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THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan Geometry (1206310) Adopted Instructional Materials: 1-1 Transformations and Congruence 1-2 Lines, Angles, & Triangles 1-3 & 2-1 Triangles Houghton Mifflin Harcourt Geometry 2-2 Quadrilaterals 2-3 Similarity 2-4 & 3-1 Trigonometry 2-4 & 3-1 Trigonometry 3-2 Circles 3-3 Solids 4-1 Probability FLORIDA STATEWIDE ASSESSMENT April 17–May 5, 2017 4-2 Polynomials (optional) The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences from the middle grades. Students explore more complex geometric situations and deepen their explanations of geometric relationships, moving towards formal mathematical arguments. Important differences exist between this Geometry course and the historical approach taken in Geometry classes. For example, transformations are emphasized early in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found in the high school standards. The Standards for Mathematical Practice apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas, organized into five units are as follows. Unit 1-Congruence, Proof, and Constructions: In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They use triangle congruence as a familiar foundation for the development of formal proof. Students prove theorems—using a variety of formats—and solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work. Unit 2- Similarity, Proof, and Trigonometry: Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean Theorem. Students develop the Laws of Sines and Cosines in order to find missing measures of general (not necessarily right) triangles, building on students’ work with quadratic equations done in the first course. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many triangles. Unit 3- Extending to Three Dimensions: Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a twodimensional object about a line. Unit 4- Connecting Algebra and Geometry Through Coordinates: Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines, which relates back to work done in the first course. Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola. Unit 5-Circles With and Without Coordinates: In this unit students prove basic theorems about circles, such as a tangent line is perpendicular to a radius, inscribed angle theorem, and theorems about chords, secants, and tangents dealing with segment lengths and angle measures. They study relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations, which relates back to work done in the first course, to determine intersections between lines and circles or parabolas and between two circles. Page 1 of 28 Updated: June 8, 2016 State Assessment Information FSA Portal Training Tests Site Calculator & Reference Sheet Policy Reference Sheet Packet Online Testing Scientific Calculator for FSA Geometry FSA Test Item Specifications Professional Development Page 2 of 28 Math Practices by Grade Level Build Relationships: Teach More Than ‘Just Math’ Sorting Equations Video: Research shows that formative assessments have a significant impact on student learning gains. This video is just one example of using formative assessment to inform instruction. CPALMS MFAS Training Research around formative assessment shows that students make greater learning gains when they are accountable for their own learning and the learning of their peers. The video, Facilitating Peer Learning, is a good example of a math classroom where students are engaged with one another. Five “Key Strategies” for Effective Formative Assessment Helpful Websites Teaching Channel: Videos and Best Practices https://www.teachingchannel.org/ Illustrative Mathematics: Performance Tasks https://www.illustrativemathematics.org/ Inside Mathematics: Videos and Best Practices http://www.insidemathematics.org/ Khan Academy: Practice by Grade Level Standards https://www.khanacademy.org/commoncore/map Shmoop: Math videos & Lesson Resources http://www.shmoop.com/video/math-videos CK12.org http://www.ck12.org/ Math Open Reference http://www.mathopenref.com/ Math Interactives (LearnAlberta) http://www.learnalberta.ca/content/mejhm/index.html?l=0 iXL Math practice (20 problems free per day) http://www.ixl.com/standards/florida/math/high-school Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 1-1 Geometry (1206310) Adopted Instructional Materials: Houghton Mifflin Harcourt Geometry Big Idea: Transformations and Congruence Standards Math Content Standards Suggested Literacy & English Language Standards MAFS.912.G-CO.1: Experiment with transformations in the plane. MAFS.912.G-CO.1.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. MAFS.912.G-CO.1.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance MAFS.912.G-CO.1.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. MAFS.912.G-CO.1.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. MAFS.912.G-CO.1.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. MAFS.912.G-CO.2: Understand congruence in terms of rigid motions. MAFS.912.G-CO.2.6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. MAFS.912.G-CO.2.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. MAFS.912.G-CO.3: Prove geometric theorems. MAFS.912.G-CO.3.9: Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding LAFS.910.WHST.1.1: Write arguments focused on disciplinespecific content. a. Introduce precise claim(s), distinguish the claim(s) from alternate or opposing claims, and create an organization that establishes clear relationships among the claim(s), counterclaims, reasons, and evidence. b. Develop claim(s) and counterclaims fairly, supplying data and evidence for each while pointing out the strengths and limitations of both claim(s) and counterclaims in a disciplineappropriate form and in a manner that anticipates the audience’s knowledge level and concerns. c. Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims. d. Establish and maintain a formal style and objective tone while attending to the norms and conventions of the discipline in which they are writing. e. Provide a concluding statement or section that follows from or supports the argument presented. Page 3 of 28 Suggested Mathematical Practice Standards MAFS.K12.MP.5.1: Use appropriate tools strategically. What tools do you have available to make constructions? How do constructions help make sense of the problem? Updated: June 8, 2016 angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segments endpoints. MAFS.912.G-CO.4: Make geometric constructions. MAFS.912.G-CO.4.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; MAFS.912.G-GPE.2: Use coordinates to prove simple geometric theorems algebraically. MAFS.912.G-GPE.2.4: Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point MAFS.912.G-MG.1: Apply geometric concepts in modeling situations. MAFS.912.G-MG.1.3: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). ★ MAFS.K12.MP.7.1: Look for and make use of structure. What information helps you determine how to solve the problem? How can you use what you know to explain why this works? Are there any patterns in the problems you’re doing that help you make generalizations? Essential Outcome Question(s) How can you use tools of the trade, or concepts, to describe the construction of geometric figures? How do transformations help identify and prove congruent figures? Aligned Learning Goals District Adopted Materials Define and identify basic geometric concepts including, point, line, line segment, ray, angle, plane, distance along a line, perpendicular lines, parallel lines, linear pair, bisect, perpendicular bisector, and angle bisector Houghton Mifflin Modules 1, 2, & 3 Use various tools to create line segments and angles, including a ruler, compass, and technology Perform a formal geometric construction to copy and bisect a line segment and angle using a variety of tools, which could include a compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software Know and use midpoint formula and distance formula to locate points and find lengths on a coordinate plane and apply these formulas when solving problems and writing coordinate proofs Know and use vocabulary associated with transformations, including transformation, image, preimage, rotation, reflection, translation, dilation, clockwise, counterclockwise, isometry, and rigid motion Page 4 of 28 Supplemental Resources Strategies for Differentiation INTERVENTION MAFS.912.GCO.1.1: Intro to Plane Geometry MAFS.912.GCO.1.1: Sage N Scribe activity using precise vocabulary MAFS.912.GGPE.2.6: Lesson plan segment partitioning MAFS.912.GCO.1.1: Angles MAFS.912.GCO.2.6: Transformations Updated: June 8, 2016 Identify the properties of rigid motions Describe transformations, both rigid and non-rigid, using coordinate notation Understand and use inductive reasoning, deductive reasoning, counterexamples, postulates, and conditional statements to write simple proofs about lines and angles Use postulates and theorems about points, lines, planes, and angles to solve problems Define and use vocabulary and notation associated with vectors, including vector, initial point, and terminal point, to describe translations Perform translations and reflections on a coordinate plane and apply knowledge of translations and reflections to identify coordinates, translation vectors, a line of reflection, center of rotation, location of a figure, image, and preimage on a coordinate plane Perform rotations using various tools both on and off a coordinate plane and apply knowledge to identify the center of rotation and angle of rotation Use symmetry and related vocabulary, including line of symmetry, line symmetry, and rotational symmetry, to describe and identify properties of transformed figures Describe a series of rotations and reflections that will carry a figure onto itself Apply two or more transformations, either rigid or non-rigid, to a given figure to draw a transformed figure, and predict beforehand the effect of the transformation Use rigid motion to determine if two figures on a coordinate plane are congruent Explain how CPCTC follows from figures being congruent and identify congruent parts of congruent figures MAFS.912.GCO.2.6: Lesson transformations by rigid motion MAFS.912.GCO.4.12: Constructions ENRICHMENT MAFS.912.GCO.2.6: Exploring rigid transformations MAFS.K12.MP.3.1: Deductive Arguments Formative Assessment Options: MFAS Tasks G-CO.1.1: Definition of Angle Definition of Perpendicular Lines Definition of Parallel Lines Definition of Line Segment Definition of a Circle Page 5 of 28 MFAS Tasks G-CO.1.3: Transformations of Parallelograms and Rhombi Transformations of Rectangles and Squares Transformations of Regular Polygons Transformations of Trapezoids MFAS Tasks G-CO.1.4: Define a Reflection Define a Rotation Define a Translation MFAS Tasks G-CO.2.6: Repeated Reflections and Rotations Transform This Congruent Trapezoids MFAS Tasks G-CO.2.7: Basketball Goal Congruence Implies Congruent Corresponding Parts Proving Congruence Using Corresponding Parts Updated: June 8, 2016 FSA Item Specifications: MAFS.912.GCO.1.1 Page 6 of 28 MAFS.912.GCO.1.2 MAFS.912.GCO.1.5 MAFS.912.GCO.2.6 MAFS.912.GCO.3.9 MAFS.912.GCO.4.12 MAFS.912.GGPE.2.4 MAFS.912.GMG.1.3 Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 1-2 Geometry (1206310) Adopted Instructional Materials: Houghton Mifflin Harcourt Geometry Big Idea: Lines, Angles, & Triangles Standards Math Content Standards Suggested Literacy & English Language Standards MAFS.912.G-CO.2: Understand congruence in terms of rigid motions. MAFS.912.G-CO.2.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. MAFS.912.G-CO.2.8: Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions. MAFS.912.G-CO.3: Prove geometric theorems. MAFS.912.G-CO.3.9: Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segments endpoints. MAFS.912.G-CO.4: Make geometric constructions. MAFS.912.G-CO.4.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. MAFS.912.G-CO.4.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. MAFS.912.G-GPE.2: Use coordinates to prove simple geometric theorems algebraically. MAFS.912.G-GPE.2.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). MAFS.912.G-SRT.2: Prove theorems involving similarity. MAFS.912.G-SRT.2.5: Use congruence and similarity criteria for triangles to solve LAFS.910.SL.1.2: Integrate multiple sources of information presented in diverse media or formats (e.g., visually, quantitatively, orally) evaluating the credibility and accuracy of each source. LAFS.910.RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words. Page 7 of 28 ELD.K12.ELL.SI.1: English language learners communicate for social and instructional purposes within the school setting. Suggested Mathematical Practice Standards MAFS.K12.MP.3.1: Construct viable arguments and critique the reasoning of others. Do you agree with that answer? Explain. Repeat what he/she said in your own words. How do you know what you are saying is true? MAFS.K12.MP.2.1: Reason abstractly and quantitatively. What does the given information in the problem represent? How does the information help to solve the problem? Updated: June 8, 2016 problems and to prove relationships in geometric figures. Essential Outcome Question(s) How does algebraic knowledge of lines help to solve problems about geometric concepts? Aligned Learning Goals Identify and define various angle pairs, including complementary, supplementary, vertical, adjacent, and linear pair Apply theorems about lines and angles to solve mathematical and real world problems Identify angle pairs formed by a transversal intersecting two coplanar lines, including corresponding angles, alternate interior angles, and same-side interior angles Use properties and theorems of angles formed by a transversal intersecting parallel lines to solve mathematical and real world problems Write proofs justifying congruent angle pairs and parallel lines Perform formal geometric constructions of parallel line, perpendicular lines, and perpendicular bisectors and explain/justify each step in a proof Prove perpendicular bisector theorem and theorems about right angles Identify properties and make generalizations for the slopes of parallel and perpendicular lines Find the equation of a line parallel or perpendicular to a given line and solve geometric problems involving parallel and perpendicular lines Use rigid motion and congruent parts of congruent triangles (CPCTC) to prove triangles are congruent Know and apply the triangle postulates (SSS, SAS, ASA, AAS and HL) and use to prove two triangles congruent Solve mathematical and real world problems using congruence theorems for triangles Page 8 of 28 District Adopted Materials Supplemental Resources Houghton Mifflin Modules 4, 5, & 6 MAFS.912.GGPE.2.5: Parallel Lines Strategies for Differentiation ENRICHMENT MAFS.912.GGPE.2.5: Parallel & Perpendicular Lines Investigation MAFS.912.GCO.4.12: Constructions MAFS.912.GCO.2.7: Congruence Updated: June 8, 2016 Formative Assessment Options: MFAS Tasks G-CO.2.8: Justifying SSS Congruence Justifying SAS Congruence Justifying ASA Congruence MFAS Tasks G-CO.3.9: Proving Vertical Angles Congruent Proving Alternate Interior Angles Congruent Equidistant Points MFAS Tasks G-CO.4.12: Constructing a Congruent Segment Constructing a Congruent Angle Bisecting a Segment and Angle Constructions for Parallel Lines Constructions for Perpendicular Lines MAFS.912.G-CO.4.12 MAFS.912.G-GPE.2.5 MFAS Tasks G-GPE.2.5: Proving the Slope Criterion for Parallel Lines 1 Proving Slope Criterion for Parallel Lines 2 Proving the Slope Criterion for Perpendicular Lines 1 Proving the Slope Criterion for Perpendicular Lines 2 Writing Equations for Parallel Lines Writing Equations for Perpendicular Lines FSA Item Specifications: MAFS.912.G-CO.3.9 Page 9 of 28 MAFS.912.G-SRT.2.5 Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 1-3 & 2-1 Geometry (1206310) Adopted Instructional Materials: Houghton Mifflin Harcourt Geometry Big Idea: Triangles Standards Math Content Standards MAFS.912.G-CO.3: Prove geometric theorems. MAFS.912.G-CO.3.10: Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180° triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. MAFS.912.G-CO.4: Make geometric constructions. MAFS.912.G-CO.4.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. MAFS.912.G-SRT.2: Prove theorems involving similarity. MAFS.912.G-SRT.2.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. MAFS.912.G-C.1: Understand and apply theorems about circles. MAFS.912.G-C.1.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. MAFS.912.G-GPE.2: Use coordinates to prove simple geometric theorems algebraically. MAFS.912.G-GPE.2.4: Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). Page 10 of 28 Suggested Literacy & English Language Standards LAFS.910.SL.2.4: Present information, findings, and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning and the organization, development, substance, and style are appropriate to purpose, audience, and task. Suggested Mathematical Practice Standards MAFS.K12.MP.6.1: Attend to precision. What labels do you need to help make sense of the model/diagram? What information is needed in order to answer the problem accurately? MAFS.K12.MP.1.1: Make sense of problems and persevere in solving them. What is this problem asking? Could someone else understand how to solve the problem based on your explanation? Updated: June 8, 2016 MAFS.912.G-GPE.2.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Essential Outcome Question(s) How do triangle properties assist in solving real world problems? Aligned Learning Goals District Adopted Materials Understand and demonstrate conceptually that the interior angles of a triangle add to 180o and this information can be used to find the interior angles of any polygons Use theorems about triangles to solve problems involving missing interior and exterior angle measures, missing side lengths, and possible side lengths Houghton Mifflin Modules 7&8 Construct midsegments in triangles, and use the midsegment theorem to solve problems about triangle side lengths Fluently use vocabulary associated with triangles and other polygons, including vertex, side, base angle, base, leg, isosceles, equilateral, and regular Use constructions to make sense of theorems about triangles Supplemental Resources INTERVENTION MAFS.912.GCO.3.10: Triangles MAFS.912.GCO.3.10: Lesson Intersecting medians and resulting ratios o Prove theorems about triangles, including, interior angles sum to 180 , the triangle inequality theorem, exterior angle theorem, isosceles triangle theorem, equilateral triangle theorem, and base angles of isosceles triangles are congruent Understand and define vocabulary associated with inscribing and circumscribing a circle about a polygon, including circumscribed, inscribed, circumcircle, circumcenter, incircle, and incenter Describe the relationship between a circumscribed or inscribed circle and special segments’ point of concurrency Create formal geometric constructions of a circumscribed circle about a triangle and an inscribed circle within a triangle Create formal geometric constructions of a triangle’s perpendicular bisectors, angle bisectors, medians, and altitudes and describe their relationship to a point of concurrency, centroid, or orthocenter Page 11 of 28 Strategies for Differentiation MAFS.912.GCO.3.10: Lesson Midsegments MAFS.912.GCO.4.12: Perpendicular Bisectors MAFS.912.G-C.1.3: Triangle Circumcenter MAFS.912.G-C.1.3: Triangle Incenter MAFS.912.GCO.4.12: Constructions ENRICHMENT MAFS.912.GCO.3.10: Centroid of a Triangle Updated: June 8, 2016 Formative Assessment Options: MFAS Tasks G-CO.3.10: Triangle Sum Proof Isosceles Triangle Proof Triangle Midsegment Proof Median Concurrency Proof MFAS Tasks G-SRT.2.5: Basketball Goal County Fair Similar Triangles 1 Prove Rhombus Diagonals Bisect Angles Similar Triangles 2 MFAS Tasks G-C.1.3: Circumscribed Circle Construction Inscribed Circle Construction Inscribed Quadrilaterals MAFS.912.G-SRT.2.5 MAFS.912.G-GPE.2.4 FSA Item Specifications: MAFS.912.G-CO.3.10 Page 12 of 28 MAFS.912.G-CO.4.12 MAFS.912.G-C.1.3 MAFS.912.G-GPE.2.5 Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 2-2 Geometry (1206310) Adopted Instructional Materials: Houghton Mifflin Harcourt Geometry Big Idea: Quadrilaterals Standards Math Content Standards Suggested Literacy & English Language Standards MAFS.912.G-CO.3: Prove geometric theorems. MAFS.912.G-CO.3.11: Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. MAFS.912.G-SRT.2: Prove theorems involving similarity. MAFS.912.G-SRT.2.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. MAFS.912.G-GPE.2: Use coordinates to prove simple geometric theorems algebraically. MAFS.912.G-GPE.2.4: Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). MAFS.912.G-GPE.2.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). MAFS.912.G-GPE.2.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. LAFS.910.RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text. ELD.K12.ELL.MA.1: English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. Suggested Mathematical Practice Standards MAFS.K12.MP.7.1: Look for and make use of structure. How can you use what you know to explain why this works? What patterns do you see? MAFS.K12.MP.6.1: Attend to precision. How are you showing the meaning of the quantities in the problem? How could you test your solution to see if your answer is accurate? Essential Outcome Question(s) How can you determine the similarities and differences between various quadrilaterals? Page 13 of 28 Updated: June 8, 2016 Aligned Learning Goals District Adopted Materials Identify properties of parallelograms’ sides, angles, and diagonals and use when solving problems about parallelograms, verifying parallelograms, and when identifying quadrilaterals Houghton Mifflin Modules 9 & 10 Define and identify by their properties and special conditions, rectangle, rhombus, and square and explain how they are related to one another and to the larger group, parallelograms Define and use associated vocabulary and symbols when sketching, identifying, and describing parallelograms, trapezoids, and kites Prove theorems about parallelograms, including opposite sides are congruent, opposite angles are congruent, diagonals bisect each other, rectangle diagonals are congruent, and diagonals of a rhombus are perpendicular using slope and/or distance formula when necessary Define and identify by their properties and special conditions, kites, trapezoids, and isosceles trapezoids, and explain how they are related to the larger group, quadrilaterals Use properties of special quadrilaterals to solve problems Prove parallel lines have the same slope and perpendicular lines have slopes that are opposite reciprocals Use coordinate geometry to solve problems involving polygons, including find the perimeter of polygons and area of triangles and quadrilaterals Understand and demonstrate how to write a coordinate proof Supplemental Resources Strategies for Differentiation INTERVENTION MAFS.912.GCO.3.11: Quadrilateral Quest MAFS.912.GMG.1.3: Lesson linking geometry to real-life problem solving Formative Assessment Options: MFAS Tasks G-CO.3.11: Proving Parallelogram Side Congruence Proving Parallelogram Angle Congruence Proving Parallelogram Diagonals Bisect Proving A Rectangle A Parallelogram Proving Congruent Diagonals MFAS Tasks G-GPE.2.4: Describe the Quadrilateral Diagonals of a Rectangle Midpoints of Sides of a Quadrilateral Type of Triangle MFAS Tasks G-GPE.2.7: Pentagon’s Perimeter Perimeter and Area of a Rectangle Perimeter and Area of a Right Triangle Perimeter and Area of an Obtuse Triangle FSA Item Specifications: MAFS.912.G-CO.3.11 Page 14 of 28 MAFS.912.G-SRT.2.5 MAFS.912.G-GPE.2.4 MAFS.912.G-GPE.2.5 MAFS.912.G-GPE.2.7 Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 2-3 Geometry (1206310) Adopted Instructional Materials: Houghton Mifflin Harcourt Geometry Big Idea: Similarity Standards Math Content Standards Suggested Literacy & English Language Standards MAFS.912.G-SRT.1: Understand similarity in terms of similarity transformations. MAFS.912.G-SRT.1.1: Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. MAFS.912.G-SRT.1.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and proportionality of all corresponding pairs of sides. MAFS.912.G-SRT.1.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. MAFS.912.G-SRT.2: Prove theorems involving similarity. MAFS.912.G-SRT.2.4: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. MAFS.912.G-SRT.2.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. MAFS.912.G-GPE.2: Use coordinates to prove simple geometric theorems algebraically. MAFS.912.G-GPE.2.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio. ★ MAFS.912.G-C.1: Understand and apply theorems about circles. MAFS.912.G-C.1.1: Prove that all circles are similar. MAFS.912.G-CO.4: Make geometric constructions. MAFS.912.G-CO.4.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic LAFS.910.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9–10 texts and topics. Page 15 of 28 ELD.K12.ELL.SI.1: English language learners communicate for social and instructional purposes within the school setting. Suggested Mathematical Practice Standards MAFS.K12.MP.5.1: Use appropriate tools strategically. What math tools are available for finding the solution? MAFS.K12.MP.8.1: Look for and express regularity in repeated reasoning. Is this always true, sometimes true, or never true? What generalizations can you make? Updated: June 8, 2016 geometric software, etc.). Essential Outcome Question(s) How can you determine if two figures are similar? Aligned Learning Goals District Adopted Materials Define and use vocabulary associated with dilations, including dilation, center of dilation, and scale factor Understand and explain how a dilation (along with reflections, translations, and rotations) is a similarity transformation Know the properties of a dilation and use them when solving problems and writing proofs involving dilations and similar figures Demonstrate how to find the center of dilation and scale factor for a given dilation Dilate polygons and line segments Identify corresponding parts of similar figures Show/prove two figures are similar by identifying their similarity transformations Prove that all circles are similar Use transformations to explain why the AA criterion is sufficient to show two triangles are similar Use similarity criteria to set up proportions and solve algebraic problems involving similarity Supplemental Resources Houghton Mifflin Modules 11 & 12 Strategies for Differentiation INTERVENTION MAFS.912.GSRT.1.1: Dilations MAFS.912.GGPE.2.7: Lesson distance and Pythagorean Theorem ENRICHMENT MAFS.912.GSRT.1.1: Dilating a Line MAFS.912.GGPE.2.6: Partitioning a segment MAFS.912.GSRT.1.1: Dilations Use properties of similar triangles to prove the triangle proportionality theorem, the geometric mean theorems, and the Pythagorean Theorem Partition a segment on a coordinate plane into a given ratio, and determine the point that partitions a segment in a given ratio Determine the missing endpoint when given a ratio, one endpoint, and a point on a partitioned line segment Formative Assessment Options: MFAS Tasks G-SRT.2.4: Converse of the Triangle Proportionality Theorem Page 16 of 28 MFAS Tasks G-GPE.2.6: Partitioning a Segment MFAS Tasks G-SRT.1.1: Dilation of a Line: Factor of Two Dilation of a Line Segment MFAS Tasks G-SRT.1.3: Describe the AA Similarity Theorem Updated: June 8, 2016 Pythagorean Theorem Proof Triangle Proportionality Theorem Dilation of a Line: Center on the Line Dilation of a Line: Factor of One Half Justifying a Proof of the AA Similarity Theorem Prove the AA Similarity Theorem FSA Item Specifications: MAFS.912.G-SRT.1.1 Page 17 of 28 MAFS.912.G-SRT.1.2 MAFS.912.G-SRT.1.3 MAFS.912.G-SRT.2.5 MAFS.912.GGPE.2.6 MAFS.912.G-C.1.1 MAFS.912.GCO.4.12 Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 2-4 & 3-1 Geometry (1206310) Adopted Instructional Materials: Houghton Mifflin Harcourt Geometry Big Idea: Trigonometry Standards Math Content Standards Suggested Literacy & English Language Standards MAFS.912.G-SRT.3: Define trigonometric ratios and solve problems involving right triangles. MAFS.912.G-SRT.3.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. MAFS.912.G-SRT.3.7: Explain and use the relationship between the sine and cosine of complementary angles. MAFS.912.G-SRT.3.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★ MAFS.912.G-GPE.2: Use coordinates to prove simple geometric theorems algebraically. MAFS.912.G-GPE.2.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. LAFS.910.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research. ELD.K12.ELL.MA.1: English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. Suggested Mathematical Practice Standards MAFS.K12.MP.2.1: Reason abstractly and quantitatively. What does the given information in the problem represent? How does the information help to solve the problem? MAFS.K12.MP.4.1: Model with mathematics. What other ways could you use to model the situation mathematically? What connections can you make between different representations of the situation? Essential Outcome Question(s) How can you find a missing side or angle of a triangle? Aligned Learning Goals District Adopted Materials Supplemental Resources Strategies for Differentiation Understand and explain the connection between similar right triangles and sine, cosine, and tangent ratios Identify the sides of a right triangle that are opposite and adjacent to a given acute angle. MAFS.912.GSRT.3.8: Trigonometry INTERVENTION Houghton Mifflin Module 13 Page 18 of 28 Updated: June 8, 2016 Identify and use trigonometric ratios needed to find missing sides of right triangles and the inverse of trigonometric ratios to find missing angle measures Explain the relationship between the sine and cosine of complementary angles (sin A = Cos B and Sin B = Cos A) Discover relationships in special right triangles, 45° − 45° − 90° and 30° − 60° − 90° Use the Pythagorean Theorem, trigonometric ratios, and special right triangle relationships to solve mathematical and real world problems involving right triangles Derive the formula for the area of any triangle with a missing height using trigonometric ratios and use the area formula to solve problems MAFS.912.GSRT.3.8: Intro to Trigonometry MAFS.912.GSRT.3.8: Trig Functions MAFS.912.GSRT.3.7: 45-45-90 Triangles MAFS.912.GSRT.3.7: 30-60-90 Triangles ENRICHMENT MAFS.912.GSRT.3.8: Trig Mini Golf Formative Assessment Options: MFAS Tasks G-SRT.3.6: The Sine of 57 The Cosine Ratio MFAS Tasks G-SRT.3.7: Patterns in the 30-60-90 Table Finding Sine Right Triangle Relationships Sine and Cosine MFAS Tasks G-SRT.3.8: Will It Fit? TV Size River Width Washington Monument Holiday Lights Step Up Perilous Plunge Lighthouse Keeper FSA Item Specifications: MAFS.912.G-SRT.3.8 Page 19 of 28 MAFS.912.G-GPE.2.7 Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 3-2 Geometry (1206310) Adopted Instructional Materials: Houghton Mifflin Harcourt Geometry Big Idea: Circles Standards Math Content Standards Suggested Literacy & English Language Standards MAFS.912.G-C.1: Understand and apply theorems about circles. MAFS.912.G-C.1.2: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. MAFS.912.G-C.1.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. MAFS.912.G-C.2: Find arc lengths and areas of sectors of circles. MAFS.912.G-C.2.5: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. MAFS.912.G-GMD.1: Explain volume formulas and use them to solve problems. MAFS.912.G-GMD.1.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieris principle, and informal limit arguments. MAFS.912.G-MG.1: Apply geometric concepts in modeling situations. MAFS.912.G-MG.1.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). ★ MAFS.912.G-GPE.1: Translate between the geometric description and the equation for a conic section. MAFS.912.G-GPE.1.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. MAFS.912.G-CO.4: Make geometric constructions. MAFS.912.G-CO.4.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. LAFS.910.SL.1.1: Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grades 9–10 topics, texts, and issues, building on others’ ideas and expressing their own clearly and persuasively. a. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well-reasoned exchange of ideas. b. Work with peers to set rules for collegial discussions and decision-making (e.g., informal consensus, taking votes on key issues, presentation of alternate views), clear goals and deadlines, and individual roles as needed. c. Propel conversations by posing and responding to questions that relate the current discussion to broader themes or larger ideas; actively incorporate others into the discussion; and clarify, verify, or challenge ideas and conclusions. Page 20 of 28 Suggested Mathematical Practice Standards MAFS.K12.MP.3.1: Construct viable arguments and critique the reasoning of others. Do you agree with ______’s answer? Updated: June 8, 2016 Essential Outcome Question(s) How can similarity and proportion be used to describe relationships among geometric figures related to circles? Aligned Learning Goals District Adopted Materials Define and identify vocabulary associated with circles, including inscribed angle, radius, diameter, arc, major arc, minor arc, adjacent arcs, intercepted arc, semicircle, central angle, inscribed angle, circumscribed angle, tangent, point of tangency, secant, chord, radian measure, and concentric circles Houghton Mifflin Modules 15, 16, & 17 Identify relationships among segments in a circle, such as radius and an inscribed angle or a radius and tangent segment Houghton Mifflin Know and apply the Inscribed Angle Theorem, Circumscribed Angle Theorem, Chord-Chord Product Theorem, Secant-Secant Product Theorem, Secant-Tangent Product Theorem, Intersecting Chords Angle Measure Theorem, Tangent-Secant Interior Angle Theorem, Tangent-Secant Exterior Angle Theorem, and Tangent-Radius Theorem to solve problems with circles Prove and apply the Inscribed Quadrilateral Theorem to solve problems involving quadrilaterals inscribed in circles Construct a circle inscribed in a triangle Construct a circle circumscribed about a triangle and a quadrilateral Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle Construct a tangent line from a point outside a given circle to the circle. Justify and apply the formulas for the circumference and area of a circle Derive the formula for arc length and apply it in problem solving Convert the measure of a central angle in degrees to radian measure and vice versa Identify the constant of proportionality when converting degrees to radian measure Derive the formula for the area of a sector and use it to solve problems Derive the equation of a circle using the Pythagorean Theorem and use it to graph and solve problems Find the center and radius of a circle by completing the square Page 21 of 28 Supplemental Resources INTERVENTION (omit 17.2) Ancillary Material Ellipses & Hyperbolas Strategies for Differentiation MAFS.912.GCO.4.13: Lesson Inscribing Regular Polygons in a Circle Lesson MAFS.912.GCO.4.13: Lesson Inscribing Hexagons and Equilateral Triangles in a Circle Lesson MAFS.912.GCO.4.13: Lesson inscribing polygons in a circle MAFS.912.GGPE.1.1: Equation for a circle using the Pythagorean Theorem MAFS.912.G-C.1.3: Geometric constructions MAFS.912.G-C.1.1: Proving circles similar MAFS.912.GGPE.1.3: Ellipses MAFS.912.G-C.1.3: Lesson circles and triangles MAFS.912.G-C.1.3: Lesson inscribing and circumscribing right triangles Updated: June 8, 2016 Derive the equation of a parabola given a focus and directrix. Derive the equations of ellipses and hyperbolas given the foci and directrices. MAFS.912.GGPE.1.1: Lesson equation of a circle MAFS.912.G-C.2.5: Lesson sector area MAFS.912.GGMD.1.1: Lesson area and circumference Formative Assessment Options: MFAS Tasks G-C.1.2: Central and Inscribed Angles Circles with Angles Inscribed Angle on Diameter Tangent Line and Radius MFAS Tasks G-C.1.3: Inscribed Circle Construction Circumscribed Circle Construction Inscribed Quadrilaterals MFAS Tasks G-C.2.5: Arc Length Sector Area Arc Length and Radians Deriving the Sector Area Formula MFAS Tasks G-GPE.1.1: Complete the Square for Center-Radius Complete the Square for Center-Radius 2 Derive the Circle - General Points Derive the Circle - Specific Points FSA Item Specifications: MAFS.912.G-C.1.2 Page 22 of 28 MAFS.912.G-C.2.5 MAFS.912.G-GMD.1.1 MAFS.912.G-GPE.1.1 MAFS.912.G-MG.1.1 MAFS.912.G-C.1.3 Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 3-3 Geometry (1206310) Adopted Instructional Materials: Houghton Mifflin Harcourt Geometry Big Idea: Solids Standards Math Content Standards Suggested Literacy & English Language Standards MAFS.912.G-GMD.1: Explain volume formulas and use them to solve problems. MAFS.912.G-GMD.1.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieris principle, and informal limit arguments. MAFS.912.G-GMD.1.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★ MAFS.912.G-GMD.2: Visualize relationships between two-dimensional and threedimensional objects. MAFS.912.G-GMD.2.4: Identify the shapes of two-dimensional cross-sections of threedimensional objects, and identify three-dimensional objects generated by rotations of twodimensional objects. MAFS.912.G-MG.1: Apply geometric concepts in modeling situations. MAFS.912.G-MG.1.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). ★ MAFS.912.G-MG.1.2: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). ★ MAFS.912.G-MG.1.3: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★ MAFS.912.G-GPE.2: Use coordinates to prove simple geometric theorems algebraically. MAFS.912.G-GPE.2.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. LAFS.910.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. ELD.K12.ELL.S.1: English language learners communicate for social and instructional purposes within the school setting. Suggested Mathematical Practice Standards MAFS.K12.MP.4.1: Model with mathematics. What do you already know about the problem? What information is important in solving this problem? What other ways could you use to model the situation mathematically? Essential Outcome Question(s) How can you use three-dimensional geometric figures to model real-world objects and to solve real-world problems? Page 23 of 28 Updated: June 8, 2016 Aligned Learning Goals Define vocabulary associated with three-dimensional figures, including net, prism, pyramid, cylinder, right prism, right cylinder, oblique prism, oblique cylinder, cone, base of a figure, height, sphere, cross-section, lateral area, lateral surface, slant height, and volume Explain how to derive the formula for the volume of a cylinder, pyramid, sphere, and cone Find and use volume of prisms, pyramids, cones, spheres, and cylinders to solve problems, including finding a missing dimension, finding dimensions when volume changes, and volume of composite figures Explain Cavalieri’s Principle and apply it when finding volume Identify the shape of a two-dimensional cross-section from a three-dimensional object and explain that the shape will depend on how the cross-section was taken (vertical, horizontal, etc.) Demonstrate how to generate a three-dimensional figure from rotating a two-dimensional figure or composite figure and identify figures generated Understand and demonstrate how to find the surface area of prisms, cylinders, cones, and pyramids, including composite figures, for the purpose of solving real-world problems Understand and explain density and population density and apply these concepts to geometry and real-world problem solving Understand constraints and apply geometric methods to solve real-world design problems District Adopted Materials Supplemental Resources Strategies for Differentiation Houghton Mifflin Modules 18, 19, & 20 MAFS.912.GGMD.1.2: Cavalieri’s Principle Video/Visual INTERVENTION MAFS.912.GGMD.2.4: Solids MAFS.912.GMG.1.2: Lesson apply concepts of density MAFS.G-GMD.2.4: cross-sections and three dimensional objects Formative Assessment Options: MFAS Tasks G-GMD.1.1: Volume of a Cone Volume of a Cylinder Volume of a Pyramid MFAS Tasks G-GMD.1.3: Sports Drinks Snow Cones Do Not Spill the Water! The Great Pyramid MFAS Tasks G-GMD.2.4: 2D Rotations of Triangles 2D Rotations of Rectangles Working Backwards – 2D Rotations Slice It Slice of a Cone Inside the Box MFAS Tasks G-MG.1.2: How Many Trees? Mudslide Population of Utah MFAS Tasks G-MG.1.3: Land for the Twins The Sprinters’ Race Softball Complex The Duplex MAFS.912.G-GPE.2.7 MAFS.912.G-GMD.2.4 FSA Item Specifications: MAFS.912.G-GMD.1.1 Page 24 of 28 MAFS.912.G-MG.1.1 MAFS.912.G-MG.1.2 MAFS.912.G-MG.1.3 Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 4-1 Geometry (1206310) Adopted Instructional Materials: Houghton Mifflin Harcourt Geometry Big Idea: Probability (Optional: Preparing for the Next Level) Standards Math Content Standards MAFS.912.S-CP.1.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). ★ MAFS.912.S-CP.1.2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. ★ MAFS.912.S-CP.1.3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. ★ MAFS.912.S-CP.1.4: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. ★ MAFS.912.S-CP.1.5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. ★ MAFS.912.S-CP.2.6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. ★ MAFS.912.S-CP.2.7: Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. ★ MAFS.912.S-CP.2.8: Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. ★ Page 25 of 28 Suggested Literacy & English Language Standards LAFS.910.SL.1.3: Evaluate a speaker’s point of view, reasoning, and use of evidence and rhetoric, identifying any fallacious reasoning or exaggerated or distorted evidence. ELD.K12.ELL.MA.1: English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. Suggested Mathematical Practice Standards MAFS.K12.MP.8.1: Look for and express regularity in repeated reasoning. Is this always true, sometimes true, or never true? What generalizations can you make? Updated: June 8, 2016 MAFS.912.S-CP.2.9: Use permutations and combinations to compute probabilities of compound events and solve problems. ★ Essential Outcome Question(s) How can you use probability to judge the likelihood of events occurring in real life and make fair decisions? Aligned Learning Goals Understand and use vocabulary and symbols associated with set notation, including set, element, universal set, empty set, Venn diagram, union, intersection, subset, compliment Find the theoretical probability and conditional probability of an event Describe the Fundamental Counting Principle and use it to solve problems Define and find permutations and use factorials to find permutations when using permutations to find probability Find a probability using combinations Find a probability of mutually exclusive events and overlapping events Find a probability from a two-way table Find the probability of independent and dependent events Use probability to make a fair decision Analyze decisions using probability Page 26 of 28 District Adopted Materials Supplemental Resources Strategies for Differentiation Houghton Mifflin Modules 21, 22, & 23 Updated: June 8, 2016 THE SCHOOL DISTRICT OF LEE COUNTY Academic Plan 4-2 Geometry (1206310) Adopted Instructional Materials: McGraw-Hill Algebra 1 Big Idea: Polynomials (Optional: Preparing for Algebra 2) Standards Math Content Standards Suggested Literacy & English Language Standards MAFS.912.A-APR.1: Perform arithmetic operations on polynomials. MAFS.912.A-APR.1.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. MAFS.912.A-APR.2: Understand the relationship between zeros and factors of polynomials. MAFS.912.A-APR.2.3: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. MAFS.912.A-SSE.1: Interpret the structure of expressions. MAFS.912.A-SSE.1.1: Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret as the product of P and a factor not depending on P. MAFS.912.A-SSE.1.2: Use the structure of an expression to identify ways to rewrite it. For example, see 𝑥 4 − 𝑦 4 as (𝑥²)² – (𝑦²)², thus recognizing it as a difference of squares that can be factored as (𝑥² – 𝑦²)(𝑥² + 𝑦²). MAFS.912.A-SSE.2: Write expressions in equivalent forms to solve problems. MAFS.912.A-SSE.2.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. MAFS.912.F-IF.3: Analyze functions using different representations. MAFS.912.F-IF.3.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. LAFS.910.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. ELD.K12.ELL.MA.1: English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. Page 27 of 28 Suggested Mathematical Practice Standards MAFS.K12.MP.7.1: Look for and make use of structure. What patterns do you see? Can you look at the individual parts/terms of the polynomials to help solve the problem? MAFS.K12.MP.8.1: Look for and express regularity in repeated reasoning. Are there generalizations you can make about multiplying binomials? Why are some products of binomials referred to as special cases? Updated: June 8, 2016 Essential Outcome Question(s) When can a polynomial function be used to model and solve a real-world problem? Aligned Learning Goals Identify parts of polynomial expressions Perform operations on polynomial expressions Use polynomial identities to rewrite quadratic expressions Solve quadratic equations by factoring District Adopted Materials McGraw-Hill Algebra 1 Chapter 8, including all Exploration Algebra Labs Supplemental Resources Strategies for Differentiation INTERVENTION MAFS.912.AAPR.1.1: Math Is Fun Polynomial Review MAFS.912.AAPR.1.1: Adding, Subtracting, Multiplying, Dividing Prezi ENRICHMENT MAFS.912.F-IF.3.8: Teach Me How to Factor Math Rap Page 28 of 28 Updated: June 8, 2016