Math 3 Unit 2 Geometric Proofs
Approximate Time Frame: 4 – 5 Weeks
Connections to Previous Learning:
In Grade 8, students applied the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons. In Math II, students revisited the Pythagorean Theorem, using it to solve right triangles in applied problems. In Math II, students also developed a precise definition of similarity in terms of similarity transformations and used this to determine if geometric objects were similar. In both Math I and II, students used the logical structure behind conditional statements to prove various relationships among geometric objects. Properties of congruent and of similar triangles will be used extensively in these proofs. Students may apply algebraic techniques developed in Math I, Math II, and earlier in Math III to study geometric objects in the coordinate plane. They may also make use of their previous work with algebraic equations, with functions, and with distance in the coordinate plane to analyze geometric objects that arise from these equations and functions.
Focus of this Unit:
Much of the geometry encountered by students so far has been synthetic (non-coordinate) geometry. In this unit, students will explore analytic (coordinate) geometry, which they were first introduced to in the unit on circles in Math II. This unit will use coordinates to connect algebra and geometry and provide students with additional analysis and problem solving techniques. Here, the graph of the solution set of an equation becomes a geometric curve, which makes it possible for students to visualize algebraic work. At the same time, geometric shapes can be described by equations, which make it possible for students to then use algebraic work as an aid in developing and strengthening geometric understanding and in modeling problem situations. While studying the connection geometric objects and algebraic equations have, students will revisit their earlier work with circles, re-analyzing them through the lens of analytic and synthetic geometry (as was done in Math II). The correspondence between coordinates and geometric points is also used as a means for proving some geometric theorems by using algebra in the coordinate plane. Along with physical models, dynamic geometry environments will provide students with tools for investigating, experimenting with, conjecturing about, and modeling geometric phenomena.
Connections to Subsequent Learning:
Understanding how to use coordinates to establish geometric results is important to the development of numerous mathematical concepts. For example, students' work with the Pythagorean Theorem and with analytic (coordinate) geometry will aid in their study of conics. Geometric representations in the coordinate plane will prove to be a useful tool in modeling various problem situations and physical phenomena. Proof continues to be a spiraling concept throughout subsequent units and courses as it is used in both geometric and non-geometric settings. The analytic geometry problem solving techniques that will be used to solve problems from this unit will help students deal with similar types of problems that they will see in later units and, particularly, in the units which involve modeling.
Desired Outcomes
Standard(s):Use coordinates to prove simple geometric theorems algebraically.
· G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
· G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point.
· G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
· G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
Understand and apply theorems about circles.
· G.C.1 Prove that all circles are similar.
· G.C.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.
· G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Find arc lengths and areas of sectors of circles.
· G.C.5 Derive, using similarity, the fact that the length of the arch intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant or proportionality. Derive the formula for the area of a sector.
Translate between the geometric description and the equation for a conic section.
· G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem. Complete the square to find the center and radius of a circle given by an equation.
· G.GPE.2 Derive the equation of a parabola given a focus and directrix.
Make geometric constructions.
· G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
· G.CO.13 Construct an equilateral triangle, a square, and a rectangular hexagon inscribed in a circle.
Transfer:
Students will apply concepts and procedures regarding theorems about circles in order to construct, identify, and prove relationships among circles, their parts, and other polygons.
Example: Given a point and a radius, students will transform a circle into a circle centered at the origin with a radius of 1.
Understandings: Students will understand that …
· Geometric figures can be represented in the coordinate plane.
· Algebraic properties (including those related to the distance between points in the coordinate plane) may be used to prove geometric relationships.
· The algebraic relationship between the slopes of parallel lines and the slopes of perpendicular lines.
· Relationships between geometric objects represented in the coordinate plane may be determined or proven through the use of similarity transformations.
· The geometric relationships that come from proving triangles congruent or from proving triangles similar may be used to prove relationships between geometric objects represented in the coordinate plane.
· The distance formula may be used to determine measurements related to geometric objects represented in the coordinate plane (e.g., the perimeter or area of a polygon).
· Different relationships among inscribed angles, radii, and chords of a circle, and between the angles of a quadrilateral inscribed in a circle are provable using previously proven relationships between geometric objects.
Essential Questions:
· How are basic geometric figures constructed in order to maintain their properties using a variety of tools?
· What is the relationship between the slopes of parallel lines and of perpendicular lines?
· How are dilations used to partition a line segment into two segments whose lengths form a given ratio?
· Given a polygon represented in the coordinate plane, what is its perimeter and area?
· How can geometric relationships be proven through the application of algebraic properties to geometric figures represented in the coordinate plane?
· What are the different relationships among inscribed angles, radii, and chords of a circle, and of the angles of a quadrilateral inscribed in a circle?
· What is the relationship between the length of the arc of a circle, the central angle of the circle that intercepts the arc, and the radius of the circle?
· What is the area of a sector of a circle?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
1. Make sense of problems and persevere in solving them. Students will recognize the hypothesis and conclusion in a proof statement and be able to generate the requisite proof using the given information in the proof statement, along with known facts, definitions, postulates, and theorems.
2. Reason abstractly and quantitatively. Students will be able to use figures and information pertaining to a specific geometric object as an aid in reasoning about that geometric object in general.
3. Construct viable arguments and critique the reasoning of others. Students will be able to create and present proofs and be able to critiques the proofs and deductive reasoning of others.
4. Model with mathematics. Students will be able to solve a variety of problems that either involve or can be modeled with triangles by applying the properties of congruent and of similar triangles.
5. Use appropriate tools strategically. Students will be able to use physical models, drawings, and dynamic geometry environments to form conjectures about geometric objects and to reason from information about the geometric object provided by these tools.
6. Attend to precision. Students will recognize that incorrect initial attempts at definitions, conjectures, and theorems may be corrected through a process of refinement.
7. Look for and make use of structure. Students will be able to use the structure of geometric objects to gain insights into, make conjectures about, and create proofs pertaining to these objects.
8. Look for and express regularity in repeated reasoning.
Prerequisite Skills/Concepts:
Students should already be able to:
· Identify and describe geometric objects such a point, line, angle, triangle, parallelogram, circle, etc.
· Plot points and sketch graphs in a coordinate plane that satisfy specific conditions.
· Apply the distance formula to determine the distance between points in a coordinate plane.
· Perform specific dilations to a given object.
· Calculate the slope of a line (segment) given any two points on that line (segment).
· Recall various geometric relationships proven in previous courses.
· Provide statements for informal proof.
The following are taught in the circles unit in Math II:
· Prove that all circles are similar.
· 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; and the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
· Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
· Derive using similarity the fact that the length of the arch intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant or proportionality; and derive the formula for the area of a sector.
· Derive the equation of a circle of given center and radius using the Pythagorean Theorem, and complete the square to find the center and radius of a circle given by an equation. / Advanced Skills/Concepts:
Some students may be ready to:
· Students will recognize how counterexamples can be used to refute conjectures.
· Prove more complex geometric theorems (e.g., the diagonals of a parallelogram bisect each other; the median of a trapezoid is the average of the bases; the coordinates of the centroid of a triangle is the average of the coordinates of the vertices).
Knowledge: Students will know…
· The definition of various geometric objects such as circle, angle, triangle, parallel lines, perpendicular lines, parallelogram, and so on.
· The Pythagorean Theorem.
· The formula for calculating the distance between two points in a coordinate plane.
· Formulas for the perimeter of a polygon and the area of a triangles and rectangles. / Skills: Students will be able to …
· Given two geometric objects, recognize if one can be transformed to the other through a sequence of rigid motions combined with a dilation.
· Sketch a figure that represents specific given information.
· Construct a conditional statement that represents a given conjecture.
· Use coordinates to prove simple geometric theorems algebraically.
· Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
· Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
· Construct the inscribed and circumscribed circles of a triangle.
· Prove properties of angles for a quadrilateral inscribed in a circle.
· Derive the formula for the area of a sector.
· Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, (e.g., using the distance formula).
· Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
· Construct an equilateral triangle, a square, and a rectangular hexagon inscribed in a circle.
· Derive the equation of a parabola given a focus and directrix.
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas, and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
· explicit instruction in geometry vocabulary.
· experiences with tactile and virtual manipulative tools for constructing and analyzing geometric figures.
· guidance connecting algebraic, tabular, and visual representations of geometric figures in the coordinate plane to the language used to describe each.
Academic Vocabulary:
Critical Terms:
Circle
Radius
Diameter
Arc
Chord
Tangent
Central angles
Inscribed angle
Circumscribed angle
Intercepted arc
Radian
Sector of a circle
Coordinate plane
Parallel
Perpendicular
Slope
Polygon / Supplemental Terms:
Conditional statement
Hypothesis
Conclusion
Proof
Necessary Conditions
Sufficient Conditions
Postulate
Theorem
Length
Angle measure
Perimeter
Area
Assessment
Summative Assessments
Summative Tasks
Pre-Assessment / Formative Assessments / Self-Assessments
Parallel and Perpendicular Lines / Geometric Definition of a Parabola
Parallel and Perpendicular Lines
Midpoint Madness / Lesson 3 Self-Assessment
Sample Lesson Sequence
1. G.CO.12, G.CO.13, G.C.3 Constructions including inscribed and circumscribed shapes and different methods (compass and straight edge, paper-folding, etc.) (1.5 weeks)
2. G.GPE.6, G.C.1, G.C.2 and G.C.5: A cohesive set of lessons involving similarity, circles, and partitions of a segment. It would also include the definition of a radian as a constant of proportionality (See Rolling Into Radians http://illuminations.nctm.org/Lesson.aspx?id=3151 ) (1 week)
3. G.GPE.4, G.GPE.5, G.GPE.7 Proofs as related to coordinate geometry (1.5 weeks) Model lesson is 3 to 4 days of this segment.
4. G.GPE.1 and G.GPE.2: Make connections to equations of a circle and parabola. (1 week)
11/17/2014 10:54:08 AM Adapted from UbD® framework Page 7