ITHS 2013 – 2014 Geometry: MGS-42 MATHEMATICS Curriculum Map Common Core State Standards Pierre-Max Foucault-Room 340 Mathematics Department ITHS CCSS Geometry Curriculum Map Common Core State Standards Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. (MP.1) Solving a mathematical problem involves making sense of what is known and applying a thoughtful and logical process which sometimes requires perseverance, flexibility, and a bit of ingenuity. 2. Reason abstractly and quantitatively. (MP.2) The concrete and the abstract can complement each other in the development of mathematical understanding: representing a concrete situation with symbols can make the solution process more efficient, while reverting to a concrete context can help make sense of abstract symbols. 3. Construct viable arguments and critique the reasoning of others. (MP.3) A well-crafted argument/critique requires a thoughtful and logical progression of mathematically sound statements and supporting evidence. 4. Model with mathematics. (MP.4) Many everyday problems can be solved by modeling the situation with mathematics. 5. Use appropriate tools strategically. (MP.5) Strategic choice and use of tools can increase reliability and precision of results, enhance arguments, and deepen mathematical understanding. 6. Attend to precision. (MP.6) Attending to precise detail increases reliability of mathematical results and minimizes miscommunication of mathematical explanations. 7. Look for and make use of structure. (MP.7) Recognizing a structure or pattern can be the key to solving a problem or making sense of a mathematical idea. 8. Look for and express regularity in repeated reasoning. (MP.8) Recognizing repetition or regularity in the course of solving a problem (or series of similar problems) can lead to results more quickly and efficiently. Mathematics Department ITHS CCSS Geometry Curriculum Map Geometry: Common Core State Standards 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 CCSS. The Mathematical Practice Standards 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. 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. 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. 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 crosssections and the result of rotating a two-dimensional object about a line. 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. 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. Mathematics Department ITHS CCSS Geometry Curriculum Map Unit 7- Properties of Polygons and Quadrilaterals G-CO.3.11 G-GPE.2.4 G-GPE.2.5 SSA Unit 8- Similarity G-SRT.1.2 G-SRT.1.3 Unit 9- Right Triangles and Trigonometry G-SRT.3.6 G-SRT.3.7 G-SRT.3.8 G-SRT.4.10 G-SRT.4.11 Unit 10- Transformational Geometry G-CO.1.4 G-CO.1.5 G-CO.1.2 G-CO.1.3 G-CO.2.6 G-SRT.1.1 Unit 11- Two-Dimensional Measurements G-MG.1.1 G-MG.1.2 G-MG.1.3 G-GPE.2.7 Mathematics Department ITHS CCSS Geometry Curriculum Map Unit 12- Properties of Circles G-GPE.1.1 G-C.1.1 G-C.1.2 G-C.1.3 G-C.1.4 G-C.2.5 The following English Language Arts CCSS should be taught throughout the course: RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements or performing tasks, attending to special cases or exceptions defined in the text. RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in context and topics. RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form and translate information expressed visually or mathematically into words. L.1.1: Initiate and participate effectively in a range of collaborative discussions with diverse partners. L.1.2: Integrate multiple sources of information presented in diverse media or formats evaluating the credibility and accuracy of each source. L.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. SL.2.4: Present information, findings and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning. Mathematics Department ITHS CCSS Geometry Curriculum Map WHST.1.1: Write arguments focused on discipline-specific content. WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research. Unit 7-Properties of Quadrilaterals Standard The students will: G-CO.3.11 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.) MP #2, #3 Mathematics Department ITHS Essential Question(s): How are quadrilaterals precisely classified? Learning Goals I can: apply theorems, postulates, or definitions to prove theorems about parallelograms, including: a) Prove opposite sides of a parallelogram are congruent; b) Prove opposite angles of a parallelogram are congruent; c) Prove the diagonals of a parallelogram bisect each other; d) Prove that rectangles are parallelograms with congruent diagonals. CCSS Remarks The definition of a parallelogram includes two pairs of opposite sides parallel and congruent. This includes rectangles, squares, rhombi, kites, and trapezoids. Resources http://www.shmoop.com/com mon-corestandards/handouts/gco_worksheet_11.pdf http://www.shmoop.com/com mon-corestandards/handouts/gco_worksheet_11_ans.pdf http://ccssmath.org/?page_id =2311 Geometry Curriculum Map 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).) MP #3, #7 represent the vertices of a figure in the coordinate plane using variables. connect a property of a figure to the tool needed to verify the property. use coordinates and the right tool to prove or disprove a claim about a figure. For example: a) Use slope to determine if sides are parallel, intersecting, or perpendicular; b) Use the distance formula to determine if sides are congruent. c) Use the midpoint formula or the distance formula to decide if a side has been bisected. Important formulas for coordinate geometry include distance formula, slope formula, midpoint formula, and definitions of quadrilaterals. https://www.cohs.com/editor/ userUploads/file/Meyn/321%2 0Student%20Workbook.pdf http://www.shmoop.com/com mon-corestandards/handouts/g-gpeworksheet_4.pdf http://www.shmoop.com/com mon-corestandards/handouts/g-gpeworksheet_4_ans.pdf Unit 7 – Properties of Quadrilaterals Mathematics Department ITHS CCSS Geometry Curriculum Map Standard The students will: 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). MP #3, #8 Essential Question(s): How are quadrilaterals precisely classified? Learning Goals I can: determine if lines are parallel or perpendicular using their slopes. Remarks Apply these concepts to the characteristics of special quadrilaterals. Resources http://www.google.com/url?sa =t&rct=j&q=ggpe.4&source=web&cd=1&ca d=rja&ved=0CDIQFjAA&url=h ttp%3A%2F%2Flearnzillion.c om%2Flessons%2F286prove-whether-a-point-is-onacircle&ei=V01kUcPTMI629gS GkYD4Cg&usg=AFQjCNEdTl gXqxqTVyWyaIKtVmFbRE9Xg&b vm=bv.44990110,d.eWU http://neaportal.k12.ar.us/inde x.php/2012/02/perpendicularand-parallel-lines/ http://map.mathshell.org/mate rials/download.php?fileid=703 Unit 8 - Similarity Mathematics Department ITHS CCSS Geometry Curriculum Map Course: Geometry Unit 8- Similarity Essential Question(s): How might the features of one figure be useful when solving problems about a similar figure? Standard Learning Goals I can: Remarks Resources The students will: G-SRT.1.2 Students think of similarity Tasks and TI-nspire define similarity as a composition of rigid Given two figures, use the and congruence as separate lessons: motions followed by dilations in which angle definition of similarity in terms of and distinct categories. http://ccssmath.org/?pa measure is preserved and side length is similarity transformations to Remind students that ge_id=2275 proportional. decide if they are similar; explain congruent figures are just identify corresponding sides and using similarity transformations similar figures with a scale “How Tall is the corresponding angles of similar triangles. the meaning of similarity for factor of 1:1. School’s Flagpole”: demonstrate that in a pair of similar triangles as the equality of all http://alex.state.al.us/les triangles, corresponding angles are corresponding pairs of angles and son_view.php?id=1669 congruent (angle measure is preserved) and the proportionality of all corresponding sides are proportional. corresponding pairs of sides. Similarity in Right determine that two figures are similar by SMP #3 Triangles task: verifying that angle measure is preserved http://alex.state.al.us/les and corresponding sides are proportional. son_view.php?id=26341 G-SRT.1.3 Students often confuse the show and explain that when two angle Use the properties of similarity triangle congruence theorems measures are known (AA), the third angle Videos, practice and transformations to establish the with the triangle similarity measure is also known (Third Angle assessments: AA criterion for two triangles to be theorems. It may be Theorem). http://fabienneriesen.co similar. conclude and explain that AA similarity is a necessary to go back and m/Geometry%3ASMP #3 review SSS, SAS, ASA and sufficient condition for two triangles to be the HL theorem to distinguish tutorials---chaptersimilar. 11.php them from AA~, SAS~ and SSS~. “Solving Geometry G-SRT.2.4 Students often cannot use theorems, postulates, or definitions to Prove theorems about triangles. visualize corresponding parts Problems: Floodlights”: prove theorems about triangles, including: http://map.mathshell.org (Theorems include: a line parallel of similar overlapping a) A line parallel to one side of a triangle /materials/download.php to one side of a triangle divides triangles. Have them divides the other two proportionally. ?fileid=1257 the other two proportionally, and separate and redraw the b) If a line divides two sides of a triangle conversely; the Pythagorean triangles. proportionally, then it is parallel to the Theorem proved using triangle third side. similarity.) c) The Pythagorean Theorem proved using SMP #3 triangle similarity. Mathematics Department CCSS Geometry Curriculum Map ITHS Course: Geometry Unit 9- Right Triangles and Trigonometry Standard The students will: 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. MP #2, 7 G-SRT.3.7 Explain and use the relationship between the sine and cosine of complementary angles. MP #2 G-SRT.3.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. MP #1, #4 Mathematics Department ITHS Essential Question(s): How can right triangles be used to solve application problems? Learning Goals I can: Remarks Resources http://ccssmath.org/?p Some students believe that right demonstrate that within a right triangle, line age_id=2283 triangles must be oriented a segments parallel to a leg create similar triangles by particular way or they do not realize “Trig River”: angle-angle similarity. http://www.teachengin that opposite and adjacent sides use characteristics of similar figures to justify the eering.org/view_activit need to be identified with reference trigonometric ratios. y.php?url=http://www.t to a particular acute angle in a right eachengineering.org/c define the following trigonometric ratios for acute triangle. angles in a right triangle: sine, cosine, and tangent. ollection/cub_/activitie Extension: Use division and the s/cub_navigation/cub_ Pythagorean Theorem (a2 + b2 = navigation_lesson03_ c2) to prove that sin2A + cos2A = 1. activity2.xml define complementary angles. http://fabienneriesen.c calculate sine and cosine ratios for acute angles in a om/Geometry.php right triangle when given two side lengths. “Temple Geometry”: use a diagram of a right triangle to explain that for a http://map.mathshell.o pair of complementary angles A and B, the sine of rg/materials/tasks.php angle A is equal to the cosine of angle B and the ?taskid=261&subpage cosine of angle A is equal to the sine of angle B. =apprentice “Pythagorean use angle measures to estimate side lengths and vice Includes the Triangle Inequality Theorem”: Theorems and Hinge Theorem versa. http://map.mathshell.o Some students believe that the solve right triangles by finding the measures of all rg/materials/lessons.p trigonometric ratios defined in this sides and angles in the triangles using Pythagorean hp?taskid=419&subpa cluster apply to all triangles, but Theorem and/or trigonometric ratios and their ge=concept they are only defined for acute inverses. angles in right triangles. draw right triangles that describe real world problems and label the sides and angles with their given measures. solve application problems involving right triangles, including angle of elevation and depression, navigation, and surveying. CCSS Geometry Curriculum Map Course: Geometry Unit 9- Right Triangles and Trigonometry (cont) Standard The students will: G-SRT.4.10 Prove the Laws of Sines and Cosines and use them to solve problems. MP #1, #2, #7 G-SRT.4.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). MP #1, #4 Essential Question(s): How can right triangles be used to solve application problems? Learning Goals I can: Remarks derive the Law of Sines by drawing an altitude in a triangle, using the sine function to find two expressions for the length of the attitude and simplifying the equation that results from setting these expressions equal. derive the Law of Cosines using the Pythagorean Theorem, two right triangles formed by drawing an altitude, and substitution. generalize the Law of Cosines to apply to each included angle. apply the Law of Sines and Cosines to solve real world problems. use the triangle inequality and side/angle relationships to estimate the measure of unknown sides and angles. distinguish between situations that require the Law of Sines and Law of Cosines. apply and use the Law of Sines and Cosines to find unknown side lengths and unknown angle measures in right and non-right triangles. represent real world problems with diagrams of right and non-right triangles and use them to solve for unknown side lengths and angle measures. Unit 10 – Transformational Geometry Mathematics Department ITHS CCSS Geometry Curriculum Map Resources Course: Geometry Unit 10- Transformational Geometry Essential Question(s): In what ways can congruence be useful? How might the features of one figure be useful when solving problems about a similar figure? Standard Learning Goals Remarks I can: The students will: The terms “mapping” and G-CO.1.4 construct the definition of reflection, translation, and “under” are used in special Develop definitions of rotation. rotations, reflections, and ways when studying construct the reflection definition by connecting any point translations in terms of on the preimage to its corresponding point on the reflected transformations. angles, circles, perpendicular image and describing the line segment’s relationship to the Students sometimes confuse lines, parallel lines, and line line of reflection. the terms “transformation” segments. construct the translation definition by connecting any and “translation.” MP #6 point on the preimage to its corresponding point on the Remind students that that translated image, and connecting a second point on the corresponding vertices have preimage to its corresponding point on the translated to be listed in order so that image, and describing how the two segments are equal in corresponding sides and length, point in the same direction, and are parallel. angles can be easily construct the rotation definition by connecting the center identified and that included of rotation to any point on the preimage and to its sides or angles are corresponding point on the rotated image, and describing apparent. the measure of the angle formed and the equal measures of the segments that formed the angle as part of the definition. Mathematics Department ITHS CCSS Resources “Translations, Reflections and Rotations” task: http://www.shodor.org/int eractivate/lessons/Transl ationsReflectionsRotatio ns/ Tessellation based Quilt design: http://alex.state.al.us/les son_view.php?id=29240 Exploring Transformations on a TI84 graphing calculator: http://alex.state.al.us/les son_view.php?id=29240 Dilations Applet: Geometry Curriculum Map 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. MP #5 Mathematics Department ITHS Course: Geometry Unit 10- Transformational Geometry draw a specific transformation when given a geometric figure and a rotation, reflection, or translation. predict and verify the sequence of transformations (a composition) that will map a figure onto another. CCSS Students may confuse rotations and reflections and be unable to differentiate the two. Allowing them the opportunity to physically manipulate the shapes (such as with cut-outs or patty paper) can clear up misconceptions. http://nlvm.usu.edu/en/n av/frames_asid_296_g_ 4_t_3.html Geometry Curriculum Map Course: Geometry Unit 10- Transformational Geometry Essential Question(s): In what ways can congruence be useful? How might the features of one figure be useful when solving problems about a similar figure? Standard Learning Goals I can: Remarks The students will: G-CO.1.2 Rigid transformations draw transformations of reflections, rotations, translations, Represent transformations in the preserve distance and and combinations of these using graph paper, plane using, e.g., transparencies angle measure transparencies, and/or geometry software. and geometry software; describe (reflections, rotations, determine the coordinates for the image (output) of a transformations as functions that figure when a transformation rule is applied to the preimage translations, or take points in the plane as inputs combinations of those). (input). and give other points as outputs. distinguish between transformations that are rigid and Compare transformations that those that are not. preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). MP #6 G-CO.1.3 This is a discussion of describe and illustrate how a rectangle, parallelogram, Given a rectangle, parallelogram, and isosceles trapezoid are mapped onto themselves using symmetry. trapezoid, or regular polygon, transformations. describe the rotations and calculate the number of lines of reflection symmetry and reflections that carry it onto itself. the degree of rotational symmetry of any regular polygon. MP #7 G-CO.2.6 Students may believe define rigid motions as reflections, rotations, translations Use geometric descriptions of rigid that all transformations, and combinations of these, all of which preserve distance motions to transform figures and to including dilations, are and angle measure. predict the effect of a given rigid rigid motions or that any define congruent figures as figures that have the same motion on a given figure; given two two figures that have shape and size and state that a composition of rigid figures, use the definition of the same area motions will map one congruent figure onto the other. congruence in terms of rigid represent a rigid predict the composition of transformations that will map a motions to decide if they are transformation. Provide figure onto a congruent figure. congruent. counterexamples. determine if two figures are congruent by determining if MP #3 rigid motions will turn one figure into the other. Mathematics Department ITHS CCSS Geometry Curriculum Map Resources Tessellation Applet: http://www.shodor. org/interactivate/act ivities/Tessellate/ Online Transformation Games: http://www.onlinem athlearning.com/tra nsformationgame.html Tasks and TInspire activities: http://ccssmath.org/ ?page_id=2245 Course: Geometry Unit 10- Transformational Geometry (cont) Essential Question(s): In what ways can congruence be useful? How might the features of one figure be useful when solving problems about a similar figure? Standard Learning Goals I can: Remarks The students will: G-SRT.1.1 Some students have prior define dilation. Verify experimentally the properties perform a dilation with a given center and scale factor on a knowledge of dilations from the concept of “dilated figure in the coordinate plane. of dilations given by a center and a pupils” which may lead them verify that when a side passes through the center of scale factor: to believe that dilation refers dilation, the side and its image lie on the same line. a. A dilation takes a line not only to objects getting larger. verify that corresponding sides of the preimage and passing through the center of Similarly, they may have images are parallel. the dilation to a parallel line, difficulty figuring out when to verify that a side length of the image is equal to the scale and leaves a line passing multiply versus divide by the factor multiplied by the corresponding side length of the through the center unchanged. scale factor. preimage. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. MP #6, #8 Unit 11 – Two-Dimensional Measurements Mathematics Department ITHS CCSS Geometry Curriculum Map Resources Course: Geometry Unit 11- Two-Dimensional Measurements Standard The students will: G-GPE.2.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. MP #1 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). MP #4 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). MP #1, #4 Mathematics Department ITHS Essential Question(s): In what ways can geometric figures be used to understand real-world situations? Learning Goals I can: Remarks Graphing the given coordinates of use the distance formula to compute the the vertices may help students perimeter and area given the coordinates of visualize the polygon in order to find vertices of a polygon. the perimeter and area. represent real-world objects as geometric figures. estimate measures (circumference, area, perimeter, volume) of real-world objects using comparable geometric shapes or threedimensional figures. apply the properties of geometric figures to comparable real-world objects. Example: The spokes of a wheel of a bicycle are equal lengths because they represent the radii of a circle. Students may have issues with estimating (rounding, conceptual, not being exact, etc.). decide whether it is best to calculate or estimate the area or volume of a geometric figure and perform the calculation or estimation. break composite geometric figures into manageable pieces. convert units of measure. apply area and volume to situations involving density. Example: Determine the population in an area. Students have difficulty converting units of area and volume due to the difference in scale factors. CCSS Resources Patchwork Task: http://map.mathshell .org/materials/downl oad.php?fileid=754 Square Task: http://map.mathshell .org/materials/downl oad.php?fileid=792 Security Camera Task: http://map.mathshell .org/materials/downl oad.php?fileid=798 Location, Location, Location Task: http://illuminations.n ctm.org/LessonDetai l.aspx?id=L660 Geometry Curriculum Map Course: Geometry Unit 11- Two-Dimensional Measurements (cont) Standard The students will: 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). MP #1, #4 Essential Question(s): In what ways can geometric figures be used to understand real-world situations? Learning Goals I can: Remarks create a visual representation of a design problem. Mathematical modeling involves solving problems in which the path to the solve design problems using a geometric model solution is not obvious. A challenge for (graph, equation, table, formula). teaching modeling is finding problems interpret the results and make conclusions based that are interesting and relevant to high on the geometric model. school students and, at the same time, solvable with the mathematical tools at the students’ disposal. Resources Definition of pi investigation: http://illuminations. nctm.org/LessonD etail.aspx?id=L575 Circles and Triangles: http://map.mathsh ell.org/materials/le ssons.php?taskid= 222&subpage=pro blem - Mathematics Department ITHS CCSS Geometry Curriculum Map Course: Geometry Unit 11- Two-Dimensional Measurements (cont) Standard The students will: 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, Cavalieri’s principle, and informal limit arguments.) MP #3, #7 Essential Question(s): In what ways can geometric figures be used to understand real-world situations? Learning Goals I can: Remarks Circumference of a Circle An informal survey of students from elementary school through define π (pi) as the ratio of a circle’s circumference to its college showed the number pi diameter. to be the mathematical idea use algebra to demonstrate that because of the definition of about which more students π (pi), the formula for a circumference must be C = π d. were curious than any other. Area of a Circle There are at least three facets inscribe a regular polygon in a circle and break it into many to this curiosity: the symbol π congruent triangles to find its area. itself, the number 3.14159…, explain and use the dissection method on regular polygons and the formula for the area of to generate an area formula for regular polygons A = ½ a circle. All of these facets can apothem perimeter (A = aP). be addressed here, at least briefly. calculate the area of a regular polygon A = aP. G-CO.4.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. MP #5, #6 Mathematics Department ITHS use pictures to explain that a regular polygon with many sides is nearly a circle, its perimeter is nearly the circumference of a circle, and that its apothem is nearly the radius of a circle. substitute the “nearly” values of a many sided regular polygon into A = aP to show that the formula for the area of a circle is A = πr2. define inscribed polygons (the vertices of the figure must be points on the circle). construct an equilateral triangle inscribed in a circle. construct a square inscribed in a circle. construct a hexagon inscribed in a circle. explain the steps to constructing an equilateral triangle, a square, and a regular hexagon inscribed in a circle. CCSS Emphasize the need for precision and accuracy when doing constructions. Stress the idea that a compass and straightedge are identical to a protractor and ruler. Explain the difference between measurement and construction. Geometry Curriculum Map Resources Rolling Cup Lesson: http://map.mat hshell.org/mat erials/downloa d.php?fileid=1 254 Evaluating Statements about Lengths and Areas: http://map.mat hshell.org/mat erials/downloa d.php?fileid=6 75 Course: Geometry Unit 12 – Properties of Circles Essential Question(s): How can the properties of circles be useful when solving geometric problems? Standard Learning Goals I can: Remarks The students will: G-GPE.1.1 Students will need to learn identify the center and radius of a circle Derive the equation of a circle of how to complete the given its equation. given center and radius using the square. draw a right triangle with a horizontal leg, a Pythagorean Theorem; complete Students forget to change vertical leg, and the radius of a circle as its the square to find the center and the sign of (h,k) to find the hypotenuse. radius of a circle given by an coordinates of the center. use the distance formula (Pythagorean equation. Theorem), the coordinates of a circle’s MP #2, #3, #7 center, and the circle’s radius to write the equation of a circle. convert an equation of a circle in general (quadratic) form to standard form by completing the square. G-C.1.1 The definition of similarity prove that all circles are similar by showing Prove that all circles are similar. and dilation will need to be that for a dilation centered at the center of a MP #3 reviewed with students. circle, the preimage and the image have Online applets can be equal central angle measures. helpful in seeing this relationship. G-C.1.2 Students may think they identify central angles, inscribed angles, Identify and describe relationships can tell by inspection circumscribed angles, diameters, radii, among inscribed angles, radii, and whether a line intersects a chords, and tangents. chords. (Include the relationship circle in exactly one point. describe the relationship between a central between central, inscribed, and It may be beneficial to angle, inscribed angle, or circumscribed circumscribed angles; inscribed formally define a tangent angle and the arc it intercepts. angles on a diameter are right line as the line recognize that an inscribed angle whose angles; the radius of a circle is perpendicular to a radius sides intersect the endpoints of the diameter perpendicular to the tangent where at the point where the of a circle is a right angle and that the radius the radius intersects the circle.) radius intersects the circle. of a circle is perpendicular to the tangent MP #1, #6 where the radius intersects the circle. Mathematics Department ITHS CCSS Resources Sectors of Circles Task: http://map.mathshell.org/m aterials/lessons.php?taski d=441&subpage=concept Deriving equations of Circles: Part 1: http://map.mathshell.org/m aterials/lessons.php?taski d=406&subpage=concept Part 2: http://map.mathshell.org/m aterials/lessons.php?taski d=425&subpage=concept Inscribed and Circumscribed Circles Task: http://map.mathshell.org/m aterials/download.php?filei d=1194 Geometry Curriculum Map Standard The students will: 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. MP #5 G-C.1.4 Construct a tangent line from a point outside a given circle to the circle. MP #5 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. MP #6, #7 Mathematics Department ITHS Essential Question(s): How can the properties of circles be useful when solving geometric problems? Learning Goals I can: Remarks Students sometimes confuse define the terms inscribed, circumscribed, angle bisector, inscribed angles and central angles. and perpendicular bisector. For example they will assume that construct the inscribed circle whose center is the point of the inscribed angle is equal to the intersection of the angle bisectors (the incenter)and arc like a central angle. circumscribed circle whose center is the point of intersection of the perpendicular bisectors of each side of the triangle (the circumcenter). apply the Arc Addition Postulate to solve for missing arc measures. prove that opposite angles in an inscribed quadrilateral are supplementary. define and identify a tangent line. construct a tangent line from a point outside the circle to the circle using construction tools or computer software. define similarity as rigid motions with dilations, which preserves angle measures and makes lengths proportional. use similarity to calculate the length of an arc. define and calculate the radian measure of an angle as the ratio of an arc length to its radius. convert degrees to radians using the constant of proportionality. calculate the area of a circle. define a sector of a circle. calculate the area of a sector using the ratio of the intercepted arc measure and 360multiplied by the area of the circle. CCSS Constant of proportionality for radian measures: 2π angle measure / 360. The formulas for converting radians to degrees and vice versa are easily confused. Knowing that the degree measure of given angle is always a number larger than the radian measure can help students use the correct unit. Sectors and segments are often used interchangeably in everyday conversation. Care should be taken to distinguish these two geometric concepts. Geometry Curriculum Map Resources Essential Question(s): How can the properties of circles be useful when solving geometric problems? Standard Learning Goals I can: Remarks The students will: G-GPE.1.1 Students will need to learn identify the center and radius of a circle Derive the equation of a circle of how to complete the given its equation. given center and radius using the square. draw a right triangle with a horizontal leg, a Pythagorean Theorem; complete Students forget to change vertical leg, and the radius of a circle as its the square to find the center and the sign of (h,k) to find the hypotenuse. radius of a circle given by an coordinates of the center. use the distance formula (Pythagorean equation. Theorem), the coordinates of a circle’s MP #2, #3, #7 center, and the circle’s radius to write the equation of a circle. convert an equation of a circle in general (quadratic) form to standard form by completing the square. G-C.1.1 The definition of similarity prove that all circles are similar by showing Prove that all circles are similar. and dilation will need to be that for a dilation centered at the center of a MP #3 reviewed with students. circle, the preimage and the image have Online applets can be equal central angle measures. helpful in seeing this relationship. Mathematics Department ITHS CCSS Resources Sectors of Circles Task: http://map.mathshell.org/m aterials/lessons.php?taski d=441&subpage=concept Deriving equations of Circles: Part 1: http://map.mathshell.org/m aterials/lessons.php?taski d=406&subpage=concept Part 2: http://map.mathshell.org/m aterials/lessons.php?taski d=425&subpage=concept Inscribed and Circumscribed Circles Task: http://map.mathshell.org/m aterials/download.php?filei Geometry Curriculum Map 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.) MP #1, #6 Mathematics Department ITHS identify central angles, inscribed angles, circumscribed angles, diameters, radii, chords, and tangents. describe the relationship between a central angle, inscribed angle, or circumscribed angle and the arc it intercepts. recognize that an inscribed angle whose sides intersect the endpoints of the diameter of a circle is a right angle and that the radius of a circle is perpendicular to the tangent where the radius intersects the circle. CCSS Students may think they can tell by inspection whether a line intersects a circle in exactly one point. It may be beneficial to formally define a tangent line as the line perpendicular to a radius at the point where the radius intersects the circle. d=1194 Geometry Curriculum Map Mathematics Department ITHS CCSS Geometry Curric