Math in Upper School
Mirman’s Upper School math program is a comprehensive and flexible program that begins with Math Innovations as a foundational middle grades curriculum and progresses through Algebra, Geometry, Algebra 2, Precalculus, and Calculus. Based on their prior experience and current rates of acquisition and retention, students enter and exit the program in different places. The most typical progression is for students to take Innovations 1 in US1, Innovations 2 in US2, Algebra in US3, and Geometry or Algebra 2 in US4. Students who demonstrate readiness for courses outside of the normal progression have the opportunity to take Precalculus and Calculus before graduating from Mirman. All Mirman math classes emphasize mathematical analysis, problem solving, strong communication skills, depth and complexity, and continuous reflection about mathematical practices. Class descriptions for all current offerings and links to current math faculty are below.
Math Innovations I
Math Innovations develops a deep understanding of mathematical concepts by actively involving students in their own learning as new ideas are placed in real-world contexts that are relevant to their own lives. Math Innovations focuses on targeted areas of mathematics while presenting students with guided opportunities for mathematical analysis and problem solving.
- Explore all four basic operations with rational numbers.
- Introduce algebra and the different uses of variables in expressions, equations, inequalities, formulas and properties with real numbers.
- Represent and analyze linear relationships using tables, graphs, equations, words and diagrams.
- Derive and create recursive and explicit rules for patterns identified in tables, graphs and drawings.
- Understand combinatorics and how to calculate permutations.
- Classify solids and analyze vertices, edges and faces of polyhedrons.
- Solve problems involving complementary, supplementary, and vertical angles as well as the sum of angle measurements in a triangle.
Math Innovations II
Math Innovations 2 continues the integrated curriculum and builds upon students’ understanding of the concepts introduced in Math Innovations I. Students remain actively engaged by applying the concepts to “real-world” contexts, working through word problems, examining the reasoning behind the computations, and exploring the relevancy of each topic as it may apply other concepts and situations. As in Innovations I, each unit in Math Innovations 2 focuses on a particular area of mathematics while presenting students with guided opportunities for mathematical analysis and problem solving.
- Understand ratios, rates, and unit rates.
- Use proportions to solve real world problems.
- Solve complex equations by using the order of operations and combining like terms.
- Solve and graph inequalities on a number line.
- Evaluate expressions using the rules of exponents.
- Graph linear functions and explore various types of linear equations.
- Find measures of angles formed by parallel lines and a transversal.
- Understand and use properties of similar triangles using dilations, rotations, reflections and translations.
- Work with and develop formulas for the volume of cylinders, cones and spheres.
- Discover and apply Pascal’s triangle and the Pythagorean theorem.
Students delve deep into the foundations of Algebra using Pearson’s Algebra I textbook and The Art of Problem Solving: Introduction to Algebra as core texts. Students develop advanced problem-solving skills, perseverance, and creativity through explorations of progressively complex mathematical concepts.
- Identify arithmetic, geometric, and exponential patterns.
- Use algebraic properties to justify mathematical proofs
- Apply and model rules of exponents.
- Graph linear and quadratic equations and inequalities.
- Explore linear, absolute value, exponential, rational and radical functions.
- Factor polynomials.
- Solve one-variable equations and linear systems with multiple variables.
Students focus on plane geometry, solid geometry, and the foundations of trigonometry.. Using The Art of Problem Solving: Introduction to Geometry as a core text, students continue to utilize advanced problem-solving skills, perseverance, and creativity through projects and explorations of mathematical topics. This integrated “hands-on” course enables students to explore design-thinking and STEM-based projects and ideas into their learning.
- Explore and use the characteristics and properties of triangles, quadrilaterals and circles to solve problems.
- Find the area of composite shapes.
- Explore different types of formal proof. Use deductive reasoning to construct proofs.
- Build on an understanding of 2-dimensional geometry to expand problem solving to figures in 3 dimensions.
- Explore trigonometry, including right triangle trigonometry and the law of sines and cosines.
Advanced Algebra delves deeper into the Algebra 1 content and begins to explore Algebra 2 concepts and foundations. Students explore and relate the concepts to real world situations through various projects and applications.
- Graph linear equations and inequalities and solve systems of linear equations and inequalities.
- Factor polynomials.
- Explore and graph quadratic, exponentials, logarithms, and other functions.
- Analyze complex numbers.
- Solve radical and rational expressions and equations.
Algebra 2 is an extension of the work students have done in Algebra 1 and Geometry. The core focus of Algebra 2 is to continue to develop advanced algebra techniques, problem solving skills and critical thinking skills necessary to advance to Precalculus and ultimately to prepare students for Calculus. The primary texts for this class are The Art of Problem Solving: Intermediate Algebra textbook and the Pearson Algebra 2 textbook.
- Systems of equations, linear algebra and matrices,
- Advanced topics in polynomials and rational functions, including finding roots, sketching curves and examining the broader properties of general polynomial functions and rational function
- Exponential and logarithmic functions
- Introductory trigonometry
- Sequences, series, probability and the binomial distribution.
The precalculus curriculum is designed to prepare students for the rigors of an AP level calculus class. The course is broadly divided into three sections:
- Functions, Polynomials, Rational Functions, Exponential and Logarithmic Functions
- Linear Algebra, Sequences and Series, Probability, The Binomial Theorem and Limits.
The Linear Algebra section goes beyond the standard course curriculum. Linear Algebra is at the heart of most modern STEM disciplines.
Students require tools from all three sections to be able to solve challenging problems in a Calculus curriculum. An emphasis is placed on practicing techniques and methods. The course has regular homework that is discussed in class, as well as examinations determined by the pacing of the course that is required for the students to have demonstrated mastery of the material.
The course has a secondary objective of providing a broader perspective on mathematics at the university level so that students can begin to develop a context in which to place these techniques and skills. Some youtube videos, side lectures and challenge problems are sprinkled in to keep the students engaged and aware of how the skills they are developing now will be useful in later courses.
By the end of the course students should be prepared for any AP/university level calculus course, as well as the SAT math subject tests, which are largely based on this curriculum. The textbook for the course is Precalculus by Blitzer.
Calculus 2 is an extension of the first year Calculus 1 course required for all STEM majors at the university level. The course can be broadly thought of as being two sections:
- The introduction of integration and the Fundamental Theorem Of Calculus. Students will develop a familiarity with basic integration techniques, and then develop an arsenal of integration techniques for various functions including integration by parts, trigonometric integrals and substitution techniques, partial fraction decomposition, computer simulations and other techniques.
- Applications of integration, including solving basic differential equations, exploring parametric equations, polar coordinates, and analyzing sequences and series using techniques of integration.
This course is also an introduction to physics. Calculus was originally invented in order to solve physics problems, and so it is logical to use physics as the primary example problems for mastering calculus. Most physics applications will be from Classical Mechanics, although some problems from Electrodynamics and modern physics may also appear. The textbook is Calculus by Stewart, 8th edition.