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Curriculum analysis
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- 2. Common Core StateStandards Understand and apply theorems about circles CCSS.MATH.CONTENT.HSG.C.A.1 • Prove that all circles are similar. CCSS.MATH.CONTENT.HSG.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. CCSS.MATH.CONTENT.HSG.C.A.3 • Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. CCSS.MATH.CONTENT.HSG.C.A.4 • Construct a tangent line from a point outside a given circle to the circle. Find arc lengths and areas of sectors of circles CCSS.MATH.CONTENT.HSG.C.B.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.
- 3. Mathematical Practices Addressed • Standard1: Make sense of problems and persevere in solving them • Standard 2: Reason abstractly and quantitatively • Standard 3: Construct viable arguments and critique the reasoning of others • Standard 4: Model with mathematics • Standard 5: Use appropriate tools strategically • Standard 6: Attend to precision • Standard 7: Look for and make use of structure • Standard 8: Look for and express regularity in repeated reasoning
- 4. Textbooks Examined 1. Dietiker,Leslie et al. Geometry Connections. Vol 2. CPM 2007 2. Serra, Michael. Discovering Geometry: An Inductive Approach. Key Curriculum Press 1997 3. Jergensen, Ray et al. Geometry. Houghton Miffin 1990
- 5. 1. Geometry Connections- CPM • CPM Connections texts are fully aligned with the CCSS Standards for Mathematical Practice. • Curriculum is based solidly on the following: – methodological research in teaching mathematics; – NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections – proficiencies of adaptive reasoning, strategic competence, conceptual understanding, procedural fluency, and productive disposition from the National Research Council’s Adding It Up • Course design is based on three principles: 1) students learn best when engaged in inquiry that leads to deep conceptual understanding of the underlying mathematics 2) exploiting each others’ insights and 3) using distributed learning to increases retention and transfer of knowledge. • integrates basic skills and procedures with conceptual understanding • Encourages students to understand ideas, see relationships between them, and apply mathematical principles to complex problems. • Students are held responsible for high academic rigor, analysis, and critical thinking, and communicate their mathematical findings in writing or in oral presentations in a clear and convincing manner.
- 6. • Major tenantof curriculum is “mastery over time.” • Development of conceptual understanding of a particular standard is not limited to the lesson cited, or even to that chapter. • Standards are regularly practiced throughout lessons and homework subsequent to the lesson cited. • More practice on the standards can be found in the Parent Guide and Extra Practice supplements available online. • The teacher notes provided at the beginning of each lesson in the CPM Teacher Edition provide clarity to educators on the purpose of some problems and how the mathematics in the standards are developed over time. • Each chapter begins with introduction, learning goals, and guiding questions (see photos) More Unique Features
- 7. • Outline ofCircle Properties Topic in Textbook: – 10.1.1 – What’s the diameter? Introduction to Chords – 10.1.2 – What’s the relationship? Angles and Arcs – 10.1.3 – What more can I learn about circles? Chords and Angles – 10.1.4 – What’s the relationship? Tangents and Chords – 10.1.5 – How can I solve it? Problem Solving with Circles – 10.2.1 – What’s the probability? Designing Spinners – 10.2.2 – What should I expect? Expected Value – 10.2.3 – What can I expect? More Expected Value – 10.3.1 – What’s the equation? – Ch. 10 Closure – What have I learned? • Chapter 10 Closure problems require students to reflect, synthesize, team brainstorm, make connections, and summarize understandings. Answer key and support is available at end of closure problems for students to check work. More Unique Features
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- 9. 2. Discovering Geometry •Discovering Geometry is an inquiry based learning program • developed before the CCSS were mandated • is unique because geometric proofs are not introduced until students are ready (prior traditional textbooks mostly teach proofs without investigations) • students work through engaging hands- on activities and develop their own conclusions to support proofs • curriculum encourages students to work in small cooperative groups on projects using take out cartons, patty paper, and string. Compass, straightedge and software are introduced later. • Students are presented with investigations, which may or may not be true.
- 10. Discovering Geometry • Outlineof Circle Properties Topic in Textbook: – Lesson 7.1 Defining Circles – Lesson 7.2 Discovering Chord Properties – Lesson 7.3 Discovering Tangent Properties – Lesson 7.4 Arcs and Angles – Lesson 7.5 The Circumference/Diameter Ratio – Lesson 7.6 Around the World – Lesson 7.7 Arc Length – Project: Racetrack Geometry – Geometer’s Sketchpad Project: Turning Wheels – Lesson 7.8 Chapter Review – Cooperative Problem Solving: Designing a Theater for Galileo • Each lesson contains investigations, take another look using dynamic geometry software, graph paper, and other math tools, and exercise sets. • Students are also asked to find what’s wrong with the picture, improve algebra skills, use graphing calculator investigations, improve reasoning skills, and improve visual thinking skills. • Real life application problems, connections to comics, and a self assessment at the end of the chapter provide a more authentic learning experience overall compared to other geometry textbooks
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- 12. 3. Geometry • Followsa more traditional format for learning geometry (skills based) • Older textbook – does not rely on investigations to acquire learning. Theorems and corollaries are given at the beginning of each section to help students answer exercises. • Includes classroom exercises, written exercises, self check tests, chapter review, and chapter test at the end • Few problems towards each of sections include real life applications – Use tangents to create formula for the distance from the observer to horizon (PH)2 = PA PB – Sketch Earth and circle latitude through city. Find radius of circle. • Explorations are included at the end of some sections that require students to use a computer program to make conjectures • Biographical note about Maria Gaetana Agnesi before 9-3 provides an authentic learning opportunity and chance to apply history to mathematics
- 13. Geometry Continued • Outlineof Circle Properties Topic in Textbook: – Tangents, Arcs, and Chords (includes objectives before begin) • 9-1 Basic Terms (classroom exercise and written exercises) • 9-2 Tangents (classroom exercise and written exercises) • 9-3 Arcs and Central Angles (classroom exercise and written exercises) • 9-4 Arcs and Chords (classroom exercise and written exercises) • Self Test 1 (checks for understanding) – Angles and Segments (includes objectives before begin) • 9-5 Inscribed Angles • 9-6 Other Angles • Self Test 2 (checks for understanding) – Chapter Summary – Chapter Review (fill in the blank and find value for x) – Chapter Test (T/F, fill in the blank, proofs) • Multiple choice chapter exams are available in the back of the textbook • Test taking strategies for students planning to attend college and take college entrance exams are located at the back of the textbook. Formulas and special triangle relationships are also available for students to refer to.
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- 15. Similarities Between Textbooks •All textbooks include real life application problems • Geometry Connections and Discovering Geometry both rely on students to use collaborative learning and teamwork strategies to solve problems. These textbooks also require students to investigate learning first before given the proof • Skills based exercise problems are available for students to develop more mastery • All three rely on using appropriate tools strategically, looking for and making use of structure, and reasoning abstractly and quantitatively • Geometry Connections and Discovering Geometry incorporate CCSS as part of curriculum (Geometry contains learning objects but doesn’t necessarily state curriculum is aligned to CCSS. However, more updated versions may have changed to reflect this) • All three textbooks incorporate vocabulary for circles, a review at the end, and topics about tangents, chords, arcs, and angles
- 16. Available Resources Geometry Connections- CPM • The teacher notes provided at the beginning of each lesson in the CPM Teacher Edition provide clarity to educators on the purpose of some problems and how the mathematics in the standards are developed over time. • Online teaching resources (http://cpm.org/) – Student resource pages for chapters (Spanish version included) – Homework help w/ hints – Extra Practice (additional problems to practice skills) – Checkpoint materials (http://cpm.org/pdfs/resCourse/CCG/CCG%20Checkpoints%20 TV.pdf) – Toolkits – Parent guides – Video models of CPM classrooms (what can I build with a circle?) – Study team strategies (http://cpm.org/pdfs/studyTeam/GC_STS.pdf) – Professional Development – workshops and articles – Technology resources (Sketchpad, SmartBoard files, Math Tech Tools)
- 17. Available Resources Discovering Geometry:An Inductive Approach – Key Curriculum Press • Online teaching resources (www.michaelserra.net) • Website provides PDFs for teachers that have correlation of Common Core to different lessons taught. PDFs chunk together various lessons in textbook according to Mathematical Practice Standard (see right) • Enrich learning online textbooks and teacher resources for discovering geometry (http://www.michaelserra.net/weblog/ discovering-geometry-online- resources/) • Breakdown of lessons and standards also available on PDF
- 18. Available Resources • Additionallinks to publications and subscriptions for similar curriculum available for following: – Discovering geometry – Mathercise series – Patty paper geometry – Pirate math – Smart moves – What’s wrong with this picture? • Discovering Geometry Video Series – Classroom: vocabulary review with string – Lectures: strategies to reduce math anxiety – Projects: reasoning relay race – Test prep: general test prep strategies • Articles and Handouts on: – Cooperative learning – Discovering geometry FAQ – Discovering geometry Intro – Proof in Dosovering Geometry – Download Articles/handouts by Michael Serra • Spanish Material – Condensed lessons available in Spanish – Discovering Geometry Video Series in Spanish • Professional Development Opportunities – FAQ – Workshop description – Testimonials • Archives from previous years (2009 – 2014) with online videos and descriptions of lessons
- 19. Available Resources • Geometry– Houghton Miffin • Publisher changed over years and website teachers may access is now available at http://www.hmhco.com • Hard to find online resources for current textbook because older version • Teachers can locate more recent publication and find resources and links to that textbook which are similar to this version • Handbook for Integrating Coordinate and Transformational Geometry available at the back of the textbook – “Deciding Which Method to Use in a Problem” provides guidance in selecting the approach that best suits a problem – Each section is intended for use after the chapter listed with its title – Students and teachers may refer to this section for hints on solving problems
- 20. And The WinnerIs…… Discovering Geometry: An Inductive Approach While both Geometry Connections and Discovering Geometry: An Inductive Approach rely on cooperative learning strategies and the students taking control of their own learning, I felt that Discovering Geometry by Michael Serra was the strongest textbook out of all three for teaching the topic on circle properties. I am currently using the Geometry Connections textbook for my geometry class and I feel the lack of illustrations (all cartoon and in black and white) sometimes cause visual learners to become disengaged with the problems. On the other hand, Discovering Geometry uses colored pictures of real life scenarios and everyday objects. I feel that this textbook does a better job of providing a more hands-on approach and “authentic” learning experience because it forces students to use various tools (dynamic geometry software, graphing paper, compass, string) while they conduct their investigations. Plus, the chapter on circles in this textbook includes more meaningful and real world connections as well as projects and performance type questions that demonstrate a summative assessment for circle properties. I also felt the online teacher and student resources were stronger for this curriculum than CPM. Even though CPM had Spanish resources available to students, it lacked Spanish versions of lessons. This is where Discovering Geometry stood out to me. The online resources even included videos and additional links for supplemental materials. Even though CPM does a good job of demonstrating mastery over time and constantly having students review topics previously learned, I’ve noticed that at times it sneaks topics we have yet to address and students get lost. I have also had to create a lot of additional “skill” type questions to differentiate for my struggling learners. Overall, I feel like Discovering Geometry applies to all learners, not just TAG students, and challenges students to inductively approach geometry topics instead of being “spoon-fed” the information.
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- 25. Final Comments andObservations • Geometry textbooks have definitely evolved over the years to include more hands-on and investigative approaches to learning about proofs. • These strategies are necessary for students to have a more authentic learning experience and conceptual understanding of how geometry connects to real world situations all around us.