operations as strategies to multiply and divide | 5.PRF.2b1 Generate or select a comparison between two graphs from a similar situation | 8.PRF.2e1 Distinguish between functions and non-functions, using equations, graphs or tables | |
| K.PRF.2a3 Extend a repeating numerical AB pattern | 4.PRF.2d3 Generate a pattern when given a rule and word problem (I run 3 miles every day, how many miles have I run in 3 days) | 6.PRF.2b2 Using provided table with numerical patterns, form ordered pairs | 8.PRF.2e2 Identify the rate of change (slope) and initial value (y-intercept) from graphs | |
| K.PRF.2b1 Create a repeating pattern using objects, pictures, or numbers | 4.PRF.2e1 Extend a numerical pattern when the rule is provided | 8.PRF.2e3 Given a verbal description of a situation, create or identify a graph to model the situation | ||
| 8.PRF.2e4 Given a graph of a situation, generate a description of the situation | ||||
| 1.PRF.2a4 Use a number line to extend the numerical patterns that grow at a constant rate (2,4,6,8) | ||||
| 1.PRF.2b2
Create a growing pattern using numbers or objects |
||||
| 1.PRF.2c1 Identify the rule of a given arithmetic pattern | ||||
| 2.PRF.2a6 Use a number line to extend the numerical patterns that grow at a constant rate (2,4,6,8) | ||||
| 2.PRF.2b3 Use a number line to extend arithmetic patterns that are decreasing | ||||
| 2.PRF.2c2 Identify the rule of arithmetic patterns that are increasing | ||||
| 2.PRF.2c3 Identify the rule of arithmetic patterns that are decreasing | ||||
| Expressions and Equations | ||||
| Grades K-2 | Grades 3-4 | Grades 5-6 | Grades 7-8 | HS |
| 6.PRF.2a2 Use variables to represent numbers and write expressions when solving real-world problems | 7.PRF.2a5 Use variables to represent two quantities in a real-world problem that change in relationship to one another | H.PRF.2a1 Translate an algebraic expression into a word problem
| ||
| 6.PRF.2a3
Use variables to represent two quantities in a real-world problem that change in relationship to one another |
7.PRF.2d1
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 |
H.PRF.2b1 Translate a real-world problem into a one variable equation | ||
| 6.PRF.2a4
Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation |
H.PRF.2b2
Solve equations with one or two variables using equations or graphs | |||
| 6.PRF.2b3
Complete a statement that describes the ratio relationship between two quantities |
||||
| 6.PRF.2b4
Determine the unit rate in a variety of contextual situations |
||||
| 6.PRF.2b5 Use ratios and reasoning to solve real-world mathematical problems (e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations) | ||||
Contents |
Grade Differentiation
Elementary School Progress Indicators
| Progress Indicator: E.PRF.2a recognizing, describing, and extending simple repeating (ABAB) and growing (A+1, A+2, A+3) patterns (e.g., colors, sounds, words, shapes, numeric – counting, odd, even) | ||
| Core Content Connectors: K | CCSS Domain/Cluster | Common Core State Standard |
| K.PRF.2a1 Describe or select the repeating pattern using objects or pictures (AB or ABC) | No CCSS linked | |
| K.PRF.2a2 Extend a repeating pattern using objects or pictures (AB or ABC) | No CCSS linked | |
| K.PRF.2a3 Extend a repeating numerical AB pattern | No CCSS linked | |
| Progress Indicator: E.PRF.2b creating and explaining repeating and growing patterns using objects or numbers | ||
| Core Content Connectors: K | CCSS Domain/Cluster | Common Core State Standard |
| K.PRF.2b1 Create a repeating pattern using objects, pictures, or numbers | No CCSS linked | |
| Progress Indicator: E.PRF.2a recognizing, describing, and extending simple repeating (ABAB) and growing (A+1, A+2, A+3) patterns (e.g., colors, sounds, words, shapes, numeric – counting, odd, even) | ||
| Core Content Connectors: 1 | CCSS Domain/Cluster | Common Core State Standard |
| 1.PRF.2a4 Use a number line to extend the numerical patterns that grow at a constant rate (2,4,6,8) | No CCSS linked | |
| Progress Indicator: E.PRF.2b creating and explaining repeating and growing patterns using objects or numbers | ||
| Core Content Connectors: 1 | CCSS Domain/Cluster | Common Core State Standard |
| 1.PRF.2b2 Create a growing pattern using numbers or objects | No CCSS linked | |
| Progress Indicator: E.PRF.2c extending and analyzing simple numeric patterns with rules that involve addition and subtraction | ||
| Core Content Connectors: 1 | CCSS Domain/Cluster | Common Core State Standard |
| 1.PRF.2c1 Identify the rule of a given arithmetic pattern | No CCSS linked | |
| Progress Indicator: E.PRF.2a recognizing, describing, and extending simple repeating (ABAB) and growing (A+1, A+2, A+3) patterns (e.g., colors, sounds, words, shapes, numeric – counting, odd, even) | ||
| Core Content Connectors: 2 | CCSS Domain/Cluster | Common Core State Standard |
| 2.PRF.2a6 Use a number line to extend the numerical patterns that grow at a constant rate (2,4,6,8) | No CCSS linked | |
| Progress Indicator: E.PRF.2b creating and explaining repeating and growing patterns using objects or numbers | ||
| Core Content Connectors: 2 | CCSS Domain/Cluster | Common Core State Standard |
| 2.PRF.2b3 Use a number line to extend arithmetic patterns that are decreasing | No CCSS linked | |
| Progress Indicator: E.PRF.2c extending and analyzing simple numeric patterns with rules that involve addition and subtraction | ||
| Core Content Connectors: 2 | CCSS Domain/Cluster | Common Core State Standard |
| 2.PRF.2c2 Identify the rule of arithmetic patterns that are increasing | No CCSS linked | |
| 2.PRF.2c3 Identify the rule of arithmetic patterns that are decreasing | No CCSS linked | |
| Progress Indicator: E.PRF.2d representing and analyzing patterns and rules (e.g., doubling, adding 3) using words, tables, graphs, and models | ||
| Core Content Connectors: 3 | CCSS Domain/Cluster | Common Core State Standard |
| 3.PRF.2d1 Identify multiplication patterns in a real world setting | Operations and Algebraic Thinking
3 OA Solve problems involving the four operations, and identify and explain patterns in arithmetic. |
3.OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. |
| 3.PRF.2d2 Apply properties of operations as strategies to multiply and divide | Operations and Algebraic Thinking
3 OA Understand properties of multiplication and the relationship between multiplication and division. |
3.OA.5 Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) |
| Progress Indicator: E.PRF.2d representing and analyzing patterns and rules (e.g., doubling, adding 3) using words, tables, graphs, and models | ||
| Core Content Connectors: 4 | CCSS Domain/Cluster | Common Core State Standard |
| 4.PRF.2d3 Generate a pattern when given a rule and word problem (I run 3 miles every day, how many miles have I run in 3 days) | Operations and Algebraic Thinking
4 OA Generate and analyze patterns. |
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. |
| Progress Indicator: E.PRF.2e extending, translating, and analyzing numeric patterns and their rules using addition, subtraction, multiplication, and division | ||
| Core Content Connectors: 4 | CCSS Domain/Cluster | Common Core State Standard |
| 4.PRF.2e1 Extend a numerical pattern when the rule is provided | Operations and Algebraic Thinking
4 OA Generate and analyze patterns. |
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. |
| Progress Indicator: M.PRF.2a representing, analyzing, extending, and generalizing a variety of patterns using tables, graphs, words, and symbolic rules | ||
| Core Content Connectors: 5 | CCSS Domain/Cluster | Common Core State Standard |
| 5.PRF.2a1 Generate a pattern that follows the provided rule | Operations and Algebraic Thinking
4 OA Generate and analyze patterns. |
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. |
| Progress Indicator: M.PRF.2b relating and comparing different forms of representation and identifying functions as linear or nonlinear | ||
| Core Content Connectors: 5 | CCSS Domain/Cluster | Common Core State Standard |
| 5.PRF.2b1 Generate or select a comparison between two graphs from a similar situation | Operations and Algebraic Thinking
5 OA Analyze patterns and relationships. |
5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. |
Middle School Progress Indicators
| Progress Indicator: M.PRF.2a representing, analyzing, extending, and generalizing a variety of patterns using tables, graphs, words, and symbolic rules | ||
| Core Content Connectors: 6 | CCSS Domain/Cluster | Common Core State Standard |
| 6.PRF.2a2 Use variables to represent numbers and write expressions when solving real-world problems | Expressions and Equations
6 EE Reason about and solve one-variable equations and inequalities. |
6.EE.6 Use variables to represent numbers and write expressions when solving real-world or mathematical problems; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. |
| 6.PRF.2a3 Use variables to represent two quantities in a real-world problem that change in relationship to one another | Expressions and Equations
6 EE Represent and analyze quantitative relationships between dependent and independent variables. |
# 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. |
| 6.PRF.2a4 Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation | Expressions and Equations
6 EE Represent and analyze quantitative relationships between dependent and independent variables. |
# 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. |
| Progress Indicator: M.PRF.2b relating and comparing different forms of representation and identifying functions as linear or nonlinear | ||
| Core Content Connectors: 6 | CCSS Domain/Cluster | Common Core State Standard |
| 6.PRF.2b2 Using provided table with numerical patterns, form ordered pairs | Operations and Algebraic Thinking
5 OA Analyze patterns and relationships. |
5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. |
| Progress Indicator: M.PRF.2b relating and comparing different forms of representation and identifying functions as linear or nonlinear | ||
| Core Content Connectors: 6 | CCSS Domain/Cluster | Common Core State Standard |
| 6.PRF.2b3 Complete a statement that describes the ratio relationship between two quantities | Ratios and Proportional Relationships
6 RP Understand ratio concepts and use ratio reasoning to solve problems. |
6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes." |
| 6.PRF.2b4 Determine the unit rate in a variety of contextual situations | Ratios and Proportional Relationships
6 RP Understand ratio concepts and use ratio reasoning to solve problems. |
6.RP.2 Understand the concept of a unit rate a/b associated with a ration a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, "this recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is ¾ cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger." |
| 6.PRF.2b5 Use ratios and reasoning to solve real-world mathematical problems (e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams,
or equations) |
Ratios and Proportional Relationships
6 RP Understand ratio concepts and use ratio reasoning to solve problems. |
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 diagrams, or equations.
a) Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. b) Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? c) 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. d) Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. |
| Progress Indicator: M.PRF.2a representing, analyzing, extending, and generalizing a variety of patterns using tables, graphs, words, and symbolic rules | ||
| Core Content Connectors: 7 | CCSS Domain/Cluster | Common Core State Standard |
| 7.PRF.2a5 Use variables to represent two quantities in a real-world problem that change in relationship to one another | Expressions and Equations
6 EE Represent and analyze quantitative relationships between dependent and independent variables. |
# 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. |
| Progress Indicator: M.PRF.2d solving linear equations and formulating and explaining reasoning about expressions and equations | ||
| Core Content Connectors: 7 | CCSS Domain/Cluster | Common Core State Standard |
| 7.PRF.2d1 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 | Expressions and Equations
7 EE Solve real-life and mathematical problems using numerical and algebraic expressions and equations. |
7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
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. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. |
| Progress Indicator: M.PRF.2c relating and comparing different forms of representation and identifying functions as linear or nonlinear | ||
| Core Content Connectors: 8 | CCSS Domain/Cluster | Common Core State Standard |
| 8.PRF.2c1 Given two graphs, describe the function as linear and not linear | Functions
8 F Define, evaluate, and compare functions. 8 F Use functions to model relationships between quantities. |
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. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
8.F.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. |
| Progress Indicator: M.PRF.2e using functions to describe quantitative relationships | ||
| Core Content Connectors: 8 | CCSS Domain/Cluster | Common Core State Standard |
| 8.PRF.2e1 Distinguish between functions and non-functions, using equations, graphs or tables | Functions
8 F Define, evaluate, and compare functions. |
8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. |
| 8.PRF.2e2 Identify the rate of change (slope) and initial value (y-intercept) from graphs | Functions
8 F Use functions to model relationships between quantities. |
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. |
| 8.PRF.2e3 Given a verbal description of a situation, create or identify a graph to model the situation | Functions
8 F Use functions to model relationships between quantities. |
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. |
| 8.PRF.2e4 Given a graph of a situation, generate a description of the situation | Functions
8 F Use functions to model relationships between quantities. |
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. |
| 8.PRF.2e5 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. | Functions
8 F Define, evaluate, and compare functions.
|
8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. |
High School Progress Indicators
| Progress Indicator: H.PRF.2a interpreting and rewriting a variety of expressions or functions to solve problems | ||
| Core Content Connectors: 9-12 | CCSS Domain/Cluster | Common Core State Standard |
| H.PRF.2a1 Translate an algebraic expression into a word problem | Seeing Structure in Expressions
A SSE Interpret the structure of expressions. |
A.SSE.1 Interpret expressions that represent a quantity in terms of its context.
a) Interpret parts of an expression, such as terms, factors, and coefficients. b) Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. |
| H.PRF.2a2 Factor a quadratic expression. | Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems |
A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.\*
a) Factor a quadratic expression to reveal the zeros of the function it defines. |
| H.PRF.2a3 Given a quadratic expression, explain the meaning of the zeros graphically. That is for an expression (x – a) (x – c), a and c correspond to the x-intercepts (if a and c are real). | Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems |
A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.\*
a) Factor a quadratic expression to reveal the zeros of the function it defines. |
| H.PRF.2a4 Use the formula to solve real world problems such as calculating the height of a tree after n years given the initial height of the tree and the rate the tree grows each year. | Seeing Structure in Expressions
Write expressions in equivalent forms to solve problems |
A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.\* |
| H.PRF.2a5 Rewrite rational expressions, a(x)/b(x), in the form q(x) + r(x)/b(x) by using factoring, long division, or synthetic division. | Arithmetic with Polynomials and Rational Expressions
Rewrite rational expressions |
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. |
| H.PRF.2a6 Write and use a system of equations and/or inequalities to solve a real world problem. | Creating Equations
Create equations that describe numbers or relationships |
A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. |
| Progress Indicator: H.PRF.2b creating equations and inequalities (in one or two variables) and use them to solve problems and graph solutions | ||
| Core Content Connectors: 9-12 | CCSS Domain/Cluster | Common Core State Standard |
| H.PRF.2b1 Translate a real-world problem into a one variable equation | Creating Equations
A CED Create equations that describe numbers or relationships. |
A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. |
| H.PRF.2b2 Solve equations with one or two variables using equations or graphs | Reasoning with Equations
and Inequalities A REI Understand solving equations as a process of reasoning and explain the reasoning. A REI Solve equations and inequalities in one variable. Creating Equations A CED Create equations that describe numbers and relationships. |
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.
A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. |
| H.PRF.2b3 Transform a quadratic equation written in standard form to an equation in vertex form (x - p) = q 2 by completing the square. | Reasoning with Equations
and Inequalities A REI Solve equations and inequalities in one variable.
|
A.REI.4 Solve quadratic equations in one variable.
a) Use the method of completing the square to transform and quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
|
| H.PRF.2b4 Derive the quadratic formula by completing the square on the standard form of a quadratic equation. | Reasoning with Equations
and Inequalities A REI Solve equations and inequalities in one variable.
|
A.REI.4 Solve quadratic equations in one variable.
a) Use the method of completing the square to transform and quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
|
| H.PRF.2b5 Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and completing the square. | Reasoning with Equations
and Inequalities A REI Solve equations and inequalities in one variable.
|
A.REI.4 Solve quadratic equations in one variable.
b) Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and 'b'. |
| H.PRF.2b6 Solve systems of equations using the elimination method (sometimes called linear combinations). | Reasoning with Equations
and Inequalities Solve systems of equations |
A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. |
| H.PRF.2b7 Solve a system of equations by substitution (solving for one variable in the first equation and substitution it into the second equation). | Reasoning with Equations
and Inequalities Solve systems of equations |
A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. |
| H.PRF.2b8 Solve systems of equations using graphs. | Reasoning with Equations
and Inequalities Solve systems of equations |
A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. |
| H.PRF.2b9 Solve a system containing a linear equation and a quadratic equation in two variables graphically and symbolically. | Reasoning with Equations
and Inequalities Solve systems of equations |
A.REI.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. |
| H.PRF.2b10 Understand that all solutions to an equation in two variables are contained on the graph of that equation. | Reasoning with Equations
and Inequalities Represent and solve equations and inequalities graphically |
A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). |
| H.PRF.2b11 Graph the solutions to a linear inequality in two variables as a half-plane, excluding the boundary for non-inclusive inequalities. | Reasoning with Equations
and Inequalities Represent and solve equations and inequalities graphically |
A.REI.12 Graph the solutions to a linear inequality in two variables as a halfplane (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. |
| H.PRF.2b12 Graph the solution set to a system of linear inequalities in two variables as the intersection of their corresponding half-planes. | Reasoning with Equations
and Inequalities Represent and solve equations and inequalities graphically |
A.REI.12 Graph the solutions to a linear inequality in two variables as a halfplane (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. |
| Progress Indicator: H.PRF. 2c using trends that follow a pattern and are described mathematically to make generalizations or predictions | ||
| Core Content Connectors: 9-12 | CCSS Domain/Cluster | Common Core State Standard |
| H.PRF. 2c1 Make predictions based on a given model (for example, a weather model, data for athletes over years) | Linear, Quadratic, and Exponential Models
F LE Construct and compare linear, quadratic, and exponential models and solve problems. |
F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. |
| Explanations and clarifications: | ||
| Progress Indicator: H.PRF. 2d: analyzing functions (using technology) by investigating significant characteristics (e.g. intercepts, asymptotes) | ||
| Core Content Connectors: 9-12 | CCSS Domain/Cluster | Common Core State Standard |
| H.PRF. 2d1 Explain why the intersection of y = f(x) and y = g(x) is the solution of f(x) = g(x) for any combination of linear or exponential. Find the solution(s) by: Using technology to graph the equations and determine their point of intersection, Using tables of values, or Using successive approximations that become closer and closer to the actual value. | Reasoning with Equations
'and Inequalities' Represent and solve equations and inequalities graphically |
A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.\* |
