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Hackettstown HACKETTSTOWN, NEW JERSEY Geometry 9-11 CURRICULUM GUIDE FINAL DRAFT August 2015 Mr. David C. Mango, Superintendent Ms. Nadia Inskeep, Director of Curriculum & Instruction Developed by: Michelle DeFilippis Suzanne Sloan This curriculum may be modified through varying techniques, strategies and materials, as per an individual student’s Individualized Education Plan (IEP). Approved by the Hackettstown Board of Education At the regular meeting held on 8/19/2015 And Aligned with the New Jersey Core Curriculum Content Standards And Common Core Content Standards Hackettstown Table of Contents Philosophy and Rationale: 3 Mission Statement: 3 Units: 3-19 NJ Content Standards: 20 21st Century Skills: 20 Hackettstown Philosophy and Rationale The mission of the Hackettstown Mathematics Department is to design and implement a mathematics curriculum that stresses the key ideas identified in the Common Core State Standards. This goal is to be achieved by continually returning to the organizing principles of mathematics and the laws of algebraic and geometric structure to reinforce those concepts. Through a variety of real world scenarios and applications, our students will be challenged to think critically, apply, evaluate, and communicate the ideas and concepts presented in each course as it relates to the field of mathematics. Students will also be challenged to analyze their performance in an effort to become independent problem solvers. Our students’ performance will be assessed using formative techniques such as homework, quizzes, and tests. Additionally, our students will be assessed using state and national assessment tools based upon the branch of mathematics being studied. Mission Statement Building on Tradition and success, the mission of the Hackettstown School District is to educate and inspire students through school, family and community partnerships so that all become positive, contributing members of a global society, with a life-long commitment to learning. Unit Rationale: This unit introduces various topics in the study of geometry and focuses on the areas of visualization, reasoning, and measurement. Hackettstown Stage 1: Desired Results Topic: Tools of Geometry Unit: 1 Common Core State Standards G.CO.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 G.CO.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 G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Essential Questions Enduring Understandings What are the building blocks of geometry? Definitions establish meanings and remove How can you describe the attributes of a segment or angle? possible misunderstandings. Attributes such as length, area, volume, and angle measure are measurable. Knowledge and Skills: By the end of the unit, students will be able: to understand basic terms and postulates of geometry to name and use points, lines, and planes to find and compare lengths of segments and measures of angles (based on knowledge from 8th grade) to identify special angle pairs and use their relationships to find angle measures (based on knowledge from 7th grade) to find the midpoint of a segment to make basic constructions using a straightedge and a compass to use physical models to test conjectures Learning Expectations/Objectives Stage 2: Evidence of Understanding Hackettstown Benchmarks: Students will: understand basic terms and postulates of geometry name and use points, lines, and planes find and compare lengths of segments and measures of angles identify special angle pairs and use their relationships to find angle measures find the midpoint of a segment make basic constructions using a straightedge and a compass use physical models to test conjectures Assessment Methods: Formative: Assignment: A) Construct, measure, and label a line segment and an angle. B) Is it possible to construct a triangle with side measures of 5", 6", and 12'? Use the geometric tools as appropriate. Write a short paragraph explaining your response to the question. Summative: Unit Skills Assessment Other Evidence and Student Self-Assessment: Construct triangle ABC with A(4, 7), B(0, 0), and C(8, 1). a. Which sides are congruent? How do you know? b. Construct the bisector of angle B. Mark the intersection of the ray and AC as D. c. What do you notice about AD and CD? Hackettstown Stage 3: Learning Plan DIFFERENTIATION: based on quiz results, group students homogeneously to complete constructions that are differentiated by difficulty. Information Literacy Access and Evaluate Information Access information efficiently (time) and effectively (sources) Evaluate information critically and competently Activity: Students will read about an antiquated system of measurement, explain its limitations and compare it to the American Standard System as well as the Metric System. Activity: Students will read directions to a buried treasure and interpret the directions to locate it using the tools of geometry. Work Creatively with Others Develop, implement and communicate new ideas to others effectively Be open and responsive to new and diverse perspectives; incorporate group input and feedback into the work Demonstrate originality and inventiveness in work and understand the real world limits to adopting new ideas View failure as an opportunity to learn; understand that creativity and innovation is a long-term, cyclical process of small successes and frequent mistakes Activity: Explore angle pairs with a partner using a clock as a base to gauge understanding by being able to successfully estimate the time shown by the various tasks. Time Allotment: 13 class periods Resources: Student Materials: textbook, notebook, calculator, pencil Technology: smartboard, Graphing calculator Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources Teaching Resources: compass, ruler, graph paper, protractor Unit Rationale: This unit focuses on lines and their relationships and the student’s ability to differentiate between characteristics of various line pairs. Stage 1: Desired Results Unit: 2 Topic: Linear Relationships Common Core State Standards G.CO.9 Prove theorems about lines and angles. 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 segment’s endpoints. G.CO.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. G.GPE.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 Questions Enduring Understandings Who was Euclid and what did he/she do in mathematics? The special angles formed by parallel lines and a How do you prove that two lines are parallel or transversal are congruent, supplementary, or both. perpendicular? Hackettstown You can use certain angle pairs to determine whether lines are parallel. You can graph a line and write its equation when you know certain facts about the line, such as its slope and a point on the line. Knowledge and Skills: By the end of the unit, students will be able: to know who Euclid was and to understand his contribution to the development of geometry to identify angles formed by two lines and a transversal (based on knowledge from 8th grade) to prove theorems about parallel lines to use properties of parallel lines to find angle measures (based on knowledge from 8th grade) to use parallel lines to prove a theorem about triangles to determine whether two lines are parallel to relate slope to parallel and perpendicular lines to find the measures of the angles of a triangle (based on knowledge from 8th grade) to identify and measure interior and exterior angles of a triangle (based on knowledge from 8th grade) to use triangles to find the sum of the measures of the interior and exterior angles of polygons (based on knowledge from 8th grade) Learning Expectations/Objectives Stage 2: Evidence of Understanding Hackettstown Benchmarks: Students will: know who Euclid was and to understand his contribution to the development of geometry identify angles formed by two lines and a transversal prove theorems about parallel lines use properties of parallel lines to find angle measures use parallel lines to prove a theorem about triangles determine whether two lines are parallel relate slope to parallel and perpendicular lines find the measures of the angles of a triangle identify and measure interior and exterior angles of a triangle use triangles to find the sum of the measures of the interior and exterior angles of polygons Assessment Methods: Formative: Complete the following using a straight edge, a compass, and a pencil or using paper folding: a)construct the line parallel to a given line and through a given point not on the line; b) construct a quadrilateral with one pair of parallel sides; c)construct the perpendicular to a given line through a point not on the line. Summative: Unit Skills Assessment Other Evidence and Student Self-Assessment: Exploring Spherical Geometry In spherical geometry, the curved surface of a sphere is studied. A line is a great circle. A great circle is the intersection of a sphere and a plane that contains the center of the sphere. 1) You can use latitude and longitude to identify positions on the Earth. Look at the markings on a globe to answer the following: a) If you 'slice' the globe with a plane at each latitude, do any of the slices contain the center of the globe? b) If you 'slice' the globe with a plane at each longitude, do any of the slices contain the center of the globe? c) which latitudes, if any, suggest great circles? Which longitudes, if any, suggest great circles? Explain your responses. 2) Hold a string taut between any two points on a tennis ball (sphere). The string forms a 'segment' that is part of a great circle. Connect three such segments to form a triangle. Trace your triangle on the tennis ball. a) what is Hackettstown the sum of the angle measures in your triangle? b) Try this experiment several times to explain how the Triangle Angle-Sum Theorem in Euclidean geometry differs from your results in spherical geometry. Stage 3: Learning Plan Produce Results Demonstrate additional attributes associated with producing high quality products including the abilities to: Work positively and ethically Manage time and projects effectively Multi-task Participate actively, as well as be reliable and punctual Present oneself professionally and with proper etiquette Collaborate and cooperate effectively with teams Respect and appreciate team diversity Be accountable for results Sample activity: Students will complete a walk-around activity in pairs. Students will solve equations and search the room for the answer. If students complete all of the equations correctly and check their answers, they will arrive back at their original problem. Interact Effectively with Others Know when it is appropriate to listen and when to speak Conduct themselves in a respectable, professional manner Activity: Student pairs will work together to create a diagram of parallel and perpendicular roads to answer a relationship question. Think Creatively Use a wide range of idea creation techniques (such as brainstorming) Create new and worthwhile ideas (both incremental and radical concepts) Elaborate, refine, analyze and evaluate their own ideas in order to improve and maximize creative efforts Activity: Construct a path to connect a trail to a parking lot. Determine the cost of the path and if there is a more cost effective path. Time Allotment: 11 class periods Resources: Student Materials: textbook, notebook, calculator, pencil Technology: smartboard, Graphing calculator Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources Teaching Resources: compass, ruler, graph paper, protractor Unit Rationale: Making a miniature, a model, or other construction requires skill and knowledge of how to find measures of similar figures. Students will develop an understanding of similar triangles and other polygons and will use their knowledge to solve problems. Hackettstown Stage 1: Desired Results Topic: Congruence Unit: 3 Common Core State Standards G.CO.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. G.CO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Essential Questions Enduring Understandings What is congruence? Congruent triangles have specific characteristics. How do you classify triangles? Two triangles can be proven to be congruent How does coordinate geometry allow you to find without having to show that all corresponding relationships in triangles? parts are congruent. How do you show that two triangles are congruent? Angles and sides of particular triangles have How do you identify corresponding parts of congruent special relationships. triangles? What makes shapes alike and different? Knowledge and Skills: By the end of the unit, students will be able: to recognize congruent figures and their corresponding parts to prove triangles congruent to apply corresponding parts of congruent triangles to triangle proofs to prove right triangles congruent to use and apply properties of isosceles and equilateral triangles Learning Expectations/Objectives Stage 2: Evidence of Understanding Hackettstown Benchmarks: Students will: recognize congruent figures and their corresponding parts prove triangles congruent apply corresponding parts of congruent triangles to triangle proofs prove right triangles congruent use and apply properties of isosceles and equilateral triangles Assessment Methods: Formative: The altitude to the hypotenuse of a right triangle divides the hypotenuse into segments 2 cm and 8 cm long. Find the length of the altitude to the hypotenuse. Use a ruler to make an accurate drawing of the right triangle. Describe how you drew the triangle above. Summative: Unit Skills Assessment Other Evidence and Student Self-Assessment: Homework checks, quizzes, Journal entries, exit slips Stage 3: Learning Plan Critical Thinking and Problem Solving Reason Effectively Use various types of reasoning (inductive, deductive, etc.) as appropriate to the situation Activity: Station activity to construct triangles and develop conjectures Hackettstown Learning and Innovation Skills Creativity and Innovation Think Creatively Use a wide range of idea creation techniques (such as brainstorming) Create new and worthwhile ideas (both incremental and radical concepts) Elaborate, refine, analyze and evaluate their own ideas in order to improve and maximize creative efforts Activity: Writing task to relate AAS congruence to ASA congruence. Time Allotment: 11 class periods Resources: Student Materials: textbook, notebook, calculator, pencil Technology: smartboard, Graphing calculator Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources Teaching Resources: compass, ruler, graph paper, protractor Unit Rationale: Students will build on their understanding of angles and triangles using visualization techniques and constructions. Stage 1: Desired Results Unit: 4 Topic: Triangle Relationships Content Standards G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; 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. G.SRT.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 Essential Questions Enduring Understandings How does coordinate geometry allow you to find There are two special relationships between a relationships in triangles? midsegment and the third side of the triangle. How do you solve problems that involve measurements of For any triangle, certain sets of lines are always triangles? concurrent. The angles and sides of a triangle have special relationships that involve inequalities. Knowledge and Skills: Students will be able: to use properties of midsegments to solve problems to use the Triangle Inequality Theorem to use and apply properties of perpendicular bisectors, angle bisectors, altitudes, and medians to solve problems to find the four points of concurrency in a triangle to formulate conjectures about the relationships between the angles of a triangle and the lengths of the sides to use the Side Splitter Theorem and the Triangle Angle Bisector Theorem Learning Expectations/Objectives Stage 2: Evidence of Understanding Hackettstown Benchmarks: Students will: use properties of midsegments to solve problems use the Triangle Inequality Theorem use and apply properties of perpendicular bisectors, angle bisectors, altitudes, and medians to solve problems find the four points of concurrency in a triangle formulate conjectures about the relationships between the angles of a triangle and the lengths of the sides use the Side Splitter Theorem and the Triangle Angle Bisector Theorem Hackettstown Assessment Methods: Formative: Complete the following using a straight edge, a compass, and a pencil or using paper folding: a)construct the line parallel to a given line and through a given point not on the line; b) construct a quadrilateral with one pair of parallel sides; c)construct the perpendicular to a given line through a point not on the line. Summative: Unit Skills Assessment Other Evidence and Student Self-Assessment: Homework checks, quizzes, Journal entries, exit slips Stage 3: Learning Plan Financial, Economic, Business and Entrepreneurial Literacy Using entrepreneurial skills to enhance workplace productivity Activity: Use the perpendicular bisector to place a t shirt stand equidistant from two attractions, and then 3 attractions. Creativity and Innovation Work Creatively with Others Demonstrate originality and inventiveness in work and understand the real world limits to adopting new ideas Activity: Locate a home that meets given requirement Time Allotment: 13 class periods Resources: Student Materials: textbook, notebook, calculator, pencil Technology: smartboard, Graphing calculator Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources Teaching Resources: compass, ruler, graph paper, protractor Unit Rationale: Trigonometry began as a computational component of geometry. Students will learn that trigonometry depends on angle measurement and quantities determined by the measure of an angle. . In order to understand trigonometry, students must be familiar with and calculate using the Pythagorean Theorem. Stage 1: Desired Results Unit: 5 Topic: Right Triangles and Trigonometry Common Core State Standards +F.TF.A.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. +F.TF.A.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π - x in terms of their values for x, where x is any real number. +F.TF.B.7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.* G.SRT.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. G.SRT.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. G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ +G.SRT.9 (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Hackettstown Essential Questions How can I use the Pythagorean Theorem to determine side length in right triangles? What is important about the study of right triangle trigonometry? What are angles of elevation and depression and how are they used? Enduring Understandings Right triangles have constant ratios of of side lengths. Construction, surveying, and aeronautics use applications of trigonometry involving indirect measurement. Knowledge and Skills: Students will be able: to find and use similarities in right triangles to use the Pythagorean Theorem and its converse (based on knowledge from 8th grade) to use the properties of special right triangles to use trigonometric ratios to determine side lengths and angle measures in right triangles to use angle of elevation and angle of depression in trigonometric ratios to create the unit circle in degrees based on special right triangles to use the unit circle to solve problems in degrees to find area of regular polygons using trigonometry Learning Expectations/Objectives Stage 2: Evidence of Understanding Hackettstown Benchmarks: Students will: find and use similarities in right triangles use the Pythagorean Theorem and its converse use the properties of special right triangles use trigonometric ratios to determine side lengths and angle measures in right triangles use angle of elevation and angle of depression in trigonometric ratios create the unit circle in degrees based on special right triangles use the unit circle to solve problems in degrees find area of regular polygons using trigonometry Assessment Methods: Formative: TASK A: You are creating a design for your website. The design consists of a 6-inch circle with a square inside of it. You want the square to be as large as possible and still be inside the circle. Determine the length of the sides of the square that will fit inside the circle. Explain your solution. TASK B: Work with a partner to answer this question. Be prepared to explain your solution to the class. A farmer's conveyor belt carries bales of hay from the ground to the barn loft as shown in the diagram. The conveyor belt moves at 100 ft/minute. How many seconds does it take for a bale of hay to go from the ground to the barn loft? Summative: Unit Skills Assessment Other Evidence and Student Self-Assessment: Homework checks, quizzes, Journal entries, exit slips Stage 3: Learning Plan Hackettstown CRITICAL THINKING AND PROBLEM SOLVING Reason Effectively Use various types of reasoning (e.g., inductive, deductive, etc.) as appropriate to the situation Activity: Draw two triangles of different sizes each with your given angles. Use a protractor. Measure the sides of each triangle to the nearest tenth of a centimeter. Use a calculator to find the ratios of the lengths of each pair of corresponding sides. What conclusion can you hypothesize about the relationship of the two triangles? Use Systems Thinking Analyze how parts of a whole interact with each other to produce overall outcomes in complex systems Activity: Similarity in right triangles Time Allotment:15 class periods Resources: Student Materials: textbook, notebook, calculator, pencil Technology: smartboard, Graphing calculator Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources Teaching Resources: compass, ruler, graph paper, protractor Unit Rationale: Students will examine the hierarchy of polygons and quadrilaterals. They will build a classification system built on the attributes of various shapes. Stage 1: Desired Results Unit: 6 Topic: Quadrilaterals and Similarity Common Core State Standards G.CO.11 Prove theorems about parallelograms. 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. G.GPE.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). G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★ G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Essential Questions Enduring Understandings How can you classify quadrilaterals? Parallelograms have special properties regarding How can you use coordinate geometry to prove their sides, angles, and diagonals. relationships? A quadrilateral can be proven to be a parallelogram by its sides, angles, and diagonals. A rhombus, rectangle, square, and trapezoid have specific properties for identification. Variables can be used to name the coordinates of a figure in the coordinate plane. Proportionality involves a relationship in which the ratio of two quantities change. Proportions involve multiplicative rather than additive comparisons. Congruent shapes are also similar, but similar shapes may not be congruent. Knowledge and Skills: Hackettstown Students will be able: to use relationships among sides, angles, and diagonals of parallelograms to determine whether a quadrilateral is a parallelogram to use and apply properties of special quadrilaterals including square, rectangle, rhombus, trapezoid, and kite. to classify polygons in the coordinate plane using coordinate proofs to identify and apply similar polygons to use similarity to find indirect measurements to write ratios and proportions to write and solve proportions to write an extended ratio a:b:c to find the geometric mean to know the Means-Extremes Property of proportions to write similarity statements and identify scale factor to prove triangles similar to use similarity to find indirect measurement Learning Expectations/Objectives Stage 2: Evidence of Understanding Hackettstown Benchmarks: Students will: use relationships among sides, angles, and diagonals of parallelograms determine whether a quadrilateral is a parallelogram use and apply properties of special quadrilaterals including square, rectangle, rhombus, trapezoid, and kite. classify polygons in the coordinate plane using coordinate proofs identify and apply similar polygons use similarity to find indirect measurements write ratios and proportions (based on knowledge from 7th grade) write and solve proportions (based on knowledge from 7th grade) write an extended ratio a:b:c find the geometric mean know the Means-Extremes Property of proportions write similarity statements and identify scale factor prove triangles similar use similarity to find indirect measurements Assessment Methods: Formative: TASK 1: ABCDEF is a regular hexagon. Use your understanding of geometry to precisely classify quadrilateral GBHE. How do you know? What are the measures of the interior angles of GBHE? Respond in writing using proper mathematical terminology. Summative: Unit Skills Assessment Other Evidence and Student Self-Assessment: Homework checks, quizzes, Journal entries, exit slips Stage 3: Learning Plan Hackettstown COMMUNICATION AND COLLABORATION Collaborate with Others Demonstrate ability to work effectively and respectfully with diverse learners Assume shared responsibility for collaborative work, and value the individual contributions made by each team member Activity: Properties of Special Parallelograms VISUAL LITERACY Demonstrate the ability to interpret, recognize, appreciate, and understand information presented through visible actions, objects and symbols, natural or man-made2 Activity: Counter examples to false special parallelogram statements to introduce kites and expand on trapezoids. Time Allotment: 12 class periods Resources: Student Materials: textbook, notebook, calculator, pencil Technology: smartboard, Graphing calculator Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources Teaching Resources: compass, ruler, graph paper, protractor Unit Rationale: Students will explore the complexity of circles and use their knowledge to solve problems. Stage 1: Desired Results Unit: 7 Topic: Circles Common Core State Standards G.CO.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. G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.C.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. G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle G.C.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. G.A.2 Derive the equation of a parabola given a focus and directrix. +G.C.4 (+) Construct a tangent line from a point outside a given circle to the circle Essential Questions Enduring Understandings How can you prove relationships between angles and arcs You can use information about congruent parts of in a circle? a circle to find information about other parts of the When lines intersect a circle or within a circle, how do you circle. find the measures of resulting angles, arcs, and segments? Geometric relationships in circles involve angles, What geometric relationships can be found in circles? arcs, and segments. What is the equation of a parabola? Angles formed by intersecting lines have a special relationship to the arcs the intersecting lines intercept. There is a relationship between the equation of a circle and the equation of a parabola. Knowledge and Skills: Hackettstown Students will be able: to use properties of a tangent to a circle to use properties of chords in a circle to find angle measures in circles including central, inscribed, and angles formed by lines to find segment lengths in circles to write the equation of a circle to write the equation of a parabola Learning Expectations/Objectives Stage 2: Evidence of Understanding Hackettstown Benchmarks: Students will: use properties of a tangent to a circle use properties of chords in a circle find angle measures in circles including central, inscribed, and angles formed by lines find segment lengths in circles write the equation of a circle write the equation of a parabola Assessment Methods: Formative: TASK A: A 2-ft-wide circular track for a camera dolly is set up in a studio to film a movie scence. The two rails of the track form concentric circles. The radius of the inner circle is 8 ft. How much further does a wheel on the outer rail travel than a wheel on the inner rail of the track in one turn? Include a diagram in your solution. TASK B: Work with a partner to solve this problem. Be prepared to share your answer in class. A floating dock sits in a lake. The dock is an 8 ft x 10 ft. rectangle. The bow of a canoe is attached to one corner of the dock with a 10 ft. rope. Sketch a diagram to show the region in which the bow of the canoe can travel. Then determine the area of that region. Round your answer to the nearest square foot. Summative: Unit Skills Assessment Other Evidence and Student Self-Assessment: Homework checks, quizzes, Journal entries, exit slips Stage 3: Learning Plan Hackettstown PRODUCTIVITY AND ACCOUNTABILITY Manage Projects Set and meet goals, even in the face of obstacles and competing pressures Prioritize, plan, and manage work to achieve the intended result Activity: "Pizza eating Contest" SCIENTIFIC AND NUMERICAL LITERACY Demonstrate the ability to evaluate the quality of scientific and numerical information on the basis of its sources and the methods used to generate it Demonstrate the capacity to pose and evaluate scientific arguments based on evidence and to apply conclusions from such arguments appropriately Demonstrate ability to reason with numbers and other mathematical concepts Activity: Develop the inscribed angle theorem by following a set of directions. INITIATIVE AND SELF-DIRECTION Work Independently Monitor, define, prioritize, and complete tasks without direct oversight Activity: on student's own time, verify length relationships found by the intersection of a tangent and a secant, two secants, or two chords Time Allotment: 13 class periods Resources: Student Materials: textbook, notebook, calculator, pencil Technology: smartboard, Graphing calculator Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources Teaching Resources: compass, ruler, graph paper, protractor Unit Rationale: Students will gain experience in drawing, explaining, and sketching transformations of shapes, including translations, reflections, rotations, and scaling. Tessellations provide an engaging, artistic way to explore geometry. Stage 1: Desired Results Unit: 8 Topic: Transformations Common Core State Standards G.CO.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 and angle to those that do not (e.g., translation versus horizontal stretch). G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.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. G.CO.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. G.SRT.1ab 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. Hackettstown G.SRT.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 the proportionality of all corresponding pairs of sides. G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Essential Questions Enduring Understandings How can coordinates be used to describe and analyze Shapes may have reflection (line) symmetry or geometric objects? rotation symmetry or neither. How can transformations be described mathematically? An object's location on a plane or in space can be described quantitatively. The position of any point on a surface can be specified by two numbers. Computations with these numbers allow us to describe and measure geometric objects. Knowledge and Skills: Students will be able: to translate, rotate, reflect, and dilate figures (based on knowledge from 8th grade) to write mathematical rules to describe transformations to graph transformations on the coordinate plane (based on knowledge from 8th grade) to identify symmetry in a figure, including line symmetry and rotational symmetrry to perform composite transformations Learning Expectations/Objectives Stage 2: Evidence of Understanding Hackettstown Benchmarks: Students will: translate, rotate, reflect, and dilate figures write mathematical rules to describe transformations graph transformations on the coordinate plane identify symmetry in a figure, including line symmetry and rotational symmetry perform composite transformations Assessment Methods: Formative: Graph A (5, 2). Graph B, the image of A for a 90-degree rotation about the origin O. Graph C, the image of A for a 180-degree rotation about O. Graph D, the image of A for a 270-degree rotation about O. What type of quadrilateral is ABCD Summative: Unit Skills Assessment Other Evidence and Student Self-Assessment: Homework checks, quizzes, Journal entries, exit slips Stage 3: Learning Plan Hackettstown CRITICAL THINKING AND PROBLEM SOLVING Use Systems Thinking Analyze how parts of a whole interact with each other to produce overall outcomes in complex systems Activity: Work with a partner to incorporate the review of reflections with a refresher in linear equations. CRITICAL THINKING AND PROBLEM SOLVING Solve Problems Solve different kinds of non-familiar problems in both conventional and innovative ways Activity: Rotations in astronomy. Rotate each star a given amount of degrees, then innovate an equivalent solution by grouping the stars first and then rotating the whole group. CREATIVITY AND INNOVATION Think Creatively Use a wide range of idea creation techniques (such as brainstorming) Create new and worthwhile ideas (both incremental and radical concepts) Elaborate, refine, analyze, and evaluate ideas in order to improve and maximize creative efforts Demonstrate imagination and curiosity Activity: Explain why it would make sense to categorize reflections and glide reflections as odd isometries and translations and rotations as even isometries. CREATIVITY AND INNOVATION Work Creatively with Others Develop, implement, and communicate new ideas to others effectively Be open and responsive to new and diverse perspectives; incorporate group input and feedback into the work Demonstrate originality and inventiveness in work and understand the real world limits to adopting new ideas View failure as an opportunity to learn; understand that creativity and innovation is a long-term, cyclical process of small successes and frequent mistakes Activity: Twisted Domino FLEXIBILITY AND ADAPTABILITY Adapt to Change Adapt to varied roles, job responsibilities, schedules, and contexts Work effectively in a climate of ambiguity and changing priorities Be Flexible Incorporate feedback effectively Activity: Four in a Row Time Allotment: 11 class periods Resources: Student Materials: textbook, notebook, calculator, pencil Technology: smartboard, Graphing calculator Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources Teaching Resources: compass, ruler, graph paper, protractor Unit Rationale: Students will calculate the lateral, surface area, and volume of various shapes. They will see that surface area is greater than lateral area. They will derive formulas for measuring 3-dimensional shapes and use reasoning and logic to analyze calculations as applied to problem solving. They will apply the concepts of ratio and proportion to various measurements of shapes. Stage 1: Desired Results Hackettstown Unit: 9 Topic: Surface area and volume Common Core State Standards G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★ G.GMD.4 Identify the shapes of two-dimensional cross-sections of three dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. G.MG.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).★ +G.GMD.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. Essential Questions Enduring Understandings What is surface area? What is volume? Three dimensional figures can be analyzed by How do you find surface area and volume of solids? connecting properties of the real objects with How do you find surface area and volume of similar solids? drawings of these objects. How can ratio be used compare similar solids? Length, area, volume, and angles are measurable attributes of geometric figures. Similarity among geometric figures can be established. Knowledge and Skills: Students will be able: to find the area of kites, rhombi, trapezoids, and regular polygons to find the measures of central angles and arcs to find arc length to find area of sectors and segments of a circle to define polyhedron to accurately sketch 3-dimensional shapes and identify parts to find the surface area of prisms, cylinders, cones, pyramids, and spheres to find the volume of prisms, cylinders, cones, pyramids, and spheres (based on knowledge from 7th and 8th grade) to use formulas appropriately when analyzing volume and surface area of 3-dimensional figures to find the ratios between similar figures and their perimeters, areas, surface areas, and volume Learning Expectations/Objectives Stage 2: Evidence of Understanding Hackettstown Benchmarks: Students will: find the area of kites, rhombi, trapezoids, and regular polygons find the measures of central angles and arcs find arc length find area of sectors and segments of a circle define polyhedron accurately sketch 3-dimensional shapes and identify parts find the surface area of prisms, cylinders, cones, pyramids, and spheres find the volume of prisms, cylinders, cones, pyramids, and spheres use formulas appropriately when analyzing volume and surface area of 3-dimensional figures find the ratios between similar figures and their perimeters, areas, surface areas, and volume Hackettstown Assessment Methods: Formative: Complete the following table. Each figure must be accurately sketched or drawn. Present your table on a 12" x 18" piece of construction paper and be prepared to share your work in class. Type of Figure Sketch of Figure Formula for Surface Area Formula for Volume Cylinder Cone Square Pyramid Triangular Pyramid Rectangular Prism Hexagonal Prism Sphere B. Explain in writing how you can justify the formula for the area of a kite (rhombus) using the area of a triangle. Summative: Unit Skills Assessment Other Evidence and Student Self-Assessment: Homework checks, quizzes, Journal entries, exit slips Stage 3: Learning Plan INITIATIVE AND SELF-DIRECTION Manage Goals and Time Set goals with tangible and intangible success criteria Balance tactical (short-term) and strategic (long-term) goals Utilize time and manage workload efficiently Activity: develop formulas for volume of a cone and a pyramid based on physical relationships to corresponding cylinders and prisms. Time Allotment: 16 class periods Resources: Student Materials: textbook, notebook, calculator, pencil Technology: smartboard, Graphing calculator Teaching Materials: Text, online textbook, teacher created resources, additional purchased resources Teaching Resources: compass, ruler, graph paper, protractor Hackettstown New Jersey Core Curriculum and Common Core Content Standards http://www.state.nj.us/education/cccs/ Integration of 21st Century Theme(s) The following websites are sources for the following 21st Century Themes and Skills: http://www.nj.gov/education/code/current/title6a/chap8.pdf http://www.p21.org/about-us/p21-framework . http://www.state.nj.us/education/cccs/standards/9/index.html 21st Century Interdisciplinary Themes (into core subjects) • Global Awareness • Financial, Economic, Business and Entrepreneurial Literacy • Civic Literacy • Health Literacy • Environmental Literacy Learning and Innovation Skills • Creativity and Innovation • Critical Thinking and Problem Solving • Communication and Collaboration Information, Media and Technology Skills • Information Literacy • Media Literacy • ICT (Information, Communications and Technology) Literacy Life and Career Skills • Flexibility and Adaptability • Initiative and Self-Direction • Social and Cross-Cultural Skills • Productivity and Accountability • Leadership and Responsibility Integration of Digital Tools Classroom computers/laptops Technology Lab FM system Other software programs