Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Rational trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Analytic geometry wikipedia , lookup
Multilateration wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Algebraic geometry wikipedia , lookup
History of trigonometry wikipedia , lookup
Line (geometry) wikipedia , lookup
Geometrization conjecture wikipedia , lookup
Willmar Public Schools Curriculum Map Subject Area Mathematics—Senior High Course Name Geometry-9(Prentice Hall Mathematics) Date April 20, 2010 Geometry 9 and Geometry parallel each other in content and time. Geometry 9 students are assigned more advanced problems for homework and for formal assessments. Week 1-2 Content Tools of Geometry (Chapter 1) Standards Addressed Understand the concept of function, and identify important features of functions and other relations using symbolic and graphical methods where appropriate. Construct logical arguments, based on axioms, definitions and theorems, to prove theorems and other results in geometry. Know and apply properties of geometric figures to solve real-world and mathematical problems and to logically justify results in geometry. Solve real-world and mathematical geometric problems using algebraic methods. Skills/Benchmarks Essential Questions 9.2.2.4(1.1) Express the terms in a geometric sequence recursively and by giving an explicit (closed form) formula, and express the partial sums of a geometric series recursively. How does having a common language, of basic geometric vocabulary, assist in further discussions in class and in life? 9.3.2.1 (1.3) Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. 9.3.2.3 (1.1) Assess the validity of a logical argument and give counterexamples to disprove a statement. 9.3.2.5 (1.1-1.7) Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. 9.3.3.1 (1.7) Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. 9.3.3.2 (1.6) Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. 9.3.4.4 (1.5, 1.8) Use coordinate What are the tools of geometry? Assessments Tests and quizzes including performance assessment questions. In-class “lab” discovery assignments using constructions, geo-sketch pad, and or graphing calculators to reveal geometric concepts. Willmar Public Schools Curriculum Map 3 Reasoning and Proof (Chapter 2) Generate equivalent algebraic expressions involving polynomials and radicals; use algebraic properties to evaluate expressions. Construct logical arguments, based on axioms, definitions and theorems, to prove theorems and other results in geometry. Know and apply properties of geometric figures to solve real-world and mathematical problems and to logically justify results in geometry. geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. 9.2.3.7 Justify steps in generating equivalent expressions by identifying the properties used. Use substitution to check the equality of expressions for some particular values of the variables; recognize that checking with substitution does not guarantee equality of expressions for all values of the variables. 9.3.2.2 (2.1 - 2.2) Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. 9.3.2.3 (2.1) Assess the validity of a logical argument and give counterexamples to disprove a statement. 9.3.2.4 (2.5) Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 9.3.3.2 (2.5) Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. How do we use patterns to study geometry? Tests and quizzes including performance assessment questions. How are statements known as conditional, biconditional, and various definitions used for geometric proof? Do group proof. Use cooperative learning to first understand proofs/ Partner proofs, then individual proofs. How can doing proofs help with making other logical connections throughout geometry and life? Patty paper investigations to discover geometric concepts. Willmar Public Schools Curriculum Map 3-4 Parallel and Perpendicular lines (Chapter 3) Understand the concept of function, and identify important features of functions and other relations using symbolic and graphical methods where appropriate. Construct logical arguments, based on axioms, definitions and theorems, to prove theorems and other results in geometry. Know and apply properties of geometric figures to solve real-world and mathematical problems and to logically justify results in geometry. 9.2.1.6 (3.6) Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. 9.2.2.3 (3.6) Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. 9.3.2.4 (3.1) Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 9.3.2.5 (p. 126) Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. 9.3.3.1 (3.1) Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. 9.3.3.2 (3.1,3.2) Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. 9.3.3.3 (3.4) Know and apply properties of equilateral, isosceles and scalene triangles to solve What are the relationships of parallel and perpendicular lines and how can on apply these concepts to everyday life? How are parallel and perpendicular lines used with triangles and other polygons with or without a coordinate plane? How can using parallel lines and a transversal help to determine angle measures in a real-life setting? How can doing proofs help with making other logical connections throughout geometry and life? Tests and quizzes including performance assessment questions. In class activity with geo-sketch pad with exploration style proofs and construction tools. p. 126 and p.156 Willmar Public Schools Curriculum Map problems and logically justify results. 9.3.3.7 (3.7) Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. 4-6 Congruent Triangles (chapter 4) Construct logical arguments, based on axioms, definitions and theorems, to prove theorems and other results in geometry. Know and apply properties of geometric figures to solve real-world and mathematical problems and to logically justify results in geometry. 9.3.2.4 (4.3) Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 9.3.2.5 (Pre 4.4) Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. What makes shapes congruent? Where are congruent figures and their properties used in real life? How do you prove two triangles are congruent without show all part are congruent? Tests and quizzes including performance assessment questions. In class activity with geo-sketch pad and construction tools. (p.220) In class discovery with patty paper. (p. 227) What impact does the triangle congruence properties have on proving triangular shapes are congruent in every day situations? Flow chart proofs. (p.247) What are the properties of a triangles and how does “if…then” vs. “if and only if” affect the properties? Tests and quizzes including performance assessment questions. 9.3.3.3 (4.5) Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. 9.3.3.6 (4.6-4.7) Know and apply properties of congruent and similar figures to solve problems and logically justify results. 6-7 Relationship within triangles (chapter 5) Construct logical arguments, based on axioms, definitions and theorems, to prove theorems and other results in geometry. Construct logical arguments, based on axioms, definitions and theorems, to prove theorems and other results in geometry. 9.3.2.2 (5.4) Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. 9.3.2.4 (5.4) Construct logical arguments and write proofs of Where are concurrent lines used in real life? How can inductive reasoning be used for determining unknown concepts in geometry and in the real world? In-class activity with geo-sketch pad and construction tool. ( P. 258, 271) Paper folding activities to help discover concepts. Willmar Public Schools Curriculum Map theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. 9.3.2.5 (pre5.1) Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. 8 9 - 10 Quadrilaterals (chapter 6) Similarity (Chapter 7) Understand the concept of function, and identify important features of functions and other relations using symbolic and graphical methods where appropriate. 9.3.3.7 (6.1, 6.3) Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. What are the different types of quadrilaterals? Tests and quizzes including performance assessment questions. How do the properties of various quadrilaterals compare? Geo-sketch pad lesson (p. 342) Know and apply properties of geometric figures to solve real-world and mathematical problems and to logically justify results in geometry. 9.3.4.4 (6.1, 6.7) Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. How do the properties help in proving coordinate geometry proofs? 9.3.3.6 (7.2, 7.4) Know and apply properties of congruent and similar figures to solve problems and logically justify results. What is the relationship of similar triangles? Solve real-world and mathematical geometric problems using algebraic methods. Know and apply properties of geometric figures to solve real-world and mathematical problems and to logically justify results in geometry. Solve real-world and mathematical geometric problems using algebraic methods. 11 - 12 Right triangles and trigonometry (chapter 8) Know and apply properties of geometric figures to solve real-world and mathematical problems and to logically justify results in geometry. 9.3.4.7 (7.3, 7.4) Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. 9.3.3.4 (8.1) Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. How can students apply these properties of congruent figures to those that are just similar? Tests and quizzes including performance assessment questions. Why are ratios and proportions needed to find unknown lengths and angles? How can we apply the concept of similarity to right triangles and the basic trigonometric ratios? How can similarity and scale changes be useful for everyday life situations? What is the Pythagorean theorem and where can it be used for real world problems? Tests and quizzes including performance assessment questions. Hands on investigations dealing with Willmar Public Schools Curriculum Map Solve real-world and mathematical geometric problems using algebraic methods. How do the basic trigonometric ratios apply to geometry and where can they be used in real-life applications? the disco very of how Pythagorean theorem is created. 9.2.1.9 (All chapter 9) Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations. What are some types of transformations and where are they found in the real world? Tests and quizzes including performance assessment questions. 9.3.1.4 (9.5) Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. Where are transformations used in art? 9.3.3.5 (8.2) Know and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems and logically justify results. 9.3.4.1 (8.4) Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. 9.3.4.2 (8.4 – 8.6) Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. 9.3.4.3 (8.5) Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. 13 Transformations (chapter 9) Understand the concept of function, and identify important features of functions and other relations using symbolic and graphical methods where appropriate. Calculate measurements of plane and solid geometric figures; know that physical measurements depend on the choice of a unit and that they are approximations. Solve real-world and mathematical geometric problems using algebraic methods. 9.3.4.6 (9.1-9.3, 9.5) Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the How are transformations used in real life? Tessellation Project which includes at least three different styles of transformations with the template. Willmar Public Schools Curriculum Map origin by multiples of 90˚, to solve problems involving figures on a coordinate grid. 14 – 15 Area (chapter 10) Calculate measurements of plane and solid geometric figures; know that physical measurements depend on the choice of a unit and that they are approximations. Know and apply properties of geometric figures to solve real-world and mathematical problems and to logically justify results in geometry. 9.3.1.2 (10.1 to 10.3) Compose and decompose two- and threedimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. 9.3.1.3 (throughout chapter 10) Understand that quantities associated with physical measurements must be assigned units; apply such units correctly in expressions, equations and problem solutions that involve measurements; and convert between measurement systems. How are areas and perimeters calculated? Tests and quizzes including performance assessment questions. How can area be applied to congruent and similar figures? In-class assignment of 574. When can area and perimeter be used throughout life? How does area apply to finding the volume of a three-dimensional figure? 9.3.3.8 (10.6) Know and apply properties of a circle to solve problems and logically justify results. 16 – 17 Surface Area and Volume (chapter 11) Calculate measurements of plane and solid geometric figures; know that physical measurements depend on the choice of a unit and that they are approximations. 9.3.1.1 (11.3, 11.5, 11.6) Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. What is the relationship between length, area and volume? Tests and quizzes including performance assessment questions. What types of figures have volume and surface area? Geo-labs on finding the relationship between volumes of a cone/pyramid to a cylinder/prism. Know and apply properties of geometric figures to solve real-world and mathematical problems and to logically justify results in geometry. 9.3.1.2 (11.2-11.6) Compose and decompose two- and threedimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. How are length, area, and volume different in real life situations? Solve real-world and mathematical geometric problems using algebraic methods. 18 Circles (chapter 12) 9.3.1.4 (11.7) Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. 9.3.3.8 (12.1 – 12.3) Know and apply properties of a circle to solve problems and logically justify results. What are the basic definitions associated with circles and the lines that intersect circles? How do arcs and angles relate within a Tests and quizzes including performance assessment questions. Willmar Public Schools Curriculum Map 9.3.4.5 (12.5) Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. circle? How can properties of circles help with future math courses?