2012 Summer Programs Catalog
Math Courses

Mathematics can be described as a language, a tool, a science, and an art. CTY math courses aim to give students the skills, knowledge, and perspective to understand the multifaceted nature of mathematics and, by doing so, enrich and accelerate their explorations of advanced math content. Students move beyond basic skills to gain a greater understanding of both the underlying meanings of mathematical concepts and the intriguing ways math can be applied and extended to a wide range of contexts.

CTY math courses offer a variety of opportunities for students to investigate advanced mathematical concepts through a process of discovery and engagement that promotes a lifelong interest in the discipline. From developing multiple strategies for solving complex mathematical problems to using patterns and algebraic reasoning for making predictions, CTY math courses allow students to build a strong foundation to enrich or accelerate their studies in mathematics.

Students who wish to seek credit or placement for CTY courses need to work closely with their home schools before the summer to determine the appropriate steps to take.

Please refer to our Eligibility web page. Sample syllabi for all courses are also available. The following math courses are listed below:

Math Problem Solving
Geometry and Spatial Sense
Numbers: Zero to Infinity
Great Discoveries in Mathematics
Inductive and Deductive Reasoning
Data and Chance
Introduction to Robotics

Math Problem Solving

Can five 20' by 18' carpets lying flat with no overlap fit in a 40' by 50' room? How can you precisely measure two liters of water using only a four-liter pitcher and a three-liter pitcher? How many different ways can you add up four even, positive numbers to get a sum of 16?

Problem solving in mathematics is far more complex than translating a word problem into numbers and symbols and applying an established method. It involves finding a path to a solution when there is no clear place to begin. In this course, students learn general strategies for solving problems that involve a wide range of mathematical concepts. Challenging problems lead students to use varied approaches such as drawing diagrams and graphing, making lists, estimating, eliminating unreasonable possibilities, identifying patterns, guessing and checking, and manipulating variables.

Working individually or in small groups, students learn to ask precise and thought-provoking questions, to match appropriate strategies to particular problems, and to effectively communicate their thought processes along the way. Demonstrations, activities, games, and explorations are incorporated to nurture students as critical thinkers and creative problem solvers, strengthening their mathematical-reasoning abilities and preparing them for future study in discrete math, probability, cryptology, and other growing fields of mathematics.

By the end of the course, students are able to demonstrate several methods for determining that the carpets can’t fit, to measure out exactly two liters of water, and to show that there are five different ways to get 16 by adding four even, positive numbers.

Sample text: Materials compiled by the instructor.

Students must have completed grades: 2 or 3

Session 1: Sandy Spring
Session 2: Alexandria, Sandy Spring

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Geometry and Spatial Sense

Spatial understanding is necessary for interpreting, understanding, and appreciating our inherently geometric world. Many tasks a person performs—whether rearranging furniture in a room, solving a jigsaw puzzle, or laying out a garden—require good spatial reasoning abilities.

Through interactive investigation and discussion, students discover mathematical relationships such as congruence, symmetry, and reflection. They learn geometric formulas to calculate area, surface area, perimeter, circumference, and volume and are asked to explain their findings using geometric terminology. Students explore the geometric properties of regular polygons and create their own polyhedra using paper-folding techniques. Tessellations, similarity, measurement, polygons, polyhedra, and curved shapes are investigated through two- and three-dimensional drawings and physical models. Students also construct classic geometric figures using a straight edge and a compass as they are introduced to the essential elements of geometry. Through explorations and demonstrations, students leave this course with greater spatial sense as well as a solid understanding of foundational geometric concepts. 

Sample text: Materials compiled by the instructor.

Students must have completed grades: 3 or 4

Session 1: Brooklandville, La Jolla, Pasadena, Sandy Spring, Windward
Session 2: Alexandria, Pasadena

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Numbers: Zero to Infinity

What does a subatomic particle measured in femtometers have in common with a galaxy measured in light years? Both are a part of the uniquely human effort to quantify the world around us. In this course, students explore numbers, from the very small to the unimaginably large, and learn how numeric representations help to explain natural phenomena such as time, distance, and temperature.

Moving beyond traditional arithmetic, this course centers on hands-on activities that develop understanding of the scope and scale of numbers. Students consider such questions as: if your dog were the size of a dinosaur, how much dog food would you need? They develop approximation and computational strategies for learning scientific notation, and determine whether answers to problems are reasonable. In examining the diversity of measurement systems, students learn the origins of some familiar and unfamiliar methods of measurement, and invent their own units of measurement. Additionally, students use dimensional analysis to investigate conversions between different scales or systems of measurement. They apply concepts of ratio and proportion by constructing and analyzing scale models of our solar system, the human body, and other objects in our natural world.

Note: For many aspects of this course, students are asked to work without a calculator. Calculators are used only when extensive computations are needed.

Sample texts: Materials compiled by the instructor; a supplemental text such as Gulliver’s Travels, Swift.

Students must have completed grades: 4 or 5

Session 1: Brooklandville, La Jolla, Sandy Spring, Windward
Session 2: Alexandria, Sandy Spring, Windward

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Great Discoveries in Mathematics

From ancient to modern times, mathematics has been instrumental in the development of science, engineering, and philosophy. In this math course, students consider the questions and problems that have fascinated humans across cultures since the beginning of recorded history.

Throughout the course, students explore number systems and mathematical concepts first considered by early cultures, including the Egyptians, Greeks, Mayans, Babylonians, and Chinese. Additionally, they consider this newfound conceptual knowledge in its historical context. Students examine number systems with different bases, and in doing so, gain a much deeper understanding of the base ten number system. Through hands-on explorations, they learn about great mathematical discoveries throughout time, such as Pascal’s Triangle, the Pythagorean Theorem, the Golden Ratio, pi, and Zeno’s paradoxes.

By examining the historical development of major mathematical ideas, students leave the course with a greater awareness of a wide range of topics within mathematics, including number theory, algebra, and geometry. They acquire a solid background in mathematical concepts they will encounter in more advanced course work.

Sample text: Materials compiled by the instructor.

Students must have completed grades: 5 or 6

Session 1: Palo Alto
Session 2: Palo Alto

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Inductive and Deductive Reasoning

Reasoning, logic, and critical thinking skills are the building blocks of intellectual inquiry. This course focuses on developing these skills through problem solving, puzzles, and exposure to a wide range of topics in mathematics. Students learn to distinguish between inductive and deductive reasoning and examine the roles played by each in mathematics.

What is the next term of the sequence 1, 5, 12, 22, 35? How do these numbers relate to triangular and square numbers? The students’ introduction to inductive reasoning begins with a search for patterns in data and creating recursive and explicit formulas to describe those patterns. Students master material by considering puzzles, algebraic and geometric concepts, patterns, and real-world questions that can be answered using induction.

As they move on to topics in deductive reasoning, students learn to use a system of logic to draw conclusions from statements that are accepted as true. Students encounter a variety of classic problem types as they explore symbolic logic, truth tables, axiomatic systems, matrix logic, syllogisms, Venn diagrams, knights and knaves problems, and Euler circuits. Emphasis is placed on the importance of proving conclusions using valid arguments and developing the ability to recognize fallacious reasoning.

Sample text: Materials compiled by the instructor; a supplemental text such as The Number Devil, Enzensberger.

Students must have completed grades: 5 or 6

Session 1: All residential sites, Brooklandville, La Jolla, Pasadena, Sandy Spring
Session 2: All residential sites, Alexandria, Pasadena, Sandy Spring

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Data and Chance

Behind only one of three doors is a fabulous prize. After you choose door #1, the host reveals door #2 has nothing behind it. She then offers you the opportunity to change your selection. Should you switch to door #3? This classic example of conditional probability is not as simple as it seems.

In this course, students develop a greater understanding of data and chance, two areas of mathematics that easily transfer from the classroom to the real world. Students conduct experiments and generate data which they display in graphs, charts, and tables in order to compare the effects of particular variables. For example, students might analyze data to examine how various design characteristics of a paper airplane, such as weight or length, affect the distance it will travel. In addition, students consider other data sources, including newspapers and journals, and identify examples of incorrectly gathered or misrepresented data that have been used to mislead consumers or influence voters.

Students also explore probability, the study of chance, to learn how to use numerical data to predict future events. Students examine permutations and combinations; develop strategies for calculating the number of possible outcomes for various events; calculate probabilities of independent, dependent, and compound events; and learn to distinguish between theoretical and experimental probability.

By the way, you should switch.

Sample text: Materials compiled by the instructor.

Students must have completed grades: 5 or 6

Session 1: Chestertown, Palo Alto, Windward
Session 2: Chestertown, Palo Alto, Thousand Oaks, Windward

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Introduction to Robotics

In the field of robotics, computer science and engineering come together to create machines that can perform a variety of tasks from manufacturing microchips to exploring Mars.

In this course, students develop familiarity with computer science concepts. For example, they explore topics such as algorithms, sequential control flow, and Boolean operators. Students also survey basic principles of mechanical engineering, such as simple machines and locomotion; and basic principles of electrical engineering, such as circuits and sensor feedback. Using LEGO® robotics equipment, they work together to construct, program, and test their robots in an object-oriented programming environment.

For their culminating project, students design, build, and program robots to complete a complex task. The project demonstrates the basic computer science and engineering principles that underlie everything from the space shuttle to the average home toaster. Students gain a foundation in computer programming and engineering that will become increasingly important in the highly technical twenty-first century.

Sample text: Materials compiled by the instructor.

Lab Fee: $100 (This fee is greater than for other courses due to higher material and equipment costs.)

Students must have completed grades: 5 or 6

Session 1: Chestertown, South Hadley, Thousand Oaks, Brooklandville, Sandy Spring
Session 2: Chestertown, South Hadley, Thousand Oaks, Sandy Spring

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