Functions Abound

by Shelly Rogers

Introduction

Concept Map

Project Calendar

Lesson Plans

Letter to Parents

Assessments

Resources

Modifications

Grant

Grant Proposal for the Functions Abound Project

Abstract:

The Functions Abound Project is a refreshing approach to the learning and understanding of functions. Instead of resorting to the traditional textbook routine, students will be “learning by doing and creating.” Rather than the students completing various textbook questions that are easily forgotten, their experiences throughout this project will promote long-term retention of the concepts. Through this project, students will be able to:

(1) recognize different types of functions that can be used to describe real-world images,

(2) formulate equations of functions that best describe particular relationships,

(3) apply their knowledge of how to manipulate functions in order to reconstruct real-world images, and

(4) develop connections between mathematics and the real-world.

Using digital cameras and computer software, students will work together to produce products that require extensive creative and mathematical consideration. Moreover, because the products themselves are recreations of images found in the real-world, the students will gain an appreciation of mathematics, realizing its presence in every-day life.

Rationale:

When parents ask, “What is the difference between my child's mathematics class and the math I grew up with?,” Austin Independent School District strives to be able to respond as follows:

Traditionally, the goal of mathematics has been to teach students how to solve various types of mathematics problems. More emphasis was placed on "how to" solve specific problems and less emphasis was placed on building an understanding of underlying number concepts and number relationships. Today's mathematics curriculum is shifting to a more challenging emphasis of helping students become mathematical thinkers - reasoning through mathematics, solving complex problems, becoming efficient in their use of mathematics. Students are involved in making sense of their world and learning to use mathematics with skill and understanding. […] In a good mathematics classroom, students are making sense of numbers and how they relate by using facts they already know to learn new facts. Building on numbers they know and using facts to solve meaningful and complex problems helps students value mathematics and understand the need for its use. (http://www.austinisd.org/academics/ curriculum/subjects/math/acme_faq01.phtml)

In order to uphold this outlook for the present and future mathematics curriculum, we must find new approaches for getting students to learn and understand. More specifically, an area within mathematics that students typically have difficultly understanding is the concept of “the function,” and if they understand the concept, they still may have difficulty understanding how to desirably manipulate functions. Therefore, especially since the understanding of functions serves as the foundation for more advanced mathematical understandings, new ways that will better cultivate and develop the student’s mathematical knowledge are essential.

Through the Functions Abound project, students will search for geometric, two-dimensional relationships that exist in the world around them and gain experience in formulating equations of functions representative of these relationships. They will develop their ability to manipulate basic mathematic functions in such a way that their manipulations are then applicable to a real-world context. In doing so, this project will be fashioning students who understand that mathematics is a useful tool for describing the world in which we live. Furthermore, by shaping such students, bearing in mind that they are the future of our community, we can hope to see a community that advances due to its new breed of worker.

Description:

Throughout this project the students will spend several days studying specific types of functions. For instance, there will be a class period completely devoted to the parabola. Initially, the teacher will guide the students to form a connection between linear functions, studied the day before, and a parabola in order that the students can build upon their prior knowledge. The students will then use a software program called Gizmo to explore the parabola; the program is designed to help students conceptualize the cause and effect relationship of changing certain aspects of a parent function and the resulting manipulated function. Afterwards, the students will be given images that consist predominately of parabolic outlines and instructed to recreate the images using Graphmatica, another software program. The rest of the period will be spent in discussion and presentation of the images that they recreated. The class periods devoted to the other types of functions will proceed similarly.

Upon completing the class periods designated for the study of particular types of functions, the students will spend the remaining project days searching for images around them that can be recreated using many of the different types of functions. They will collect these images using the digital cameras, so that they can be saved on the computers and so that the students will then have the images with them while they use the Gizmo and Graphmatica programs to do the image-reconstructions. For the final assignment, the students will be required to find an image that they especially like and follow the same process as before in order to reconstruct it using functions.

Throughout the unit the students will generate a portfolio of their recreated images, notes as to how they constructed their images, and reflections on what they learned during the process, particularly of how they learned to manipulate functions. To conclude the Functions Abound Project, the students will present their portfolios to their fellow classmates.

To accompany the students’ work, one day during the project, a guest speaker will discuss his or her career with the students and explain how the use of functions is relevant within their occupation. Through this speaker and the students’ experiences relating functions to real-world images, the students will see mathematics in a different light. They will understand that math is intrinsic within their lives. They will also tangibly learn math, rather than trying to learn about functions and the manipulations of functions through an often distant and un-relatable textbook.

Calendar:

Meets everyday							Block Schedule: A-Day/B-Day
(~45 minute classes)							(~1.5 hour classes)
S	M	T	W	T	F	S	S	M	T	W	T	F	S
	1		2		3			1		2		3
		4		5					4		5
	6		7		8			6		7		8
	9							9
		10							10
	1	-	Intro to Graphmatica and Exploration of Transformations of Linear Functions
	2	-		Parabolic
	3	-		Polynomial and Exponential					Deeper Exploration
	4	-		Logarithmic and Power					into Various Types
	5	-		Trigonometric					of Functions
	6	-		Polar and Parametric
	7	-	Guest Speaker and/or Video
	8	-	Hunt for Functions
	9	-	Final Assignment & Portfolio Lab Days
	10	-	Presentations

Budget:

Item Description Unit Price Quantity Total Cost

Computers $700.00 5 $3,500.00

Gizmo: ExploreLearning

Software Subscription $799.00 1 $799.00

Graphmatica Software $0.00 $0.00

Digital Cameras $400.00 5 $2,000.00

Memory Cards $70.00 5 $350.00

Photo Printer $200.00 1 $200.00

Total Project Cost $6,849.00

Evaluation Plan:

The students’ work and portfolios will provide tangible evidence of whether or not the students are truly understanding the concepts. If a student is unable to create an image that remotely resembles the original image, then the student apparently is having difficultly understanding the unit. Therefore, obtaining a superficial evaluation of the students can be done easily. However, because the students are required to include notes as to how they reconstructed their images, as well as reflections on what they have learned, the teacher easily can evaluate the students’ understanding in much further depth.

This evaluation only pertains to short-term analysis however. Long-term analysis of the worthiness of this project will be much more beneficial, and there are several ways to assess this. The TAKS scores (specifically related to relative math sections) of the students who participate in this project can be compared to those who do not participate in this project. Also, since the project will involve students at the Algebra I level, these students can be evaluated in comparison to other students as they advance to higher level math courses.

Letters of Support:

(told not to worry about this section)

Vitae:

(A hard copy of my resume will be provided to the instructor.)