Oxford High School
Courses in Mathematics
# 232 Geometry and Measurement - Academic (10-12) Full Year 1.0 Credit
In Geometry, students will develop reasoning and problem solving skills as they study topics such as congruence and similarity, and apply properties of lines, triangles, quadrilaterals, and circles. Students will also develop problem solving skills by using length, perimeter, area, circumference, surface area, and volume to solve real world problems. In addition to geometry concepts, this course includes numerous examples and exercises involving algebra, data analysis and probability. This course is an important part of CAPT and SAT preparation, therefore, instruction, practice, and assessments will be presented in a variety of formats – multiple choice, short answer, grid-in and open ended that students will encounter on standardized testing. There will also be extensive use of technology including the graphing calculator throughout the year.
1. Enduring Understandings (broad ideas, usually grounded in the discipline):
Mathematics as Calculation – performing calculations. Knowing when exact verses generalized calculation is appropriate and develop insight into a mathematical theory when they see how it relates to actual calculations.
Mathematics as a Field of study – using mathematical vocabulary. Communicate clearly by using the formal vocabulary and rules of mathematics.
Mathematics as a Modeling Language – using mathematical models. Understand the various types of models used in mathematics and how to translate real life situations to those mathematical models to obtain a solution and then to translate those solutions back into the context of the real-life application.
Mathematics as Logical Thought – thinking logically. Think logically in a world that requires careful thought as a reasoning process
2. Course Specific Goals (aligned with Content Standards):
Understanding congruence and similarity
Applying properties of lines, triangles, quadrilaterals and circles
Understanding and solving problems using length, perimeter, area, circumference, surface area and volume.
Building problem solving and communications skills through reasoning and proof.
3. Expected Performances (aligned with state Expected Performances)
Essentials of Geometry
Reasoning and proof
Parallel and perpendicular lines
Congruent triangles and relationships with triangles
The concept of similarity
Right triangles
Quadrilaterals
Properties of transformations
Measuring length and area
Surface area and volume of solids
Properties of circles
Geometry and Measurement: Scope and Sequence Calendar
First Semester Material to be Covered Prior to the Mid-Term Exam
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Points, Lines, and Planes |
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1. Essentials of Geometry – students will name and sketch geometric figures, use postulates to identify congruent segments, find lengths of segments in the coordinate plane, and find the midpoint of a segment. Students also will name, measure and classify angles, identify complementary and supplementary angles, and classify polygons. Finally, they will find circumference and area of circles, and area an perimeter of rectangles. 2. Reasoning and Proof- students will describe patterns, including visual and number patterns, and use inductive reasoning to make and test conjectures. They will analyze conditional statements and write the converse, inverse, and contrapositive of a conditional statement. They will explore how conditional and biconditional statements are used to state definitions. Students will use deductive reasoning, the Law of Detachment, and the Law of Syllogism to develop simple logical arguments. Students will learn what can and cannot be assumed from a diagram. Finally, they will use properties of equality and the laws of logic to prove basic theorems about congruence, supplementary angles, complementary angles and vertical angles. 3. Parallel and Perpendicular Lines – students will classify angle pairs formed by three intersecting lines, study angle pairs formed by a line that intersects two parallel lines, and use angle relationships to prove lines parallel. They will investigate slopes of lines and study the relationship between slopes of parallel and perpendicular lines. Students will find equations of lines. Finally, they will use theorems about perpendicular lines and find the distance between parallel lines in the coordinate plane.
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Triangles |
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1. Congruent Triangles – students will classify triangles, find measures of angles of triangles, identify congruent figures, and prove triangles congruent. They will also use theorems about isosceles and equilateral triangles and perform transformations. 2. Relationships within Triangles – students will use properties of mid-segments to find lengths of segments in triangles. They then learn to write a coordinate proof. They explore perpendicular bisectors and use the concurrency of perpendicular bisectors of a triangle to solve problems. They use angle bisectors to find distance relationships and explore the concurrency of angle bisectors of a triangle. Students use medians of a triangle to find the centroid and to find segment lengths, and they use altitudes of a triangle to find and explore the orthocenter. Students relate side length and angle measures of a triangle, find possible side lengths for the third side of a triangle and use inequalities to make comparisons in two triangles. Finally, students learn to write indirect proofs. 3. Similarity – students use ratios, proportions, and geometric means to solve geometry problems. They use ratios to find the scale of a drawing and then use the scale to find the actual distance on a map or the actual height of a building. They use proportions to identify similar polygons and find the scale factor between two polygons, they use a scale factor to find corresponding lengths in similar polygons, and they use the AA Similarity Postulate, the SSS Similarity Theorem, or the SAS Similarity Theorem to determine whether two triangles are similar. Finally, students use proportions and the Triangle Proportionality Theorem or its converse to find the lengths of segments related to triangles or parallel lines.
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Figures in a Plane |
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Second Semester material to be covered prior to the final exam: 1. Right Triangles – students investigate side lengths and angles in triangles. They start by using the Pythagorean theorem to find the length of the third side in a right triangle, then use the Converse of the Pythagorean Theorem, and other theorems to decide if three given side lengths form an acute, right or obtuse triangle. Finally, students explore ratios of lengths formed by an altitude to the hypotenuse of a right triangle and use the ratios of side lengths for a 45o-45o-90o triangle and a 30o-60o-90o triangle. 2. Quadrilaterals – students will find angle measures in polygons. They will investigate properties of parallelograms and learn what information they can use to conclude that a quadrilateral is a parallelogram. Students will also study special quadrilaterals such as rhombuses, rectangles, squares, trapezoids, and kites. 3. Properties of Transformations – students will perform translations with vectors and algebra. They will reflect figures in a give line, rotate figures about a point and identify line and rotational symmetry.
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Circles and Measurement |
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1. Measuring Length and Area – students develop and use formulas for the area of triangles, parallelograms, trapezoids, and other polygons. They use ratios to find areas of similar polygons, and they use ratio of areas to find missing lengths in similar figures. Students develop and use a formula for the area of a regular polygon. 2. Surface Area and Volume of Solids – students find the surface areas of spheres and cylinders. They use nets to find the surface area of cylinders. Finally, they find the volume of prisms, cylinders, cones and spheres. 3. Properties of Circles - students investigate aspects of circles. They start by drawing tangents to circles and seeing how a tangent to a circle is related to the radius at the point of tangency. They use intercepted arcs of circles to measure angles formed by chords in a circle and to measure angles formed by secants and tangents to a circle. They explore relationships between segment lengths of chords that intersect in a circle, and they investigate relationships between segment lengths of secants and tangents to a circle.
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