Embedded Files
LESSON DETAILS
LESSON DETAILS
Lesson Goals
Lesson Goals
By the end of the lesson, students should be able to:
- identify, label and define a range of circle features from a diagram
- recognise and construct example diagrams of newly introduced circle terminology: secant, subtend, standing on the same arc, angle at the centre or circumference, angle in a segment
- show, with the aid of hand-drawn or dynamic diagrams, that tangents are perpendicular to a radius drawn to the point of contact
Related Topics
Related Topics
Angles
Length
Two-Dimensional Space
Content
Content
Prove and apply angle and chord properties of circles (ACMMG272)
Students:
- identify and name parts of a circle (centre, radius, diameter, circumference, sector, arc, chord, secant, tangent, segment, semicircle)
- use terminology associated with angles in circles, e.g. subtend, standing on the same arc, angle at the centre, angle at the circumference, angle in a segment
- identify the arc on which an angle at the centre or circumference stands
- demonstrate that at any point on a circle there is a unique tangent to the circle, and that this tangent is perpendicular to the radius at the point of contact
Working Mathematically
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)
- construct (using compass-and-straight-edge) and label circle diagrams from worded or verbal descriptions (Fluency, Communication)
Resources
Resources
- Assessment Notification Copies
- Classroom Decorations (see Lesson Setup)
- Poster Activity Handout (A4 Size)
- Whiteboard & Whiteboard Markers
- GeoGebra Resources (Laptops + Projector)
- PowerPoint (Blank)
- Blank A4 Paper
- Exit Slips (Printed)
- Mobile Phones (Cameras)
- Google Classroom
- Spare Stationary: Pencils, Pens, Erasers, Sharpeners, Compasses, Rulers, Protractors
Strategies Used
Strategies Used
- Literacy Activity: Poster Activity + Group Work
- ICT Usage (Terminology Demonstrations + Tangent-and-Radius Property)
- Teacher Demonstration, Monitoring and Questioning
- Student-Created Examples (Share & Explain)
- Group Hands-on Activity (w/ ICT)
- Think-Share-Pair + Class discussion
- Exit Slips
- Open-Ended Homework
LESSON PLAN
LESSON PLAN
Lesson Setup
Lesson Setup
Prior to the lesson, change the classroom around to stimulate students' thoughts towards circles.
- Goal: students should be able to guess what the new topic is without having to be explicit
- E.g. decorate the classroom with circle-related posters & place circular objects around the classroom like coins, buttons, counters
- Optional: For this unit of work, leave space around the classroom walls to display student work, students will also have opportunities to upload photos of their work to Google Classroom
- Optional: Give table groups circle-related nicknames, e.g. Wheel, Button, Pie, Coin, Moon; or allow students to name their table groups something circle-related
For each table group, prepare a laptop with the lesson's GeoGebra files loaded (either offline or online). Print off the Poster Activity Handouts and copies of the Assessment Notification. 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 four (4).
HAND OUT A COPY OF THE ASSESSMENT NOTIFICATION TO EACH STUDENT AS THEY ENTER THE CLASSROOM. READ THROUGH THE ASSESSMENT NOTIFICATION AS A CLASS BEFORE PROCEEDING WITH THE LESSON. FURTHER QUESTIONS SHOULD BE ASKED AFTER CLASS OR POSTED ONTO GOOGLE CLASSROOM.
Link to: Assessment Notification
Link to: Assessment Notification
Orientation
Orientation
Poster Activity - Recap (LIT) [10 minutes]
WM: Understanding, Fluency, Reasoning
The teacher hands out the Poster Activity handout to each table group and provide the following instructions:
- In front of each group is a poster handout with a circle diagram
- The goal of this activity is to work together as a table group to identify and label as many features as they recognise (5 minutes)
- Extension: If students finish quickly, they should be encouraged to design another circle diagram with as many features included as possible
While students are working, monitor group discussion and encourage students to recall what they remember from their prior study of circles.
- Can support students by asking them to redraw certain parts of the diagram isolated from the rest of the poster
After five minutes, ask each table group to share 1-2 features they recognised (AFL).
- Extend students by following-up their responses with further questions about the nature of different circle features (see Questions)
- Instruct students to fill in any missing information on their group worksheet
- Ask students to hold up their worksheet for the teacher to check before moving on
- Students should photograph their poster for their portfolio (AOL)
With EAL/D students in the classroom: encourage students to also note down the names of circle features in their primary language or dialect; use these names with the class when sharing with the class.
Additional question: which terminology were we familiar with, and which were new to us? (evaluate students' confidence in their vocabulary knowledge - remind students of definitions when they are reintroduced in future activities).
Circle Investigation Poster
Circle Investigation Poster
Can be used to demonstrate the following features:
- centre, radius, diameter, circumference, sector, arc, chord, tangent, segment, semicircle
Students can be asked to identify and label parts of the circle in groups, as well as construct their own posters that have these features (if they finish quickly).
The poster could also optionally be used to encourage students to do length and angle measurements, as well as for practising notation.
Created in GeoGebra: https://www.geogebra.org/geometry/d8qss3kq
Can use GeoGebra file to change the diagram and add/remove features
Inspiration: https://i.pinimg.com/originals/7a/7e/11/7a7e11699d5e7469f7a8d440e1390f64.jpg
Can use GeoGebra file to change the diagram and add/remove features
Inspiration: https://i.pinimg.com/originals/7a/7e/11/7a7e11699d5e7469f7a8d440e1390f64.jpg
Questions
Questions
- Is the diameter a chord?
- Is the circumference an arc?
- Is a semicircle a segment of a circle? If so, major or minor? (or both?)
- What are some possible definitions of a circle?
- Note: could include points a fixed distance away, or involve Pythagoras Theorem, or a single-sided shape, etc.
Introduction
Introduction
For definitions, see Vocabulary
New Terminology (LIT + ICT) [20 minutes]
WM: Understanding, Communicating
Goal for the introduction: introducing and providing visual examples of new circle geometry terminology:
- Secant, subtend, standing on the same arc, angle at the centre, angle at the circumference, angle in a segment
The teacher explicitly outlines the goal for the introduction (vocabulary focus), then goes through definitions (blank PowerPoint, type in definitions for each then upload the PowerPoint to Google Classroom) and visual examples (whiteboard drawing) of each new terminology. Students are encouraged to write both into their workbooks, as well as to use their table's laptop to explore the GeoGebra dynamic drawings after each new terminology is introduced.
- Workbook drawings should be labelled and annotated, e.g. "Angle ACB standing on minor arc AB"
- The teacher can spend time during the first few examples checking students remember how to properly draw circles using compass-and-straight-edge, and provide assistance if needed (by demonstrating the procedure to the class) (AFL)
For the last examples (e.g. angle at the centre), the teacher provides a worded definition, then invites a student to the whiteboard to do a joint-construction of the corresponding visual representation (AFL). The class should follow along in their workbooks.
- Students should be advised to complete a list or table of the definitions with accompanying diagrams for their portfolios (AOL)
Questions
Questions
(For the whole class or when monitoring students' work)
- How might we describe the angle in this diagram?
- e.g. angle AOB - angle at the centre, standing on the minor arc AB
- Note: focus is not on students memorising how to interpret diagrams but to understand how the terminology translates visually
- If I were to describe an angle, could you draw an example of what it might look like?
- Note: Multiple valid interpretations can be given by students
- What is the key difference between a secant, a tangent, and an external line?
- Show me how to draw circles using a compass
- Note: May need to reteach students
Body
Body
Group Task [15 minutes]
WM: Understanding, Communication
In their table groups, students are provided a blank A4 piece of paper to draw circle diagrams on, allowing them to explore the tangent-and-radius theorem by drawing tangents to the circle from different external points, then drawing radii from the centre of the circle to the point of contacts.
- The teacher monitors students' constructions and provides guidance if students are unsure about how to proceed (e.g. where they should draw points) (AFL)
- Once their diagrams are complete, students use protractors to measure the angle between the tangent and the radius at the point of contact (to verify the property)
- Students upload a photo/scan of their A4 paper to Google Classroom (with names included) as evidence of their participation (AOL)
- When students are done, they should use the GeoGebra file to explore the tangent-and-radius property
Afterwards, students engage in think-pair-share to propose descriptions for the tangent-and-radius theorem.
- The teacher selects one and types it onto a PowerPoint slide for students to write down into their workbooks before concluding the lesson, e.g. "the angle between where the radius and tangent meet is always 90 degrees"
Tangent-and-Radius Property
Tangent-and-Radius Property
If students have access to laptops/tablets, can give students access here: https://www.geogebra.org/geometry/vzvczz2e
Alternatively, can ask students to construct the diagram themselves using available tools (circle with centre, point, tangent, angle)
Questions
Questions
- Without using our protractors, what do we notice about the angle between the radius and the tangent?
- Use students' responses as a prediction, then allow them to check using protractors to confirm
- Can also confirm using GeoGebra resource
- Does this work for any tangent we draw to the circle?
- This is known as the tangent-and-radius property. How might we describe this property in our own words, without a picture?
Conclusion
Conclusion
Exit Slip [5 minutes]
WM: Fluency, Communicating
Students are provided an exit slip (AFL) to complete before leaving the classroom (see Exit Slip Question Examples).
- Follow-up on the results of the exit slips next lesson if a significant number of students submit incorrect solutions
Exit Slip Question Example
Exit Slip Question Example
Describe how the highlighted angle(s) can be described using the circle terminology we learned today
Example Answers:
- Angles standing on the major arc AB
- Angles standing on the minor arc AB
- Angle at the centre standing on chord AB
- Angle at the circumference standing on chord AB
Example Exit Slip Questions
Homework
Homework
Homework Task (ICT)
Students identify real-world examples of circles, upload photos of them to Google Classroom and comment on visible circle features (e.g. chords, arcs) (AOL).
- Students are encouraged to draw on their photos to highlight key features, as well as to identify angles in the circle
- At the beginning of the next lesson, highlight creative/interesting student responses
- Optional: based on the quality of students' examples, their images should serve as the background to future GeoGebra dynamic drawings as a starting point when introducing new properties
Questions
Questions
- What circle features are visible in this photo?
- If there is an angle in this photo, write a description of the angle using the new terminology we learned this lesson.
Example Submissions
Lesson Evaluation Questions
Lesson Evaluation Questions
Orientation
- How effective was the poster activity in testing students' abilities to recognise circle feature?
- Would students prefer to draw separate diagrams & write definitions into their workbook as an alternate introduction task/homework?
Introduction
- Was the goals of the lesson clearly understood by students?
- Were students able to connect the written definitions with the visual diagrams?
- How well do students respond to being invited to the whiteboard to draw diagrams for the class?
- Were you able to cover all of the new terminology in 20 minutes?
Body
- Did students enjoy using GeoGebra to manipulate dynamic diagrams (as opposed to solely hand-drawing the diagram)?
- Were students able to quickly transition to using GeoGebra, or did they need preparation?
- Did students need additional scaffolding to construct circle diagrams based on verbal or written instructions?
Conclusion
- What gaps in understanding were revealed from the exit slip responses?
- Would students prefer instead to summarise/share what they learned in group discussions?
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