Students connect polynomial arithmetic to computations with whole numbers and integers. Students learn that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. This unit helps students see connections between solutions to polynomial equations, zeros of polynomials, and graphs of polynomial functions. Polynomial equations are solved over the set of complex numbers, leading to a beginning understanding of the fundamental theorem of algebra. Application and modeling problems connect multiple representations and include both real world and purely mathematical situations.
Module 2 builds on students previous work with units and with functions from Algebra I, and with trigonometric ratios and circles from high school Geometry. The 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. This 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. Students graph sinusoidal and other trigonometric functions, and use the graphs to help in modeling and discovering properties of trigonometric functions. The 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. They 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). They explore (with appropriate tools) the effects of transformations on graphs of exponential and logarithmic functions. They notice that the transformations on a graph of a logarithmic function relate to the logarithmic properties (F-BF.B.3). Students identify appropriate types of functions to model a situation. They 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. The description of modeling as, the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions, is at the heart of this module. In 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. The 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. Students make inferences and justify conclusions from sample surveys, experiments, and observational studies. Data is used from random samples to estimate a population mean or proportion. Students calculate margin of error and interpret it in context. Given 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 this module, students reconnect with and deepen their understanding of statistics and probability concepts first introduced in Grades 6, 7, and 8. Students develop a set of tools for understanding and interpreting variability in data, and begin to make more informed decisions from data. They work with data distributions of various shapes, centers, and spreads. Students build on their experience with bivariate quantitative data from Grade 8. This module sets the stage for more extensive work with sampling and inference in later grades.
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.
As students will have previous exposure to the historical themes and factual information about the attacks on Pearl Harbor, the United States involvement in WWII, and the internment of Japanese in camps throughout the western United States, this lesson exemplar will allow students to participate in critical discussion of two stories that illuminate important, yet divergent, experiences of war and conflict. This lesson exemplar will push students to think critically about the experience of wartime as felt by both soldiers and civilians as they navigated specific trials that were a part of their direct or peripheral involvement in WWII. This close reading exemplar is intended to model how teachers can support their students as they undergo the kind of careful reading the Common Core State Standards require. Teachers are encouraged to take these exemplars and modify them to suit the needs of their students.
The goal of the Listening and Learning Strand is for students to acquire language competence through listening, specifically building a rich vocabulary, and broad knowledge in history and science by being exposed to carefully selected, sequenced, and coherent read-alouds. The 9 units (or domains) provide lessons (including images and texts), as well as instructional objectives, core vocabulary, and assessment materials. The domain topics include: Different Lands, Similar Stories; Fables and Stories; The Human Body; Early World Civilizations; Early American Civilizations; Astronomy; Animals & Habitats; Fairy Tales; and History of the Earth.
The Listening and Learning Strand consists of a series of read_alouds organized by topics (called domains), many of which are informational in nature. The goal of the Listening and Learning Strand is for students to acquire language competence through listening, specifically building a rich vocabulary, and broad knowledge in history and science by being exposed to carefully selected, sequenced, and coherent read_alouds. The 9 units (or domains) provide lessons (including images and texts), as well as instructional objectives, core vocabulary, and assessment materials. The domain topics include: Fighting for a Cause; Fairy Tales and Tall Tales; Cycles in Nature; Insects; Ancient Greek Civilizations; Greek Myths; Charlotte's Web; and Immigration.
The Skills Strand teaches the mechanics of reading_Ŕstudents are taught systematic and explicit phonics instruction as their primary tool for decoding written English. By the end of grade 2, students have learned all of the sound_spelling correspondences in the English language and are able to decode written material they encounter. In addition to phonics, students also are taught spelling, grammar, and writing during the Skills Strand. A downloadable story "Jump!" with illustrations is provided for instruction.
This module uses literature and informational text such as "My Librarian Is a Camel" to introduce students to the power of literacy and how people around the world access books. This module is intentionally designed to encourage students to embrace a love of literacy and reading. There are 3 units in this module. Unit 1 explores the question ĺăĄWhy do people seek the power of reading?ĺăĺ In unit 2 students explore their own ĺăĄpowers of readingĺăĺ that help them access text. And unit 3 explores how geography impacts readersĺăĚă access to books.
This module ensures that students read, write, listen and speak to learn the history and contributions of Native Americans in New York State, particularly the Iroquois Confederacy. It focuses on reading and listening to primary and secondary sources to gather specific details and determine central ideas, and to reinforce reading fluency and paragraph writing. Students will read literature to develop an understanding of setting, characterization and theme, and informational writing.
The goal of the Listening and Learning Strand is for students to acquire language competence through listening, specifically building a rich vocabulary, and broad knowledge in history and science by being exposed to carefully selected, sequenced, and coherent read_alouds. The 9 units (or domains) provide lessons (including images and texts), as well as instructional objectives, core vocabulary, and assessment materials. The domain topics include: Nursery Rhymes and Fables; Five Senses; Stories; Plants; Farms; Kings and Queens; Seasons and Weather; Colonial Towns; and Taking Care of the Earth.
The Skills Strand teaches the mechanics of reading. Students are taught systematic and explicit phonics instruction as their primary tool for decoding written English. By the end of grade 2, students have learned all of the sound spelling correspondences in the English language and are able to decode written material they encounter. In addition to phonics, students also are taught spelling, grammar, and writing during the Skills Strand. A downloadable story "Kits Hats" with illustrations is provided for instruction.
Module 1 embodies critical changes in Geometry as outlined by the Common Core. The heart of the module is the study of transformations and the role transformations play in defining congruence. The topic of transformations is introduced in a primarily experiential manner in Grade 8 and is formalized in Grade 10 with the use of precise language. The need for clear use of language is emphasized through vocabulary, the process of writing steps to perform constructions, and ultimately as part of the proof-writing process.
Just as rigid motions are used to define congruence in Module 1, so dilations are added to define similarity in Module 2. To be able to discuss similarity, students must first have a clear understanding of how dilations behave. This is done in two parts, by studying how dilations yield scale drawings and reasoning why the properties of dilations must be true. Once dilations are clearly established, similarity transformations are defined and length and angle relationships are examined, yielding triangle similarity criteria. An in-depth look at similarity within right triangles follows, and finally the module ends with a study of right triangle trigonometry.
Module 3, Extending to Three Dimensions, builds on students understanding of congruence in Module 1 and similarity in Module 2 to prove volume formulas for solids. The student materials consist of the student pages for each lesson in Module 3. The copy ready materials are a collection of the module assessments, lesson exit tickets and fluency exercises from the teacher materials.
In this module, students explore and experience the utility of analyzing algebra and geometry challenges through the framework of coordinates. The module opens with a modeling challenge, one that reoccurs throughout the lessons, to use coordinate geometry to program the motion of a robot that is bound within a certain polygonal region of the planethe room in which it sits. To set the stage for complex work in analytic geometry (computing coordinates of points of intersection of lines and line segments or the coordinates of points that divide given segments in specific length ratios, and so on), students will describe the region via systems of algebraic inequalities and work to constrain the robot motion along line segments within the region.
This module brings together the ideas of similarity and congruence and the properties of length, area, and geometric constructions studied throughout the year. It also includes the specific properties of triangles, special quadrilaterals, parallel lines and transversals, and rigid motions established and built upon throughout this mathematical story. This module's focus is on the possible geometric relationships between a pair of intersecting lines and a circle drawn on the page.