Vernier Software & Technology

# Stepping to the Greatest Integer: The Greatest Integer Function

## Introduction

Not all mathematical functions have smooth, continuous graphs. In fact, some of the most interesting functions contain jumps and gaps. One such function is called the greatest integer function, written as y = int x. It is defined as the greatest integer of x equals the greatest integer less than or equal to x. For example, int 4.2 = 4 and int 4 = 4, while int 3.99999 = 3.

In this activity, you will create a function similar to the greatest integer function graph by having a group of students stand in a line in front of a Motion Detector and then step aside one by one. The equation for this graph, in the general form, is

${\text{y = A int (Bx) + C}}$

You can find appropriate values for the parameters A, B, and C so that the model fits the data.

## Objectives

• Use a Motion Detector to collect position data showing evenly-spaced jumps in value.
• Model the position data using the greatest integer function.

## Sensors and Equipment

This activity features the following Vernier sensors and equipment.

### Option 2

You may also need an interface and software for data collection. What do I need for data collection?

## Real-World Math with Vernier

See other experiments from the lab book.

 1 Walk the Line - Straight Line Distance Graphs 2 Making Cents of Math: Linear Relationship between Weight and Quantity 3 Pool Plunge - Linear Relationship between Water Depth and Pressure 4 Funnel Volumes - Volume and Weight 5 Keep It Bottled Up - Rates of Pressure Increase 6 Mix It Up - Mixing Liquids of Different Temperatures 7 Spring Thing - Newton's Second Law 8 Stretch It to the Limit - The Linear Force Relation for a Rubber Band 9 What Goes Up - Position and Time for a Cart on a Ramp 10 That's the Way the Ball Bounces - Height and Time for a Bouncing Ball 11 Walk This Way - Definition of Rate 12 Velocity Test - Interpreting Graphs 13 From Here to There - Applications of the Distance Formula 14 Under Pressure - The Inverse Relationship between Pressure and Volume 15 Light at A Distance - Distance and Intensity 16 Chill Out: How Hot Objects Cool 17 Charging Up, Charging Down - Charging a Capacitor 18 Bounce Back - The Pattern of Rebound Heights 19 Sour Chemistry - The Exponential pH Change 20 Swinging Ellipses - Plotting an Ellipse 21 Lights Out! - Periodic Phenomena 22 Tic, Toc: Pendulum Motion 23 Stay Tuned: Sound Waveform Models 24 Up And Down: Damped Harmonic Motion 25 How Tall? Describing Data with Statistical Plots 26 And Now, the Weather - Describing Data with Statistics 27 Meet You at the Intersection: Solving a System of Linear Equations 28 Titration Curves: An Application of the Logistic Function 29 Clock Design: Period and Length of a Simple Pendulum 30 Graph It in Pieces: Piecewise Defined Functions 31 Stepping to the Greatest Integer: The Greatest Integer Function 32 Crawling Around: Parametric Plots

### Activity 31 from Real-World Math with Vernier Lab Book

#### Included in the Lab Book

Vernier lab books include word-processing files of the student instructions, essential teacher information, suggested answers, sample data and graphs, and more.

Dev Reference: VST0350