Functional Graphs

Abstract

In this lesson, students will graph input/output pairs from a simple linear function in order to gain an understanding of basic linear functions.

Objectives

Upon completion of this lesson, students will:

  • understand the graphical nature of linear functions
  • understand the translation of functions from equations to graphical representations

Standards Addressed

Grade 3

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions.
    • The student demonstrates algebraic thinking.

Grade 4

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences.
    • The student demonstrates algebraic thinking.

Grade 5

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences.
    • The student demonstrates algebraic thinking.

Grade 6

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences.
    • The student demonstrates algebraic thinking.

Grade 7

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 8

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 9

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 10

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Sixth Grade

  • Expressions and Equations

    • Apply and extend previous understandings of arithmetic to algebraic expressions.
    • Represent and analyze quantitative relationships between dependent and independent variables.

Seventh Grade

  • Expressions and Equations

    • Use properties of operations to generate equivalent expressions.

Functions

  • Building Functions

    • Build a function that models a relationship between two quantities

  • Interpreting Functions

    • Understand the concept of a function and use function notation

  • Linear, Quadratic, and Exponential Models

    • Construct and compare linear, quadratic, and exponential models and solve problems

Grades 3-5

  • Algebra

    • Represent and analyze mathematical situations and structures using algebraic symbols

Grades 6-8

  • Algebra

    • Understand patterns, relations, and functions

  • Data Analysis and Probability

    • Develop and evaluate inferences and predictions that are based on data

Grade 6

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 5: The learner will demonstrate an understanding of simple algebraic expressions.

3rd Grade

  • Algebra

    • Content Standard 2.0 The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety of contexts and problems using graphs, tables, and equations.

Grade 6

  • Number, Operation, and Quantitative Reasoning

    • 2. The student adds, subtracts, multiplies, and divides to solve problems and justify solutions.

  • Patterns, Relationships, and Algebraic Thinking

    • 4. The student uses letters as variables in mathematical expressions to describe how one quantity changes when a related quantity changes.

Grade 7

  • Number, Operation, and Quantitative Reasoning

    • 2. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions.

Grade 8

  • Number, Operation, and Quantitative Reasoning

    • 2. The student selects and uses appropriate operations to solve problems and justify solutions.

Textbooks Aligned

Book 1

  • Module 4 - Mind Games

    • Section 5: Graphing on the Coordinate Grid
    • Section 5: Expressions and Equations Multiples

6th

  • Module 4 - Statistical Safari

    • Section 4: Stem-and-Leaf Plots, Decimal Division, and Graphing

Student Prerequisites

  • Arithmetic: Students must be able to:
    • perform integer arithmetic
    • plot ordered pairs in the Cartesian coordinate system
    • draw lines connecting points in the Cartesian coordinate system
  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click, and drag
    • use a browser for experimenting with the activities

Teacher Preparation

  • Access to a java-enabled browser
  • Pencil and paper
  • A copy of the worksheet for each student

Key Terms

coordinate plane

A plane with a point selected as an origin, some length selected as a unit of distance, and two perpendicular lines that intersect at the origin, with positive and negative direction selected on each line. Traditionally, the lines are called x (drawn from left to right, with positive direction to the right of the origin) and y (drawn from bottom to top, with positive direction upward of the origin). Coordinates of a point are determined by the distance of this point from the lines, and the signs of the coordinates are determined by whether the point is in the positive or in the negative direction from the origin

coordinates

A unique ordered pair of numbers that identifies a point on the coordinate plane. The first number in the ordered pair identifies the position with regard to the x-axis while the second number identifies the position on the y-axis

function

A function f of a variable x is a rule that assigns to each number x in the function's domain a single number f(x). The word "single" in this definition is very important

graph

A visual representation of data that displays the relationship among variables, usually cast along x and y axes.

line

A continuous extent of length containing two or more points

linear

An equation or graph is linear if the graph of the equation is a straight line

linear function

A function of the form f(x) = mx + b where m and b are some fixed numbers. The names "m" and "b" are traditional. Functions of this kind are called "linear" because their graphs are straight lines

Lesson Outline

  1. Focus and Review

    Give students a simple problem:

    • There's a "number cruncher" that takes a number and changes it somehow by following a rule.
    • If I put a 1 into the number cruncher, it spits out a 2. What do you think the rule is?
    • It might be doubling the number or it could be adding one. Those are good guesses, but we need more information.
    • I put a 2 into the same number cruncher, and it spits out a 3. What do you think the rule is now?

  2. Objective

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today, class, we are going to learn more about rules like this and how they relate to graphs.
    • We are going to use the computers to learn more about this, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first.

  3. Teacher Input

    Introduce students to the Whole Number Cruncher. Explain that this is just like the number cruncher from before. Demonstrate how it works by asking students what input you should use and observing the output. Ask students questions such as the following:

    • How many numbers do you have to put in before you know what the formula is? Are you sure you don't need more? Could you figure it out with fewer inputs?
    • What numbers can you use as input?
    • Are there numbers that make it easier to figure out the formula? Why or why not?

  4. Guided Practice

    Have students work in groups to develop their own input/output list, using the Whole Number Cruncher. Have all students use the same function by using seed random. Remind all students to stay on that function.

    Once students have developed lists of input/output pairs, use a projector to show the students GraphIt. As a class, plot input/output pairs on the graph.

    Ask students to solve the formula in Whole Number Cruncher. When students have the computer confirm the right answer, type that formula into GraphIt. Ask students questions such as the following:

    • Does the graph confirm that you were correct? How do you know?
    • What shape is the graph of the formula?

  5. Independent Practice

    Ask students to make predictions about the graphs they're going to look at. Ask questions such as the following:

    • Do you think the graph of the formula goes through every input/output pair? Are there any exceptions?
    • Do you think all functions have graphs that look like lines? Why or why not?

    Have students work in groups to complete the Functional Graphs Worksheet.

  6. Closure

    Lead a class discussion of the findings of the lesson. Ask questions such as the following:

    • Why do you think it's important that the graph is "linear"?
    • Can that help us at all in making predictions? How?

    Take a moment to address the misconception that all functions are lines. Explain that the functions we looked at are all called "linear" functions, which is why they all made lines. There are many different functions out there, and only some of them are linear.

Alternate Outline

If only one computer is available for the classroom, this lesson can be rearranged in the following way:

  • The teacher may do this activity as a demonstration. Have students volunteer different inputs for the Whole Number Cruncher. Then graph those input/output pairs as a class using GraphIt.
  • As a class, analyze the different graphs that are created and discuss the results.

Suggested Follow-Up

Students can continue to explore the nature of functions by considering the following lesson plans: