Lesson 5

Out of the Box / Angle Properties (Cyclic Quadrilaterals)

LESSON DETAILS

Lesson Goals

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

use physical materials to construct circle diagrams of the Angle on the Same Arc property
develop a proof for the Angle on the Same Arc property using the Angle at the Centre/Circumference property
recognise cyclic quadrilaterals and identify the interior and exterior angles of cyclic quadrilaterals
verify and prove the interior and exterior angle properties of cyclic quadrilaterals using previous angle properties and relationships
solve for unknown lengths or angles using chord and angle properties
recall and draw diagrams of chord and angle properties

Content

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

Students:

prove the following angle properties of circles:
- angles at the circumference, standing on the same arc, are equal
- the opposite angles of cyclic quadrilaterals are supplementary
- an exterior angle at a vertex of a cyclic quadrilateral is equal to the interior opposite angle
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 angle relationships to proofs and problems involving circles (Fluency, Problem Solving, Reasoning)
construct circle diagrams using compass-and-straight-edge and dynamic geometry software (Fluency, Communication)
use geometric properties of triangles and quadrilaterals to solve problems involving unknown lengths or angles (Understanding, Fluency, Problem Solving)

Resources

Circular Paper Plates
String
Tape/Blu-Tac
Card Game Handout
A4 Paper
Mobile Phones (Camera)
GeoGebra Resources (Laptop & Projector)
Whiteboard & Whiteboard Markers
Google Classroom
Spare Stationary: Pencils, Pens, Erasers, Sharpeners, Compasses, Rulers, Protractors

Strategies Used

Physical construction + group work
Think-pair-share, class discussion
ICT (verifying properties), Predict-Observe-Explain
Literacy activity (card game, problem-solving), pair-work
Exit Task - consolidation, literacy

LESSON PLAN

Lesson Setup

Prior to the lesson, prepare laptops with GeoGebra activity files on each table group, as well as circular plates, string and blu-tac (one set per table to minimise resource cost). Print out and prepare the Card Game handout (one set designed for pairs).

Welcome students into the classroom, instruct students to take out their workbooks and stationary.

Orientation

Address questions from self-assessment, direct students to resources if they are curious about anything mathematics related (e.g. Byrne’s Elements)

Ask students to take out their homework and to compare it against the solutions on the whiteboard, check if anyone had any difficulties before moving on

Check if anyone attempted the research task, ask them to share something they found interesting

Physical modelling [10 minutes]

WM: Reasoning, Communication

In small groups (of 4), students use circular paper plates, string, tape/blu-tac & protractors to verify the angles on the same arc property.

Think-pair-share: Students consider and suggest a description for the property, then work together in small groups to formulate a proof.

Teacher conducts a class discussion afterwards to summarise the proof and evaluate the students' understanding (AFL)
When presenting their proofs, students should utilise their circle constructions (appropriate labelling/construction lines drawn on)
Students should take a photo of their circle construction and upload it to Google Classroom (AOL)

Plate/String Activity

Inspiration: https://2.bp.blogspot.com/-sHGN1UsWAak/VGJ2PRs2WFI/AAAAAAAADCQ/3w9cqEgbt_o/s1600/hula%2Bhoop%2B2.JPG

Introduction

Cyclic Quadrilaterals (ICT) [20 minutes]

WM: Understanding, Reasoning

Teacher demonstrates cyclic quadrilaterals using whiteboard drawings (quadrilateral inscribed in a circle)

Predict-Observe-Explain: Students construct examples in their workbooks, write down predictions of cyclic quadrilateral properties, explore (using their drawings and GeoGebra) interior and exterior angles of cyclic quadrilaterals, and individually prove both properties

Teacher should monitor students' work and check students' proofs for cyclic quadrilateral properties (AFL)
If students are struggling, the teacher should suggest appropriate construction lines (connecting points of the cyclic quadrilateral to the centre) and advise students to use the angle at the centre/circumference property from the previous lesson
Students should save a copy of the proofs for each cyclic quadrilateral property in their workbooks for their portfolios (AOL)

Supplementary Angles

https://www.geogebra.org/geometry/pjsdf735

Exterior Angle

https://www.geogebra.org/geometry/pkqexarz

Body

Circle Geometry Card Game (time dependent, LIT) [15 minutes]

WM: Fluency, Problem Solving, Reasoning

For students who finish the introduction task quickly: provide pairs with a set of cards (Circle Geometry Card Game handout) with different circle diagrams (featuring chords, angles, arc, radii)

Students compete against one another by comparing who has the better card in a chosen category (e.g. bigger angle)

Students take turns playing their card first, and the other students tries to select a card they think can win in that category (cannot play the same card until all cards have been played)

In order to play a card, students must solve for all the missing information on their card (which requires applying previously learnt properties and algebra)
When playing a card with missing information, students must justify how they determined what the information (e.g. based on circle properties)
The teacher should demonstrate a quick round of the activity before leaving students to compete. Near the end of the lesson, check in with students to see who won the most and ask for their strategy

Example:

Student A selects a card and states that they think their card has a bigger angle than any of Student B's cards
Student B responds by selecting the card in their hand that has the largest angle
Both students reveal their cards. Student B's card has the bigger angle, and so they win that round.

Card Game Example

In pairs, students are each given 5 cards with different circle diagrams. Each diagram has a set of missing information students will need to solve (using angle and chord properties) before they are able to use them in the game.

Students take turns picking a category they want to compete in (e.g. smallest angle, longest chord, shortest radius). Once a category has been selected, students pick a card and reveal it to their partner. Students track how many times they win, and reuse their cards (or swap).

Inspiration: https://en.wikipedia.org/wiki/Top_Trumps

Conclusion

Exit Task (LIT) [5 minutes]

WM: Understanding, Communicating

Students are given a blank A4 sheet of paper to redraw and label at least five properties they remember about chords and angles.

The diagrams should be sufficiently large and clearly labelled

Students submit their drawings to the teacher before leaving the classroom

Students are encouraged to take a photo of their final drawing before submitting
After marking, return students' drawings for their portfolio (AOL)
Optionally, allow students to create a summary chart at home and upload a photo via Google Classroom

Homework

Students review the work they have completed in class (regarding chord and angle properties) and create a problem for others to solve on Google Classroom (AOL).

The problem can either be a proof question (e.g. prove two lengths are equal) or a problem-solving question (e.g. find the unknown angle)
Students should either create a diagram in GeoGebra, or hand-draw a circle diagram and upload a photo of it to Google Classroom

After uploading their problem, students should also solve three different questions from their peers and reply to their posts with a complete solution (AOL).

Students should ensure that every question posted on Google Classroom has at least one attempted solution
Students are responsible for providing feedback regarding solutions to their peers
Questions that are interesting or challenging should be shown at the start of the next lesson; students who attempted the question are asked to share their insights

Lesson Evaluation Questions

Orientation:

Was the use of physical construction materials engaging or novel for students?
Were students able to easily use the string to dynamically change their diagrams?
Would it be better to scale the activity up (i.e. table-sized circles and longer string)?
Was the angles on the same arc property easy to verify using protractors?
What alternative resources could be used?

Introduction:

Was the use of GeoGebra useful for helping students to identify cyclic quadrilateral properties (specifically, using the angle tool)?
Did students prefer to work individually, in pairs or groups, or as a class?

Body:

Were all students actively participating in the activity?
Are the circle problems too difficult to solve in the allocated time?
Did students prefer to work in pairs to solve the problems before playing the game?
Did students require additional cards?
Was the game quick to finish?

Conclusion:

Did students need to refer back to their workbooks?
Was the goal of the activity clear to students?
Was there enough time to draw five different properties?

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