Embedded Files
LESSON DETAILS
LESSON DETAILS
Lesson Goals
Lesson Goals
By the end of the lesson, students should be able to:
- recognise and recall circle features they have seen in previous lessons
- verify and prove the Intersecting Chords & Intersecting Secants properties using similarity tests
- solve for unknown pronumeral values by applying the Intersecting Chords & Intersecting Secants properties
- draw circle diagrams when given circle property prompts
- recognise circle properties based on circle diagrams
Related Topics
Related Topics
Angle + Angle Relationships
Two-Dimensional Space
Congruent Triangles
Algebraic Techniques
Similar Triangles
Content
Content
Students:
- prove the following tangent and secant properties of circles:
- the products of the intercepts of two intersecting chords of a circle are equal
- the products of the intercepts of two intersecting secants to a circle from an external point are equal
- 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)
- apply the properties of similar triangles to proofs and numerical exercises involving circles (Fluency, Problem-Solving, Reasoning)
- apply circle properties to deduce unknown lengths and angles (Understanding, Fluency, Problem Solving)
Resources:
Resources:
- A4 Circle Diagram Handouts
- Pyramid Puzzle Handout
- GeoGebra Resources (Laptop & Projector)
- Whiteboard & Whiteboard Markers
- Group-testing Handout
- Spare Stationary: Pencils, Pens, Erasers, Sharpeners, Compasses, Rulers, Protractors
Strategies Used
Strategies Used
- Progressive brainstorm (literacy activity), class discussion
- ICT (verifying property), think-pair-share
- Teacher demonstration
- Puzzle activity, teacher monitoring + student-developed problems
- Group Testing
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 off Pyramid Puzzle handouts for each student, and group-testing handout for each table group.
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
Progressive Brainstorm (LIT) [10 minutes]
WM: Fluency, Problem Solving
Each table group of students are given a circle diagram with some features (excluding secant) on an A4 handout.
Groups are given a minute to identify and label circle features, and calculate missing information by applying different circle properties.
- After one minute has passed, students pass their paper clockwise to the next group
- Students should write down all their working to help other groups complete the task
After the diagrams have been passed around the classroom and returned to the original tables, each group presents their example (AFL) and the class discusses what feature they did not see in any of the examples (secant)
- Focus for the final lessons - secants
Progressive Brainstorm
Progressive Brainstorm
Alternative Orientation Task: students given a revision question involving similar triangles to complete, can consult peers; see Similarity Ratios for ideas
Introduction
Introduction
Demonstration (ICT) [15 minutes]
WM: Reasoning, Understanding, Fluency
Students use workbook drawings and GeoGebra Geometry diagrams to explore two properties (intersecting chords & intersecting secants).
Students are given time to think-pair-share in their groups what strategies they could use to prove the two properties.
The class discusses what strategies could be used, then the teacher demonstrates how to use similar triangles to prove one property (intersecting chords). Students prove the other property using the same technique.
- Teacher monitors each table group and checks if they remember how to use similar triangles (if not, the teacher should provide a recap) (AFL)
- When students are done, the teacher selects a table group to write their proof on the whiteboard and explain their solution to the class (AFL)
- Students should save a copy of the proofs for both properties in their workbooks for their portfolios (AOL)
Body
Body
Pyramid Puzzle [15 minutes]
WM: Fluency, Problem Solving
Students are given a worksheet with secant length problems, arranged in such a way that the missing pronumeral value is the sum of the two pronumerals above (check resource)
Students solve these problems independently or in pairs, then design their own problems for their peers to solve (if time permitting)
- Teacher monitors students' work and provides feedback on their progress (AFL).
- Teacher also assists students in understanding the chord/secant property using diagrams and modelling working out with students
- When students are done, they should check their solutions by applying the property, as well as checking the sum of the two pronumeral values above are equal to the pronumeral value below; afterwards, students can paste their worksheet into their workbooks (AOL)
Pyramid Puzzle Example
Pyramid Puzzle Example
Created in GeoGebra Geometry
Inspiration: https://i.pinimg.com/originals/c2/49/76/c2497672004a096b9b666e09f41a01f9.png
Conclusion
Conclusion
Group Testing [10 minutes]
WM: Fluency, Communicating
For each table group, print off a complete list of circle properties covered in class so far and cut the list into individual cards. Place the cards into a small bowl or hat, and instruct students to draw one at a time. After students read their strip, they should attempt to draw a diagram representing the property and wait for the rest of the table group to identify which property they are drawing.
Students should continue this activity until the end of the lesson.
- Alternatively, this activity could be done as a class: select one student to go to the whiteboard and draw a card; they draw on the whiteboard until someone in the class correctly guesses; the student with the correct guess replaces the drawer, repeat until the class is over (students cannot draw again but can help their peers if requested)
- If struggling, students should invite another peer to the whiteboard to assist them (but not take over)
- BONUS CHALLENGE: combine two cards together into a single diagram
The teacher should inform students about their upcoming end-of-unit mini-project (decide which format they want to present their work in e.g. presentation, video, poster).
group testing list.docxCreated in Word
Numbers indicate which lesson the property was covered in. For future usage, the handout could be laminated and kept in zip-lock bags.
Homework/Follow-Up
Homework/Follow-Up
Students are reminded to work on Part B of their assessment for the unit of work (will be due in a few lessons).
Use the link below to see the assessment notification.
Link to: Assessment
Link to: Assessment
Lesson Evaluation Questions
Lesson Evaluation Questions
Orientation:
- Did the activity prompt group discussion?
- Were the goals of the activity clear to students?
- Did students engage in the classroom discussion?
- Was there enough time to work on each example?
Introduction:
- Was GeoGebra useful for verifying the intersecting chords & secants properties
- Are there alternative resources or strategies that could be used if GeoGebra is unavailable?
- Were students comfortable with teaching the class?
Body:
- Were the instructions/format of the activity easy to understand?
- Was the activity too easy or difficult for students?
- Did students require extra scaffolding/examples?
- Were students motivated to complete the entire worksheet?
Conclusion:
- Was the activity fun and engaging for students?
- Were most students able to participate?
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