MATHEMATICS To the reader: This document presents our mathematics standards through the lens of Understanding by Design. We began by identifying the important broad understandings and the essential questions for mathematics. These particular understandings and questions allow us to answer the major question: What are the most significant ideas that we want all students to appreciate after completing the study of mathematics in the Sharon school system? We also determined the broad understandings for number sense, algebra and geometry. After that, we worked on the knowledge and skills that students K-12 need to learn. For many, this change in emphasis on the understandings of mathematics, rather than on memorizing steps is new. However, with depth of understanding, students will be more able to apply what they learn to new situations - which is truly the essential goal of education. We will revisit this document as we use it in our classrooms and consider modifications. Two appendices occur at the end of the document. The first appendix indicates the materials that are used in the K-5 classrooms for our Everyday Mathematics program. The second appendix indicates the vocabulary and important phrases that students are expected to learn in grades K-8. K-12 MATHEMATICS STANDARDS The following list of standards represents the major understandings and questions that underlie all classes in mathematics. We would like you to keep these in mind throughout your reading because they are an important part of each grade. Problem solving, making connections, estimation, patterns, and appropriate use of technology are essential ingredients throughout our program. Through modeling events in the real world, mathematics is the foundation for many aspects of society. How do we use each type of mathematics to model situations in the real world? How do we create, test and validate a model? What mathematics can be used to obtain a desired product or outcome? What models produce the best solutions for a given problem? What are the benefits & limitations of modeling situations? Successful problem solving involves flexible thinking, asking questions and taking risks. Additionally, a variety of problem solving strategies may be used to arrive at appropriate solutions. What are the most appropriate problem solving strategies for a given situation? How are visual and physical models of a mathematical idea helpful for representing concepts and solving problems?
What information is relevant or superfluous? Is estimation more appropriate than finding an exact answer? Is the proposed solution reasonable? Patterns & relationships can be used to represent information, interpret data and make predictions. What patterns or relationships do we see in each type of mathematics? What are the different ways to represent the patterns or relationships? What different interpretations can be obtained from a particular pattern or relationship? What predictions can the patterns or relationships support? How can we use or test our predictions? Are they valid? Are they significant? Mathematics is a language for communicating ideas. How do we translate verbal ideas to the language of mathematics? How do we translate the mathematics into English? What are the different ways of communicating mathematics with clarity? What is the most appropriate way of communicating a mathematical idea in a particular situation? Connections exist between current learning and prior understanding. Also, connections can be made to various strands of mathematics, as well as to other disciplines. What previous learning helps us to understand the new material? How does what we are learning relate to other areas of mathematics? How does what we are learning relate to other disciplines? Properties, postulates and theorems form a foundation for the structure of mathematics. Where did the properties, postulates and theorems originate? How did the properties, postulates and theorems originate? Who were some of the people responsible for developing the different theories of mathematics? Why is a universally accepted set of properties, postulates and theorems essential?
How are the properties, postulates and theorems used in mathematics? Technology and other resources can be used to support mathematical investigations. What technological tools or other resources are available? Which tool(s) is (are) most effective in a particular situation? How do we determine the reasonableness of technological results? When is it appropriate to use technology GRADE 12 COURSE: CALCULUS The understandings, knowledge and skills gained through the algebra, geometry and precalculus courses continue to be needed throughout calculus. In the college preparatory calculus, some minimal review of skills, functions and their graphs will occur. However, students in the advanced placement classes will be expected to be able to recall their prior mathematics without this review. Students in the AP classes will also be required to complete problems equivalent to those on the AP exams. *AB & BC do these topics in addition to what is expected of the standard classes **BC only Understanding The concept of a limit is the foundation for calculus What is meant by the concept of a limit? How is the concept of a limit connected to continuity? How is the concept of a limit connected to a derivative? How is continuity connected to differentiability? Methods for finding a limit Meaning of right hand and left hand limits Rules and theorems related to finding limits L Hopital s Rule Conditions for continuity at a point Theorems related to continuity (IVT and Extreme Value) Limit definition of a derivative Connection between continuity and Explain the process of finding a limit. Find limits graphically, numerically and analytically Determine the continuity/discontinuity of a function analytically and graphically Find a derivative using the definition of a limit Set up the limit of a Reimann
Understanding The derivative is a function that represents the instantaneous rate of change of one quantity with respect to another quantity. How is the concept of a limit connected to an integral? Why is the derivative a function? How is the graph of a derivative function related to the graph of the original function? How does the derivative represent an instantaneous rate of change? How do you find a derivative function without the definition? When and why are methods for approximating the value of a derivative at a point useful? What are the applications for derivatives? differentiability Connection among limit, the slope of a secant line, slope of a tangent line and the derivative Connection between the integral and the limit of a Reimann sum Connection between the graph of a function and the area between the curve and the x-axis Vocabulary and notation related to derivatives Graphic/geometric and application interpretation of the derivative Differences between average and instantaneous rates as well as between speed and velocity Methods for finding the derivative as a function (shortcut rules, chain rule, parametric rule, implicit differentiation) Methods for approximating the value of the derivative (graphically and from a table) Procedures for finding the value of a derivative using the calculator Results and meaning when finding a second derivative Use of a slope field sum to indicate an integral. Find derivatives of polynomial, rational exponential logarithmic and trigonometric functions Find derivatives (including 1 st, 2 nd, and higher derivatives) using the shortcut rules, chain rule, parametric rule, implicit differentiation Find the derivative of an inverse function without finding the inverse* Approx. derivatives graphically and numerically Find and compare average and instantaneous rates of change Use differentials to approximate small changes or errors in an application* Determine the equation of a tangent line to a curve at a point Use tangent line approximations
Determine whether a function: is incr. or decr.: is concave up or down; has horiz. or vert. tangents; has local or global extrema; has points of inflection, from the 1 st or 2 nd derivatives of the function Determine the graph of a function from the graph of its derivative and vice versa and justify the result* Understanding The integral is a function that can be used to determine the summation of an infinite set. Why is the indefinite integral represented by a family of functions? Explain the different results achieved when antidifferentiating and integrating. How does the integral Vocabulary and notation related to integrals Graphic/geometric and application interpretation of the integral Different results possible when antidifferentiating or integrating Procedures for finding a definite Find the derivatives of polar and vector functions** Use differentiation for an application such as extrema, distance-velocityacceleration, related rates and any rate of change Write equations involving derivatives from verbal descriptions Sketch or interpret a slope field Approximate solutions of differential equations using Euler s Method** Find the antiderivative or integral of a function using the basic rules, substitution, by parts* or partial fractions** Find specific antiderivatives using initial conditions Approximate the value of an integral by left/right endpoint or midpoint
represent the summation of an infinite set? What are the applications for integrals? integral using the calculator evaluation, trapezoidal rule, geometrically and from a table Find the integral of a polar function** Find a definite integral (from a to x or a to b) Understanding There are significant theorems which form the basis for calculus When and why are methods for approximating integrals useful? How do you translate the meaning of the theorem/ Why are the theorems important? Explain how and integral is a summation Use integration for applications such as acceleration-velocity-distance, area between curves, volumes of solids of revolution and solids built on a base, economic situations, biological situations, exponential growth and decay*, average value of a function, length of a curve** Solve differential equations using integration Find an improper integral** Solve logistic differential equations and use them in modeling** Mathematicians who were responsible for some of the development and important theorems in calculus IVT, MVT, Sandwich or Squeeze Theorem, FT of C, Extreme Value Theorem Explain the significance of theorems and their graphic interpretation where appropriate Explain the connection between differentiation and integration Apply the theorems to given information or problems
Understanding Differentiation and integration can be used with a function that is represented as a series.** (This entire category is required of BC calculus students only.) How closely do series represent a function? How are series that represent functions formed? Vocabulary related to sequences and series, convergence and divergence Types of series: geometric, harmonic, alternating, Maclaurin and Taylor Tests for proving that series converge or diverge Maclaurin series for f (x)=sin x, f (x)=cos x, f(x)=e x, 1 f(x)=, f(x)=ln(1+x), and 1 x f(x)=arctan x Methods for creating series Connection between the graph of increasing number of terms of a series and its convergence or divergence Determine the convergence or divergence of a series using a variety of tests such as the p-test, ratio test, integral test Determine the error for an alternating series Use the Lagrange error bound for Taylor polynomials Determine the radius and interval of convergence of power series Differentiate and integrate series