Algebra 1 Unit 3: Systems of Equations s3

PreCalculus Unit 2 Analytic Geometry

Enduring understanding (Big Idea): Students will understand that the characteristics of quadratic relations and their representations (both algebraically and graphically are useful in solving real-world problems.

Essential Questions: 1. What are the conic sections and why are they called conic sections? 2. What are the parts of each conic? How do they relate algebraically and graphically? 3. Why do we study conic sections? 4. How are conic sections applied to the real world?

BY THE END OF THIS UNIT: Students will know… Students will be able to… Standard form of equations for conic sections:  Describe a conic sections as a intersection of a cone and plane Circle: (x – h)2 + (y – k)2 = r2  Identify equations of and graph conic sections in Cartesian form – Parabola: y = a(x – h)2 + k including circle, ellipse, hyperbola and parabola. (x  h)2 (y  k)2  Define, analyze and write the equations of conic sections – Ellipse:   1 a 2 b 2 including circle, ellipse, hyperbola and parabola.\ (x  h)2 (y  k) 2 (y  k)2 (x  h)2  Use completing the square to write equations of conic sections in Hyperbola:   1 and   1 standard form. a 2 b 2 a 2 b 2  Write the equation of and graph a translated conic section. Vocabulary: Circle, Parabola, Ellipse, Hyperbola, Eccentricity, Major axis, minor axis, directrix, center, focus, vertices, conic section,  Model practical problems involving conic sections and solve real- Line of symmetry, asymptotes world problems.(e.g. orbits of planets)

Unit Resources Mathematical Practices in Focus: Learning Task https://www.georgiastandards.org/Frameworks/GSO%20Frameworks/Math%20III%20Unit%205%20SE4. Model with mathematics %205-25-10.pdf 5. Use appropriate tools strategically Performance Task: Project: Graphing Project 7. Look for and make use of structure Unit Review Game:

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. PreCalculus Unit 2 Analytic Geometry

CORE CONTENT Cluster Title: Standard NCSCOS 1.02 Use the quadratic relations (parabola,circle,ellipse,and hyperbola) to model and solve problems; justify a) Solve using tables, graphs, and algebraic properties. b) Interpret the constants and coefficients in the context of the problem. Concepts and Skills to Master  Write the equation and describe the characteristics of a circle, ellipse, parabola and hyperbola algebraically and graphically.  Graph and identify graphs of the conic sections  Describe the properties of circles, ellipses, parabolas, and hyperbolas.  Use completing the square to write conics in standard form. SUPPORTS FOR TEACHERS Critical Background Knowledge Completing the square; factoring; Vertex form; translations Academic Vocabulary Circle, Parabola, Ellipse, Hyperbola, eccentricity, major and minor axis, directrix, center, focus, vertices, conic section, line of symmetry, asymptotes Suggested Instructional Strategies Resources  Have students research several types of media: videos,  Textbook Correlation: 8.1, 8.2, 8.3 journals, magazines, internet web sites, to find real-life  Conics – Circle formulas (skill based) uses for conics. Through that research they should also  Conics – Ellipse formulas (skill based) discover how each conic section is derived algebraically  Conics – Parabola formulas (skill based) and graphically.  Conics – more circles, ellipses, and parabolas (skill  Use learning tasks to encourage students to investigate based) conic sections.  Conics – Hyperbola formulas (skill based)  Conics test and review samples (skill based)

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes. PreCalculus Unit 2 Analytic Geometry

Sample Formative Assessment Tasks Skill-based task Problem Task

Write in standard form and identify the conic According to Kepler’s first law of planetary motion, each represented by the equation: planet moves in an ellipse with the sun at one focus. Assume 2x2 + 4y2 – 6x + 8y – 7 = 5 that one focus (the Sun) has coordinates (0,0) and the major axis of each planetary ellipse is the x-axis on a cosmic coordinate system (one unit = one billion kilometers). The minimum and maximum distances for Neptune are 4.456 and 4.537 billion kilometers. The minimum and maximum distances for Pluto are 4.425 and 7.375 billion kilometers. For each planet determine the coordinates of the center and second focus; write an equation which represents the orbit; and determine the eccentricity. Graph and describe the orbits.

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.