Just as rigid motions are used to define congruence in Module 1, so dilations are added to define similarity in Module 2.
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Module 3, Extending to Three Dimensions, builds on student's understanding of congruence in Module 1 and similarity in Module 2 to prove volume formulas for solids.
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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 planešthe 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 describe the region via systems of algebraic inequalities and work to constrain the robot motion along line segments within the region.
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With geometric intuition well established through Modules 1, 2, 3, and 4, students are now ready to explore the rich geometry of circles. This module brings together the ideas of similarity and congruence studied in Modules 1 and 2, the properties of length and area studied in Modules 3 and 4, and the work of geometric construction studied throughout the entire 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.
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In this module, students draw on their foundation of the analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi-digit integers and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. Students explore the role of factoring, as both an aid to the algebra and to the graphing of polynomials. Students continue to build upon the reasoning process of solving equations as they solve polynomial, rational, and radical equations, as well as linear and non-linear systems of equations. The module culminates with the fundamental theorem of algebra as the ultimate result in factoring. Students pursue connections to applications in prime numbers in encryption theory, Pythagorean triples, and modeling problems.
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Module 2 builds on student's previous work with units and with functions from Algebra I, and with trigonometric ratios and circles from high school Geometry.
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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 and then extend their work with these functions to include solving exponential equations with logarithms. They use appropriate tools to explore the effects of transformations on graphs of exponential and logarithmic functions. They notice that the transformations of a graph of a logarithmic function relate to the logarithmic properties. 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, students acquire the insight that the same mathematical or statistical structure can sometimes model seemingly different situations.
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The concepts of probability and statistics covered in Algebra II build on student's previous work in Grade 7 and Algebra I. In Topic A, fundamental ideas from Grade 7 are revisited and extended to allow students to build a more formal understanding of probability. More complex events are considered (unions, intersections, complements). Students calculate probabilities based on two-way data tables and interpret them in context. They also see how to create "hypothetical 100" two-way tables as a way of calculating probabilities. Students are introduced to conditional probability, and the important concept of independence is developed. The final lessons in this topic introduce probability rules.
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Module 1 sets the stage for expanding student's understanding of transformations by first exploring the notion of linearity in an algebraic context ("Which familiar algebraic functions are linear?"). This quickly leads to a return to the study of complex numbers and a study of linear transformations in the complex plane.
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In Module 1, students learned that throughout the 1800s, mathematicians encountered a number of disparate situations that seemed to call for displaying information via tables and performing arithmetic operations on those tables. One such context arose in Module 1, where students saw the utility of representing linear transformations in the two-dimensional coordinate plane via matrices. Students viewed matrices as representing transformations in the plane and developed an understanding of multiplication of a matrix by a vector as a transformation acting on a point in the plane. This module starts with a second context for matrix representation, networks.
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Students encountered the fundamental theorem of algebra, that every polynomial function has at least one zero in the realm of the complex numbers, in Algebra II Module 1. Topic A of this module brings students back to the study of complex roots of polynomial functions. Students first briefly review quadratic and cubic functions and then extend familiar polynomial identities to both complex numbers and to general polynomial functions. Students use polynomial identities to find square roots of complex numbers. The binomial theorem and its relationship to Pascal's triangle are explored using roots of unity. Topic A concludes with student's use of Cavalieri's principle to derive formulas for the volume of the sphere and other geometric solids.
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Trigonometry was introduced in Geometry through a study of right triangles. In Algebra II, work was conducted on extending basic trigonometry to the domain of all real numbers via the unit circle. This module revisits, unites, and further expands those ideas and introduces new tools for solving geometric and modeling problems through the power of trigonometry.
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In this module, students build on their understanding of probability developed in previous grades. In Topic A, the multiplication rule for independent events introduced in Algebra II is generalized to a rule that can be used to calculate the probability of the intersection of two events in situations where the two events are not independent. In this topic, students are also introduced to three techniques for counting outcomesšthe fundamental counting principle, permutations, and combinations. These techniques are then used to calculate probabilities, and these probabilities are interpreted in context.
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In this first module of Grade 1, students make significant progress towards fluency with addition and subtraction of numbers to 10 as they are presented with opportunities intended to advance them from counting all to counting on, which leads many students then to decomposing and composing addends and total amounts. In Kindergarten, students achieved fluency with addition and subtraction facts to 5. This means they can decompose 5 into 4 and 1, 3 and 2, and 5 and 0. They can do this without counting all. They perceive the 3 and 2 embedded within the 5.
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Module 2 serves as a bridge from problem solving within 10 to work within 100 as students begin to solve addition and subtraction problems involving teen numbers. In Module 1, students were encouraged to move beyond the Level 1 strategy of counting all to the more efficient counting on. Now, they go beyond Level 2 to learn Level 3 decomposition and composition strategies, informally called make ten or take from ten.
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Grade 1 Module 3 opens in Topic A by extending student's Kindergarten experiences with direct length comparison to the new learning of indirect comparison whereby the length of one object is used to compare the lengths of two other objects.
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Module 4 builds upon Module 2's work with place value within 20, now focusing on the role of place value in the addition and subtraction of numbers to 40 .

The module opens with Topic A, where students study, organize, and manipulate numbers within 40. Having worked with creating a ten and some ones in Module 2, students now recognize multiple tens and ones. Students use fingers, linking cubes, dimes, and pennies to represent numbers to 40 in various ways--from all ones to tens and ones. They use a place value chart to organize units. The topic closes with the identification of 1 more, 1 less, 10 more, and 10 less as students learn to add or subtract like units.
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Throughout the year, students have explored part_whole relationships in many ways, such as their work with number bonds, tape diagrams, and the relationship between addition and subtraction. In Module 5, students consider part_whole relationships through a geometric lens.
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In this final module of the Grade 1 curriculum, students bring together their learning from Module 1 through Module 5 to learn the most challenging Grade 1 standards and celebrate their progress.

In Topic A, students grapple with comparative word problem type. While students solved some comparative problem types during Module 3 and within the Application Problems in Module 5, this is their first opportunity to name these types of problems and learn to represent comparisons using tape diagrams with two tapes.
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Module 1 sets the foundation for students to master sums and differences to 20. Students subsequently apply these skills to fluently add one-digit to two-digit numbers at least through 100 using place value understanding, properties of operations, and the relationship between addition and subtraction. In Grade 1, students worked extensively with numbers to gain fluency with sums and differences within 10 and became proficient in counting on (a Level 2 strategy). They also began to make easier problems to add and subtract within 20 and 100 by making ten and taking from ten (Level 3 strategies)
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