Lesson 6

Going Off On Tangents / Tangent Properties (Introduction)

LESSON DETAILS

Lesson Goals

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

use physical construction materials to create a circle diagram with tangents from an external point drawn to the circle
develop a proof demonstrating that tangents from an external point to a circle are equal in length using congruency tests
correct a congruency proof demonstrating that the centres and the point of contact of two touching circles are collinear
identify errors in solutions to circle problems and suggest corrections, citing proper reasoning
recall chord, angle and tangent circle properties when given circle diagram examples

Content

Prove and apply tangent and secant properties of circles

Students:

prove the following tangent and secant properties of circles:
- the two tangents drawn to a circle from an external point are equal in length
- when two circles touch, their centres and the point of contact are collinear

Working Mathematically

Students:

apply tangent properties of circles to deduce unknown lengths and angles (Understanding, Fluency, Problem Solving)
construct circle diagrams using string (Fluency, Communication)
identify faults and suggest corrections for proofs or solutions to circle problems (Understanding, Reasoning)

Resources:

Chalk, String
Google Classroom
Faulty Proof (write on whiteboard/worksheet)
A4 paper
Mobile Phones (Camera)
GeoGebra Resources (Laptop & Projector)
Whiteboard & Whiteboard Markers
PowerPoint (see below)
Flashcard Revision Handout
Spare Stationary: Pencils, Pens, Erasers, Sharpeners, Compasses, Rulers, Protractors

Strategies Used

Consolidation/review (revisiting students' past work, chart construction and presentation, literacy activity)
Physical construction (hands-on) + group discussion
Incorrect proofs/working + group discussion/sharing with class, peer-evaluation
Flashcard revision
Student-developed questions (w/ ICT)

LESSON PLAN

Lesson Setup

Prior to the lesson, prepare chalk, string and rulers for each table group. Connect a laptop to the classroom projector and load the lesson's PowerPoint file. Print the Flashcard Revision handout for each table group.

Welcome students into the classroom and instruct them to unpack their workbooks and stationery. Advise students that they will be exiting the classroom soon.

Check if any student has completed a full summary chart of the chord & angle circle properties. If so, find a space on the classroom wall to display it, and ensure a photo of it is uploaded to Google Classroom for students to refer to.

Orientation

Can spend some time looking at questions from the previous lesson's homework task

Revision (ICT) [3 minutes]

Students are reintroduced to their drawings from Lesson 1 (via Google Classroom) and are instructed to redraw the tangent-and-radius property into their workbooks. The teacher informs students that for the remainder of the unit of work, they will be exploring tangent and secant properties.

Triangle Chart (LIT) [7 minutes]

WM: Fluency, Communicating

The teacher provides each table group a blank A4 sheet of paper to construct a simple diagram describing the different triangle types. Working together, each group is given 3 minutes to draw and write descriptions for each triangle.

Afterwards, the teacher selects members from each group to describe to the class something they included in their chart, as well as pose follow-up questions (AFL).

Tangent Perpendicular to Radius

https://www.geogebra.org/geometry/vzvczz2e

Example Questions

What are the two ways in which we can classify triangles?
What is the difference between scalene, isosceles and equilateral triangles?
What is the difference between acute, right and obtuse angle triangles?
Can a triangle have more than one right or obtuse angle? Why or why not?
How many angles in an acute triangle have to be acute?
Can an acute triangle also be a: scalene triangle, isosceles triangle, equilateral triangle?
Can a right triangle also be a: scalene triangle, isosceles triangle, equilateral triangle?
Can an obtuse triangle also be a: scalene triangle, isosceles triangle, equilateral triangle?

Introduction

Chalk Drawing [15 minutes]

WM: Fluency, Reasoning, Understanding

The teacher selects one student from each table group to carry their table’s chalk, string and ruler supplies outside. These students are in charge of returning the equipment after finishing the lesson. The remainder of the class are instructed to leave their bags inside and to bring their workbooks and a pen outside.

The teacher demonstrates how to construct a circle using chalk and string on the ground, then instructs each group to construct their own circles. Once drawn, students are then instructed to draw a point outside of the circle and to draw tangents from that point to the circle.

When this is done, the teacher should ask students if they notice anything abut the length of the tangents, then to encourage them to verify their predictions using their rulers.

For the remainder of the time outside (until 30 minutes into the lesson), groups are tasked with developing a proof for the tangent property. Students are encouraged to continue using their chalk to draw additional construction lines to assist in their proofs, or to otherwise use their workbooks. Before returning to class, students should take a photo of their chalk drawing to upload to Google Classroom (AOL).

After collecting the chalk, string and rulers from each table group, discuss the proof of the tangent property as a class, and ask a student to write their proof on the board (the class should provide feedback) (AFL).

Body

Faulty Proof [15 minutes]

WM: Reasoning, Communicating, Understanding

The teacher demonstrates the touching-circle property using GeoGebra, then displays an incorrect proof of the property for students to individually analyse and identify corrections that should be made.

In their table groups, students discuss the corrections they would make. The teacher selects students to share their reasoning behind their fixes with the class (AFL).

For the remainder of the lesson, students are given incorrect working out (in terms of incorrect reasoning, labelling etc.) for a series of questions to fix independently.

When students are finished, they are instructed to share their fixes with their table partners for peer-evaluation. The teacher should also be monitoring peer-feedback and assessing the quality of their answers (AFL).

Students should save a copy of their work for their portfolio (AOL)

Faulty Proof Example

faulty proof example.pptx

Ask students to deduce what the actual tangent property is, and to make corrections to the proof (students should also provide reasoning behind their corrections)

Conclusion

Flashcard Revision [10 minutes]

WM: Fluency

NOTE: PRINT BOTH SIDES (LONG EDGE)

In table groups of 4, with the flashcards face up (picture facing students), students shuffle the cards around then take 3 cards each.

Note: there is an additional page with the remaining circle properties; this activity can be reattempted at the end of the unit with the complete set of circle properties

Students take turns selecting one of their cards and attempting to recall the property. After making a guess, students should privately check the back of the card to see if they were correct. If successful, they gain a point; otherwise, everyone else at the table has a chance to identify the property.

Fastest student to identify the property can gain the point instead
The game ends when all of the cards have been exhausted
The winner of the game is whoever has the most points

circle geometry flashcards (1).pdf

Homework

Peer developed questions (ICT)

Students create a circle geometry problem for their peers to solve based on their chalk drawings and posts it on Google Classroom (AOL). Students also solve problems based on two other groups' chalk drawings, and reply to their posts with their solutions (AOL).

Problems should be typed or written on paper and photographed

BONUS: Using the problem they have designed, students write a potential solution for their problem, but change one step to be incorrect. Peers are tasked with rechecking the solution, identifying the error, and suggesting a fix.

Lesson Evaluation Questions

Orientation:

Was re-introducing the tangent perpendicular to radius property useful for orientating students for the introduction activity?
Did students engage in discussion when creating their triangle charts?
Were all students participating at their table groups?
Did the follow-up questions prompt deeper discussion?

Introduction:

Did the chalk drawing activity take too long to set-up (incl. going outside)?
What alternative resources could be used (e.g. if it is raining on the day)?
Was the use of chalk to draw circle diagrams engaging for students?
Did the activity spark students' participation in discussion?
After the activity, was the outside area too messy?

Body:

Was correcting other (fictional) students' mistakes more effective than writing the solutions?
Did students work mostly independently or engage in group discussion at their tables?
Was it simple or difficult to determine if students' corrections were correct?

Conclusion:

Did students find the activity engaging and/or challenging?
Were the cards easy to read?
Were students motivated to complete the activity without cheating?

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