Abstract

This lesson is designed to introduce students to graphing functions.

Objectives

Upon completion of this lesson, students will:

• have been introduced to plotting functions on the Cartesian coordinate plane
• seen several categories of functions, including lines and parabolas

Standards

The activities and discussions in this lesson address the following NCTM standards:

Algebra

Understand patterns, relationships and functions

• represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules;
• relate and compare different forms of representation for a relationship;
• identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations.

• develop an initial conceptual understanding of different uses of variables;
• explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope;
• use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships;
• recognize and generate equivalent forms for simple algebraic expressions and solve linear equations

• model and solve contextualized problems using various representations, such as graphs, tables, and equations

• use graphs to analyze the nature of changes in quantities in linear relationships.

Student Prerequisites

• Arithmetic: Students must be able to:
  • perform integer and fractional arithmetic
  • plot points on the Cartesian coordinate system
  • read the coordinates of a point from a graph
• Algebraic: Students must be able to:
  • work with very simple algebraic expressions
• Technological Students must be able to:
  • perform basic mouse manipulations such as point, click and drag
  • use a browser such as Netscape for experimenting with the activities

Teacher Preparation

Students will need:

• Access to a browser
• pencil and graph paper
• Copies of supplemental materials for the activities:
  • Graph Sketcher Activity Worksheet

Key Terms

constant function constants coordinate coordinate plane

function graph negative number

Lesson Outline

These activities can be done individually or in teams of as many as four students. Allow for 2-3 hours of class time for the entire lesson if all portions are done in class

1. Focus and Review

   Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

   • Can someone tell me what a function is?
   • Will someone give me an example of a function?
   • Will someone give me anexample of something that is not a function?

2. Objectives

   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 function.
   • We are going to use the computers to learn more about functions, 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

   • Lead a discussion how functions and graphs are related.

4. Guided Practice
   • Have the students try plotting points for several simple functions to ensure that they have some skill at plotting by hand. Even if graphing calculators are available, have the students plot points on graph paper - this is a skill that is important to practice by hand. Here are a few functions that might be assigned:
     1. y = 3x - 2
     2. y = x^2
     3. y = 3 - 4x
     4. y = 4 - x^2
   • Practice the students' function plotting skills by having them check their work from the previous activity by plotting the same functions using the Graph Sketcher Tool.
   • Have the students investigate functions of the form y = _____ x + ____ using the Graph Sketcher Tool to determine what kinds of functions come from this form, and what changing each constant does to the function. Be sure to have them keep track of what they try and record their hypotheses and observations.
   • Relate these graphs to the lesson on Linear Functions to demonstrate the rationale for the terms m = slope and b = intercept in the formula Y = m * X + b.

5. Independent Practice
   • e have the students repeat the previous activity with functions of the form y = ____ x^2 + ____.

6. Closure

   • You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outlines

This lesson can be rearranged in several ways.

• Replace all Graph Sketcher activities with graphing calculator activities. Note: Depending on the graphing calculator, you might have to spend some additional time discussing setting the window ranges.
• Replace all Graph Sketcher activities with Simple Plot activities. Simpleplot is a point plotting activity, which requires that the students create tables of values for the functions before plotting.
• Limit investigations to functions with one operation as in the Function Machines lesson and/or to linear functions as in the Linear Functions lesson.

Suggested Follow-Up

After these discussions and activities, students will have more experience with functions and graphing. The next lesson, Reading Graphs, shows the students that graphs can be used to convey lots of information about a given situation.