Introduction

Liz Berlinger, Chris Copeland, Janie McMillin, Meagan Vickers

Target Audience: Pre-Calculus Honors Students

Project Description:

Our project will foster a rich learning environment in which students will be using inquiry-based methods to learn how to create a sundial. In doing so, they will be learning about the development of math as a necessity and they will be connecting mathematics with real world applications in such a way that it will ignite interest in math. In some cases, students will be doing math without even knowing it, which when they reflect upon these experiences, they will see a greater relevance of math in their lives. Students will soon become very interested in the project when they discover that they will be building their own sundial. This will motivate students to learn because they know that they will have to use the information in class to design and build their sundial. When students use the Sketchpad program and are physically designing and constructing their sundial, they will be excited to be learning because it provides a new, unique experience and there are a lot of opportunities for the students to be creative. Students will develop an understanding in geometry and trigonometry that will allow them to make connections between real world applications and concepts in math.

The project will consist of a wide variety of activities that aim to provide unique insight into mathematics. Initially students will be learning the history of the sundial and how certain mathematical advances were necessary to even make telling time via the sun possible. Then students will engage in online activities and resources that will aid them in their understanding and learning of the concepts essential to making sundials work. Once a thorough knowledge of the content has been established, students will start to see the newly learned mathematical concepts put into practice in designing a sundial in the Geometer’s Sketchpad program. Using their work from this as a blueprint, students will finally begin to construct their own well thought-out sundial after being properly trained in safety for building. By the end of the project, students will not only have a working knowledge of concepts in trigonometry, geometry, and astronomy, but they will also have had directly applied mathematics to real world scenarios. More than just the mathematical knowledge, this project offers a chance to spark interest in learning and will hopefully inspire the students involved to see the endless possibilities behind math, science, and education.

Driving Question: How do you build your own sundial?

Overall Goals of the Project:

Our goal when creating this unit was not just to impact the students involved, but to also impact the study of effective project-based mathematics in the classroom. We strongly believe that until we, as educators, can move away from the traditional methods of teaching and towards the use of problem-based systems, that the potential of our students to become investigative thinkers will continue to be limited.

With said theory in mind, we have designed this unit to help students bring mathematics into a real-world setting. In this way we hope to increase our students’ abilities to make connections and think about mathematics in ways that are useful them. We want students to learn, and more importantly understand mathematics on a deeper level than we feel they do currently. We hope to bridge disciplines, and show students that mathematics can be learned through history and English as well as science. Through our studies of how people learn, we have found that students benefit most when they are forced to make discoveries and think for themselves. These skills are not limited to the mathematics classroom, and will affect how our students learn for the rest of their lives!

In this way, we believe that our unit will not only affect those of us involved in the short-term aspects of the project, but then go on impact the community and the education system as a whole. Parents can learn from their children’s problem-solving skills and then coworkers can learn from each other. The skills taught in the classroom will have an outreaching and all encompassing effect as the number of interactions between investigative thinkers grows exponentially. If we can change the way that people learn to think about problems and challenge themselves, then we have been successful and that success will continue to snowball into greater things.

Project Objectives:

Students will:

1) recognize relationships and properties between functions and inverse functions

2) be able to graph inverse functions

3) determine the limitations on the domain and range of inverse trig. functions

4) be capable of solving for an equation involving inverse. Trig. functions

5) explore real life applications of mathematical concepts

6) use technology to increase competence with specific topics

7) build sundials

8) Incorporate historical mathematical concepts with present day concepts

9) Understand the relationship of math to everyday needs and the need for math in society

10) Familiarize with using on-line resources to complete tasks

11) Be able to determine True North as well as understand the significance of position on the globe in relation to how the sundial is built.

12) use Geometer’s Sketchpad to construct a blueprint for their sundial.

13) learn how to use sketchpad to explore concepts in geometry and pre-calculus.

Rationale:

Knowledge of our past helps us move into the future. Often times, teachers in the classrooms hear students ask “When am I ever going to use this?” or “How does this apply to the real world?” For most teachers, these nagging questions are hard to respond to until the students are able to independently explore and discover the connections for themselves.

Current teaching methods such as lecture style and textbook instruction, have shown to be effective for a small number of students and sufficient for exposing the students to a breadth of knowledge in the subject area. While these techniques adhere strictly to the curriculum and can be beneficial to student understanding, there is much to be said for a project-based mathematics classroom. In such a classroom, curriculum will focus more on depth instead of breadth. This method also falls within the guidelines of the curriculum and NCTM standards/recommendations, but encourages and allows students to develop higher level thinking skills. The consensus among educational researchers has been that problem based classrooms force the students to make stronger connections, creating a lasting impact. Funding for the startup costs of this type of classroom will allow the students to leave their footprints on the community and enhance the world they live in.

Important to the creation of a project-based classroom are computers, software programs such as Geometer’s Sketchpad, and program licenses. Students will be constructing sundials, which will require building materials ranging fromplywood to power tools. Educational researcher Morris Kline lays “total blame for the downfall in mathematics education on the separation of mathematics from science through axiomatic training of mathematical researchers.” Thus, we are working to bridge disciplines amongst math, computer science, history, and shop class that will create for the students a rich understanding and deep appreciation for math and its applications in real world settings. Students will also be writing journals to sort out thoughts, possible solution methods, and preliminary designs for their projects.

As educators, we need to raise interest in mathematics. Research shows that students who are more interested in math and the sciences are prone to enter levels of higher education. Having more students in those fields leads to higher capabilities for conducting research which will propel the community to an overall higher level of education and advancement. This grant will enable us to help students move into those fields so that they can learn to apply mathematics to everyday life through research and exploration. Students will be able to create answers to problems that, prior to intense investigation, had no answers. Not only will our students learn important math skills, they will learn leadership, teamwork, and other character–building skills.

Background:

There are traditional ways of teaching trigonometric functions, but we have devised a unit, with the use of sundials, that goes beyond these traditional methods and employs project-based mathematics in the classroom. Sundials may not at first seem to encompass the use of trigonometry, but their construction requires a thorough knowledge of these concepts. Sundials date back to ancient times when Mayans, an intellectually advanced civilization for their time period, made breakthroughs in time telling technology. Determining the size and placement of the gnomon with respect to the size of the base is the key to a properly functioning sundial. The gnomon is a part of the sundial placed on top of the base, used to create the shadow from the sun. The Law of Sines and Cosines is used here to determine the height of the gnomon with respect to the angle created by the sun in terms of your geographic latitude. Each hour on the sundial face is placed in equal increments around the gnomon, which requires the students to have prior knowledge of angles and the unit circle. Many of the concepts required date back to Geometry including, the Pythagorean Theorem, properties of circles, and properties of triangles. Geometer’s Sketchpad is a wonderful tool that the students can use to build a template for the construction of their sundials. This program requires the knowledge of geometric concepts such as parallel and perpendicular lines, the construction of circles, and much more. The students will use this program to create templates or blueprints for the constructions of their sundials.

Books to Read:

Albert Waugh: Sundials, Their Theory and Construction

Newton and Margaret Mayall: Sundials: Their Construction and Use

René Rohr: Sundials: History, Theory, and Practice

Websites: (http://www.sundials.org)

What is a Sundial? - The Franklin Institute Online

Sundials - History, Theory, and Types of sundials from The Royal Observatory, Greenwich

Sundials - Sundial history From the National Institute of Standards and Technology

A Glossary of Sundial Terms - By John Davis

Really great resources for using Geometer’s Sketchpad:

http://mathforum.org/sketchpad/sketchpad.html

Standards Addressed:

TEKS:

(b) Introduction.

(1) In Precalculus, students continue to build on the K-8, Algebra I, Algebra II, and Geometry foundations as they expand their understanding through other mathematical experiences. Students use symbolic reasoning and analytical methods to represent mathematical situations, to express generalizations, and to study mathematical concepts and the relationships among them. Students use functions, equations, and limits as useful tools for expressing generalizations and as means for analyzing and understanding a broad variety of mathematical relationships. Students also use functions as well as symbolic reasoning to represent and connect ideas in geometry, probability, statistics, trigonometry, and calculus and to model physical situations. Students use a variety of representations (concrete, numerical, algorithmic, graphical), tools, and technology to model functions and equations and solve real-life problems.

(2) As students do mathematics, they continually use problem-solving, language and communication, connections within and outside mathematics, and reasoning. Students also use multiple representations, applications and modeling, justification and proof, and computation in problem-solving contexts

(1) The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, radical, exponential, logarithmic, trigonometric, and piecewise-defined functions. The student is expected to:

(A) describe parent functions symbolically and graphically, including y = xn, y = ln x, y = loga x, y = , y = ex, y = ax, y = sin x, etc.;

(B) determine the domain and range of functions using graphs, tables, and symbols;

(2) The student interprets the meaning of the symbolic representations of functions and operations on functions within a context. The student is expected to:

(B) perform operations including composition on functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, and graphically; and

(C) investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties.

(3) The student uses functions and their properties to model and solve real-life problems. The student is expected to:

(A) use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data;

(d) solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas.

NCTM:

In grades 9^th through 12^th students should:

Numbers and Operations:

-develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases.

Algebra:

-understand the meaning of equivalent forms of expressions, equations, inequalities, and relations;

-write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases;

-judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.

Geometry:

-use trigonometric relationships to determine lengths and angle measures;

-use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations;

-draw and construct representations of two- and three-dimensional geometric objects using a variety of tools;

-use geometric models to gain insights into, and answer questions in, other areas of mathematics;

-use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.

Measurement:

-make decisions about units and scales that are appropriate for problem situations involving measurement;

-understand and use formulas for the area, surface area, and volume of geometric figures, including cones, spheres, and cylinders;

Problem Solving:

-build new mathematical knowledge through problem solving;

-solve problems that arise in mathematics and in other contexts;

-apply and adapt a variety of appropriate strategies to solve problems.

Communication:

-organize and consolidate their mathematical thinking through communication;

-communicate their mathematical thinking coherently and clearly to peers, teachers, and others;

-analyze and evaluate the mathematical thinking and strategies of others;

-use the language of mathematics to express mathematical ideas precisely.

Connections:

-recognize and use connections among mathematical ideas;

-understand how mathematical ideas interconnect and build on one another to produce a coherent whole;

-recognize and apply mathematics in contexts outside of mathematics.

Representations:

-use representations to model and interpret physical, social, and mathematical phenomena.

National Technology Standards:

Description of Formative and Summative Assessments:

Throughout this unit, we teachers have very high expectations for our students. We want them to perform at their highest level of ability and be able to articulate that which they have learned to others.

Each week, students will learn important mathematical procedures that can be drawn from textbooks and then will use those procedures and other vital information to begin the construction phases of their group sundial. Students will be responsible for creating blueprints and templates that will be used to create their sundial. Since the each phase of the building has an important mathematical concept tied to it, there are many assessments teachers will perform each calendar day to assure the students’ understanding. Modifications to project and student expectations can be made along the way to ensure understanding of concepts and incorporate optimal learning strategies. With techniques ranging from Misconception/Preconception Checks to Concept Maps to Course-Related Self-Confidence Surveys, students will be held accountable for not only their building progress, but for the accumulation of knowledge they are expected to gain from this project and the real world applications that follow.

To a large extent, students’ final grades will be a result of their functional, final sundials as well as a test of their knowledge of the math concepts. Students are expected to understand the fundamental concepts of the project and be able to apply them. Success will be measured by determining that the students have a deeper understanding for math concepts when they are able to apply them to activities. Project-based mathematics classrooms are built on this idea. Evaluating the students’ progress along the way through the aforementioned strategies will provide educators with data to compare with traditional teaching methods. In order to assure successful mastery of concepts, teachers can use the ongoing assessments to make changes to meet students’ needs. The goal of this project is to instill the importance of mathematics in our students, as well as the drive to further their competence in the field. We will consider ourselves successful when students make connections between math concepts and other disciplines through the use of project-based mathematics over traditional teaching methods.

Breakdown of the grades:

Participation / Group work 15%

Classwork / Handouts 10%

Weekly quizzes 10%

Tests 30%

Sundial – templates 5%

Sundial – completed product 10%

Sundial – final presentation 20%