Embedded Files
LESSON DETAILS
LESSON DETAILS
Lesson Goals
Lesson Goals
By the end of the lesson, students should be able to:
- provide and/or follow verbal instructions for drawing a circle diagram of the Angle in the Alternate Segment property
- verify and prove the Angle in the Alternate Segment property using previous circle properties
- create circle geometry problems involving chord, angle or tangent properties that their peers can solve
Related Topics
Related Topics
Angle + Angle Relationships
Two-Dimensional Space
Properties of Geometrical Figures
Trigonometry
Congruent Triangles
Algebraic Techniques
Content
Content
Prove and apply tangent and secant properties of circles
Students:
- prove the following tangent and secant properties of circles:
- the angle between a tangent and a chord drawn to the point of contact is equal to the angle in the alternate segment
- apply tangent and secant properties of circles to find unknown angles and lengths in diagrams
Working Mathematically
Working Mathematically
Students:
- use dynamic geometry software to investigate tangent and secant properties of circles (Problem Solving, Reasoning)
- 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 circle properties to deduce unknown lengths and angles (Understanding, Fluency, Problem Solving)
Resources
Resources
- Running Dictation Handout
- A4 Paper
- Post-it Notes
- Mobile Phones (Camera)
- GeoGebra Resources (Laptop & Projector)
- Whiteboard & Whiteboard Markers
- Google Classroom
- Self-Evaluation Handout
- Spare Stationary: Pencils, Pens, Erasers, Sharpeners, Compasses, Rulers, Protractors
Strategies Used
Strategies Used
- Running dictation (literacy activity, group task)
- ICT (verifying property)
- Teacher monitoring + suggestions
- Class discussion + peer-teaching (student invited to teach class)
- Problems + student-developed questions (for peers to solve, provide feedback on)
- Self-evaluation
- Student video-recording (problem-solving)
LESSON DETAILS
LESSON DETAILS
Lesson Setup
Lesson Setup
For each table group, prepare a laptop with the lesson's GeoGebra files loaded (either offline or online). Print self-evaluation handouts for each student.
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
Orientation
Running Dictation (LIT) [10 minutes]
WM: Communication, Fluency
In groups (of 3), students practise their drawing and communication skills by constructing an angle in an alternate segment diagram
A copy of the diagram is kept on one side of the classroom, while a blank A4 sheet is kept on the opposite side of the classroom
One student acts as a reader who gives instructions, one as a drawer, and one as a runner (who describe the reader's instruction to the drawer)
- The runner and drawer should not see the diagram during the activity, they may be asked to restart if their final diagram does not match
Running Dictation
Running Dictation
Students should be given the image only. Students who are assigned as the 'reader' are required to consider how they might give instructions to the runner/drawer, e.g.:
- "Draw a point first and label it O, then draw a circle around that point. Afterwards, at the bottom of the circle draw a horizontal tangent. The point where the tangent meets the circle should be labelled A. On the tangent to the right of A, label a point D."
- "Next, on the top left of the circle, label a point C, and draw the chord AC. On the top right of the circle, label a point B, and draw the chords BC and AB. Draw the angle BCA and BAD."
Created in GeoGebra Geometry
After a group has successfully drawn the diagram, the activity should be stopped and the class should discuss what they notice about the diagram (AFL):
- Matching angles
- Chords and segments
- Tangent
Introduction
Introduction
Property (ICT) [15 minutes]
WM: Understanding, Reasoning
Teacher uses the whiteboard & GeoGebra to demonstrate the angle in the alternate segment property to the class.
Students are given time to construct the diagram in their workbooks and to explore the property using GeoGebra. Afterwards, students determine a proof for the property themselves (independently or in small groups).
- The teacher monitors students' discussion and progress (AFL), and provides feedback/suggestions if students are struggling (discussion between peers is encouraged before relying on teacher input)
- E.g. "If there's not enough information, how could we add more information?" (construction lines)
Class discussion is held after students have derived a proof. A pair of students are selected to teach the proof to the class (AFL).
- Support students with joint construction and asking for input from peers
- Students should recheck their proofs before moving on to the next activity
- Students should also save a copy of the proof in their workbooks for their portfolios (AOL)
Angle in the Alternate Segment
Angle in the Alternate Segment
Can be accessed here: https://www.geogebra.org/geometry/f8a7qjta
Note: The teacher can add construction lines to the dynamic drawing when discussing the proof of the angle in the alternate segment property.
Body
Body
Student-Developed Questions (ICT) [20 minutes]
WM: Understanding, Communicating
Students are instructed to design questions for their peers to solve and provide feedback on.
- The teacher should advise students to review their workbook for inspiration
- Students are encouraged to develop more than one question (unless their question is difficult and utilises multiple properties)
- To ease students into the question-designing process, they should start with questions that involve one property, then increase the number of properties as they become more comfortable
- Another strategy for designing questions is to create a diagram with complete information, then to remove pieces of information for others to deduce based on the remaining information
The teacher monitors students' efforts and provides suggestions (e.g. properties to use, types of questions). Students are advised that for homework, they need to upload photos of the questions they have created onto Google Classroom, and to solve 3 of their peers' questions by replying to their photos with solutions.
- Five minutes before the conclusion, select one problem from a student to re-draw on the whiteboard
- Invite a table group to the whiteboard to discuss the problem between themselves before sharing a solution with the class (AFL); the rest of the class should discuss the problem in their table groups while waiting, then compare their proofs afterwards
Conclusion
Conclusion
Self-evaluation [5 minutes]
WM: Fluency
The teacher provides each students with a list of circle geometry properties that they have covered in the last 7 lessons. Students rate their understanding of each property either 1, 2 or 3 (AFL):
- I do not understand or recall ever learning this property
- i remember this property but need further help with understanding this property
- I am confident in my understanding of this property
If there is time before the lesson is over, gather comments from students about the unit of work/their understanding.
Generate a spreadsheet using the results.
- Low total average indicates revisiting circle properties through lesson time or online tasks would be helpful
- Return students' self-evaluation sheets for their portfolios (AOL)
self evaluation questions.docxHomework/Follow-Up
Homework/Follow-Up
HW Task (ICT)
- Students upload photos (or GeoGebra drawings) of the problems they have developed to Google Classroom (AOL).
- Students also select at least three of the problems designed by their peers and record themselves demonstrating how they should be solved (AOL).
- Students are assigned one student to review their solutions and provide feedback before the unit of work is finished (AOL).
Lesson Evaluation Questions
Lesson Evaluation Questions
Orientation:
- Was the activity useful for getting students to practise their communication skills?
- Were students able to effectively use circle geometry terminology?
- Did the activity take too long to complete?
- Was there enough space to do the activity on the whiteboard?
- What alternative resources or locations could be used to do the activity?
Introduction:
- Was GeoGebra useful for demonstrating the angle in the alternate segment property?
- Were students working independently or taking advantage of group discussion?
- Were students engaged in listening to their peers teach the class?
- Regarding the pair of students selected to peer-teach, were they comfortable?
Body:
- Did students prefer creating their own problems or solving pre-made questions?
- Was the activity engaging for students?
- Did students collaborate with each other to create questions?
- Was it necessary to provide scaffolding and/or example questions?
Conclusion:
- Were students honest in their self-assessment?
- Did students recognise the importance of reflecting on their current understanding?
- Are there alternative ways to poll student understanding (e.g. online polling)?
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