St. Mary's College High School Geometry Points, Lines, and Planes and Angles What notations represent different sets of points? Why are some important geometric terms left undefined? How is the distance calculated between two points on a coordinate plane? What is a polygon? What is congruence? Is it the same as equals? Learning the meaning and use of various geometric terms. Reviewing the distance formula, and is application to set the foundation for future uses. Learn to identify the different polygons, and how to find their perimeter. Learn the difference and similarities of congruence and equals, and then apply to proofs in the future. A1. Identify and Model points, lines, and planes. A2. Identify collinear and coplanar points, and intersecting lines and planes in space. B1. Measure segments and determine accuracy of measurements. B2. Determine two line segments are congruent. C1. Calculate the distance between two points using the distance formula. C2. Find the midpoint of a line segment using the midpoint formula. D1. Measure and classify angles. D2. Identify congruent angles and angle bisectors. E1. Identify and use special pairs of angles. E2. Identify perpendicular lines. F. Identify, name, and find the perimeter of polygons. G. Distinguish the difference between congruence and equals. Write a short essay on the meaning of Geometry and how it used in everyday life. Diagnostic check on some basic Algebra skills (Formative Review vocabulary terms, and use a vocabulary builder. (Formative Review nightly homework assignments. during class. Short quiz. Reasoning and Proofs 1 of 9
A. Make conjectures based on Short basic skills quiz. inductive reasoning, and find counterexamples. How are conditional statements related to theorems in geometry? What is the role of postulates and theorems in building geometry as a mathematical system? What is a Proof? Why is Proof important? How is proof used in everyday life? We use conditional statements to establish a logical process to explain the use of Proofs. - By learning the difference between postulates and theorems, students will be able to determine their use of building a Proof. Parallel and Perpendicular Lines B1. Determine truth values of conjunctions and disjunctions. B2. Construct truth tables. C1. Analyze and write statements in if-then form. C2. Write the converse, inverse, and contrapositive of if-then statements. D. Determine valid conclusions, using the Law of Detachment and the Law of Syllogism. E. Identify and use basic postulates and theorems to write paragraph proofs. F1. Use algebra to write twocolumn proofs. F2. Use properties of equality and definitions in geometric proofs. G. Write proofs involving segment addition and segment congruence. H1. Write proofs involving supplementary and complementary angles. H2. Write proofs involving congruent and right angles. Review nightly homework. during class. In-class examples, practice and Chapter 1 test, with open-ended problems, with critical thinking problems. Geometry crossword puzzle. Logic Matrix. Construction problem, going around a lake, to create a line extension using perpendicular lines and bisectors. Chapter 2 Test with open-ended and logic problems, two-column proofs, and critical thinking problems. (Summative What does parallel mean and how is it different from perpendicular? What is a transversal? A1. Identify the relationships between two lines and two planes. A2. Name angles formed by a pair of lines and a transversal. B1. Use the properties of parallel lines and transversals to determine congruent and supplementary 2 of 9
The knowledge of parallel angles. and perpendicular lines are B2. Use Algebra to find angle demonstrated by the measures. definition of slope on a coordinate plane. C. Find slopes of lines, and use it to identify parallel and perpendicular lines. D1. Write an equation in slopeintercept form, of a line, given information about its points, slope or its graph. D2. Solve problems by writing equations. Parallel and Perpendicular LInes How do you prove that two lines are parallel? How is the distance between two parallel lines determined? To calculate the distance between two parallel lines, the distance formula, and knowledge of slope, and perpendicular must be combined. CONGRUENT TRIANGLES E1. Recognize angle conditions that occur with parallel lines. E2. Prove that two lines are parallel based on given angle relationships. F. Apply the distance formula, find the distance between a point and a line, and between two parallel lines. A. Identify and classify triangles by sides and angles. B. Apply the Angles Sum Theorem and the Exterior Angle Theorems. Review nightly homework assignments. during class. Short basic skills quiz. In-class critical thinking problems. Chapter 3 test, with open-ended, multiple choice, and critical thinking problems. Chapter 4 test with open-ended problems, two-column proofs, and critical thinking problems. 3 of 9
What is a triangle, and how are they classified? What is necessary to prove congruence between two different triangles? C1. Name and label corresponding parts of congruent triangles. C2. Identify congruence transformation. Triangle play a significant role in geometry, so being able to identify and classify them is essential. Proving triangles are congruent involves the use of theorems, properties, and definitions. D. Use the SSS, the SAS, the ASA, and the AAS postulates and theorems to test for triangle congruence. E. Understand that Corresponding Parts of Congruent Triangles are Congruent. (CPCTC) F. Apply properties of isosceles and equilateral triangles. The understanding of congruent parts of a triangle is essential when proving congruence. Congruent Triangles What is a triangle, and how are they classified? What is necessary to prove congruence between two different triangles? Triangles play a significant role in geometry, so being able to identify them is A. Identify and classify triangles by sides and by angles. B. Apply the Angle Sum and the Exterior Angle Theorems. C1. Name and label corresponding parts of congruent triangles. C2. Identify congruence transformation. D. Use the SSS, the SAS, the ASA, and the AAS postulates and theorems to test for triangle congruence. Review nightly homework assignments. during class. Short basic skills quiz. Utilize constructions in class to deepen the understanding of congruent triangles. 4 of 9
essential. Proving triangles are congruent involves the use of theorems, properties, and definitions. The understanding of congruent parts of a triangle is essential in proving congruence. E. Understand that Congruent Parts of Congruent Triangles are Congruent. (CPCTC). F. Apply properties of isosceles and equilateral triangles. Relationships in Triangles What are bisectors, medians, and altitudes, and how are they determined? How to determine if a figure is a triangle. A. Identify and apply perpendicular bisectors, angle bisectors, altitudes, and medians in triangles. B. Recognize and apply properties of inequalities to the measure of angles of a triangle, and to the relationships between angles and sides of a triangle. C. Use indirect proofs with Algebra and Geometry Chapter 5 Test with multiple choice and open-ended problems, twocolumn proofs, and critical thinking problems. (Summative Cumulative Final Exam on chapters 1-5, with open-ended problems, two-column proofs, and critical thinking problems. Understand and be able to find the bisectors, medians, and altitudes of a triangle. Application of each to locate the centroid, the circumcenter, and the orthocenter of a triangle. D1. Apply the Triangle Inequality Theorem. D2. Determine the shortest distance between a point and a line in a coordinate plane. E. Apply the SAS and the SSS inequality theorems. F. Practice problems for the entire first trimester. 5 of 9
Proportions and Similarity What is a proportion, and how does it relate to triangles? What is similar, and how is similat different from congruent? Understand how similar triangles are in proportion. Utilize this understranding to solve problems involving parts of triangles. Right Triangles & Trigonometry ESSENTIAL QUESTION: What is a right triangle, and what special properties does it have? How is proportion related to the geometric mean? What is the Pythagorean Thereom, and how is it applicable to triangles? What is trigonometry, and how can it be used in real life? Understand and apply all special properties of a right triangle. Understand the use of trigonometry in the real world. A1. Write and solve ratios. A2. Solve proportions by using cross products. B. Identify similar polygons and solve problems involving scale factors. C. Identify similar triangles, and apply them to solve problems. D. Divide segments into proportional parts to determine segment lengths. E. Recognize and apply corresponding perimeters, altitudes, angle bisectors, and medians of similar triangles to solve problems. A. Find the geometric mean between two numbers, and apply the relationship to parts of a right triangle. B. Use the Pythagorean Thereom and its converse to find missing sides of a triangle. C. Use the properties of a 45-45- 90, and 30-60-90 right triangles. D. Find trigonometric rations in right triangles, and apply them to solve problems. Review nightly assignments. during class. Use of daily warm ups, often in groups. (Formative Assessments) Assign Geometry Story to be written, in order to introduce students to upcoming Geometry terms. (Formative assessment) Perform measuring activity demonstrating triangle similarity. Weekly short basic skills quizzes. Chapter 6 test, with several multiple choice, open-ended and critical thinking questions. Use trigonmetric ratios to calculate the height of various structures, or trees. Continued instruction with the calculator, needed to determine trig ratios. Chapter 7 test, with open-ended, and critical thinking problems. 6 of 9
Quadrilaterals What are the various quadrilaterals, and their properties? How are the properties similar, and how are they different? Learn and understand the various properties of each quadrilateral. A. Find the sum of the interior and exterior angles of polygons. B. Recognize and apply the properties of the sides, angles, and diagonals of parallelograms. C. Recognize and apply the properties of a rectangle, rhombus, square, kite and trapezoid. D. Graph vertices of quadrilaterals on a coordinate plane, and be able to identify them. Review nightly homework assignments. during class. Use of daily warm ups, often in groups. Create quadrilateral family tree. Weekly short class basic skills quizzes. Chapter 8 test, with multiple choice, open-ended and critical thinking questions. Circles What is circumference? What are central and inscribed angles? How are arcs measured? What are tangents and secants? What is the equation for a circle, and how can it be graphed on a coordinate plane? Learn the components of a circle, and what they mean. Learn how to calculate arc measures and arc lengths. A. Find the sum of the intrerior A1. Identify and use parts of circles. A2. Solve problems involving the circumference of a circle. B. Recognize major arcs, minor arcs, central angles, inscribed angles, and their measures. C. Use properties of tangents and secants to solve problems involving circumscribed polygons. D. Find measures of segments that intersect in the interior and exterior In-class examples, paractice and warm up worksheets and activties. Mainly group work. (Formative during class and homework review. (Formative and Summative Several short class quizzes on basic skills material. (Summative. Geometry Activity to create understanding of circumference. 7 of 9
Know the difference between a secant and a tangent, and learn how to use them to solve problems. Understand how to graph a circle on a coordinate plane, and be able to write the equation for a circle. of a circle. A. Find the perimeters and ares of parallelograms. Chapter 10 Test with multiple choice, open-ended, and critical thinking questions. (Summative Areas of Polygons and Circles What are the various formulas used to determine the area of polygons and circles? How can these formulas be used to determine the area of irregular figures? Learn all the formulas to calculate the area of pplygons and circles, and how to apply them. Review trigonometric functions and the special right triangle properties to aid in the determination. Use the radius to calculate the area of circles. B. Find the area of triangles, trapezoids, and rhombi. C. Find the area of regulat polygons and circles using the apothem measurement. D. Learn to calculate the area of irregular figurs by separating them into triangles, trapezoids, rhombi, or rectangles. Chapter 11 test, with multiple choice, open-ended, and critical thinking questions. Surface Area: What are three dimensional figures, and how can they be sketched on paper? A1. Use the orthogonal drawings of three-dimensional figures to make models. A2. Identify and use threedimensional models. warm-up during class, and daily assignment review. 8 of 9
B. Draw two-dimensional What is a net, and how can it be used to calculate the surface area of various shapes? What are prisms, cylinders, pyramids, cones, and spheres, and how are their surface areas calculated? modesl for three-dimensional figures, and calculate the surface area of the figures. C. Find the lateral areas and the total surface areas of prisms. Weekly short quizzes. Learn how to sketch threedimensional on special orthogonal paper. Learn the formulas for the surface areas of prisms, cylinders, pyramids, cones, spheres, & how to apply them. D. Find the lateral areas and the total surface areas of cylinders. E. Find the lateral areas and the total surface areas of regular pyramids. F. Find the lateral areas and the total surface areas of cones. Demonstrations of prisms, cylinders, pyramids, and cones in class. Chapter 11 Test, with open-ended and critical thinking problems. Cumulative Final Exam for chapters 6-11, with multiple choice, open-ended, word problems, and critical thinking questions. 9 of 9