Students connect polynomial arithmetic to computations with whole numbers and integers. åÊStudents learn that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. åÊThis unit helps students see connections between solutions to polynomial equations, zeros of polynomials, and graphs of polynomial functions. åÊPolynomial equations are solved over the set of complex numbers, leading to a beginning understanding of the fundamental theorem of algebra. åÊApplication and modeling problems connect multiple representations and include both real world and purely mathematical situations.
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Module 2 builds on studentsåÕ previous work with units and with functions from Algebra I, and with trigonometric ratios and circles from high school Geometry.åÊThe heart of the module is the study of precise definitions of sine and cosine (as well as tangent and the co-functions) using transformational geometry from high school Geometry.åÊThis precision leads to a discussion of a mathematically natural unit of rotational measure, a radian, and students begin to build fluency with the values of the trigonometric functions in terms of radians.åÊStudents graph sinusoidal and other trigonometric functions, and use the graphs to help in modeling and discovering properties of trigonometric functions.åÊThe study of the properties culminates in the proof of the Pythagorean identity and other trigonometric identities.
In this module, students synthesize and generalize what they have learned about a variety of function families. åÊThey extend the domain of exponential functions to the entire real line (N-RN.A.1) and then extend their work with these functions to include solving exponential equations with logarithms (F-LE.A.4). åÊThey explore (with appropriate tools) the effects of transformations on graphs of exponential and logarithmic functions. åÊThey notice that the transformations on a graph of a logarithmic function relate to the logarithmic properties (F-BF.B.3). åÊStudents identify appropriate types of functions to model a situation. åÊThey adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. åÊThe description of modeling as, åÒthe process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions,åÓ is at the heart of this module. åÊIn particular, through repeated opportunities in working through the modeling cycle (see page 61 of the CCLS), students acquire the insight that the same mathematical or statistical structure can sometimes model seemingly different situations.
Students build a formal understanding of probability, considering complex events such as unions, intersections, and complements as well as the concept of independence and conditional probability. åÊThe idea of using a smooth curve to model a data distribution is introduced along with using tables and techonolgy to find areas under a normal curve. åÊStudents make inferences and justify conclusions from sample surveys, experiments, and observational studies. åÊData is used from random samples to estimate a population mean or proportion. åÊStudents calculate margin of error and interpret it in context. åÊGiven data from a statistical experiment, students use simulation to create a randomization distribution and use it to determine if there is a significant difference between two treatments.
In earlier grades, students define, evaluate, and compare functions and use them to model relationships between quantities. In this module, students extend their study of functions to include function notation and the concepts of domain and range. They explore many examples of functions and their graphs, focusing on the contrast between linear and exponential functions. They interpret functions given graphically, numerically, symbolically, and verbally; translate between representations; and understand the limitations of various representations.
In earlier modules, students analyze the process of solving equations and developing fluency in writing, interpreting, and translating between various forms of linear equations (Module 1) and linear and exponential functions (Module 3). These experiences combined with modeling with data (Module 2), set the stage for Module 4. Here students continue to interpret expressions, create equations, rewrite equations and functions in different but equivalent forms, and graph and interpret functions, but this time using polynomial functions, and more specifically quadratic functions, as well as square root and cube root functions.
This module consolidates and expands upon studentsåÕ understanding of equivalent expressions as they apply the properties of operations to write expressions in both standard form and in factored form.åÊ They use linear equations to solve unknown angle problems and other problems presented within context to understand that solving algebraic equations is all about the numbers.åÊ Students use the number line to understand the properties of inequality and recognize when to preserve the inequality and when to reverse the inequality when solving problems leading to inequalities.åÊ They interpret solutions within the context of problems.åÊ Students extend their sixth-grade study of geometric figures and the relationships between them as they apply their work with expressions and equations to solve problems involving area of a circle and composite area in the plane, as well as volume and surface area of right prisms.
In Module 4, students extend what they already know about unit rates and proportional relationships to linear equations and their graphs.åÊ Students understand the connections between proportional relationships, lines, and linear equations in this module.åÊ Students learn to apply the skills they acquired in Grades 6 and 7, with respect to symbolic notation and properties of equality to transcribe and solve equations in one variable and then in two variables.
In order to assist educators with the implementation of the Common Core, the New York State Education Department provides curricular modules in P-12 English Language Arts and Mathematics that schools and districts can adopt or adapt for local purposes. The full year of Algebra I curriculum is available below.