Spatial Sense

Experiential Learning, Collaborative Learning, Talk for Learning
Learning Experience: Learners in a mixed-ability group apply the idea of distance between two points and its equation to derive the equation of a circle.
Activity 1: Learners in a mixed-ability group discuss the equation derived through construction with ICT tools like GeoGebra. Learners in groups brainstorm the equation derived. Learners in mixed-ability groups brainstorm on how the equation of the circle can be derived.
Activity 2: Learners in a mixed-ability group discuss how to derive the standard form of the equation of a circle. Learners in a mixed-ability group construct a circle and label the centre as 𝑂(ℎ, 𝑘) and any point 𝑃(𝑥, 𝑦)on the circumference of the circle.
- Learners then apply their understanding of distance between two points to state the equation of distance between 𝑂 and 𝑃, which is the radius of the circle.

- Learners in their groups derive the equation 𝑂 and 𝑃 as 𝑟 = w(𝑥 − ℎ) ) − (𝑦 − 𝑘) )
- Learners in their groups simplify this expression as 𝑟 ) = (𝑥 − ℎ) ) − (𝑦 − 𝑘) )
- Learners compare the equation derived from the distance between 𝑂 and the equation of the circle derived by GeoGebra to establish that the two equations are the same.
- Learners in their mixed-ability groups recognise the equation of a circle in standard form as 𝑟 ) = (𝑥 − ℎ) ) − (𝑦 − 𝑘) ) where (h, k) is the centre and (x, y) is a point on the circumference.
Activity 3: Learners in a mixed-ability group discuss how to deduce the equation of a circle with a centre at the origin.
- Learners in their groups investigate the behaviour of the equation of a circle if the centre is at the origin 𝑂(0,0).

E.g., At (0,0), we have 𝑟 ) = (𝑥 − ℎ) ) − (𝑦 − 𝑘) ) 𝑟 ) = (𝑥 − 0) ) − (𝑦 − 0) ) Therefore, the equation of the circle turns to 𝑟 ) = 𝑥 ) + 𝑦 ) .
Activity 4: Learners in their mixed-ability group model practical problems for other groups to derive the equation of a circle.
Example 1: A goat is tethered by a rope of length 2 meters away from a peg. If the peg is at the point (3, 2). Find the equation to describe the area the goat can go round the peg.
Example 2: Write the standard form of the equation of a circle with centre (2,4) and radius of 10.
Solution: (𝑥 − 2) ) + (𝑦 − 4) ) = 10 )
Activity 5: Learners in their mixed-ability group derive the general equation of a circle. Learners in their groups expand the standard form of the equation of a circle to derive the general equation of a circle. That is, expanding 𝑟 ) = (𝑥 − ℎ) ) − (𝑦 − 𝑘) ) to obtain 𝑥 ) + 𝑦 ) − 2ℎ𝑥 − 2𝑘 + ℎ ) + 𝑘 ) − 𝑟 ) = 0 Hence, the general equation of the circle is 𝑥 ) + 𝑦 ) + 2𝑔 + 2𝑓 + 𝑐 = 0, Where g replaces −ℎ with 𝑓 replaces −𝑘 and 𝑐 = ℎ ) + 𝑘 ) − 𝑟 ) since ℎ, 𝑘 𝑎 𝑟 are numbers.
Example 1: Write the general equation of a circle with a centre (2,4) and a radius of 10.
Solution: 𝑥 ) + 𝑦 ) − 4𝑥 − 8𝑦 − 80 = 0
Activity 6: Learners create and solve practical problems.
Example: Learners establish that the general equation of a circle is 𝑥 ) + 𝑦 ) + 2𝑔 + 2𝑓 + 𝑐 = 0 with centre (−𝑔, −𝑓) and radius 𝑟 ) = 𝑔 ) + 𝑓 ) − 𝑐 Learners in their mixed-ability groups create and solve practical problems for other groups to derive the equation of a circle.
Example 1: The point (4,6) is on a circle whose centre is (1,2). Write a standard equation of the circle.
Solution: 𝑟 = w(4 − 1) ) + (6 − 2) ) 𝑟 = 5
Using 𝑟 = 5 and centre (1,2), the equation of the circle is (𝑥 − 1) ) + (𝑦 − 2) ) = 5 )
Example 2: Find the centre and the radius of the circle with the equation 𝑥 ) + 𝑦 ) − 14𝑥 + 16𝑦 − 12 = 0
Learners apply their knowledge in completing the square or any other method to solve the question.
Solution: Using completing the square method: 𝑥 ) + 𝑦 ) − 14𝑥 + 16𝑦 − 12 = 0 𝑥 ) − 14𝑥 + 𝑦 ) + 16𝑦 − 12 = 0 (𝑥 − 7) ) − 7 ) + (𝑦 + 8) ) − 8 ) − 12 = 0 (𝑥 − 7) ) + (𝑦 + 8) ) = 125 Therefore, the centre is (7, −8), and the radius is 𝑟 ) = 125 = 5√5
Alternative method: 2𝑔 = −14, ∴ 𝑔 = −7 2𝑓 = 16, ∴ 𝑓 = 8 Since the centre of the circle is (−𝑔, −𝑓), the centre of this circle is (7, −8) The radius of the circle is found by 𝑟 ) = 𝑔 ) + 𝑓 ) − 𝑐 𝑟 = 7 ) + (−8) ) + 12 ) 𝑟 = √125
Teaching and Learning Resources:
- Worksheets
- Scientific Calculator
- Technological tools, apps, etc.
- SHS Additional Mathematics Curriculum
Assessment (2.2.1.AS.2). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding.