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Lesson 3: Contracts, Domain, and Range

Overview

Contracts provide a way for students to better understand and discuss functions. Through this lesson, students will look at known functions and come up with the contracts that describe those functions.

Agenda

Getting Started

Activities

Wrap-up

Anchor Standard

Common Core Math Standards

  • F.IF.1 - Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Objectives

Students will be able to:

  • Describe a function in terms of its name, domain, and range.
  • Create contracts for arithmetic and image-producing functions.

Links

For the Teacher

For the Students

Vocabulary

  • Contract - A statement of the name, domain, and range of a function.
  • Domain - The type of data that a function expects.
  • Range - The type of data that a function produces.

Teaching Guide

Getting Started

What’s in a Function?

You’ve already seen several functions that take in two Numbers, such as +, and -. Other functions like “star”, take in a Number and two Strings. Different functions take in different inputs, and we need a way to keep track of the requirements for each function.

  • What does the ‘+’ function do?
    • What does it take as input?
    • What does it return as output?
  • How about the ‘triangle’ function?
  • What do these different functions have in common?

Let’s look at a simple way to describe any function, it’s called a “contract”

  • What is a Contract?
    • A formal agreement
    • A description of expected behavior
  • What do Contracts tell us?
    • What a function should do
    • What inputs a function needs
    • What a function returns

Encourage students to think about contracts in the real world. What purpose do they serve? If a contract is signed, do we expect it to be followed?

Contracts have three distinct parts

  1. Name
  2. Domain
  3. Range

The Name of a function briefly describes what the function does.

The Domain of a function is the data that the function expects.

The Range of a function is the data that the function produces.

By keeping a list of all the functions in a language, and their Domains, programmers can easily look up how each function is used. However, it’s also important to keep track of what each function produces! For example, a program wouldn’t use “star” if they were trying to produce a Number, because star only produces Images.

Domains and Ranges help programmers write better code, by preventing silly mistakes and giving themselves hints about what to do next. A programmer who wants to use “star” can look up the Domain and immediately know that the first input has to be a Number (like 100), without having to remember it each time. Instead of writing a single value there, a programmer could write a whole expression, like (25 * 4). We know this code will return an appropriate value (Number) by looking at the Range for *; therefore, the result of * can be used in place of any Number value.

When programmers write down the Domains and Ranges of each function, they write what are called contracts, to keep track of what each function needs.

Activities

Reading Contracts

Let’s look at a few example contracts - for each contract we’ll identify the Name, Domain, and Range

  • +: Number Number -> Number
  • add function
  • triangle: Number String String -> Image
  • triangle function
  • rotate: Number Image -> Image
  • rotate function

Writing Contracts

Let’s see if we can come up with contracts for some of the functions you’ve already seen. You’ll want to make sure that you’ve got your Contract Log, as this is where you’ll keep a running document of all contracts you write - both for existing functions and ones of your own creation.

  • We’ll start with contracts for simple arithmetic functions
  • +, -, *, /

Those were pretty easy as arithmetic functions only deal in Numbers. When it comes to writing functions that deal with multiple data types, looking at the Evaluation Block can give us some helpful clues.

  • The Name of each function is at the top
  • There will be a slot for each Domain element
  • The color of each slot tells you Domain type
  • The color of the whole block tells you Range
  • Color codes: Number String Image

Lesson Tip

Common mistakes when students first write down contracts include: writing values (such as “red”) instead of types (such as “String”) and forgetting arguments. Read your students’ contracts carefully, as they often indicate misconceptions that will persist and affect them later on.

Display each of the following Evaluation Blocks and ask students:

  • What is the Name of this function?
  • What is the Domain of this function?
  • What is the Range of this function?
  • Add this function’s contract to your reference

rectangle function
string-append function
text function

Wrap-up

Keep up your Contracts

As you continue programming, make sure that you document a contract for every new function you encounter or write. In the next unit, you’ll learn how to create your own functions to save work in writing expressions (this will turn out to be an essential part of writing a game). You’ll also start customizing your game with images for the elements in your game design.

Standards Alignment

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 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.
  • 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 by an algebraic expression, determine which function has the greater rate of change.
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 corresponding to the input x. The graph of f is the graph of the equation y = f(x).
  • 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.
MP - Math Practices
  • MP.1 - Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
  • MP.2 - Mathematically proficient students make sense of quantities and their relationships in problem situations. They 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.
  • MP.3 - Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
  • MP.4 - 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.
  • MP.5 - Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
  • MP.6 - Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
  • MP.7 - 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 x2 + 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.
  • MP.8 - Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x - 1) = 3. Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x2 + x + 1), and (x - 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
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 times as large as 18932 + 921, without having to calculate the indicated sum or product.
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).