# Math Courses

## Why Take a Math Course?

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 in CTY math courses 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 their studies in mathematics.

## Math Courses

Math courses require a minimum score on the quantitative sections of the designated tests. Learn more about course eligibility.

For second and third graders, offered in the day program only:

For third and fourth graders, offered in the day program only:

For fourth and fifth graders, offered in the day program only:

For fifth and sixth graders:

Please note that the sample syllabi are meant to provide an idea of the level of the course and whether or not the content area interests you. CTY instructors are given guidelines within which they each develop their own syllabi. Choice and sequencing of topics within the content area, as well as specific activities, labs, and assignments, will vary.

## Math Course Descriptions and Syllabi

### 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, making lists, 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, and other growing fields of mathematics.

Sample text: Materials compiled by the instructor.

Students must have completed grades: 2 or 3

Session 1: Not offered
Session 2: Alexandria

### Geometry and Spatial Sense

Spatial understanding is necessary for interpreting, understanding, and appreciating our inherently geometric world. Many tasks a person performs—whether designing a treehouse, 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: New York, Portola Valley
Session 2: Alexandria, New York

### 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 to 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.

Students must have completed grades: 4 or 5

Session 1: Baltimore, Hong Kong, Los Angeles (day site), New York, Sandy Spring
Session 2: Alexandria, New York, Sandy Spring

### 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.

Students explore 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. Through hands-on explorations, they learn about great mathematical discoveries throughout time, such as Pascal’s Triangle, the Pythagorean Theorem, the Golden Ratio, pi, Zeno’s paradoxes, and the roots of modern mathematics.

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 coursework.

Sample text: Materials compiled by the instructor.

Students must have completed grades: 5 or 6

Session 1: Los Angeles (residential site)
Session 2: Los Angeles (residential site)

### 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 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.

The students’ introduction to inductive reasoning begins with a search for patterns and creating recursive and explicit formulas to describe those patterns. Students master material by considering algebraic and geometric concepts, patterns, and real-world questions that can be answered using inductive reasoning.

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 topics such as symbolic logic, truth tables, syllogisms, knights and knaves problems, and Euler circuits. Emphasis is placed on the importance of proving conclusions using valid arguments.

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, Baltimore, Hong Kong, La Jolla, New York, Portola Valley
Session 2: All residential sites, Alexandria, New York, Sandy Spring

### Data and Chance

You meet a new friend at CTY who teaches you a dice game. The rules are simple: if you roll a 4, you win and the game ends. If your friend rolls a 5, she wins and the game ends. You take turns rolling until one person wins. If you roll first, what is the probability that you will win the game? There are several ways to solve this problem, and the answer is not obvious.

In this course, students develop a greater understanding of probability and statistics, 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.

Sample text: Materials compiled by the instructor.

Students must have completed grades: 5 or 6

Session 1: Collegeville, San Rafael, Los Angeles (day site)
Session 2: Collegeville
, San Rafael

### 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 physics and 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 a modern programming environment.

With each project, students design, build, and program robots to complete a complex task, and reinforce a new concept. These projects demonstrate 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 21st century.

Note: CTY is committed to eliminating the gender gap in technology and engineering fields. To that end, an all-girls section of this class will be held both sessions at the Collegeville site and at the Los Angeles residential site second session. Girls may request this or a co-ed section of the class.

Sample text: Materials compiled by the instructor.

Lab Fee: \$70

Students must have completed grades: 5 or 6

Session 1: Bristol, Collegeville, Los Angeles (residential site), San Rafael, Baltimore, Hong Kong, New York, Sandy Spring
Session 2: Bristol, Collegeville, Los Angeles (residential site), San Rafael, Alexandria, New York, Sandy Spring