Mathematics with Creative Designs

By Iftikhar Husain

This unique and exceptionally creative use of TI graphing calculators enables one to Explore Algebra with Creative Designs. The designing technique encourages students to quickly see and study graphs of functions encountered in Pre-algebra through Calculus courses. The activities are designed to help students to visualize concepts, predict results, and make connections between algebra and geometry. A Mathematical Art gallery to exhibit the students' creative work, with its mathematical descriptions, can be organized. Students make their own posters, banners, and print their own T-shirt, which inspire and motivate other students in the learning process. Teachers can make their own classroom Posters and Borders demonstrating various mathematical concepts.

• Language: English
• Publisher: hhusains4ever
• Release date: Apr 28, 2018
• ISBN: 9780971864016

Iftikhar Husain

Mr. Iftikhar Husain, recipient of Governor of New Jersey Teacher Recognition Award 2005-06, nominee for the President of United State Award 2006 and author of “Mathematics with Creative Designs”. He has presented in many local, regional, state, national, and international level mathematics conferences like New Jersey Education Association AMTNJ of New Jersey, NJEA Convention, AMTNY of New York, AMTNE of MA, T3, and NCTM, New Jersey Rutgers University. Mr. Husain has Master of Science (Physics) degree from Aligarh Muslim University in India and recently retired as a mathematics teacher at the University High School in Newark, New Jersey. All of his presentations teaching approaches strongly support “Learning Mathematical Concepts Visually” which helps learners to gain the knowledge in the most natural way and for life. Mathematics with Creative Designs becomes lively to students, teachers and parents. As a teacher I try to reach to the level of individual student’s understanding of mathematical concept and work from there. I adopt various methods while teaching students based on their individual needs; it helps students when I use a visual/ animated approach. I have created many visual math activities with animation.

Book preview

Mathematics with Creative Designs

Unique Graphing Calculator Activities for Middle Schools and Above

Teaching and Learning Concepts of Algebra, Geometry, and Trigonometry by Creating Designs on a Graphing Calculator

BY

Iftikhar Husain

Retired Mathematics Educator

Newark Public Schools

Newark, New Jersey

USA

Published and distributed by:

INAZ ENTERPRISES

1701 VAN NESS TERRACE

UNION

NEW JERSEY 07083

E-mail: [e-mail]

[e-mail]

Forward

AN IDEA MAY BE BORN fresh from one’s imagination, or it may be inspired through a number of circumstances, be it times of happiness, of tragedy, or of escape. Hidden behind, the numbers, equations, and digital graphing lessons in this book, lies a very personal story that served as an inspiration for this book.

In the winter of 1999, my younger sister, Laiq Fatima visited from India with a horrible affliction – cancer.  She had come with the hope that some intervention could be made to help prolong her life. Unfortunately, much of the time she had remaining in those last several months was lying in a hospital bed.  I remember those long nights I spent with her very well.  It was a very emotional time for me that was flowing with feelings of love, fear, despair, and hope. So much at the time had made no sense to me that I would often take refuge in the world of mathematics where there are rules to follow, and a clear sense of black and white, or right and wrong. Staring at the ceiling for hours, I began to see the lines and shapes that made up that room.  I then wondered if I could change my visual perspective to a mathematical one or vice versa.  As it turns out, I could.

Indeed, many of the figures or outlines that we see in our everyday lives can be described in a mathematical way.  Furthermore, in the process of doing so, our understanding of what we perceive or more importantly, what we can describe and communicate through equations and graphs become more complete.  This book is a first step in understanding this concept.  I strongly believe it will enhance your ability to learn the art of mathematics by changing your perspective and bolstering your fundamental knowledge in algebra, geometry, trigonometry, and calculus.

My sister, Laiq, did not have a background in mathematics, though she could certainly appreciate the notion of changing one’s outlook or seeing the world around us in a different light. I have dedicated this book to her loving memory, which served as my inspiration to set free my imagination into a new field in mathematics.

Iftikhar Husain

Table of Contents

Mathematics with Creative Designs

Forward

Mathematics with Creative Designs

How is the creative design approach unique?

How does it work?

How is a design made?

Linear Designs

Linear Functions with Restricted Domain

Designs by List Options of TI-8X family

Curve Stitching for TI-8X family Calculators

Quadratic Functions

Cubic Functions

Conic Relations

Trigonometric Functions

Designs on TI-89 family and Voyage 200

Programming on TI-8X family

Useful Calculation Tips

Evaluating Equations:

Projects based Activities

Conic Sections Projects

Trigonometric Functions Activities

Solving Trigonometric Equations

TI-8X Graphing Calculator Solution:

TI-8X Calculator’s different approach:

Selected Answers

Mathematics with Creative Designs

OBJECTIVES:

The student will be able to graph and write equations of desired functions and functions with restricted domain.

The student will be able to create a design using equations of various functions on a graphing calculator.

The student will be able to make connections between algebra and geometry.

The student will be able to find an equation for a desired graph using Stat Plot and various Regression options on TI-8X family graphing calculators.

The student will be able to write mathematical programs on a graphing calculator.

The student will be able to evaluate algebraic expressions, equations, and find the shaded area between the curves using TI-8X family graphing calculators.

Introduction:

The activities provide instruction of mathematical concepts of all levels by creating geometric and non-geometric designs on a graphing calculator. This design-based graphing calculator activity serves as an interactive calculator and a worksheet program. It is designed to develop the complete conceptual knowledge of some of the functions that are included in pre-Algebra, Algebra 1, Geometry, Algebra II, Trigonometry and Calculus mathematics courses.

Furthermore, it explores the graphs of the linear, quadratic, cubic, and trigonometric functions and other functions, typically found in a high school algebra curriculum. In addition, this workbook also explores graphing non-functional equations like circles, and ellipses. Indeed, by creating an interactive tool that unites functional equations with innovative graphics, this text hopes to bridge the gap between abstract concepts such as functions with creative educational designs.

How is the creative design approach unique?

ONE OF THE GREATEST challenges in generating interest in mathematical theory is to overcome the abstract concept of a functional equation.  This book hopes to develop the learning process through visual construction.  In doing so, both conceptualizing an equation and grasping its practical application are achieved.  The activities in this book provide readers the opportunity to see the immediate practical application of various mathematical concepts.  Moreover, the unlimited creative appeal can be applied not only to students, but teachers as well.

All topics are chosen to correspond to the maturity and interest levels of different mathematics classes. Indeed, many of the activities are true exploration. By completing their classroom exercises, students are led to discover many important and fascinating facts. For example design 3, leads students to discover the relevance of the equation of a line passing through a specific point parallel to yet another line. The work is designed to be fun, and because the students are engaged in self-discovery, their understanding is deeper and their retention is longer.

How does it work?

THE STUDENT WILL STUDY graphs of elementary functions as linear or quadratic functions. A graphing calculator and its related designs will allow the student to compare, analyze, and make generalizations about the graphs. The material is created so the reader can quickly appreciate how individual coefficients affect the shape of the graph in different forms of a functional equation.

Once a student has a complete grasp of a graph and its equation, he or she can incorporate the skill to create a design. In creating a design, the purpose of writing an equation for the desired graph is immediately realized. Furthermore, designing technique can promote self-motivation and limitless innovation. By simply linking the calculator to a computer, mathematical designs may be printed. If properly applied, even a mathematical art gallery is not out of the question.

How is a design made?

THE DESIGN-BASED ACTIVITIES are concept oriented and constitute a direct application of Algebraic knowledge.  For instance, the following example relates to a regular hexagon. Students must first visualize a design such as the hexagon.  Six linear equations, one for each side, are written.  Because it is a regular hexagon, each interior angle must be 120 degrees. Students will then find the slope of each line and its y-intercept, in order to write its equation and create a basic outline.  Once the desired hexagon is plotted on a graphing calculator, students then create a more detailed image by shading the hexagon.

The Students may create different designs by shading between various equations despite having the same six basic equations for the hexagon. Consequently, the activity forms the groundwork for understanding calculus. Most of the designs deal with the slope of a line, an equation of a function, and the area between the curves.  Respectively, these are enforcing the concepts of the first derivative, equation(s) of curves, and the definite integral.

What are the types of activities?

The designing activities focus on the same graphs that appear as a standard part of algebra courses.  These include linear, quadratic, cubic, and trigonometric functions, along with functions pertaining to restricted domains and ranges. The student who can visualize graphs of various functions will be able to write the equation of a desired graph in a variety of forms.

The unlimited design patterns create an infinite number of individual or group activities.  The activities are developed for direct use with a Texas Instrument TI-8X family, TI-89, and Voyage 200 graphing calculator but can easily be adapted for use with other models. One should check the graphing calculator manual for specific differences if other models are used.

What is the teacher’s role?

The teacher’s primary role is to introduce theories that are necessary to successfully complete a designing activity and master its concept. These concepts are listed under each heading. For example students must have knowledge of various forms of linear equations before creating a design using linear functions. Designing technique motivates students in the learning process.

Another key role of the teacher is to introduce the various keys and features of the graphing calculator to the students. For instance, the next page is designed to help teachers in the instruction of multiple features; such as setting a window, shading the area between the curves, adding and deleting text, storing and recalling a design etc.

Creating a design should incorporate not only the series of commands that build the image, but the fundamental logic behind its construction. Two examples on page 11 and 13 were developed to demonstrate and stimulate interest. In example 1 various shading pattern features on the graphing calculator are discussed.

The mathematics behind the design of example 1 can be discussed based on the grade level. For example referring to the design it can be discussed that the square has eight congruent isosceles right triangles. Furthermore the class will explore the possibility of having more such right triangles. Or the length of the diagonal of the square can be calculated, using the distance formula after the coordinates of its vertices are found. Teacher can also discuss the properties of 45° – 45° – 90° special right triangle.

These demonstrations serve to bring together mathematical concepts and produce visual graphics to stimulate the mind. Often, only one or two demonstrations are necessary before a student can begin making his or her own designs by applying the very mathematical concepts.

Mathematics teachers can have their classroom decorations custom made. Classroom borders can be made displaying students’ designs, like the one shown below.

TEACHERS CAN ALSO MAKE their own classroom poster showing a design with its mathematical description.

Students can make their own T-shirt with their designs. T-shirt making motivates other students in the learning process.

Shading the area between the curves

It is important to learn how the shading is done on TI – 8X graphing calculators. Shading is the one makes a design unique, it is therefore, important to spend time to master the technique. In the following paragraphs you will learn how a distinct design is made using different shading commands, how setting the calculator’s window differently changes a design, how to write text, how to erase the unwanted portion of the design, and how to store & recall a design.

Window:

Use the window feature to set the parameters for the viewing window, including not only the range for each axis but also the value of each tick mark on the graph. The window feature allows you to adjust how much of the graph you wish to view. When you press [WINDOW] on your calculator keypad, a list of values appears as shown below:

THE ABOVE WINDOW EDITOR defines the displayed portion of the coordinate plane. xmax and xmin  mean the minimum and maximum values on the horizontal scale.

xscl represents how many units you wish each tick mark on the axis to represent. Likewise ymax, ymin and yscl represent similar values along the vertical axis. If you wish to turn off the tick marks, set xscl = 0 and