## Unit 2 - Course B

### Lesson 1: Video Games and Coordinate Planes

#### CSTA K-12 Computer Science Standards (2011)

**CT** - Computational Thinking

**CT.L1:6-01**- Understand and use the basic steps in algorithmic problem-solving (e.g., problem statement and exploration, examination of sample instances, design, implementation and testing).**CT.L2:14**- Examine connections between elements of mathematics and computer science including binary numbers, logic, sets and functions.

#### Common Core Math Standards

**MP** - Math Practices

**MP.1**- Make sense of problems and persevere in solving them**MP.2**- Reason abstractly and quantitatively**MP.3**- Construct viable arguments and critique the reasoning of others**MP.4**- Model with mathematics**MP.5**- Use appropriate tools strategically**MP.6**- Attend to precision**MP.7**- Look for and make use of structure**MP.8**- Look for and express regularity in repeated reasoning

**NS** - The Number System

**6.NS.5**- Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use posit**6.NS.8**- Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

**OA** - Operations And Algebraic Thinking

**5.OA.1**- Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.**5.OA.2**- Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three t

**Q** - Quantities

**N.Q.1**- Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.**N.Q.2**- Define appropriate quantities for the purpose of descriptive modeling.

### Lesson 2: The Big Game - Variables

#### Common Core Math Standards

**CED** - Creating Equations

**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.**A.CED.2**- Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

**EE** - Expressions And Equations

**6.EE.4**- Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardles

**IF** - Interpreting Functions

**F.IF.1**- Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f c**F.IF.2**- Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.**F.IF.3**- Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

**LE** - Linear, Quadratic, And Exponential Models★

**F.LE.1**- Distinguish between situations that can be modeled with linear functions and with exponential functions.

**MP** - Math Practices

**MP.1**- Make sense of problems and persevere in solving them**MP.2**- Reason abstractly and quantitatively**MP.3**- Construct viable arguments and critique the reasoning of others**MP.4**- Model with mathematics**MP.5**- Use appropriate tools strategically**MP.6**- Attend to precision**MP.7**- Look for and make use of structure**MP.8**- Look for and express regularity in repeated reasoning

**OA** - Operations And Algebraic Thinking

**5.OA.1**- Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.**5.OA.2**- Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three t

**Q** - Quantities

**N.Q.1**- Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.**N.Q.2**- Define appropriate quantities for the purpose of descriptive modeling.

**SSE** - Seeing Structure In Expressions

**A.SSE.1**- Interpret expressions that represent a quantity in terms of its context.**A.SSE.2**- Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

### Lesson 3: The Big Game - Animation

#### Common Core Math Standards

**BF** - Building Functions

**F.BF.1**- Write a function that describes a relationship between two quantities.**F.BF.2**- Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★

**EE** - Expressions And Equations

**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 va**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.

**F** - 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.1**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

**IF** - Interpreting Functions

**F.IF.1**- Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f c**F.IF.2**- Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.**F.IF.4**- For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercep**F.IF.5**- Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers w**F.IF.6**- Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★

**LE** - Linear, Quadratic, And Exponential Models★

**F.LE.1**- Distinguish between situations that can be modeled with linear functions and with exponential functions.**F.LE.2**- Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

**MP** - Math Practices

**MP.1**- Make sense of problems and persevere in solving them**MP.2**- Reason abstractly and quantitatively**MP.3**- Construct viable arguments and critique the reasoning of others**MP.4**- Model with mathematics**MP.5**- Use appropriate tools strategically**MP.6**- Attend to precision**MP.7**- Look for and make use of structure**MP.8**- Look for and express regularity in repeated reasoning

**NS** - The Number System

**6.NS.8**- Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

**OA** - Operations And Algebraic Thinking

**5.OA.2**- Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three t

**Q** - Quantities

**N.Q.1**- Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.**N.Q.2**- Define appropriate quantities for the purpose of descriptive modeling.

### Lesson 4: Booleans and Logic

#### Common Core Math Standards

**EE** - Expressions And Equations

**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 va

**F** - 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.1**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

**MP** - Math Practices

**MP.1**- Make sense of problems and persevere in solving them**MP.2**- Reason abstractly and quantitatively**MP.3**- Construct viable arguments and critique the reasoning of others**MP.4**- Model with mathematics**MP.5**- Use appropriate tools strategically**MP.6**- Attend to precision**MP.7**- Look for and make use of structure**MP.8**- Look for and express regularity in repeated reasoning

**NS** - The Number System

**6.NS.1**- Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use

**OA** - Operations And Algebraic Thinking

**5.OA.1**- Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.**5.OA.2**- Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three t

**Q** - Quantities

**N.Q.1**- Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.**N.Q.2**- Define appropriate quantities for the purpose of descriptive modeling.

**REI** - Reasoning With Equations And Inequalities

**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).**A.REI.3**- Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

### Lesson 5: Boolean Operators

#### Common Core Math Standards

**EE** - Expressions And Equations

**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 va**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.

**F** - 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.1**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

**MP** - Math Practices

**MP.1**- Make sense of problems and persevere in solving them**MP.2**- Reason abstractly and quantitatively**MP.3**- Construct viable arguments and critique the reasoning of others**MP.4**- Model with mathematics**MP.5**- Use appropriate tools strategically**MP.6**- Attend to precision**MP.7**- Look for and make use of structure**MP.8**- Look for and express regularity in repeated reasoning

**NS** - The Number System

**6.NS.8**- Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

**OA** - Operations And Algebraic Thinking

**5.OA.1**- Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.**5.OA.2**- Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three t

**Q** - Quantities

**N.Q.1**- Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.**N.Q.2**- Define appropriate quantities for the purpose of descriptive modeling.

### Lesson 6: Sam the Bat

#### Common Core Math Standards

**EE** - Expressions And Equations

**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 va**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.

**F** - 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.1**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

**MP** - Math Practices

**MP.1**- Make sense of problems and persevere in solving them**MP.2**- Reason abstractly and quantitatively**MP.3**- Construct viable arguments and critique the reasoning of others**MP.4**- Model with mathematics**MP.5**- Use appropriate tools strategically**MP.6**- Attend to precision**MP.7**- Look for and make use of structure**MP.8**- Look for and express regularity in repeated reasoning

**NS** - The Number System

**6.NS.8**- Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

**OA** - Operations And Algebraic Thinking

**5.OA.1**- Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.**5.OA.2**- Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three t

**Q** - Quantities

**N.Q.1**- Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.**N.Q.2**- Define appropriate quantities for the purpose of descriptive modeling.

### Lesson 7: The Big Game Booleans

#### Common Core Math Standards

**EE** - Expressions And Equations

**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 va**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.

**F** - 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.1**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

**MP** - Math Practices

**MP.1**- Make sense of problems and persevere in solving them**MP.2**- Reason abstractly and quantitatively**MP.3**- Construct viable arguments and critique the reasoning of others**MP.4**- Model with mathematics**MP.5**- Use appropriate tools strategically**MP.6**- Attend to precision**MP.7**- Look for and make use of structure**MP.8**- Look for and express regularity in repeated reasoning

**NS** - The Number System

**6.NS.8**- Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

**OA** - Operations And Algebraic Thinking

**5.OA.1**- Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.**5.OA.2**- Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three t

**Q** - Quantities

**N.Q.1**- Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.**N.Q.2**- Define appropriate quantities for the purpose of descriptive modeling.

### Lesson 8: Conditionals and Piecewise Functions

#### Common Core Math Standards

**EE** - Expressions And Equations

**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 va**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.

**F** - 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.1**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

**IF** - Interpreting Functions

**F.IF.7.b**- Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

**MP** - Math Practices

**MP.1**- Make sense of problems and persevere in solving them**MP.2**- Reason abstractly and quantitatively**MP.3**- Construct viable arguments and critique the reasoning of others**MP.4**- Model with mathematics**MP.5**- Use appropriate tools strategically**MP.6**- Attend to precision**MP.7**- Look for and make use of structure**MP.8**- Look for and express regularity in repeated reasoning

**NS** - The Number System

**6.NS.8**- Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

**OA** - Operations And Algebraic Thinking

**5.OA.1**- Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.**5.OA.2**- Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three t

**Q** - Quantities

**N.Q.1**- Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.**N.Q.2**- Define appropriate quantities for the purpose of descriptive modeling.

### Lesson 9: Conditionals and Update Player

#### Common Core Math Standards

**EE** - Expressions And Equations

**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 va**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.

**F** - 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.1**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

**MP** - Math Practices

**MP.1**- Make sense of problems and persevere in solving them**MP.2**- Reason abstractly and quantitatively**MP.3**- Construct viable arguments and critique the reasoning of others**MP.4**- Model with mathematics**MP.5**- Use appropriate tools strategically**MP.6**- Attend to precision**MP.7**- Look for and make use of structure**MP.8**- Look for and express regularity in repeated reasoning

**NS** - The Number System

**6.NS.8**- Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

**OA** - Operations And Algebraic Thinking

**5.OA.1**- Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.**5.OA.2**- Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three t

**Q** - Quantities

**N.Q.1**- Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.**N.Q.2**- Define appropriate quantities for the purpose of descriptive modeling.

### Lesson 10: Collision Detection and the Pythagorean Theorem

#### Common Core Math Standards

**EE** - 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.**8.EE.2**- Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrationa

**F** - 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.1**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

**G** - Geometry

**8.G.7**- Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.**8.G.8**- Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

**MP** - Math Practices

**MP.1**- Make sense of problems and persevere in solving them**MP.2**- Reason abstractly and quantitatively**MP.3**- Construct viable arguments and critique the reasoning of others**MP.4**- Model with mathematics**MP.5**- Use appropriate tools strategically**MP.6**- Attend to precision**MP.7**- Look for and make use of structure**MP.8**- Look for and express regularity in repeated reasoning

**NS** - The Number System

**6.NS.8**- Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

**Q** - Quantities

**N.Q.1**- Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.**N.Q.2**- Define appropriate quantities for the purpose of descriptive modeling.

### Lesson 11: The Big Game - Collision Detection

#### Common Core Math Standards

**EE** - 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.**8.EE.2**- Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrationa

**F** - 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.1**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

**G** - Geometry

**8.G.7**- Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.**8.G.8**- Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

**MP** - Math Practices

**MP.1**- Make sense of problems and persevere in solving them**MP.2**- Reason abstractly and quantitatively**MP.3**- Construct viable arguments and critique the reasoning of others**MP.4**- Model with mathematics**MP.5**- Use appropriate tools strategically**MP.6**- Attend to precision**MP.7**- Look for and make use of structure**MP.8**- Look for and express regularity in repeated reasoning

**NS** - The Number System

**6.NS.8**- Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

**Q** - Quantities

**N.Q.1**- Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.**N.Q.2**- Define appropriate quantities for the purpose of descriptive modeling.