This sub-organizer contains the following sections:
Prescribed Learning Outcomes
Suggested Instructional Strategies
Suggested Assessment Strategies
Recommended Learning Resources

PRESCRIBED LEARNING OUTCOMES

It is expected that students will develop and apply the geometric properties of circles and polygons to solve problems.

It is expected that students will:

• use technology with dynamic geometry software to confirm and apply the following properties:
  • the perpendicular from the center of a circle to a chord bisects the chord
  • the measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc
  • the inscribed angles subtended by the same arc are congruent
  • the angle inscribed in a semicircle is a right angle
  • the opposite angles of a cyclic quadrilateral are supplementary
  • a tangent to a circle is perpendicular to the radius at the point of tangency
  • the tangent segments to a circle, from any external point, are congruent
  • the angle between a tangent and a chord is equal to the inscribed angle on the opposite side of the chord
  • the sum of the interior angles of an n-sided polygon is 180(n-2)
• prove the following general properties, using established concepts and theorems:
  • the perpendicular bisector of a chord contains the center of the circle
  • the measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc (for the case when the center of the circle is in the interior of the inscribed angle)
  • the inscribed angles subtended by the same arc are congruent
  • the angle inscribed in a semicircle is a right angle
  • the opposite angles of a cyclic quadrilateral are supplementary
  • a tangent to a circle is perpendicular to the radius at the point of tangency
  • the tangent segments to a circle from any external point are congruent
  • the angle between a tangent and a chord is equal to the inscribed angle on the opposite side of the chord
  • the sum of the interior angles of an n-sided polygon is 180(n-2)
• solve problems, using a variety of circle properties, and justify the solution strategy used

SUGGESTED INSTRUCTIONAL STRATEGIES

Students will investigate, hypothesize, and verify properties of circle geometry and apply these properties to solve problems. The deductive reasoning used to solve problems involving circle geometry can be employed to make decisions in a variety of fields e.g., landscape architecture, meteorology, and civil engineering.

Note:
Although the prescribed learning outcomes do not require students to use formal two-column proof, students are nevertheless expected to logically justify conjectures regarding geometric properties and explain their solutions to the problems.

• Have groups of students experiment (e.g., investigate, hypothesize, confirm, discuss, use inductive reasoning) to identify the properties of a circle. Encourage students to verify their results through measurements, calculations, and reasoning. Possible experiments include:
  • constructing a perpendicular bisector with compass and straightedge - drawing circles of varying sizes, and using chords of different length on these circles to construct the perpendicular bisectors
    Have students discuss the importance of accuracy while presenting their conclusions to the class.
• As enrichment, have students research and report on the life and work of a famous mathematician (historical or contemporary). Discuss with students the concept of mathematics as an evolving discipline and not as a static set of rules. For example Euclid’s Elements is an in-depth set of rules that were deduced inductively.
• Have students write a logical-sequential verbal explanation of a solution to a problem or a proof.

SUGGESTED ASSESSMENT STRATEGIES

To enable them to work effectively with properties of circles, students need a solid understanding of the properties of parallel lines, similar and congruent figures, polygons, and angle relationships. Students demonstrate their learning when they solve geometric problems by applying circle properties and their converse statements.

Observe

• Before advancing in this topic, assess students’ background knowledge of the properties of parallel lines, similar and congruent figures, polygons, and angle relationships previously learned. Look for evidence of students’ recognition of properties and their abilities to solve problems using:
  • parallel lines
  • congruent triangles
  • basic geometric constructions
  • properties of polygons
• Ask students to perform experiments with circle properties, observe how effectively they can:
  • use a protractor and compass correctly and accurately
  • follow directions to draw the figures and measure the angles
  • explain orally the process of experimentation to derive conclusions
  • communicate their conclusions clearly in writing; explain the basis of their conclusions
  • identify the appropriate circle properties used to find the measurements of sides and angles

Self-/Peer Assessment

• Work with students to develop a table of specifications for a test on this unit of work. Identify for them the content categories along the vertical axis and the intellectual levels along the horizontal axis. After the weighing is identified for each cell, have students work in groups to develop questions for each cell.

RECOMMENDED LEARNING RESOURCES

Print Materials

• Mathematics 11, Western Canadian Edition
  Ch. 7 (Sections 7.1 - 7.4)
  Ch. 8 (Sections 8.1 - 8.6)
• MATHPOWER 11, Western Edition
  Ch. 7 (Sections 7.2 - 7.7)
  Ch. 8 (Sections 8.1 - 8.5)

Software

• The Geometer’s Sketchpad
• Green Globs & Graphing Equations
• Secondary Math Lab Toolkit

CD-ROM

• Geometry Blaster

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: December 4, 2000

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