West Windsor-Plainsboro Regional School District Basic Geometry Grades 9-12

Transcription

1 West Windsor-Plainsboro Regional School District Basic Geometry Grades 9-12

2 Unit 1: Basics of Geometry Content Area: Mathematics Course & Grade Level: Basic Geometry, 9 12 Summary and Rationale This unit is an introduction to the basic elements of geometry: points, lines, planes, segments, and angles. Students are shown reasoning skills which they will use throughout the course, specifically making predictions and conjectures based on patterns they observe. 16 days Standard 4.G CO Congruence Recommended Pacing State Standards 1 Know precise definitions of angle, circle, perpendicular line, parallel line, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. 12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc. Standard 4.G MG Modeling with Geometry 2 Apply concepts of density based on area and volume in modeling situations. Instructional Focus Unit Enduring Understandings Students will understand and use the basic definitions and postulates and incorporate algebra to solve problems Unit Essential Questions How can we use patterns to help make predictions? How can we use inductive reasoning to make conjectures? Objectives Students will know: Terms: conjecture, inductive reasoning, deductive reasoning, counterexample, undefined term, point, line, plane, postulate, collinear points, coplanar points, coplanar lines, segment, endpoint, ray, intersect, intersection, coordinate, distance, length, between, congruent segments, angle, sides of an angle, vertex of an angle, measure of an angle, degrees, congruent angles, acute angle, right angle, obtuse angle, straight angle. Formulas: distance and midpoint formula Students will be able to: Find patterns and use them to make predictions. Use inductive reasoning to make conjectures. Use postulates and undefined terms. Sketch simple figures and their intersections. Measure segments and add segment lengths. Measure and classify angles.

3 Add angle measures. Core Text: Geometry: Concepts & Skills, Larson Suggested Resources: Resources

4 Unit 2: Segments and Angles Content Area: Mathematics Course & Grade Level: Basic Geometry, 9 12 Summary and Rationale This unit introduces students to special angles and their properties. These facts begin setting the foundation for students introduction to proofs. Students learn about if then statements and the basics of logical thinking to prepare them for their study of proofs. 16 days Standard 4.G CO Congruence Recommended Pacing State Standards 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. 9 Prove theorems about lines and angles. 12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Instructional Focus Unit Enduring Understandings Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments. Students will work together to learn mathematics. Children learn mathematics well in cooperative, noncompetitive environments, where they can share conjectures with their classmates. Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking. Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills. Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions How do we bisect an angle and then find the coordinates of the midpoint of a segment? What is the relationship between the angles formed by two intersecting lines? Objectives Students will know: Terms: midpoint, segment bisector, bisect, angle bisector, complementary angles, complement of an angle, supplementary angles, supplement of an angle, adjacent angles, theorem, vertical angles, linear pair, if then statement, hypothesis, conclusion, deductive reasoning. The relationships between angles, segments, lines, and rays and will apply those relationships to find information using complex figures. Students will be able to: Bisect a segment and find the coordinate of the midpoint of a segment.

5 Bisect an angle. Find the measures of complimentary and supplementary angles Find the measures of angles formed by intersecting lines. Use if then statements and apply laws of logic. Use properties of equality and congruence. Core Text: Geometry: Concepts & Skills, Larson Suggested Resources: Resources

6 Unit 3: Parallel and Perpendicular Lines Content Area: Mathematics Course & Grade Level: Basic Geometry, 9 12 Summary and Rationale Students will study the angles formed when two lines are cut by a third line, called the transversal. Throughout the unit, they will be introduced to several theorems and postulates related to parallel and perpendicular lines. 16 days Standard 4.G CO Congruence Recommended Pacing State Standards 1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance around a circular arc. 3 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. Instructional Focus Unit Enduring Understandings Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments. Students will work together to learn mathematics. Children learn mathematics well in cooperative, noncompetitive environments, where they can share conjectures with their classmates. Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking. Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills. Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions How are lines related in space? What are the relationships among the angles formed when two parallel lines are cut by a transversal? Objectives Students will know: The relationship between angles, segments, lines, and rays and will apply those relationships to find information using complex figures How to use tools of geometry, including computers and calculators, to measure figures, discuss their findings

7 and make conjectures Terms: parallel lines, perpendicular lines, skew lines, parallel planes, line perpendicular to a plane, transversal, corresponding angles, alternate interior angles, alternate exterior angles, same side interior angles, converse, construction, translation, image, transformation. Students will be able to: Identify the relationships between lines Apply theorems about perpendicular lines Find the congruent angles formed when a transversal cuts parallel lines. Show that two lines are parallel. Construct parallel and perpendicular lines. Use properties of parallel and perpendicular lines. Identify and use translations. Core Text: Geometry: Concepts & Skills, Larson Suggested Resources: Resources

8 Unit 4: Triangle Relationships Content Area: Mathematics Course & Grade Level: Basic Geometry, 9 12 Summary and Rationale Students will learn to recognize a variety of triangles, and to compute angle measures and side lengths in them. Several fundamental theorems related to triangles will be explored. 16 Standard 4.G CO Congruence 9 Prove theorems about lines and angles. Recommended Pacing State Standards 10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Standard 4.G SRT Similarity, Right Triangles, and Trigonometry 4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Instructional Focus Unit Enduring Understandings Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments. Students will work together to learn mathematics. Children learn mathematics well in cooperative, noncompetitive environments, where they can share conjectures with their classmates. Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking. Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills. Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions What is the relationship among the lengths of the sides of a right triangle? What is the relationship between the side lengths of a triangle and its angle measures? Objectives Students will know: How to use the tools of geometry, including computers and calculators, to measure figures, discuss their findings and make conjectures

9 How to discover the relationships that exist within a triangle and between triangles, using these schools to solve a variety of problems Terms: Equilateral, Isosceles, Scalene, Equiangular, Acute, Right, Obtuse, Vertex, Interior angles, Exterior angles, Base Angles, Hypotenuse, Median and Centroid of a Triangle Formulas: Pythagorean Theorem, Distance Formula Students will be able to: Classify triangles by their sides and angles Find angle measures in a triangle using the sum of the angles Use the Pythagorean Theorem and the Distance Formula Apply the Pythagorean Theorem to real life situations Use the converse of the Pythagorean Theorem; Use side lengths to classify triangles Identify medians in triangles Determine the longest side of a triangle and the largest angle Resources Core Text: Geometry: Concepts & Skills, Larson Suggested Resources:

10 Unit 5: Congruent Triangles Content Area: Mathematics Course & Grade Level: Basic Geometry, 9 12 Summary and Rationale In this unit, students will explore the properties of congruent triangles and how to show that two triangles are congruent. An examination of angle bisectors and perpendicular bisectors leads to a discussion of reflections and symmetry. 16 days Standard 4.G CO Congruence Recommended Pacing State Standards 6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Focus Unit Enduring Understandings Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments. Students will work together to learn mathematics. Children learn mathematics well in cooperative, noncompetitive environments, where they can share conjectures with their classmates. Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking. Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills. Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions Can you determine that two triangles are congruent without listing all of the corresponding congruent sides and angles? Is there a side side angle congruence postulate or theorem? Objectives Students will know: How to recognize and use properly the basic vocabulary of geometry How to discover the relationships that exist within a triangle and between triangles. Students will apply those relationships to solve a variety of problems Students will apply the relationships that exist within a triangle and between triangles to solve a variety of

11 problems Terms: Corresponding Parts, Congruent Figures, Proof, Vertical angles, alternate interior angles, Equidistant, Angle Bisector, Perpendicular Bisector, Image, Reflection and Line of Symmetry Students will be able to: Identify congruent triangles and corresponding parts Show triangles are congruent using SSS and SAS Show triangles are congruent using ASA and AAS Use HL Congruence Theorem and summarize congruence postulates and theorems Show corresponding parts of congruent triangles are congruent Use angle bisectors and perpendicular bisectors Identify and use reflections and lines of symmetry Core Text: Geometry: Concepts & Skills, Larson Suggested Resources: Resources

12 Unit 6: Quadrilaterals Content Area: Mathematics Course & Grade Level: Basic Geometry, 9 12 Summary and Rationale This unit begins with a discussion of all polygons and how to identify and classify them based on their properties. Students will then examine quadrilaterals. 16 days Standard 4.G CO Congruence Recommended Pacing State Standards 11 Prove theorems about parallelograms. 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. 12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices; paper folding, dynamic geometric software, etc.). Standard 4.G MG Modeling with Geometry 1 Use geometric shapes, their measures, and their properties to describe objects. Instructional Focus Unit Enduring Understandings Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments. Students will work together to learn mathematics. Children learn mathematics well in cooperative, noncompetitive environments, where they can share conjectures with their classmates. Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking. Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills. Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions What are the properties of a parallelogram? How can you use the angle measures in a quadrilateral to show that it is a parallelogram? What are some properties of the midsegment of a trapezoid? Objectives Students will know: How to use the tools of geometry, including computers and calculators, to measure figures, discuss their findings and make conjectures How to explore the characteristics of many polygons, including quadrilaterals Terms: Polygon; Side, Diagonal, Parallelogram, Rhombus, Rectangle, Square, Trapezoid, Isosceles Trapezoid,

13 and Midsegment of a Trapezoid. Students will be able to: Identify and classify polygons Find the angle measures quadrilaterals Use the properties of parallelograms Use deductive reasoning to show that a quadrilateral is a parallelogram Use the properties of trapezoids Identify special quadrilaterals based on limited information Core Text: Geometry: Concepts & Skills, Larson Suggested Resources: Resources

14 Unit 7: Similarity Content Area: Mathematics Course & Grade Level: Basic Geometry, 9 12 Summary and Rationale The concept of similarity is explored in this unit. Students will use ratios and proportions in connection with similar polygons. 17 days Recommended Pacing State Standards Standard 4.G SRT Similarity, Right Triangles, and Trigonometry 2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. 3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Standard 4.G MG Modeling with Geometry 1 Use geometric shapes, their measures, and their properties to describe objects. Instructional Focus Unit Enduring Understandings Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments. Students will work together to learn mathematics. Children learn mathematics well in cooperative, noncompetitive environments, where they can share conjectures with their classmates. Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking. Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills. Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions When a figure is enlarged, how are corresponding angles related? How are corresponding lengths related? How can we use ratios, proportions, and similarity to solve problems? Objectives Students will know: How to apply ratio and proportion to real life situations How to identify similar polygons

15 How to apply the concept of similarity to indirect measurement How to use similarity to solve complex problems How to use the tools of geometry, including computers and calculators, to measure figures, discuss their findings and make conjectures Students will be able to: Use ratios and proportions Identify similar polygons Show two triangles similar using the AA, SSS, SAS Similarity Theorems Use the Triangle Proportionality Theorem and its converse Identify and draw dilations of polygons Core Text: Geometry: Concepts & Skills, Larson Suggested Resources: Resources

16 Unit 8: Polygons and Area Content Area: Mathematics Course & Grade Level: Basic Geometry, 9 12 Summary and Rationale Students will derive area formulas for several special polygons and use the formulas to find the areas of these polygons. 16 days Recommended Pacing State Standards Standard 4.G GPE Expressing Geometric Properties with Equations 7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g. Using the distance formula. Standard 4.G MG Modeling with Geometry 1 Use geometric shapes, their measures, and their properties to describe objects. 3 Apply geometric methods to solve design problems. Standard 4.G GMD Geometric Measurement and Dimension 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. Instructional Focus Unit Enduring Understandings Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments. Students will work together to learn mathematics. Children learn mathematics well in cooperative, noncompetitive environments, where they can share conjectures with their classmates. Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking. Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills. Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions How can we describe polygons? How can we find the area of polygons and use this to solve problems? Objectives Students will know: How to explore the characteristics of many polygons, including quadrilaterals How to apply area, perimeter, surface area and volume to a variety of real life situations How to use the tools of geometry, including computers and calculators, to measure figures, discuss their

17 findings and make conjectures Students will be able to: Describe and classify polygons Find the measures of the interior and exterior angles of polygons Find the area of squares, rectangles, triangles, parallelograms, trapezoids, and circles Find the perimeter of polygons and circumference of circles Apply their knowledge of area to real life situations and complex problems Resources Core Text: Geometry: Concepts & Skills, Larson Suggested Resources:

18 Unit 9: Surface Area and Volume Content Area: Mathematics Course & Grade Level: Basic Geometry, 9 12 Summary and Rationale In this unit, students explore solid figures, including prisms, pyramids, cylinders, cones, and spheres. 16 days Recommended Pacing State Standards Standard 4.GMD Geometric Measurement and Dimension 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. 2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. 3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Standard 4.G MG Modeling with Geometry 2 Apply concepts of density based on area and volume in modeling situations. 3 Apply geometric methods to solve design problems. Instructional Focus Unit Enduring Understandings Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments. Students will work together to learn mathematics. Children learn mathematics well in cooperative, noncompetitive environments, where they can share conjectures with their classmates. Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking. Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills. Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions How can we identify and name solid figures? How can we find the surface area of solid figures? How does the volume of a pyramid relate to the volume of a prism with the same base? Objectives Students will know: How to explore the characteristics of many polygons, including quadrilaterals How to apply area, perimeter, surface area and volume to a variety of real life situations How to use tools geometry, including computers and calculators, to measure figures, discussing their findings

19 and make conjectures Students will be able to: Identify and classify solid figures Find the surface area of prisms, pyramids, cones, cylinders and spheres Find the volume of prisms, pyramids, cones, cylinder and spheres Apply their knowledge of volume and surface area to real life situations and complex problems Resources Core Text: Geometry: Concepts & Skills, Larson Suggested Resources:

20 Unit 10: Special Right Triangles and Trigonometry Content Area: Mathematics Course & Grade Level: Basic Geometry, 9 12 Summary and Rationale This unit investigates the relationship between the side lengths of special right triangles, and the use of the three main trigonometric rations. 17 days Recommended Pacing State Standards Standard 4.G SRT Similarity, Right Triangles, and Trigonometry 6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratio for acute angles. 7 Explain and use the relationship between the sine and cosine of complementary angles. 8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 10 Prove the Laws of Sines and Cosines and use them to solve problems. Instructional Focus Unit Enduring Understandings Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments. Students will work together to learn mathematics. Children learn mathematics well in cooperative, noncompetitive environments, where they can share conjectures with their classmates. Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking. Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills. Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions Does the size of similar right triangles affect the ratio of their leg lengths? How can we use trig ratios to solve problems? Objectives Students will know: Terms: sine, cosine, tangent, radical, radicand, leg opposite an angle, leg adjacent to an angle. The relationships that exist within a triangle and between triangles, students will be able to apply to relationships to solve a variety of problems Students will be able to: Simplify square roots Find the side lengths of 45, 45, 90 triangle and a 30, 60, 90 triangle Find the sine, cosine, and tangent of an acute angle Solve a right triangle

21 Apply trigonometry to real life situations an complex problems Resources Core Text: Geometry: Concepts & Skills, Larson Suggested Resources:

22 Unit 11: Circles Content Area: Mathematics Course & Grade Level: Basic Geometry, 9 12 Summary and Rationale Students use properties involving tangents, secants, and chords of a circle. 17 days Standard 4.G C Circles Recommended Pacing State Standards 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. 3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 4 Construct a tangent line from a point outside a given circle to the circle. Instructional Focus Unit Enduring Understandings Students will understand how to use the properties of length and measure to justify segment and angle relationships through logical arguments. Students will work together to learn mathematics. Children learn mathematics well in cooperative, noncompetitive environments, where they can share conjectures with their classmates. Students will learn to communicate mathematically. Discussing ideas assists students to clarify and solidify their thinking. Students will refine their reasoning skills. Using inductive and deductive reasoning to test conjectures will enable students to analyze and hone their reasoning skills. Students will make connections between previously taught algebraic skills and the newly taught concepts of geometry. Unit Essential Questions What is the relationship between a tangent and a circle? How are inscribed angles related to central angles? Objectives Students will know: Terms: chord, secant, tangent, minor arc, major arc, arc length, inscribed angle, intercepted arc, rotation, rotational symmetry. How to recognize and use properly the basic vocabulary of geometry How to apply the relationships between circles and lines to an array of complex situations Students will be able to: Use and apply the vocabulary of circles

23 Identify segments and lines related to circles Use the properties of a tangent to a circle Use the properties of arcs of circles Use the properties of chords and circles Use the properties of inscribed angles Core Text: Geometry: Concepts & Skills, Larson Suggested Resources: Resources