## Description

- Overview:
- This lesson unit is intended to help teahcers assess how well students are able to interpret speed as the slope of a linear graph and translate between the equation of a line and its graphical representation.

- Level:
- Lower Primary, Upper Primary, Middle School, High School
- Grades:
- Kindergarten, Grade 1, Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7, Grade 8, Grade 9, Grade 10, Grade 11, Grade 12
- Material Type:
- Assessment, Lesson Plan
- Provider:
- Shell Center for Mathematical Education, U.C. Berkeley
- Provider Set:
- Mathematics Assessment Project (MAP)
- Date Added:
- 04/26/2013

- License:
- Creative Commons Attribution-NonCommercial-NoDerivs 3.0
- Media Format:
- Downloadable docs, Text/HTML

# Comments

## Standards

Cluster: Understand the connections between proportional relationships, lines, and linear equations

Standard: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

Degree of Alignment: 3 Superior (1 user)

Cluster: Work with radicals and integer exponents

Standard: Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

Degree of Alignment: Not Rated (0 users)

Cluster: Work with radicals and integer exponents

Standard: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 10^8 and the population of the world as 7 × 10^9, and determine that the world population is more than 20 times larger.

Degree of Alignment: Not Rated (0 users)

Cluster: Work with radicals and integer exponents

Standard: Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^2 × 3^(–5) = 3^(–3) = 1/(3^3) = 1/27.

Degree of Alignment: Not Rated (0 users)

Cluster: Work with radicals and integer exponents

Standard: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Degree of Alignment: Not Rated (0 users)

Cluster: Define, evaluate, and compare functions

Standard: 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.

Degree of Alignment: Not Rated (0 users)

Cluster: Define, evaluate, and compare functions

Standard: 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. (Function notation is not required in Grade 8.)

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Degree of Alignment: Not Rated (0 users)

Cluster: Analyze and solve linear equations and pairs of simultaneous linear equations

Standard: Analyze and solve pairs of simultaneous linear equations.

Degree of Alignment: Not Rated (0 users)

Cluster: Use functions to model relationships between quantities

Standard: 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.

Degree of Alignment: Not Rated (0 users)

Cluster: Analyze and solve linear equations and pairs of simultaneous linear equations

Standard: Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

Degree of Alignment: Not Rated (0 users)

Cluster: Define, evaluate, and compare functions

Standard: 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^2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

Degree of Alignment: Not Rated (0 users)

Cluster: Analyze and solve linear equations and pairs of simultaneous linear equations

Standard: Solve linear equations in one variable.

Degree of Alignment: Not Rated (0 users)

Cluster: Use functions to model relationships between quantities

Standard: 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.

Degree of Alignment: Not Rated (0 users)

Cluster: Understand the connections between proportional relationships, lines, and linear equations

Standard: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y =mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Degree of Alignment: Not Rated (0 users)

Cluster: Analyze and solve linear equations and pairs of simultaneous linear equations

Standard: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

Degree of Alignment: Not Rated (0 users)

Cluster: Analyze and solve linear equations and pairs of simultaneous linear equations

Standard: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

Degree of Alignment: Not Rated (0 users)

Cluster: Analyze and solve linear equations and pairs of simultaneous linear equations

Standard: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Degree of Alignment: Not Rated (0 users)

Cluster: Analyze and solve linear equations and pairs of simultaneous linear equations

Standard: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions

Standard: Define, evaluate, and compare functions

Indicator: 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^2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

Degree of Alignment: 3 Superior (1 user)

Learning Domain: Functions

Standard: Use functions to model relationships between quantities

Indicator: 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Expressions and Equations

Standard: Analyze and solve linear equations and pairs of simultaneous linear equations

Indicator: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions

Standard: Use functions to model relationships between quantities

Indicator: 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions

Standard: Define, evaluate, and compare functions

Indicator: 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. (Function notation is not required in Grade 8.)

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions

Standard: Define, evaluate, and compare functions

Indicator: 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Expressions and Equations

Standard: Analyze and solve linear equations and pairs of simultaneous linear equations

Indicator: Solve linear equations in one variable.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Expressions and Equations

Standard: Understand the connections between proportional relationships, lines, and linear equations

Indicator: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y =mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Expressions and Equations

Standard: Analyze and solve linear equations and pairs of simultaneous linear equations

Indicator: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Expressions and Equations

Standard: Understand the connections between proportional relationships, lines, and linear equations

Indicator: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Expressions and Equations

Standard: Analyze and solve linear equations and pairs of simultaneous linear equations

Indicator: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Expressions and Equations

Standard: Analyze and solve linear equations and pairs of simultaneous linear equations

Indicator: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Expressions and Equations

Standard: Analyze and solve linear equations and pairs of simultaneous linear equations

Indicator: Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Expressions and Equations

Standard: Work with radicals and integer exponents

Indicator: Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^2 x 3^(-5) = 3^(-3) = 1/(3^3) = 1/27.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Expressions and Equations

Standard: Work with radicals and integer exponents

Indicator: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 x 10^8 and the population of the world as 7 x 10^9, and determine that the world population is more than 20 times larger.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Expressions and Equations

Standard: Work with radicals and integer exponents

Indicator: Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that ‰ö_2 is irrational.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize"Óto abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents"Óand the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Expressions and Equations

Standard: Work with radicals and integer exponents

Indicator: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Expressions and Equations

Standard: Analyze and solve linear equations and pairs of simultaneous linear equations

Indicator: Analyze and solve pairs of simultaneous linear equations.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Linear, Quadratic, and Exponential Models

Standard: Construct and compare linear, quadratic, and exponential models and solve problems.

Indicator: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Degree of Alignment: 3 Superior (1 user)

Learning Domain: Interpreting Functions

Standard: Interpret functions that arise in applications in terms of the context.

Indicator: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Degree of Alignment: 3 Superior (1 user)

Learning Domain: Interpreting Functions

Standard: Interpret functions that arise in applications in terms of the context.

Indicator: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Degree of Alignment: 2 Strong (1 user)

Learning Domain: Linear, Quadratic, and Exponential Models

Standard: Interpret expressions for functions in terms of the situation they model.

Indicator: Interpret the parameters in a linear or exponential function in terms of a context.

Degree of Alignment: 2 Strong (1 user)

Learning Domain: Creating Equations

Standard: Create equations that describe numbers or relationships.

Indicator: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions

Standard: Define, evaluate, and compare functions.

Indicator: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions

Standard: Define, evaluate, and compare functions.

Indicator: 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.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions

Standard: Use functions to model relationships between quantities.

Indicator: Describe qualitatively the functional relationship between two quantities by analyzing a graph where the function is increasing, decreasing, constant, linear, or nonlinear. Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions

Standard: Use functions to model relationships between quantities.

Indicator: Apply the concepts of linear functions to real-world and mathematical situations.

Degree of Alignment: Not Rated (0 users)

## Evaluations

# Achieve OER

Average Score (3 Points Possible)Degree of Alignment | 2.7 (2 users) |

Quality of Explanation of the Subject Matter | 2.5 (2 users) |

Utility of Materials Designed to Support Teaching | 2.5 (2 users) |

Quality of Assessments | 3 (1 user) |

Quality of Technological Interactivity | 0 (1 user) |

Quality of Instructional and Practice Exercises | 3 (1 user) |

Opportunities for Deeper Learning | 2 (1 user) |

# Tags (10)

- Mathematics
- Algebra and Calculus
- Algebra
- CCSS
- Common Core Math
- Common Core PD
- Graphs
- Linear Equations
- ODE Learning
- Slope

on Feb 11, 03:34am Evaluation

## Quality of Technological Interactivity: Very Weak (0)

There was no application of technology

on Feb 11, 03:34am Evaluation

## Quality of Assessments: Superior (3)

There are multiple opportunities for assessments during the lesson plan.

This is a lesson/assessment tool. It has many resources for teachers. Detailed lesson plans, ppt, student worksheets and assessments, and card sets that students have to match up. It is addressing CCSS 8.EE.5 - comparing two proportional relationships represented in different ways. I like how students have to match up two graphs together because the rate of change between the graphs is related.