Lesson 2

Striking a Chord / Chord Properties (Introduction)

LESSON DETAILS

Lesson Goals

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

recall circle features and terminology
create and use hand-drawn and dynamic circle diagrams to verify the Equal Chords & Perpendicular Bisector properties
prove chord properties using congruency tests
solve for unknown side length using chord properties
BONUS: follow or provide verbal instructions for constructing circle diagrams

Content

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

Students:

prove the following chord properties of circles:

Working Mathematically

Students:

use dynamic geometry software to investigate chord properties of circles (Problem Solving, Reasoning)
use written, oral and graphical representations and conventions to describe parts of the circle (Understanding, Communicating)
apply trigonometry to solve problems involving unknown lengths and angles (Understanding, Fluency, Problem Solving)
construct (using compass-and-straight-edge) and label circle diagrams from worded or verbal descriptions (Fluency, Communication)
describe a circle in sufficient detail for it to be sketched (Fluency, Communication)
apply the properties of congruent triangles to proofs and numerical exercises involving circles (Fluency, Communicating, Reasoning)

Resources

Blank A4 Paper
Mobile Phones (Cameras)
Whiteboard & Whiteboard Markers
GeoGebra Resources (Laptop + Projector)
PowerPoint
Describe & Draw Handout
Modified Describe & Draw Homework Sheet
Google Classroom
Sticky Notes or Index Cards (Exit Task Question)
Bonus: Honeycomb Wax Picture (available on Google Images)
Spare Stationary: Pencils, Pens, Erasers, Sharpeners, Compasses, Rulers, Protractors

Strategies Used

Literacy: Group brainstorm + share with class
Step-by-step modelling
ICT usage (verifying property)
Think-pair-share
Independent/Pair-work
Teacher Monitoring & Questioning
Describe-and-draw (cooperative literacy activity, peer-teaching)
Exit Task/3-2-1 (self-assessment)

LESSON PLAN

Lesson Setup

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.

Orientation

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.

During this time, the teacher should ensure each student is participating (if not, prompts should be provided)
Students are encouraged to include diagrams in their brainstorms

After 5 minutes, the teacher selects table groups to share what they wrote with the class (AFL).

Optional: Instruct students to upload a photo of their work to Google Classroom (to share with their peers and to revisit at home for later revision)
Optional: determine which group created the best brainstorm, then hang their work on the classroom wall

Questions

What circle features do we remember discussing from last lesson?
We discussed different examples of angles in a circle, do you remember one of them?
Can you explain what (e.g. angle standing on an arc) looks like?

Introduction

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.

If students have rulers & protractors, they should also verify the property using their workbook drawings

https://www.sciencefriday.com/wp-content/uploads/2017/08/Beehive-micro_large-min.jpg
Note: If you inscribe a regular hexagon in a circle, the side lengths of the hexagon are equal to the radius
This fact can be used as an enrichment task to get students to research how bees create honeycomb whilst integrating mathematics
Question originally inspired from Signpost Mathematics 10 Advanced, 2nd Edition

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.

Support students by using the GeoGebra file to highlight matching features (for proving congruency)
Afterwards, students complete the proof for equidistant chords by themselves (they should be encouraged to discuss the proof with their table group)
The teacher monitors table groups during this time and selects one student to write the proof on the board (students compare their proof with the one on the board before moving on) (AFL)
Students should write down their proof into their workbook and save a copy for their portfolio (AOL)

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)

Equal Chords Property

Use for teacher demonstration + student exploration

If students have access to laptops/tablets, can give students access here: https://www.geogebra.org/geometry/v77ar3cd

Body

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).

If students finish early, ask them to verify with one another that their proofs are valid, then to compare their proof with an example set of solutions
Students should keep a copy of their proofs for their portfolio (AOL)

(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:

One student adopts the role of describer, while the other student adopts the role of drawer (students should swap roles after solving each question)
The describer is given a series of circle diagrams with accompanying questions. Without allowing the drawer to see the original diagram, the describer is tasked with providing a set of instructions for the drawer to follow in order to reconstruct the circle diagram
After the diagram is 'complete', the pair compare the drawing with the original diagram and reflect on the process (can retry if the diagram is not a close match)
Students use the drawings to answer the accompanying question (e.g. determine missing side/angle)

Questions/Prompts

Verify that when the chords are equal in length that the angles subtended at the centre are equal/the chords are equidistant from the centre
Pay close attention to the orientation of the triangles, and what parts correspond to each other
What strategies can we use to approach proving this property?
Can you recall the four tests we could use for proving congruency?
What parts of the diagram do we know match (sides or angles)

Describe & Draw Handout

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:

Did you manage to provide accurate instructions for the drawer?
Does your diagram match the one from the worksheet?

Conclusion

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).

Additionally, advise students to check Google Classroom for a summary of the properties they have covered so far

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).

Use the results of the sticky notes to determine whether students understand how to find missing side/angles (using the chord properties)
If not, follow-up by setting additional homework (with examples) on Google Classroom to be completed later in the week

Exit Task Question Example

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:

3 things I learned this lesson
2 things I have questions about
1 thing I want the teacher to know

Students submit their responses to the teacher before leaving (AFL).

Follow-up students' responses by preparing example questions for the next lesson

Homework

Provide students with a homework sheet as they leave the classroom (to be handed in next lesson) (AOL).

Modified describe-and-draw worksheet (individual): students follow a list of instructions for drawing, then flip the worksheet to compare their drawing with the one provided; students then complete the accompanying question(s)
Make a copy of the homework sheet available on the Google Classroom

Modified Describe & Draw Homework Sheet

describe-and-draw-homework.docx

Homework sheet created using the examples from the Describe-and-Draw activity, can modify by changing the diagrams and rewriting the drawing instructions on the first page.

Lesson Evaluation Questions

Orientation:

Did the students prefer to write down what they could remember or draw example diagrams of specific circle feature or terminology?
In each group, was each and every student contributing to the brainstorm? (Would individual brainstorms be more effective?)
Was the brainstorm activity effective in encouraging students to use circle terminology in conversation with their peers?

Introduction:

How useful did students find the teacher providing a step-by-step guide for constructing diagrams and proving properties?
Did the provision of GeoGebra diagrams prove helpful for developing student understanding?
What alternatives could be given if GeoGebra is unavailable?

Body:

Do students prefer to work independently, in pairs or in groups when developing proofs?
Is the describe-and-draw activity engaging and/or appropriately challenging for students?
Were the instructions for the activities easy to follow, or is more clarification required?
Was there enough time to run both activities?

Conclusion:

Was the exit question provided to students appropriate to their current level of understanding?
Did all students participate in the activity?
Are there alternate (and possibly more effective) ways to collect student responses (e.g. using ICT, multiple choice, etc.)

Google Sites

Report abuse