Provide at least three specific examples of how the Standards Deconstructed document supported your understanding of the Standards. In other words, what ar

HS

COMMON CORE

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HIGH SCHOOL GEOMETRY

DECONSTRUCTED for CLASSROOM IMPACT

State Standards

MATHEMATICS

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 3

MATHEMATICS

Introduction The Common Core Institute is pleased to offer this grade-level tool for educators who are teaching with the Common Core State Standards.

The Common Core Standards Deconstructed for Classroom Impact is designed for educators by educators as a two-pronged resource and tool 1) to help educators increase their depth of understanding of the Common Core Standards and 2) to enable teachers to plan College & Career Ready curriculum and classroom instruction that promotes inquiry and higher levels of cognitive demand.

What we have done is not all new. This work is a purposeful and thoughtful compilation of preexisting materials in the public domain, state department of education websites, and original work by the Center for College & Career Readiness. Among the works that have been compiled and/or referenced are the following: Common Core State Standards for Mathematics and the Appendix from the Common Core State Standards Initiative; Learning Progressions from The University of Arizona’s Institute for Mathematics and Education, chaired by Dr. William McCallum; the Arizona Academic Content Standards; the North Carolina Instructional Support Tools; and numerous math practitioners currently in the classroom.

We hope you will find the concentrated and consolidated resource of value in your own planning. We also hope you will use this resource to facilitate discussion with your colleagues and, perhaps, as a lever to help assess targeted professional learning opportunities.

Understanding the Organization

The Overview acts as a quick-reference table of contents as it shows you each of the domains and related clusters covered in this specific grade-level booklet. This can help serve as a reminder of what clusters are part of which domains and can reinforce the specific domains for each grade level.

Critical Areas of Focus is designed to help you begin to approach how to examine your curriculum, resources, and instructional practices. A review of the Critical Areas of Focus might enable you to target specific areas of professional learning to refresh, as needed.

For each domain is the domain itself and the associated clusters. Within each domain are sections for each of the associated clusters. The cluster-specific content can take you to a deeper level of understanding. Perhaps most importantly, we include here the Learning Progressions. The Learning Progressions provide context for the current domain and its related standards. For any grade except Kindergarten, you will see the domain-specific standards for the current grade in the center column. To the left are the domain-specific standards for the preceding grade and to the right are the domain-specific standards for the following grade. Combined with the Critical Areas of Focus, these Learning Progressions can assist you in focusing your planning.

Math Fluency Standards K Add/subtract within 5

1 Add/subtract within 10

2 Add/subtract within 20

Add/subtract within 100 (pencil & paper)

3 Multiply/divide within 100

Add/subtract within 1000

4 Add/subtract within 1,000,000

5 Multi-digit multiplication

6 Multi-digit division

Multi-digit decimal operations

7 Solve px + q = r, p(x + q) = r

8 Solve simple 2 x 2 systems by inspection

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4 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L, 11TH – 12TH GRADES: 1185L TO 1385L

HIGH SCHOOL GEOMETRY

For each cluster, we have included four key sections: Description, Big Idea, Academic Vocabulary, and Deconstructed Standard.

The cluster Description offers clarifying information, but also points to the Big Idea that can help you focus on that which is most important for this cluster within this domain. The Academic Vocabulary is derived from the cluster description and serves to remind you of potential challenges or barriers for your students.

Each standard specific to that cluster has been deconstructed. There Deconstructed Standard for each standard specific to that cluster and each Deconstructed Standard has its own subsections, which can provide you with additional guidance and insight as you plan. Note the deconstruction drills down to the sub-standards when appropriate. These subsections are:

� Standard Statement

� Standard Description

� Essential Question(s)

� Mathematical Practice(s)

� DOK Range Target for Learning and Assessment

� Learning Expectations

� Explanations and Examples

As noted, first are the Standard Statement and Standard Description, which are followed by the Essential Question(s) and the associated Mathematical Practices. The Essential Question(s) amplify the Big Idea, with the intent of taking you to a deeper level of understanding; they may also provide additional context for the Academic Vocabulary.

The DOK Range Target for Learning and Assessment remind you of the targeted level of cognitive demand. The Learning Expectations correlate to the DOK and express the student learning targets for student proficiency for KNOW, THINK, and DO, as appropriate. In some instances, there may be no learning targets for student proficiency for one or more of KNOW, THINK or DO. The learning targets are expressions of the deconstruction of the Standard as well as the alignment of the DOK with appropriate consideration of the Essential Questions.

The last subsection of the Deconstructed Standard includes Explanations and Examples. This subsection might be quite lengthy as it can include additional context for the standard itself as well as examples of what student work and student learning could look like. Explanations and Examples may offers ideas for instructional practice and lesson plans. A wonderful resource for explanations and examples, which we often referred to and cited as a source in this tool, is www.shmooop.com.

The high school standards specify the mathematics that all students should study in order to be college and career ready. Additional mathematics that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics is indicated by (+), as in this example:

(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers).

All standards without a (+) symbol should be in the common mathematics curriculum for all college and career ready students. Standards with a (+) symbol may also appear in courses intended for all students.

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 5

MATHEMATICS

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PRACTICE EXPLANATION

MP.1 Make sense and persevere in problem solving.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need.

Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

MP.2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Standards for Mathematical Practice in High School Mathematics Courses The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

6 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L, 11TH – 12TH GRADES: 1185L TO 1385L

HIGH SCHOOL GEOMETRY PRACTICE EXPLANATION

MP.3 Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others.

They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.

Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

MP.4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later.

They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, 2-by-2 tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions.

They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

MP.5 Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, ruler, protractor, calculator, spreadsheet, computer algebra system, statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students interpret graphs of functions and solutions generated using a graphing calculator.

They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 7

MATHEMATICS PRACTICE EXPLANATION

MP.6 Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem.

They express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other.

By the time they reach high school they have learned to examine claims and make explicit use of definitions.

MP.7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have.

Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7.

They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems.

They also can step back for an overview and shift perspective.

They can see complicated things, such as some algebraic expressions, as single objects or as composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

MP.8 Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)(x – 1) = 3.

Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series.

As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

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8 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L, 11TH – 12TH GRADES: 1185L TO 1385L

HIGH SCHOOL GEOMETRY OVERVIEW

An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts—interpreting a schematic drawing, estimating the amount of wood needed to frame a sloping roof, rendering computer graphics, or designing a sewing pattern for the most efficient use of material.

Although there are many types of geometry, school mathematics is devoted primarily to plane Euclidean geometry, studied both synthetically (without coordinates) and analytically (with coordinates). Euclidean geometry is characterized most importantly by the Parallel Postulate, that through a point not on a given line there is exactly one parallel line. (Spherical geometry, in contrast, has no parallel lines.)

During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs. Later in college some students develop Euclidean and other geometries carefully from a small set of axioms.

The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance and angles (and therefore shapes generally). Reflections and rotations each explain a particular type of symmetry, and the symmetries of an object offer insight into its attributes—as when the reflective symmetry of an isosceles triangle assures that its base angles are congruent.

In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other. This is the principle of superposition. For triangles, congruence means the equality of all corresponding pairs of sides and all corresponding pairs of angles. During the middle grades, through experiences drawing triangles from given conditions, students notice ways to specify enough measures in a triangle to ensure that all triangles drawn with those measures are congruent. Once these triangle congruence criteria (ASA, SAS, and SSS) are established using rigid motions, they can be used to prove theorems about triangles, quadrilaterals, and other geometric figures.

Similarity transformations (rigid motions followed by dilations) define similarity in the same way that rigid motions define congruence, thereby formalizing the similarity ideas of “same shape” and “scale factor” developed in the middle grades. These transformations lead to the criterion for triangle similarity that two pairs of corresponding angles are congruent.

The definitions of sine, cosine, and tangent for acute angles are founded on right triangles and similarity, and, with the Pythagorean Theorem, are fundamental in many real-world and theoretical situations. The Pythagorean Theorem is generalized to non-right triangles by the Law of Cosines. Together, the Laws of Sines and Cosines embody the triangle congruence criteria for the cases where three pieces of information suffice to completely solve a triangle. Furthermore, these laws yield two possible solutions in the ambiguous case, illustrating that Side-Side-Angle is not a congruence criterion. Analytic geometry connects algebra and geometry, resulting in powerful methods of analysis and problem solving. Just as the number line associates numbers with locations in one dimension, a pair of perpendicular axes associates pairs of numbers with locations in two dimensions. This correspondence between numerical coordinates and geometric points allows methods from algebra to be applied to geometry and vice versa. The solution set of an equation becomes a geometric curve, making visualization a tool for doing and understanding algebra. Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric understanding, modeling, and proof. Geometric transformations of the graphs of equations correspond to algebraic changes in their equations.

OVERVIEW

Dynamic geometry environments provide students with experimental and modeling tools that allow them to investigate geometric phenomena in much the same way as computer algebra systems allow them to experiment with algebraic phenomena.

Connections to Equations

The correspondence between numerical coordinates and geometric points allows methods from algebra to be applied to geometry and vice versa. The solution set of an equation becomes a geometric curve, making visualization a tool for doing and understanding algebra. Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric understanding, modeling, and proof.

Congruence (G-CO) � Experiment with transformations in the plane.

� Understand congruence in terms of rigid motions.

� Prove geometric theorems.

� Make geometric constructions

Similarity, Right Triangles, and Trigonometry (G-SRT) � Understand similarity in terms of similarity transformations.

� Prove theorems involving similarity.

� Define trigonometric ratios and solve problems involving right triangles.

� Apply trigonometry to general triangles.

Circles (G-C) � Understand and apply theorems about circles.

� Find arc lengths and areas of sectors of circles.

Expressing Geometric Properties with Equations (G-GPE) � Translate between the geometric description and the equation for a conic section.

� Use coordinates to prove simple geometric theorems algebraically.

Geometric Measurement and Dimension (G-GMD) � Explain volume and formulas and use them to solve problems.

� Visualize relationships between two-dimensional and three-dimensional objects.

Modeling with Geometry (G-MG) � Apply geometric concepts in modeling situations.

Mathematical Practices (MP) MP 1. Make sense of problems and persevere in solving them.

MP 2. Reason abstractly and quantitatively.

MP 3. Construct viable arguments and critique the reasoning of others.

MP 4. Model with mathematics.

MP 5. Use appropriate tools strategically.

MP 6. Attend to precision.

MP 7. Look for and make use of structure.

MP 8. Look for and express regularity in repeated reasoning.

COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT 9

MATHEMATICS

OVERVIEW

Dynamic geometry environments provide students with experimental and modeling tools that allow them to investigate geometric phenomena in much the same way as computer algebra systems allow them to experiment with algebraic phenomena.

Connections to Equations

The correspondence between numerical coordinates and geometric points allows methods from algebra to be applied to geometry and vice versa. The solution set of an equation becomes a geometric curve, making visualization a tool for doing and understanding algebra. Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric understanding, modeling, and proof.

Congruence (G-CO) � Experiment with transformations in the plane.

� Understand congruence in terms of rigid motions.

� Prove geometric theorems.

� Make geometric constructions

Similarity, Right Triangles, and Trigonometry (G-SRT) � Understand similarity in terms of similarity transformations.

� Prove theorems involving similarity.

� Define trigonometric ratios and solve problems involving right triangles.

� Apply trigonometry to general triangles.

Circles (G-C) � Understand and apply theorems about circles.

� Find arc lengths and areas of sectors of circles.

Expressing Geometric Properties with Equations (G-GPE) � Translate between the geometric description and the equation for a conic section.

� Use coordinates to prove simple geometric theorems algebraically.

Geometric Measurement and Dimension (G-GMD) � Explain volume and formulas and use them to solve problems.

� Visualize relationships between two-dimensional and three-dimensional objects.

Modeling with Geometry (G-MG) � Apply geometric concepts in modeling situations.

Mathematical Practices (MP) MP 1. Make sense of problems and persevere in solving them.

MP 2. Reason abstractly and quantitatively.

MP 3. Construct viable arguments and critique the reasoning of others.

MP 4. Model with mathematics.

MP 5. Use appropriate tools strategically.

MP 6. Attend to precision.

MP 7. Look for and make use of structure.

MP 8. Look for and express regularity in repeated reasoning.

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10 COMMON CORE STATE STANDARDS DECONSTRUCTED FOR CLASSROOM IMPACT

LEXILE GRADE LEVEL BANDS: 9TH GRADE: 1050L TO 1260L, 10TH GRADE: 1080L TO 1335L, 11TH – 12TH GRADES: 1185L TO 1385L

HIGH SCHOOL GEOMETRY

Domain HS Algebra I Mathematics I Geometry

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The Real Number System (RN) RN, 1, 2, 3

Quantities (Q) Q.1, 2, 3 Q.1, 2, 3

The Complex Number System (CN)

Vector Quantities and Matrices (VM)

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Seeing Structure in Expressions (SSEE) SSE.1a, 1b, 2, 3a, 3b, 3c SSE.1a, 1b

Arithmetic with Polynomials and Rational Expressions (APR) APR .1

Creating Equations (CED) CED. 1, 2, 3, 4 CED. 1, 2, 3, 4

Reasoning with Equations and Inequalities (REI)

REI. 1, 3, 4a, 4b, 5, 6, 7, 10, 11, 12 REI. 1, 3, 5, 6, 10, 11, 12

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Interpreting Functions (IF) IF, 1, 2, 3, 4, 5, 6, 7a, 7b, 7c, 8a, 8b, 9

IF. 1, 2, 3, 4, 5, 6, 7a, 7c, 9

Building Functions (BF) BF. 1a, 1b, 2, 3, 4a BF. 1a, 1b, 2, 3

Linear, Quadratic, and Exponential Models (LE) LE. 1a, 1b, 1c, 2, 3, 5 LE. 1a, 1b, 1c, 2, 3, 5

Trigonometric Functions (TF)

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Congruence (CO) CO. 1, 2, 3, 4, 5, 6, 7, 8, 12, 13 CO. 1-13

Similarity, Right Triangles, and Trigonometry (SRT) SRT. 1-11

Circle (C) C. 1-5

Expressing Geometric Properties with Equations (GPE) GEP. 4, 5, 7 GPE. 1, 2, 4-7

Geometric Measurement and Dimension (GMD) GMD. 1, 3, 4

Modeling with Geometry (MG) MG. 1, 2, 3

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