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Transcript

Polygons: Unit 4 Overview While this unit may explore many different types of polygons, the focus of the unit is on quadrilaterals. Students were introduced to quadrilaterals in elementary and middle school. In high school, students more formally express properties of quadrilaterals individually and their relationships to each other. Students will solve problems about polygons, expressing side and angle measures with variable expressions. Students may construct proofs about polygons including coordinate proofs where coordinates are variables rather than integers. Essential Ideas 1. Understand and apply properties of parallelograms (including rectangles, rhombi and squares). 2. Describe and justify hierarchical relationships among quadrilaterals. 3. Use coordinates to prove geometric theorems algebraically. Content Expectations MI: Mathematics MI: HS: Algebra Reasoning with Equations & Inequalities HSA-REI.A. Understand solving equations as a process of reasoning and explain the reasoning. HSA-REI.A.1. Explain each step in solving a simple equation as following from the equality of numb original equation has a solution. Construct a viable argument to justify a solution method. HSA-REI.B. Solve equations and inequalities in one variable. HSA-REI.B.3. Solve linear equations and inequalities in one variable, including equations with coeffi HSA-REI.B.4. Solve quadratic equations in one variable. MI: HS: Modeling Mathematical Practice MP.The Standards for Mathematical Practice describe varieties of expertise that mathematics educ MP.1. Make sense of problems and persevere in solving them. MP.2. Reason abstractly and quantitatively. MP.3. Construct viable arguments and critique the reasoning of others. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. MP.6. Attend to precision. MP.7. Look for and make use of structure. MP.8. Look for and express regularity in repeated reasoning. MI: HS: Geometry Congruence HSG-CO.C. Prove geometric theorems HSG-CO.C.10. Prove theorems about triangles. HSG-CO.C.11. Prove theorems about parallelograms. Expressing Geometric Properties with Equations HSG-GPE.B. Use coordinates to prove simple geometric theorems algebraically HSG-GPE.B.4. Use coordinates to prove simple geometric theorems algebraically. HSG-GPE.B.7. Use coordinates to compute perimeters of polygons and areas for triangles and rectang Lesson Sequence and Objectives Learning Objectives (textbook reference) The student will: 1. Optional review: Identify and name polygons. Determine whether they are concave or convex. (1-6) 2. Use coordinates to solve multistep problems about the perimeter of polygons. (1-6) 3. Understand and apply properties of parallelograms (including rectangles, rhombi and squares). 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. (6-2, 6-4, 6-5) 4. Solve multistep problems involving angle measure and side length in parallelograms, with an emphasis on problems in which angles measure or side length is written as a variable expression. (various sections) 5. Use coordinates to prove simple geometric theorems algebraically. For example, write coordinate proofs about properties of polygons. For advanced students, include problems in which coordinates are variables. (6-3, 6-4, 6-5, 6-7) 6. Describe and justify hierarchical relationships among quadrilaterals. (various sections) 7. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. (supplement: holiday ornament activity) 8. Use geometric shapes, their measures and their properties to describe objects. Use units and define quantities appropriately for the purpose of descriptive modeling. Choose an appropriate level of accuracy. (various sections) 9. Optional enrichment: Understand and apply formulas for the sum of interior angles of a polygon. (1-6, 6-1) 10. Optional enrichment: Write equations of lines through a point and parallel or perpendicular to a given line. (3-4) Order of learning objectives does not imply required sequence. Objectives may be addressed at any time throughout the semester. Vocabulary concave consecutive angles convex decagon diagonals heptagon hexagon interior angle irregular isosceles trapezoid kite median of a trapezoid nonagon octagon opposite angles opposite sides parallelogram pentagon polygons quadrilateral rectangle regular rhombus square trapezoid