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Curriculum and Instruction – Office of Mathematics 3rd Quarter GEOMETRY DRAFT Introduction In 2014, the Shelby County Schools Board of Education adopted a set of ambitious, yet attainable goals for school and student performance. The District is committed to these goals, as further described in our strategic plan, Destination2025. By 2025, 80% of our students will graduate from high school college or career ready 90% of students will graduate on time 100% of our students who graduate college or career ready will enroll in a post-secondary opportunity In order to achieve these ambitious goals, we must collectively work to provide our students with high quality, College and Career Ready standardsaligned instruction. The Tennessee State Standards provide a common set of expectations for what students will know and be able to do at the end of a grade. College and Career Ready Standards are rooted in the knowledge and skills students need to succeed in post-secondary study or careers. The TN State Standards represent three fundamental shifts in mathematics instruction: focus, coherence and rigor. Focus • • • • The Standards call for a greater focus in mathematics. Rather than racing to cover topics in a mile-wide, inch-deep curriculum, the Standards require us to significantly narrow and deepen the way time and energy is spent in the math classroom. We focus deeply on the major work of each grade so that students can gain strong foundations: solid conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the math classroom. For geometry, the major clusters, account for 53% of time spent on instruction. Supporting Content - information that supports the understanding and implementation of the major work of the grade. Additional Content - content that does not explicitly connect to the major work of the grade yet it is required for proficiency. Subject to revision Coherence Rigor Thinking across grades: • The Standards are designed around coherent progressions from grade to grade. Learning is carefully connected across grades so that students can build new understanding onto foundations built in previous years. Each standard is not a new event, but an extension of previous learning. Conceptual understanding: • The Standards call for conceptual understanding of key concepts, such as place value and ratios. Students must be able to access concepts from a number of perspectives so that they are able to see math as more than a set of mnemonics or discrete procedures. • Procedural skill and fluency: • The Standards call for speed and accuracy in calculation. While the high school standards for math do not list high school fluencies, there are suggested fluency standards for algebra 1, geometry and algebra 2. Linking to major topics: • Instead of allowing additional or supporting topics to detract from course, these concepts serve the course focus. For example, instead of data displays as an end in themselves, they are an opportunity to do grade-level word problems. Application: • The Standards call for students to use math flexibly for applications in problem-solving contexts. In content areas outside of math, particularly science, students are given the opportunity to use math to make meaning of and access content. Shelby County Schools2015/2016 Revised 12/15/15 3 of 11 Curriculum and Instruction – Office of Mathematics 3rd Quarter GEOMETRY Our collective goal is to ensure our students graduate ready for college and career. The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation and connections. Look for and express regularity in repeated reasoning Look for and make use of structure Problem Solving Connection Reasoning and Proof Make sense of problems and persevere in solving them Reason abstractly and quatitatively Mathematical Practices Attend to precision Construct viable arguments and crituqe the reasoning of others Model with mathematics Use appropriate tools strategically Subject to revision DRAFT Representation Communication The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations) procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics and sensible, useful and worthwhile, coupled with a belief in diligence and one’s own efficacy). Throughout the year, students should continue to develop proficiency with the eight Standards for Mathematical Practice. Shelby County Schools2015/2016 Revised 12/15/15 4 of 11 Curriculum and Instruction – Office of Mathematics 3rd Quarter GEOMETRY DRAFT How to Use the Mathematic Curriculum Maps This curriculum map is designed to help teachers make effective decisions about what mathematical content to teach so that ultimately our students can reach Destination 2025. To reach our collective student achievement goals, we know that teachers must change their instructional practice in alignment with the three College and Career Ready shifts in instruction for Mathematics. We should see these shifts in all classrooms: 1) Focus 2) Coherence 3) Rigor Throughout this curriculum map, you will see resources as well as links to tasks that will support you in ensuring that students are able to reach the demands of the standards in your classroom. In addition to the resources embedded in the map, there are some high-leverage resources around each of the three shifts that teachers should consistently access: The TNCore Mathematics Standards The Tennessee Mathematics Standards: Teachers can access the Tennessee State standards, which https://www.tn.gov/education/article/mathematics- are featured throughout this curriculum map and standards represent college and career ready learning at reach respective grade level. Mathematical Shifts Focus The standards are focused on fewer topics so students can http://achievethecore.org/shifts-mathematics learn more Coherence http://achievethecore.org/shifts-mathematics Topics within a grade are connected to support focus, and learning is built on understandings from previous grades Rigor http://achievethecore.org/shifts-mathematics The standards set expectations for a balanced approach to pursuing conceptual understanding, procedural fluency, and application and modeling Subject to revision Shelby County Schools2015/2016 Revised 12/15/15 5 of 11 Curriculum and Instruction – Office of Mathematics 3rd Quarter GEOMETRY DRAFT Curriculum Maps: Locate the TDOE Standards in the left column. Analyze the language of the standards and match each standard to a learning target in the second column. Consult your McGraw-Hill/Glencoe Teachers’ Edition (TE) and other cited references to map out your week(s) of instruction. Plan your weekly and daily objectives, using the standards' explanations provided in the second column. Best practices tell us that making objectives measureable increases student mastery. Carefully review the web-based resources provided in the 'Content and Tasks' column and use them as you introduce or assess a particular standard or set of standards. Review the CLIP Connections found in the right column. Make plans to address the content vocabulary, utilizing the suggested literacy strategies, in your instruction. Examine the other standards and skills you will need to address in order to ensure mastery of the indicated standard. Using your McGraw-Hill/Glencoe TE and other resources cited in the curriculum map, plan your week using the SCS lesson plan template. Remember to include differentiated activities for small-group instruction and math stations. Subject to revision Shelby County Schools2015/2016 Revised 12/15/15 6 of 11 Curriculum and Instruction – Office of Mathematics 3rd Quarter TN State Standards GEOMETRY Essential Understandings Content & Tasks DRAFT CLIP Connections Unit 4 Similarity Using Similar Triangles ( 8 days) G.SRT Similarity, Right Triangles, and Trigonometry Prove theorems using similarity G.SRT.B.4 Prove theorems about triangles. G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Lesson 7.3 Similar Triangles G.SRT Similarity, Right Triangles, and Trigonometry Prove theorems using similarity G.SRT.B.4 Prove theorems about triangles. G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Lesson 7-4 Parallel Lines and Proportional Parts (midsegments was previously covered in unit 2) G.SRT Similarity, Right Triangles, and Trigonometry Prove theorems using similarity G.SRT.B.4 Prove theorems about triangles. G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Lesson 8-1 Geometric Mean Find the geometric mean between two numbers Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse G.SRT Similarity, Right Triangles, and Lesson 7.5 Parts of Similar Triangles Subject to revision Identify similar triangles using the AA Similarity Postulate and the SSS and SAS similarity Theorems Use similar triangles to solve problems Use proportional parts within triangles Use proportional parts with parallel lines Lesson 7.3 pp. 474-483 Similarity AA Similarity SSS & SAS Similarity Applying Similar Triangles Contrast and compare the triangle congruence theorems with the triangle similarity theorems. Lesson 7-4 pp. 484 - 492 Triangle Side Splitter Theorem Dividing a Line Segment into Equal Parts Parallel Lines and Proportional Segments See Mathematics, Instructional Resources, Geometry, Task Arc: Investigating Coordinate Geometry Partitioning See Mathematics, Instructional Resources, Geometry, Task Arc: Investigating Coordinate Geometry However You Want to Slice It Comparing Shapes tncore.org/ Use multiple representations to explore angle bisectors and proportions. See p. 492, #47 Lesson 8-1 pp. 531 - 539 Special Relationships Within Right Triangles Prove the Pythagorean Theorem Using Similarity Lesson 7-5 pp. 495 -503 Utilize Tasks to include the Standards for Mathematical Practice where students have to reason, justify, explain, construct & model their thinking. What is an arithmetic mean and a geometric mean of two numbers? Are they ever equal? Justify your answer. Find a counterexample: If the measure Shelby County Schools2015/2016 Revised 12/15/15 7 of 11 Curriculum and Instruction – Office of Mathematics 3rd Quarter TN State Standards Trigonometry Prove theorems using similarity G.SRT.B.4 Prove theorems about triangles. G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. GEOMETRY Essential Understandings Recognize and use proportional relationships of corresponding angle bisectors, altitudes, and medians of similar triangles Use the Triangle Angle Bisector Theorem Content & Tasks Triangle Angle Bisector Theorem DRAFT CLIP Connections of an altitude and side of a triangle are proportional to the corresponding altitude and corresponding side of another triangle, then the triangles are similar. Unit 5 –Trigonometry Trigonometry with Right Triangles (8 days) G-SRT Similarity, Right Triangles, and Trigonometry Define trigonometric ratios and solve problems involving right triangles Lesson 8-3 – Special Right Triangles Identify and apply side ratios in 45-45-90 right triangles. Identify and apply side ratios in 30-60-90 right triangles Define trigonometric ratios and solve problems involving right triangles G-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ G-SRT.C.6 Subject to revision Discovering Special Right Triangles Learning Task Finding Right Triangles in your Environment Learning Task H.O.T. Problems p.559 #50 Explain how you can find the lengths of two legs of a 30-60-90 triangle in radical form if you are given the length of the hypotenuse. Create your own triangles Learning Task G-SRT.C.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 Similarity, Right Triangles, and Trigonometry Lesson 8.3 pp. 552-559 Utilize Tasks to include the Standards for Mathematical Practice where students have to reason, justify, explain, construct & model their thinking. Lesson 8-4 – Trigonometry Define trigonometric ratios for acute angles in right triangles Use trigonometric rations and Pythagorean Theorem to solve right triangles Use the relationship between the sine and cosine of complementary angles. Lesson 8.4 pp. 562-571 Discovering Trigonometric Ratio Relationships learning task p.22 H.O.T. Problems p.570 #65 Explain how you can use ratios of the side lengths to find the angle measures of the acute angles in a right triangle. Shelby County Schools2015/2016 Revised 12/15/15 8 of 11 Curriculum and Instruction – Office of Mathematics 3rd Quarter TN State Standards G-SRT Similarity, Right Triangles, and Trigonometry Define trigonometric ratios and solve problems involving right triangles G-SRT.C.8 GEOMETRY Essential Understandings Lesson 8-5 – Angles of Elevation and Depression Solve problems involving angles of elevation. Solve problems involving angles of depression. Content & Tasks Lesson 8.5 pp. 574-581 Find that Side or Angle Task Interstate DRAFT CLIP Connections H.O.T. Problems p.580 #25 Classify the statement below as true or false. Explain. “As a person moves closer to an object he or she is sighting, the angle of elevation increases” Unit 7 – Measurement and Modeling in Two and Three Dimensions Volume Formulas (6 days) G-GMD Geometric Measurement and Dimension Explain volume formulas and use them to solve problems G-GMD.A.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. G.GMD.A.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★ Lesson 12.4 – Volumes of Prisms and Cylinders Find volumes of prisms Find volumes of cylinders H.O.T. Problems p.853 #44 Write a helpful response to the following questions posted on an Internet garden forum. “I am new to gardening. The nursery will deliver a truckload of soil, which they say is 4 yards. I know that a yard is 3 feet, but what is a yard of soil? How do I know what to order?” Utilize Tasks to include the Standards for Mathematical Practice where students have to reason, justify, explain, construct & model their thinking. G-GMD Geometric Measurement and Dimension Explain volume formulas and use them to solve problems G-GMD.A.1 G.GMD.A.3 Lesson 12.5 – Volumes of Pyramids and Cones G-GMD Geometric Measurement and Dimension Explain volume formulas and use them to solve problems Lesson 12.6 – Volume of Spheres Find volumes of spheres Subject to revision Lesson 12.4 pp. 847-854 How much money is that? (prism) Centerpiece (cylinder) Find volumes of pyramids Find volumes of cones Lesson 12.5 pp. 857-863 Doctors Appointment (cone) Great Egyptian Pyramids (pyramid) Lesson 12.6 pp. 873-878 Guessing Gumballs Task H.O.T. Problems p.862 #40 Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. H.O.T. Problems p.870 #48 Write a ratio comparing the volume of a sphere with radius r to the volume of a Shelby County Schools2015/2016 Revised 12/15/15 9 of 11 Curriculum and Instruction – Office of Mathematics 3rd Quarter TN State Standards GEOMETRY Essential Understandings Content & Tasks G-GMD.A.1 G.GMD.A.3 DRAFT CLIP Connections cylinder with radius r and height 2r. Then describe what the ratio means. Unit 7 – Measurement and Modeling in Two and Three dimensions Modeling and Problem Solving (5 days) G-MG Modeling with Geometry Apply geometric concepts in modeling situations G.MG.A.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-MG Modeling with Geometry Apply geometric concepts in modeling situations No Glencoe Lesson Use three-dimensional figures to describe objects Subject to revision A plastic beach ball has a diameter of 15 inches. About how much plastic is used to make the beach ball? Describe what processes you used to find the answer. Utilize Tasks to include the Standards for Mathematical Practice where students have to reason, justify, explain, construct & model their thinking. No Glencoe Lesson Model and solve problems using figures and their measurements Glencoe Lesson CCSS Lab 14: Geometry Lab: Population Density (Supplement) Illustrative mathematics G-MG.A.2 multiple lessons and tasks that can be used No Glencoe Lesson Model and solve problems using figures and their measurements Illustrative mathematics G.MG.A.3 multiple lessons and tasks that can be used G-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). ★ G-MG Modeling with Geometry Apply geometric concepts in modeling situations G-MG.A.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).★ Illustrative mathematics G.MG.A.1 multiple lessons and tasks that can be used A candy factory makes peanut butter chocolate balls with a 1 inch diameter using a chocolate shell and a peanut ball center. If the diameter of the peanut butter ball is .5 inch, find the volume of chocolate used to make each ball. What percentage of each ball is peanut butter? Justify your answer. Shelby County Schools2015/2016 Revised 12/15/15 10 of 11 Curriculum and Instruction – Office of Mathematics 3rd Quarter TN State Standards GEOMETRY Essential Understandings Content & Tasks DRAFT CLIP Connections Unit 6: Properties of Circles Angles and Segments in circles (5 days) G - C Circles Understand and apply theorems about circles G.C.A.1 Prove that all circles are silimar Lesson 10-1 – Circles and Circumference Give an argument to justify the formula for the circumference of a circle. Prove that all circles are similar. Lesson 10.1 pp. Similar Circles Task All Circles are Similar Task Utilize Tasks to include the Standards for Mathematical Practice where students have to reason, justify, explain, construct & model their thinking. G-CO Congruence Experiment with transformations in the plane G.CO.A.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-GMD Geometric Measurement and Dimension G.GMD.A.1 G - C Circles Understand and apply theorems about circles G-C.A.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. Subject to revision H.O.T. Problems p.690 #54 Research and write about the history of pi and its importance to the study of geometry. Lesson 10-2 – Measuring Angles and Arcs Identify central angles, major arcs, minor arcs, and semicircles and find their measures. Lesson 10.2 pp. 692-700 Circles and their Relationships among Central Angles, Arcs and Chords Investigating Angle Relationships in Circles H.O.T. Problems p.699 #62 Describe the three different types of arcs in a circle and the method for finding the measure of each one. Shelby County Schools2015/2016 Revised 12/15/15 11 of 11 Curriculum and Instruction – Office of Mathematics 3rd Quarter TN State Standards G - C Circles Understand and apply theorems about circles G-C.A.2 Subject to revision GEOMETRY Essential Understandings Lesson 10-4 – Inscribed Angles Identify and describe relationships involving inscribed angles. Prove properties of angles for a quadrilateral inscribed in a circle. Content & Tasks Lesson 10.4 pp. 709-716 Opposite angles in a cyclic quadrilateral DRAFT CLIP Connections H.O.T. Problems p.715 #50 Compare and contrast inscribed angles and central angles of a circle. If they intercept the same arc, how are they related? Shelby County Schools2015/2016 Revised 12/15/15 12 of 11 Curriculum and Instruction – Office of Mathematics 3rd Quarter Textbook Resources ConnectED Site - Textbook and Resources Glencoe Video Lessons Hotmath - solutions to odd problems Comprehensive Geometry Help: Online Math Learning (Geometry) I LOVE MATH NCTM Illuminations New Jersey Center for Teaching & Learning (Geometry) Calculator Finding Your Way Around TI-83+ & TI-84+ (mathbits.com) Texas Instruments Calculator Activity Exchange Texas Instruments Math Nspired STEM Resources Casio Education for Teachers *Graphing Calculator Note: TI tutorials are available through Atomic Learning and also at the following link: Math Bits graphing calculator steps Some activities require calculator programs and/or applications. Use the following link to access FREE software for your MAC. This will enable your computer and TI Calculator to communicate: Free TI calculator downloads Subject to revision GEOMETRY RESOURCE TOOLBOX CCSS Common Core Standards - Mathematics Common Core Standards - Mathematics Appendix A TN Core CCSS Flip Book with Examples of each Standard Geometry Model Curriculum http://www.ccsstoolbox.org/ http://insidemathematics.org/index.php/high-school-geometry http://www.azed.gov/azcommoncore/mathstandards/hsmath/ http://learnzillion.com/common_core/math/hs http://www.livebinders.com/play/play/454480 https://www.livebinders.com/play/play?id=464831 http://www.livebinders.com/play/play?id=571735 North Carolina – Unpacking Common Core http://thegeometryteacher.wordpress.com/the-geometry-course/ http://mathtermind.blogspot.com/2012/07/common-core- geometry.html Utah Electronic School - Geometry Ohio Common Core Resources Chicago Public Schools Framework and Tasks Mathy McMatherson Blog - Geometry in Common Core Interactive Manipulatives GeoGebra – Free software for dynamic math and science learning NCTM Core Math Tools http://www.keycurriculum.com/products/sketchpad (Not free) Any activity using Geometer’s Sketchpad can also be done with any software that allows construction of figures and measurement, such as Cabri, Cabri Jr. on the TI-83 or 84 Plus, TI-92 Plus, or TI-Nspire DRAFT Videos Math TV Videos The Teaching Channel Teacher Tube Khan Academy Videos (Geometry) CLIP Resources Literacy Skills and Strategies for Content Area Teachers (Math, p. 22) Glencoe Reading & Writing in the Mathematics Classroom Graphic Organizers (9-12) (teachervision.com) Shelby County Schools2015/2016 Revised 12/15/15 13 of 11