Lesson 4

Thales' Tale / Angle Properties (Introduction)

LESSON DETAILS

Lesson Goals

By the end of the lesson, students should be able to:

recall and graphically represent angle types and angle relationships
prove Thales’ Theorem using either isosceles triangles or the angle at the centre/circumference property
prove the Angle at the Centre/Circumference property and apply the property to solve problems involving unknown angles

Content

Prove and apply angle and chord properties of circles (ACMMG272)

Students:

prove the following angle properties of circles:
apply chord and angle properties of circles to find unknown angles and lengths in diagrams

Working Mathematically

Students:

use dynamic geometry software to investigate angle properties of circles (Problem Solving, Reasoning)
apply circle theorems to prove that the angle in a semicircle is a right angle (Problem Solving, Reasoning)
apply angle relationships and geometric properties of triangles to proofs and problems involving circles (Fluency, Problem Solving, Reasoning)
construct circle diagrams using compass-and-straight-edge and dynamic geometry software (Fluency, Communication)

Resources

Post-it Notes
A3 cardboard sheet
Mobile Phones (Camera)
Thales' biography https://www.famousscientists.org/thales/
GeoGebra Resources (Laptop & Projector)
Index cards (self-assessment)
Whiteboard & Whiteboard Markers
PowerPoint (w/ Self-Assessment Questions)
Google Classroom
Spare Stationary: Pencils, Pens, Erasers, Sharpeners, Compasses, Rulers, Protractors

Strategies Used

Affinity mapping (cooperative literacy activity) + class discussion
ICT (verifying property) + class discussion
Group work + peer-teaching
Cross-curriculum connections (History)
Self-assessment

LESSON PLAN

Lesson Setup

For each table group, prepare a laptop with the lesson's GeoGebra files loaded (either offline or online), as well as a stack of Post-It notes to write on and A3 cardboard to stick the notes onto. Connect a laptop to the classroom projector and load a PowerPoint file (with the self-assessment questions for the lesson conclusion).

Welcome students into the classroom and instruct them to unpack their workbooks and stationary, as well as to sit in table groups of four (4).

Orientation

Select one student's homework response to discuss in class and to share their process.

Affinity Mapping (LIT) [10 minutes]

WM: Fluency, Communicating

The teacher provides the students with Post-It notes and displays a prompt on the whiteboard:

"Angles and Angle Relationships"

Students individually write down everything they remember about the prompt, then group together with their table neighbours and arrange their notes together according to key ideas. Students stick their Post-It notes onto an A3 cardboard sheet. Afterwards, each table group show off their work and share a couple concepts they recalled (AFL).

During this activity, the teacher selects representatives from each group to add an idea to the whiteboard (communal brainstorm)
If there are missing angles or angle relationships on the whiteboard, the teacher asks students to contribute more ideas (or fill in the missing information for students to add to their cardboard sheet)
Students photograph their cardboard sheet and upload it to Google Classroom to share with peers and for their portfolios (AOL)
Students should keep the cardboard sheet nearby for referencing during class time

Useful and informative mind-map of angles and angle relationships from Stage 4 Angle Relationships Unit of Work.

Can show students this mind-map if they have missed anything.

Angle Revision

If students need revision of any of the angle types or angle relationships:

Introduction

History Intermission (ICT) [10 minutes]

WM: Understanding, Reasoning

Teacher provides a very brief presentation on Thales' life, his role in mathematics (including geometry) and an introduction to the theorem named after him.

Instruct students to verify Thales's theorem using GeoGebra), and then to copy the diagram into their workbooks

After verification, the teacher holds a class discussion to gather suggestions for proving the property.

Students are offered time afterwards to work on a proof, allow for 5 minutes before going through the proof as a class (invite a student to the board if they have a complete proof) (AFL)
Note: there are two proofs available (using either isosceles triangles or angle at the centre/circumference property, revisit second proof at the end of the lesson)
Alt Task (if resources/space available): Bring students outside the classroom and set up a camera (on a tripod) opposite a building's facade, have students trace out the possible camera positions using chalk where the building's facaded subtends a right angle to the camera, relate the results to the life of Thales

Thales of Miletus, Biography: https://www.famousscientists.org/thales/

Body

Group work (ICT) [25 minutes]

WM: Reasoning, Problem Solving, Fluency

Teacher displays three different cases of the angle at centre/circumference property. Students spend the remainder of the lesson verifying the property using GeoGebra. Each table group decides who in their group will work on each case. Students are tasked with developing proofs for their specific case (similar to the proof for Thales' Theorem).

Teacher monitors each group's progress and provides feedback/scaffolding (e.g. revisit the proof for Thales' Theorem, what did we do after we added this construction line)
When students have finished working on their proofs, they teach their proof to their peers, visually and verbally communicating their proof to their group members; teacher monitors students' discussions during this time (AFL)
Teacher selects students who have finished their proofs to write it on the whiteboard (AFL)
If students complete their proofs early, they can then complete example problems based on these cases and compare their answers with peers (disputes in solutions should be settled through class discussion) (AFL)

Cases for Angle at Centre & Circumference

Diagrams created in GeoGebra Geometry

Conclusion

Self-assessment [5 minutes]

WM: Understanding

Teacher displays self-assessment questions on the whiteboard (PowerPoint) and provides students with index cards to respond to the questions:

What did I learn about in today's lesson?
What do I want to learn more about in the future?

Students submit their responses it to the teacher before leaving the class (AFL). Address students' responses at the start of the next lesson.

Homework

The teacher informs students that they can re-prove Thales' Theorem using the angle at the centre & circumference property (simpler proof). Students are to bring their proof to class next lesson (see notes in Introduction regarding assessment).

Bonus task: Research Euclid (Elements), discuss interesting findings at the start of the next lesson (students could also submit their findings to Google Classroom; review everyone's findings at the start of the next lesson)

https://upload.wikimedia.org/wikipedia/commons/c/cf/Title_page_of_Sir_Henry_Billingsley%27s_first_English_version_of_Euclid%27s_Elements%2C_1570_%28560x900%29.jpg

Lesson Evaluation Questions

Orientation:

To what extent did students individually participate in the affinity mapping activity?
Did students prefer to write down or draw what they remembered?
In what ways did students organise their ideas?
Did students engage with students outside of their groups for ideas?

Introduction:

Were students interested in the historical context of Mathematics?
Did students prefer to work independently or in groups to develop a proof?
Was GeoGebra helpful in developing students' understanding of the property?

Body:

Was the group-based nature of the task (developing a proof then explaining it to peers) an effective strategy for engaging students?
Were students actively communicating with one another regarding their proofs?
Was the task too difficult for students to do alone?
Did students identify different cases to those presented by the teacher? What kind of conversations developed?

Conclusion:

Do students recognise the benefit of reflecting on their progress during a unit of work? Are these benefits made obvious?
Did all students choose to participate in self-reflection?
How interested were students in learning about the historical context of mathematics?

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