Building a Catapult
by Stephen Alwon
and Sherwin Fernandes
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Description Mathematics in
AISD is taught for the most part as a series of lectures disparate from
everything else the students learn in their other classes. In using a project based around designing a
catapult, students will be able to see mathematics come to life in a manner
that brings together physics, engineering, and history in a way that they
have not seen before. The students
will participate in a series of labs during which they will construct both
the basic theory of projectile motion and the basic principles and behavior
of quadratic functions. Students will
then be able to put their knowledge to use by building a working catapult and
rationalizing how far it will launch a ball.
Through this project, the students will be able to bring together
mathematics and physics in a manner that will enable them to solve a problem
that has been around for thousands of years, giving them a historical
background for their project.
Additionally, the students will have a working knowledge of the basic
concepts of physics, allowing them to be better prepared when they enter a
physics course. Finally, the project
will be fun, and will encourage students to think deeply about mathematics
because they want to, not just in order to pass a test or so they can graduate. Driving Question What’s the most efficient way to launch a catapult? Overall
Goals/Project Objectives Our project will
attempt to get students interested in mathematics by bringing other
disciplines into the mathematics classroom.
In this way, students will be able to see how mathematics can relate
to subjects they most likely will not have seen a connection to before, such
as history. Students will be motivated
to learn the mathematics not necessarily just to learn math, pass a test, or
graduate high school, but as a way to better understand how history was
shaped for hundreds of years. Students
whose primary scholastic interests were in social studies before
participating in the project will have a different outlook on mathematics and
the ways it can be used. In this way,
these students will have a new appreciation for mathematics and will be more
excited to participate in math classes in the future. Conversely, students whose main interests
lie in the sciences will have a new outlook on history, and will be more
excited to participate in those classes.
Our project will also prepare students for when they take physics by
introducing them to some of the more basic concepts they will learn in that
class. This will enable the teacher
who has these students to save time in covering some of the basic concepts,
and instead allow them to spend more time on the concepts that the students
have not been presented previously. Rationale One of the main problems with the way mathematics is taught in AISD is
that it is taught in a vacuum, with rarely any connection being made between
the mathematics classroom and any other class they are taking. Students are made to forget everything they
learned during the previous hour and start entirely fresh when they enter
their mathematics class. With this
project, students will bring together concepts from various subjects such as
engineering, physics, and history while they are learning mathematics. The problem of how to best launch a
projectile is one that has been around for hundreds of years, and students
will be able to see this problem come to life in their world history
class. The ability to bring together
mathematics and history will encourage students who are interested in either
subject to seek out the other.
Students will discover how wars were fought for thousands of years,
making the problem of how to build a working one relevant to students who
enjoy history and social studies, while students who enjoy mathematics and
the sciences will be able to see the social studies in a new light. The project will also introduce students to
the engineering process, from learning the necessary physics and mathematics
to designing and constructing a working catapult. Background Projectile motion is an easily observable phenomenon, but requires knowledge of energy and trigonometry to understand. The building of a projectile launcher is one that is engaged in by many people in amateur competitions around the country. This project will allow students to study the mathematical and physical concepts that go into building a projectile launcher, and eventually build their own for the purpose of striking a target a set distance away. This project focuses on the most efficient way to launch a projectile, and students will eventually calculate which launcher is the most efficient (which utilized the greatest percentage of its potential energy in accomplishing the task of hitting the target). Students will begin the project by learning how to calculate stored or potential energy in springs, gravity and then kinetic energy. They will compare the energy in a spring with extension and the variation of energy with speed. They will then plot the horizontal and vertical position of a falling ball and use that to derive the equations of motion. Stroboscope pictures of a falling ball will be used to plot each component versus time. The division of a force into its vertical and horizontal components will then be investigated using similar triangles for a derivation of sine and cosine functions. A single fixed angle triangle will first be used to show that the angle is proportional to the side lengths. The angle will then be varied and the ratio recorded. The graph of the ratios versus the angle will show the sine, cosine and tangent functions. Using these
concepts, students will decide on which design of launcher would be best,
design and make predictions for it. When the design is approved, students
will construct the actual model from an erector set and test their launchers. Standards Addressed TEKS (a) (3) Functions, equations, and their relationship. The study of
functions, equations, and their relationship is central to all of
mathematics. Students perceive functions and equations as means for analyzing
and understanding a broad variety of relationships and as a useful tool for
expressing generalizations. (5)
Tools for algebraic thinking. Techniques for working with functions and
equations are essential in understanding underlying relationships. Students
use a variety of representations (concrete, numerical, algorithmic, graphical), tools, and technology, including, but not
limited to, powerful and accessible hand-held calculators and computers with
graphing capabilities and model mathematical situations to solve meaningful
problems. (b) (2) The student understands the importance of the skills required
to manipulate symbols in order to solve problems and uses the necessary
algebraic skills required to simplify algebraic expressions and solve
equations and inequalities in problem situations. Following are performance
descriptions. (A) The student uses tools including matrices,
factoring, and properties of exponents to simplify expressions and transform
and solve equations. (B) The student uses complex numbers to describe the
solutions of quadratic equations. (C) The student connects the function notation of y =
and f(x) =. (c) (1) The student connects algebraic and geometric representations
of functions. Following are performance descriptions. (A) The student identifies and sketches graphs of
parent functions, including linear (y = x), quadratic (y = x2),
square root (y = Ö x), inverse (y = 1/x), exponential (y = ax),
and logarithmic (y = logax) functions. (B) The student extends parent functions with
parameters such as m in y = mx and describes
parameter changes on the graph of parent functions (d) Quadratic and square root functions: knowledge and
skills and performance descriptions. (1) The student understands that quadratic functions can
be represented in different ways and translates among their various
representations. Following are performance descriptions. (A) For given contexts, the student determines the
reasonable domain and range values of quadratic functions, as well as
interprets and determines the reasonableness of solutions to quadratic
equations and inequalities. (B) The student relates representations of quadratic
functions, such as algebraic, tabular, graphical, and verbal descriptions. (C) The student determines a quadratic function from
its roots or a graph. (2) The student interprets and describes the effects of
changes in the parameters of quadratic functions in applied and mathematical
situations. Following are performance descriptions. (A) The student uses characteristics of the quadratic
parent function to sketch the related graphs and connects between the y = ax2
+ bx + c and the y = a(x - h)2
+ k symbolic representations of quadratic functions. (B) The student uses the parent function to
investigate, describe, and predict the effects of changes in a, h, and k on
the graphs of y = a(x - h)2 + k form of a
function in applied and purely mathematical situations. (3) The student formulates equations and inequalities based on quadratic functions, uses a variety
of methods to solve them, and analyzes the solutions in terms of the
situation. Following are performance descriptions. (A) The student analyzes situations involving
quadratic functions and formulates quadratic equations or inequalities to
solve problems. (B) The student analyzes and interprets the solutions
of quadratic equations using discriminants and
solves quadratic equations using the quadratic formula. (C) The student compares and translates between
algebraic and graphical solutions of quadratic equations. (D) The student solves quadratic equations and
inequalities. NCTM Instructional programs from prekindergarten
through grade 12 should enable all students to--
Instructional programs from prekindergarten
through grade 12 should enable all students to--
National
Technology Standards Students use technology to locate, evaluate, and
collect information from a variety of sources. Students use technology resources for solving problems
and making informed decisions. Students practice responsible use of technology
systems, information, and software. Formative and
Summative Assessments Our project will
depend on two main techniques for measuring the progress of our
students. The first will be the labs
and homeworks which will be assigned on a regular
basis. The labs will be done mostly in
class, with the students able to finish outside if necessary. Through these, the students will be able to
witness first hand the physical and mathematical concepts they are to learn,
and successful completion of the lab write-ups will depend on them thoroughly
understanding the concepts the labs are designed to exhibit. These will let us know that the students
are working during class time and that their work is productive. The homeworks
will be designed to evaluate the concepts that are more mathematical in
nature – problems such as finding quadratic roots. These will enable us to quickly judge if
the students are “getting” the mathematical concepts involved in
the project. The other evaluation
technique will be through the final project itself – each group’s
design of the catapult and their rationale of why their catapult works and
how far they expect it to shoot. This
will allow us to determine how well the students brought together all of the
concepts involved and how well they are able to communicate their knowledge
of those concepts. Below is the rubric
for the grading of the final project.
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