Lesson 3

Split Down The Middle / Chord Properties (Bisector)

LESSON DETAILS

Lesson Goals

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

reorganise a proof for a chord problem into the correct sequence
locate the circumcentre of three non-collinear points and construct a circle that touches all three points
explain why many circles can be constructed touching one or two given points, but only one circle can be constructed touching three given points
prove that the line joining the centres of two intersecting circles bisects their common chord at right angles using congruency tests

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)
apply chord properties of circles to deduce unknown lengths and angles (Understanding, Fluency, Problem Solving)
construct circle diagrams using compass-and-straight-edge and dynamic geometry software (Fluency, Communication)
apply the properties of congruent triangles to proofs and numerical exercises involving circles (Fluency, Communicating, Reasoning)

Resources

Sequencing Handout
Whiteboard & Whiteboard Markers
GeoGebra Resources (Laptop + Projector)
Map Customiser (or Map Printouts)
Google Classroom
Mobile Phones (Cameras)
PowerPoint Files (see below)
Index Cards & Exit Task Question
ALT 1: ground markers, tape measures, string
ALT 2: butchers paper, tape measures, metre-long rulers, protractors, coloured markers, string, point markers, Blu-Tac
Spare Stationary: Pencils, Pens, Erasers, Sharpeners, Compasses, Rulers, Protractors

Strategies Used

Sequencing Activity (LIT)
Class Discussion (open-ended questions asking for students' process)
Brainstorm
Joint-Construction (demonstrating process)
Physical construction
ICT Activity + Contextualised Problem
Individual/Pair Work + students writing answers on whiteboard
Exit Task
Student-Developed Question (HW Task, w/ ICT)

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 the materials required for the introduction activity. Print off copies of the Sequencing Handout. Connect a laptop to the classroom projector and load the lesson's PowerPoint file (see below).

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

Sequencing (LIT) [7 minutes]

WM: Understanding, Reasoning

At the beginning of the lesson, provide feedback to students based on their responses to the exit task or 3-2-1 task.

The teacher displays a solution to a chord problem on the whiteboard (PowerPoint), but in a jumbled state. Students work independently to rearrange the solution into the correct order, then discuss their solution with their table group.

After 3 minutes, the teacher selects students to comment on how they fixed the solution and why they chose a particular order to rearrange it (AFL).

Variation: can increase the task difficulty by leaving blanks, have students collaborate in pairs to figure it out
Variation: turn the activity into a communicative gap task by providing two versions of the proof, alternating blanks

Sequencing Activity.pptx

Diagram for Sequencing Handout Example

Show proof out-of-sequence on the interactive whiteboard/projector, get students to rewrite proof in their books in the correct order
Alternatively, students should be given the proof in shuffled paper strips, glue them into their workbooks

Introduction

Demonstration [15 minutes]

WM: Fluency, Communicating

As a class, brainstorm on the whiteboard different methods of drawing circles (e.g. using compass-and-straight-edge, string, chalk, tracing around circular objects).

Key Question:

"Suppose I drew three random points on the whiteboard. Is there an easy way to draw a circle that touches all three points?"

The teacher invites a student to the whiteboard to do a joint-construction, and instructs the remainder of the class to follow along in their workbooks.

The student follows the teacher's instructions for drawing a circle using three non-collinear points in order to demonstrate the process to the class (see Steps).

Can give students opportunity to practise by instructing them to randomly place points on a page, then to select any three to create a circle, repeating a few times with different points
Instruct students to draw their circles large so they can hold them up for the teacher to see (give students time before doing so)

Steps

Refer to image example

First, draw three points anywhere on the whiteboard/page, as long as they are non-collinear (A, B, C)
Connect the points together with line segments (AB, BC, AC)
Mark the midpoints of each line segment (E, F, G)
From each midpoint, draw a perpendicular line (can extend it as far as you want) (lines ED, FD, GD)
The three perpendicular lines should all intersect at one point, label this as the centre (D)
Using your compass-and-straight-edge, draw a circle using the centre (D) and the distance from the centre to any of the three starting points as the radius

(1) Example of circle construction given 3 non-collinear points; (2) circular garden

Bonus Task

Split the class into groups of 5 or 6 students and head outside (school oval if possible)

Each group is given 3 ground markers, string and tape measures. Place the 3 markers far apart (maximum a few metres), and use the string and tape measures to locate the circumcentre

Example scenario: creating a circular garden, want to trace out an accurate perimeter

Students should document their work with photos/video and upload them to Google Classroom (AOL)

ALTERNATE BODY ACTIVITY: For smaller classes (15 or less)

WM: Fluency, Reasoning, Communicating

Place whiteboards on top of tables in the middle of the classroom and invite students to stand around them (each student should have a marker).

Progress through the following questions (all students should write on the whiteboard to respond when appropriate) (AFL):

Starting with one point, can we draw a circle that touches this point
Can we draw a circle that touches two given points?

Split the class into three or four table groups, and guide them through the process of constructing a circle from three non-collinear points.

Example diagrams made using string, coloured markers and butchers paper

Recommended: provide metre-long rulers for constructing lines
After students have drawn their diagrams, they should take a photo to upload to Google Classroom (AOL)

Constructing Circles v4.pptx

Prepare butchers paper, tape measures or metre-long rulers, protractors, coloured markers, string and objects to serve as point markers.

Students should use the rulers + protractors to make their measurements
String should be used in lieu of compasses (emulating older circle construction techniques)
The point marker objects should be small and coloured so students can differentiate them

Body

Map Activity (ICT) [8 minutes]

WM: Fluency, Reasoning, Communicating

Direct students to open the GeoGebra Map activity on their table laptop. The teacher describes the scenario to the students:

Three students (choose three names from the class) want to meet together somewhere in the school to have lunch, but cannot decide on where to meet. After a few minutes, one student suggested meeting at the point that is of an equal distance to each of them (i.e. the circumcentre), so they would all have to travel the same distance
Task: students locate the central point equidistant to everyone

The teacher provides a demonstration for how to use the GeoGebra tools (point, segment, line, circle) to solve the example problem.

At each table group, students are tasked with choosing three points on the school map (representing each student's location) and finding the circumcentre of those 3 points

After finding the circumcentre, students should use GeoGebra (or a map scale) to calculate the (straight-line) distance each student would have to travel
The teacher should monitor the table groups and help students to reset their activity if necessary

After 10 minutes, each table group shares the work they did (i.e. where was each student, and where should they meet), and are asked additional questions by the teacher (see Questions) (AFL).

Questions

Every student would need to travel the same distance to the circumcentre, what do we call this distance?
Suppose you wanted to invite another student to lunch. Where could this student be located such that they also travel the same distance?
If we wanted our three students to meet at the canteen and travel only 50m, where can these students possibly be?
If all three students were collinear, would it be possible to find a circumcentre and thus construct a circle? Why or why not?

Map Activity

Example: https://www.geogebra.org/geometry/xtcegcgu

Requires use of https://www.mapcustomiser.com, can add markers to the map, screenshot and upload to GeoGebra as an image
Can also print out maps and get students to draw on top of them if technology is unavailable
For class purposes, the school map should be used (with a scale if possible)

Inspired by Tim Brzezinski's 'Where in NYC? (VA)' activity: https://www.geogebra.org/m/C7dutQHh#material/Za9dMJMg

Individual/Pair work [15 minutes]

WM: Reasoning, Communicating, Problem-Solving

Students are given another chord property to prove independently or in pairs:

"when two circles intersect, the line joining their centres bisects their common chord at right angles"

Prompt students by suggesting they draw construction lines from the points where the circles intersect to the centre of each circle
The teacher monitors students' work in order to determine who can provide answers on the whiteboard (AFL)
Students should save a copy of their final proofs for their portfolios (AOL)

Five students are selected by the teacher to go to the whiteboard and write a part of the answer (with teacher guidance). After each student is done, the teacher asks them to explain their part of the proof (AFL). Students are asked to re-explain it to each other at their table groups in order to check if they understand it.

If students understand, then they are tasked with attempting example problems from the whiteboard (PowerPoint) for the remainder of the lesson; otherwise, the teacher should re-explain the proof to students

Example Problems

Problems recreated in GeoGebra from Signpost Mathematics 10 Advanced, 2nd Ed

Conclusion

Exit Task [5 minutes]

WM: Fluency

The teacher displays the following question on the board, then provides each student with an index card to write their answer on:

"Summarise the procedure for drawing a circle given 3 non-collinear points"

Students submit their index card to the teacher before leaving the classroom (AFL).

Follow-up on incorrect responses by sending students a summary of the procedure via Google Classroom so that they can complete the homework task

Homework

Homework Task (ICT)

Students use GeoGebra & Map Customiser to create their own problem and solution. Students upload their work to Google Classroom, either as a video or a photo with a detailed explanation (AOL).

Instructions should be provided for how to use GeoGebra and Map Customiser, either written or by video (made available on Google Classroom)
Students who were born overseas (including EAL/D, international or refugee students) are encouraged to design their problem using maps of their home country and share a story whilst providing their explanation; alternatively, advise students to use maps of their neighbourhood to increase their familiarity with their local surroundings/community

Alternatively, the teacher can provide students a different map and coordinates to plot; students demonstrate how to find the circumcentre.

Link to: Bonus Videos

Lesson Evaluation Questions

Orientation:

Was the end-result of the activity obvious to students?
Did students have a clear strategy for reorganising the jumbled proof?
Did the activity encourage all students to participate (individually or in pairs)?

Introduction:

Which approach was more effective/engaging: whiteboard peer teaching or smaller groups?
Were all students following along/participating?
Was the (alternate) activity set up and explained well?
Did students find the task interesting?

Body:

Did students appreciate the context of the map activity?
Were students comfortable with using GeoGebra?
What alternate task or resources could be provided if GeoGebra is unavailable?
Was the transition from the GeoGebra task to the individual/pair task smooth or disjointed?
Was there enough time to run both tasks?

Conclusion:

How much detail did students include in their responses?
Was it more effective to encourage students to independently answer the exit task or to allow pair/group discussion?

Google Sites

Report abuse