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.
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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.
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.
In Module 10.1, students engage with literature and nonfiction texts and explore how complex characters develop through their interactions with each other, and how these interactions develop central ideas such as parental and communal expectations, self-perception and performance, and competition and learning from mistakes.
In this module, students read, discuss, and analyze poems and informational texts focusing on how authors use rhetoric and word choice to develop ideas or claims about human rights. Students will also explore how the nonfiction authors develop arguments with claims, evidence, and reasoning. The texts in this module offer rich opportunities to analyze authorial engagement with the struggle for human rights and to consider how an author’s rhetorical choices advance purpose.
In Module 10.3, students engage in an inquiry-based, iterative process for research. Building on work with evidence-based analysis in Modules 10.1 and 10.2, students explore topics that have multiple positions and perspectives by gathering and analyzing research based on vetted sources to establish a position of their own. Students first generate a written evidence-based perspective, which will serve as the early foundation of what will ultimately become a written research-based argument paper that synthesizes and articulates several claims with valid reasoning and relevant and sufficient evidence. Students read and analyze sources to surface potential problem-based questions for research, and develop and strengthen their writing by revising and editing.
In this module, students read, discuss, and analyze nonfiction and dramatic texts, focusing on how the authors convey and develop central ideas concerning imbalance, disorder, tragedy, mortality, and fate.
In this module, students read, discuss, and analyze literary and nonfiction texts focusing on how central ideas develop and interact within a text. Students also explore the impact of authors’ choices regarding how to develop and relate elements within a text.
In this module, students read, discuss, and analyze literary and informational texts, focusing on how authors use word choice and rhetoric to develop ideas, and advance their points of view and purposes. The texts in this module represent varied voices, experiences, and perspectives, but are united by their shared exploration of the effects of prejudice and oppression on identity construction. Each of the module texts is a complex work with multiple central ideas and claims that complement the central ideas and claims of other texts in the module. All four module texts offer rich opportunities to analyze authorial engagement with past and present struggles against oppression, as well as how an author’s rhetoric or word choices strengthen the power and persuasiveness of the text.
In Module 11.3, students engage in an inquiry-based, iterative process for research. Building on work with evidence-based analysis in Modules 11.1 and 12.2, students explore topics that have multiple positions and perspectives by gathering and analyzing research based on vetted sources to establish a position of their own. Students first generate a written evidence-based perspective, which will serve as the early foundation of what will ultimately become a written research-based argument paper. The research-based argument paper synthesizes and articulates several claims using valid reasoning and relevant and sufficient evidence to support the claims. Students read and analyze sources to surface potential problem-based questions for research, and develop and strengthen their writing by revising and editing.
In this module, students read, discuss, and analyze literary texts, focusing on the authors’ choices in developing and relating textual elements such as character development, point of view, and central ideas while also considering how a text’s structure conveys meaning and creates aesthetic impact. Additionally, students learn and practice narrative writing techniques as they examine the techniques of the authors whose stories students analyze in the module.|