Mathematics can be described as a language, tool, science, and 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 greater understanding of both the underlying meanings of mathematical concepts and the intriguing ways math can be applied and extended in a wide range of contexts. CTY young students 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 to solve complex mathematical problems to using patterns and algebraic reasoning for making predictions to exploring some of mathematics’ greatest problems and breakthroughs, CTY young students math courses allow pupils to build an enriched foundation for their studies in mathematics or accelerate their studies to allow for the pursuit of more advanced topics in their home schools. Students who wish to seek credit or placement for CTY courses need to work closely with their home schools to determine the appropriate steps to be taken. Please refer to our Eligibility web page. Sample syllabi for all courses are also available.
Math Problem Solving
If a student is sent the cryptic message L ORYH PDWK (Caesar Cipher, n=3), what response could be sent back? Can five 20' by 18' carpets lying flat with no overlap fit in a room 40' by 50'? What is EGG + EGG? 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 can respond to the cryptic message with VR GR L, have several methods for determining that the carpets can’t fit, and know that EGG + EGG is clearly PAGE. Sample text: Materials compiled by the instructor. Students must have completed grades: 2 or 3 Session 1: Brooklandville, Pasadena, Sandy Spring, Windward
Session 2: Alexandria, Sandy Spring, Windward Top
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 basic 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 Top
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 unique 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 will be asked to work without a calculator. Calculators will be 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, Sandy Spring, Windward
Session 2: Alexandria, Sandy Spring, Windward Top
Great Discoveries in Mathematics
From ancient to modern times, mathematics has been fundamental to 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, and Babylonians. They work through problems discovered in ancient texts, such as the Rhind Papyrus, the Plimpton 322 tablet, and the Rosetta Stone. Additionally, they consider their newfound conceptual knowledge in its historical context. For example, students do hands-on measurements and solve problems using the transcendental number pi. Along with this mathematical work, they look at the social and historical importance of pi, from its approximated value of three in the Bible, to its computer-calculated value up to a billion digits. 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, Thousand Oaks
Session 2: Palo Alto, South Hadley, Thousand Oaks Top
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? 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, logic problems, 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 sound arguments and developing the ability to recognize fallacious reasoning. Sample texts: 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, Sandy Spring
Session 2: All residential sites, Alexandria, Pasadena, Sandy Spring Top
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, the answer to the question is that 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, South Hadley, Windward Top
Individually Paced Math Sequence
Individually Paced Math Sequence includes courses in Arithmetic, Pre-algebra, Algebra I, Geometry, Algebra II, and pre-calculus topics including Functions, Trigonometry, Discrete Math, and Analytic Geometry. It is designed for students with the interest and ability to accelerate their study of mathematics. Inspired by research conducted by Johns Hopkins University’s Study of Mathematically Precocious Youth (SMPY), this course is designed for students with strong independent learning skills, high self-motivation, and high mathematical ability. Math Sequence allows students to work independently at a pace commensurate with their individual abilities. At the beginning of the session, instructors assess the knowledge base of each student with diagnostic testing. The results of this testing, together with information provided by students and their schools, determine the appropriate starting point for each student in CTY’s curriculum and allow students to focus their attention on unfamiliar material. For most of the day, students work on their own, solving problems from their textbooks. The instructor and assistant monitor each student’s progress carefully and offer support and individualized instruction as needed. Students must demonstrate a mastery of the concepts and skills within each subject area before moving on to the next topic. For more information, please visit the Math Sequence Page.
Students who have not consulted with their home school regarding credit and placement should not register for this course. Sample texts: Fundamental Mathematics, Bittinger; Prealgebra, Bittinger; Algebra, Smith; Geometry, Jurgensen; Algebra and Trigonometry, Smith; Advanced Mathematics, Brown; Precalculus with Trigonometry, Foerster. Students must have completed grades: 5 or 6 Session 1: Bethlehem, Palo Alto
Session 2: Bethlehem, Thousand Oaks, Alexandria Top
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. Using LEGO® robotics equipment, students work together to construct, program, and test their robots in an object-oriented programming environment. They develop familiarity with foundational concepts in this computer science course. 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. 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 be 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 Top | 
