Patterns, Relations, and Functions 2
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6 EE Represent and analyze quantitative relationships between dependent and independent variables. | 6 EE Represent and analyze quantitative relationships between dependent and independent variables. | ||
| − | |width = "834" style="background-color:#FFFFFF;"| | + | |width = "834" style="background-color:#FFFFFF;"| 6.EE.C.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. ''Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.'' |
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6 EE Represent and analyze quantitative relationships between dependent and independent variables. | 6 EE Represent and analyze quantitative relationships between dependent and independent variables. | ||
| − | |width = "834" style="background-color:#FFFFFF;"| | + | |width = "834" style="background-color:#FFFFFF;"| 6.EE.C.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. ''Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. '' |
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Revision as of 14:21, 9 October 2013
| (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.D.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.B.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.C.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. 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.C.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. 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.C.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. 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.B.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. 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.B.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.C.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. |
| 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.C.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. |
| 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.B.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. 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.A.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.A.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.A.3 Use ratio and rate reasoning to solve real‐world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
|
| 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.C.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. |
| 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.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
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| 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.A.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 | No CCSS linked | |
| 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.B.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.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. |
| 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.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. |
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.
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| 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. |
| 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: | ||
