# Math

The math program at Urban draws from both traditional and contemporary approaches and content. Our core classes are Math 1, 2, 3 and Functions. Our approach to teaching these courses is state-of-the-art and reflects the professional consensus as expressed by the National Council of Teachers of Mathematics Principles and Standards. In content, the courses are close to the traditional *Algebra 1, Geometry, Algebra 2* and *Precalculus* sequence.

We navigate between multiple representations of concepts to reach different types of learners and to provide more depth of understanding to all. This includes graphical, hands-on, technological, geometric, numerical and symbolic approaches whenever possible. Foundational concepts (such as equations, functions and trigonometry) are investigated and practiced each year at increasing levels of challenge. On a number of topics, our curriculum goes much deeper than tradition requires, and many of our lessons serve as the backbone of a syllabus for the professional development of high school math teachers.

Our elective courses include unusual offerings for high school students. We offer an exceptionally broad selection of electives, including accessible introductory courses in *Computer Science* (three levels)* *and* Statistics*, as well as more advanced theoretical courses such as *UAS Group Theory* and *UAS Infinity: Theory of Infinite Sets and Chaos Theory*. This extraordinary range of classes makes it possible for students to be exposed to a wide range of mathematics and select courses that are appropriate to their interests, abilities and priorities.

## DEPARTMENTAL REQUIREMENTS

Urban requires three years of math, taken in high school, for graduation. Students planning to apply to competitive colleges should take four years of math. Students considering a career in math, science, computer science or engineering should take four or more years of math, including *Functions* and challenging electives, such as *Analytic Geometry*, *Calculus*, *Computer Science*, *Infinity* or *Space*).

## UAS IN MATH

Urban Advanced Studies (UAS) math classes offer coursework appropriate for students preparing for advanced and college level work in mathematics or the mathematically based sciences. These UAS electives include many college level topics taught in a pace and style accessible to high school students. In these advanced electives students are expected to have mastered the content, skills and habits from the core course sequence of *Math 1, 2* and *3*, and to be able to apply these with confidence to more complex conceptual domains. UAS math courses move at a faster pace and involve more challenging, rigorous problems than our core sequence or regular elective courses. Students are expected to be able to sustain focus on concepts and problems over significant periods of time, working productively and creatively both independently and in small groups. Students are also expected to take responsibility for mastering any prerequisite skills and concepts via self-study, as class time will be reserved for exploration of new material and deeper investigation of key topics.

## MATH COURSES 2021-22

**Math 1A and 1B** helps students develop the concepts, skills and habits that form the foundation of high school mathematics. Many of the tools are algebraic, but almost all concepts are looked at in a variety of ways including geometric, numeric and verbal approaches. Basic arithmetic and algebraic operations are modeled with physical manipulatives, making a geometric and visual connection with these operations. Our goal is for students to integrate and connect these methods. Principal topics:

- Linear equations and graphs

- Introduction to the graphical representation of a variety of functions: continuous, discontinuous, step, linear, quadratic, exponential

- Equation solving by several methods: graphical intersection, numerical (systematic guess and check), “cover-up,” algebraic symbol manipulation

- Number Sense: large and small numbers, scientific notation, estimation, opposites, reciprocals

- The distributive rule and factoring

(1 credit)

**Math 2A and 2B** is an integrated course where students explore concepts through hands-on materials to create geometric conjectures, to use the language of algebra to describe some of these relationships, and to write formal proofs. Various algebraic concepts, such as simplification of square roots and variation functions, are approached geometrically. Slope and measurement are used to introduce trigonometric ratios. Writing computer programs and using dynamic geometry to create designs and figures, students deepen their understanding of geometric relationships as they experience the logic of computers. Principal topics:

- Angles, polygons, parallel lines, circles

- Linear functions and systems of equations

- Distance and the Pythagorean Theorem

- Dynamic geometry with GeoGebra software

- Scaling, proportions and variation functions

- Similarity and congruence

- Computer programming in Snap! : scripts and variables

- Sine, cosine, tangent in the right triangle

- Law of Sines and Cosines

- Transformations: isometries and dilations

- Simultaneous equations

- Quadratics

- Exponential functions

(1 credit)

**Math 3A and 3B** continues and deepens our work with algebraic manipulation and graphical representation of functions as mathematical models. The practices developed in previous courses are expected to be in place so that the focus is on understanding concepts and demonstrating mastery. In particular, we expect fluency with algebraic symbols and notation. As the last course required for all students, Math 3 rounds out the basics of mathematical literacy, intensifies the challenge for students, and provides the foundation for upper level electives. Principal topics:

- Unit Circle Trigonometry

- Logarithms

- Arithmetic and geometric sequences and series

- Composition of functions and inverse functions

- Families of functions

- Construction

- Formal Proof

- Polar coordinates and vectors

- Complex numbers

(1 credit)

Electives

**Advanced Math Applications** is appropriate for students who would benefit from more experience with, and a deeper understanding of, the key math concepts that are foundation for Functions and other upper level math and science courses. Key topics from Math 1, 2 and 3 are reviewed and extended. Emphasis is placed on numeric and algebraic fluency. The course is also appropriate for any students interested in the history of mathematics and its applications to science. Principal topics:

- Ratio, proportion and scientific notation applied to astronomical and sub-atomic scales

- Applications of linear, exponential, variation, quadratic and trigonometric functions in science

- Review and extension of logarithms including a mastery of the laws of exponents

- Applications of triangle trigonometry

- Unit circle trigonometry and radians applied to mapping, astronomy and basic physics

- Logarithms and logarithmic scale

- Mathematical language and notation, algebraic thinking, solving equations and graphing

(1/2 credit)

**Prerequisite: **Math 3

**Statistics & Probability** is an elective that concentrates on the applications of mathematics to the social and life sciences. This course is appropriate both for students who intend to go on to calculus, as well as students who do not. Students apply concepts of counting, combinations and permutations to probability problems, and to the foundations of statistics. They use appropriate tools and techniques to interpret data. The course also includes the mathematics underlying the sampling techniques used by scientists and pollsters. Principal topics:

- Use of software to demonstrate and interpret data

- Analyzing the association of two variables from graphs

- Use of least squares line to determine and evaluate models for data

- Pascal’s triangle and binomial distribution

- Simulations with dice and with software

- Sampling and sources of bias

(1/2 credit)

**Prerequisite: **Math 3

**Discrete Mathematics: Puzzles and Games** is a survey course, covering many topics in mathematics that are relevant in today’s world. Students will be introduced to and study practical applications of graphs and networks, theories about numbers, and logic. They will also be exposed to more abstract concepts deriving from these topics. Principal Topics:

- Simple graphs and trees, networks, paths and isomorphisms

- Graph coloring

- Prime numbers, prime factorizations and the greatest common divisor

- Modular arithmetic, cryptography and magic squares

- Truth tables, truth values and logic puzzles

(1/2 credit)

**Prerequisite:** Math 3

**Computer Science 1** is an introduction to programming concepts using Snap!, a computer language developed at UC Berkeley. Snap! makes it possible for students to program images, animation and interactions and learn about algorithms, data handling and other fundamentals of computer programming, in a visual context. Principal topics:

- Variables and scoping

- Passing parameters, returning values

- Functions and modularity

- Looping and conditionals

- Data structures (lists)

- List mutability

- Abstract data types

- Introduction to recursion

(1/2 credit)

**Prerequisite:** Math 3

**UAS Computer Science 2** focuses attention on the central idea of abstraction, make heavy use of the idea of functions as data, and discuss relevant topics in Computer Science such as functions as data, complexity and graph theory. It will also focus on some of the “Big Ideas” of computing such as recursion, concurrency and the limitations of computing. Principal topics:

- Algorithms, both classic and heuristic

- Algorithm Complexity: constant, linear, quadratic, exponential and logarithmic run times

- Recursion

- Higher Order Functions and using functions as data

- Graphs: paths, cycles, cliques

(1/2 credit)

**Prerequisites:** Math 3, Computer Science 1 or instructor approval.

**UAS Computer Science 3** continues the Computer Science sequence, focusing on more advanced principles of software engineering, data structures and algorithms, emphasizing computability and feasibility. Topics in computer science such as Game Theory and Machine Learning will be discussed. Principal topics:

- Fundamental dynamic data structures including linear lists, queues, trees, arrays and dictionaries

- Computational complexity of algorithms and the key difference between denotation, computability and feasibility

- The object-oriented programming paradigm and a class-based approach

- Basic ideas and techniques underlying the design of intelligent computer systems

- Differences between machine learning, deep learning and neural networks

- Decision trees, pruning, and graph algorithms

(1/2 credit)

**Prerequisite:** UAS Computer Science 2

**UAS Functions** focuses on the topics needed for calculus. It is structured around functions as models of change, emphasizing that they can be grouped into families that model real-world phenomena. One goal of this course is to begin the transition toward more text-based college-level courses and more independent student learning. Students extend and deepen their knowledge and skills of the core curriculum (Math 1-3). Principal topics:

- Functions: increased depth on composition, inverse and general fluency

- Functions: transformations, odd and even, limits and end behavior

- Trigonometric functions, equations and identities – radians

- e and natural logarithms

- Polynomial and rational functions

(1/2 credit)

**Prerequisite: **Math 3 (and in some cases, Advanced Math Applications)

**UAS ****Calculus A and B** seeks to provide students with a solid foundation for subsequent college level courses in mathematics and other disciplines. The course is focused on differentiation, integration and their relationship. The math concepts are enhanced by applications relating to geometry, physics, economics, ecology and medicine. Students are expected to take full responsibility for their learning by using the text and applying all the skills and content learned in previous courses. They are expected to navigate between graphical, numerical, analytical and verbal representations of problems and to use the graphing calculator appropriately. Principal topics:

- Differentiation

- Limits

- Integration

- Graphical analysis

- Introduction to differential equations and slope fields.

(1 credit)

**Prerequisite:** Functions

**UAS ****Analytic Geometry** introduces complex topics at the precalculus level that are challenging and useful for advanced students, but not prerequisite for the standard calculus course. The daily problems can be more substantial than the standard work in the core curriculum. There is a focus on moving fluently back and forth from a variety of algebraic forms to graphing in different coordinate systems in two and three dimensions. Students build models, do constructions, and derive equations from definitions and general principles. Central topics:

- Vectors in three dimensions

- Conic sections

- Polar coordinate equations and graphs

- Parametric equations and graphs

(1/2 credit)

**Prerequisite:** Math 3

**UAS Infinity: Theory of Infinite Sets and Chaos Theory** allows students to discuss ancient paradoxes about infinity, and learn how Georg Cantor resolved them. This discussion launches our most theoretical course. Infinity includes a strong emphasis on formal proof and an introduction to chaos theory and fractal geometry, two computer-centered branches of mathematics. Connections are made with literature and philosophy. Principal topics:

- Introduction to set theory and infinite sets

- Different-sized infinities, transfinite numbers

- Cardinals and ordinals arithmetic

- Proof by contradiction, proof by induction

- Iteration and recursion

- Fibonacci numbers

- Dynamical systems and the Mandelbrot set

(1/2 credit)

Prerequisite: Math 3

Not Offered This Year; Offered in Alternate Years

**UAS ****Space: Group Theory** is an advanced geometry course, which includes a thorough exploration of symmetry, including an introduction to group theory, and extends students’ geometric experiences into three and four dimensions. Many hands-on 3d building labs, creative projects and the reading of mathematical fiction illustrate the concepts. Principal topics:

- Geometric transformations

- Symmetry groups, tessellation, Escher – art projects

- Matrices

- Three-dimensional geometry, especially polyhedra

- Flatland and the fourth dimension

- Zome System construction kit

- Computer labs, using Cabri and Cabri 3D software.

(1/2 credit)

**Prerequisite:** Math 3