Mathematics · Grade 11 · Characteristics of Functions · Term 1

Relations vs. Functions: Core Concepts

Distinguishing between functions and relations using mapping diagrams, graphs, and sets of ordered pairs, focusing on the definition of a function.

• Ontario Curriculum Expectations
• HSF.IF.A.1
• HSF.IF.A.2

About This Topic

This topic establishes the foundation for Grade 11 Functions by distinguishing between general relations and the specific criteria that define a function. Students explore these concepts through various lenses, including mapping diagrams, sets of ordered pairs, and graphical representations. In the Ontario curriculum, this transition from Grade 10 linear and quadratic relations to formal functional notation is a critical step toward higher level calculus and advanced functions.

Understanding functions is not just about passing the vertical line test. It is about recognizing predictable patterns in data and physical phenomena. Students learn to identify independent and dependent variables, which helps them model real world scenarios like fuel consumption or population growth. This topic comes alive when students can physically model the patterns and debate whether specific input-output scenarios qualify as functions through peer explanation.

Key Questions

1. How does the vertical line test communicate the fundamental definition of a function?
2. Why is it useful to differentiate between a relation and a function in mathematical modeling?
3. Analyze how changing the domain of a relation affects its status as a function.

Learning Objectives

• Classify a given set of ordered pairs, mapping diagram, or graph as either a relation or a function.
• Explain, using the definition of a function, why a specific relation fails to meet the criteria for a function.
• Analyze how restricting or expanding the domain of a relation can change its status as a function.
• Compare and contrast the graphical representations of relations that are functions with those that are not, using the vertical line test.
• Demonstrate the relationship between independent and dependent variables in a given real-world scenario, identifying if it represents a function.

Before You Start

Introduction to Coordinate Geometry

Why: Students need to be able to plot points and interpret graphs on the Cartesian plane to understand graphical representations of relations.

Sets and Set Notation

Why: Familiarity with sets and notation is necessary for understanding domains, ranges, and sets of ordered pairs.

Basic Algebraic Expressions

Why: Students should be able to evaluate simple expressions to understand the concept of input and output values.

Key Vocabulary

Relation	A set of ordered pairs, where each pair represents a relationship between an input and an output value.
Function	A special type of relation where each input value is associated with exactly one output value.
Domain	The set of all possible input values (x-values) for a relation or function.
Range	The set of all possible output values (y-values) for a relation or function.
Vertical Line Test	A graphical method to determine if a relation is a function; if any vertical line intersects the graph at more than one point, it is not a function.

Watch Out for These Misconceptions

Common Misconception

Students often believe that every graph or equation is a function.

What to Teach Instead

Teachers should provide examples of circles or sideways parabolas. Using a physical vertical line (like a ruler) on a shared graph helps students see that multiple outputs for a single input violate the definition of a function.

Common Misconception

Confusion between the terms 'relation' and 'function'.

What to Teach Instead

Explain that all functions are relations, but not all relations are functions. Peer teaching where students categorize various sets of data into 'Relation Only' or 'Function' bins helps clarify this hierarchy.

Active Learning Ideas

Stations Rotation: Function or Relation?

Set up four stations with different representations: a set of ordered pairs, a mapping diagram, a table of values, and a graph. Small groups rotate through stations, identifying if each is a function and justifying their choice using specific vocabulary like domain and range.

40 min·Small Groups

Think-Pair-Share: The Vending Machine Analogy

Students use the analogy of a vending machine to explain functions (one button leads to one specific snack). They work in pairs to create their own real world analogies, such as a person's height over time or a social insurance number, and present them to the class.

20 min·Pairs

Inquiry Circle: Domain Constraints

Groups are given a set of physical constraints, such as the height of a ball over time, and must determine the appropriate domain and range. They then swap their scenarios with another group to see if the mathematical model holds up under peer review.

30 min·Small Groups

Real-World Connections

• In automotive engineering, the relationship between engine speed (RPM) and horsepower is often modeled as a function. Understanding this allows engineers to design engines that operate efficiently within specific performance parameters.
• Biologists use functions to model population growth over time. For instance, the number of bacteria in a culture at a given hour must be a single, specific value for the model to be predictive and useful for research.
• Financial analysts model the relationship between interest rates and loan payments. It is critical that for any given interest rate, there is only one specific monthly payment amount, making this a functional relationship.

Assessment Ideas

Quick Check

Provide students with three different representations: a set of ordered pairs, a mapping diagram, and a graph. Ask them to label each as 'Relation only' or 'Function' and write one sentence justifying their choice for at least two of them.

Discussion Prompt

Present students with the scenario: 'The number of hours you study and your test score.' Ask: 'Is this always a function? Explain your reasoning.' Then, ask: 'How might changing the domain (e.g., only considering students who studied between 1 and 3 hours) affect whether it's a function?'

Exit Ticket

Give students a graph that fails the vertical line test. Ask them to: 1. Write down two ordered pairs from the graph that demonstrate why it is not a function. 2. Sketch a slight modification to the graph that would make it a function.

Frequently Asked Questions

What is the simplest way to explain a function to a Grade 11 student?

A function is a special type of relationship where every input has exactly one output. Think of it as a predictable machine. If you put in the same number twice, you must get the same result both times. If you get two different results, it is a relation but not a function.

How can active learning help students understand representing functions?

Active learning allows students to move between different representations, graphical, algebraic, and numerical, rather than just memorizing the vertical line test. By engaging in station rotations or peer debates, students must articulate the 'why' behind a function's definition, which builds a deeper conceptual map than passive note-taking.

Why is the vertical line test used for functions?

The vertical line represents a single x-value (input). If the line hits the graph in more than one spot, it means that one input has multiple y-values (outputs), which fails the definition of a function.

How do mapping diagrams help with function notation?

Mapping diagrams visually separate the domain and range. They show the 'flow' of the function, making it easier for students to see if an element in the domain is branching out to two different spots, which clearly identifies a non-function.