By the end of the lesson, students should be able to:
Angle
Two-dimensional space
Trigonometry
Congruent Triangles
Algebraic Techniques
Prove and apply angle and chord properties of circles (ACMMG272)
Students:
Students:
For each table group, prepare a laptop with the lesson's GeoGebra files loaded (either offline or online). Print off the Describe & Draw Handouts and Homework Sheets. Connect a laptop to the classroom projector and open a blank PowerPoint.
Welcome students into the classroom and instruct them to unpack their workbooks and stationary, as well as to sit in table groups of 4.
Brainstorm - Recap (LIT) [10 minutes]
WM: Understanding, Communicating
At the beginning of the lesson, the class looks at students' responses to last lesson's homework before starting the brainstorm activity.
Students are instructed to work with their table groups to brainstorm everything they can remember related to circle geometry (i.e. features and terminology, can also include diagrams) on an A4 sheet of paper.
After 5 minutes, the teacher selects table groups to share what they wrote with the class (AFL).
Exploration (ICT) [10 minutes]
WM: Fluency, Reasoning
The teacher displays the following property on the board (PowerPoint);
"chords of equal length in a circle subtend equal angles at the centre and are equidistant from the centre"
and highlights key parts before drawing (step-by-step) the visual representation of the property on the whiteboard. Students are asked to follow along and redraw the diagram into their workbook, and then to verify the chord property using GeoGebra.
Students engage in think-pair-share to develop suggestions for how they could prove either part of the chord property (equal angles or equidistant). The teacher reintroduces the process of using congruent triangles to prove geometric facts, modelling each step for how to prove chords of equal length subtend equal angles at the centre.
Bonus: Can pose a question to students using honeycomb/hexagonal wax cells - when you form a circle around one of these hexagonal cells, why are the sides equal in length to the radius? (see image caption)
Use for teacher demonstration + student exploration
If students have access to laptops/tablets, can give students access here: https://www.geogebra.org/geometry/v77ar3cd
Pair-Work (ICT) [20 minutes]
WM: Understanding, Fluency, Reasoning
In pairs, students verify two other chord properties using pre-prepared dynamic examples (GeoGebra):
"the perpendicular from the centre of a circle to a chord bisects the chord; conversely, the line from the centre of a circle to the midpoint of a chord is perpendicular to the chord"
"the perpendicular bisector of a chord of a circle passes through the centre"
Students copy the diagrams into their workbook then work together to formulate proofs using congruent triangles. The teacher monitors pair-work and provides feedback or suggestions based on their progress (AFL) (see Questions/Prompts).
(Bonus) Describe & Draw (LIT)
WM: Communicating, Problem Solving
Students who have completed their work early pair up with another student (from their table group) and are given the Describe & Draw activity handout to complete for the remainder of the lesson.
Summary:
Student A are given a diagram and are tasked with providing instructions for drawing the diagram
Student B follows Student A's instructions and constructs the diagram using compass-and-straight-edge (the diagram should be labelled)
Afterwards, students work together (or independently) to solve the question
Reflection Questions:
Exit Task [10 minutes]
WM: Problem Solving, Fluency
Recap each property using PowerPoint and GeoGebra, then ensure students have written the three chord properties into their workbook (AFL).
The teacher displays a question (PowerPoint) for students to solve. Each student is provided a sticky note to write down their solution on and to hand in before leaving (AFL).
Find the length of DF given that AE = 6 cm and EO = OF = 3 cm. (provide reasons)
Alternative Conclusion Task: 3-2-1
Students are provided with an index card to write responses to the following prompts:
Students submit their responses to the teacher before leaving (AFL).
Provide students with a homework sheet as they leave the classroom (to be handed in next lesson) (AOL).
Orientation:
Introduction:
Body:
Conclusion: