Spatial Sense

Learning Experience: Learners in group members recollect how to find the gradient of a line and apply it to find the equation of a line in various forms.
Activity 1: Think, pair and share the concept of the gradient of a line.
Learners work in pairs to recollect and state the formula for finding the gradient of a line. A *A A *A The formula is 𝑚 = ! $ 𝑜 $ ! . 0 ! *0 $ 0 $ *0 !
Learners in pairs solve examples of direct and indirect questions about the gradient of a line and share their solutions with the whole class.
Example 1: Find the gradient of the line passing through the points 𝐴(6, −4) and 𝐵(4,2).
Activity 2: Use collaborative groups to find the Equation of a line passing through two points.
Learners in mixed-ability recollect the equation of a line as 𝑦 = 𝑚 + 𝑐.
Learners in mixed-ability groups, manipulate the formula for finding a gradient to deduce the equation of a line given two points, say 𝐴 and 𝐵.
Learners discover that:
- If 𝐴(𝑥 ! , 𝑦 ! ) and 𝐵(𝑥 ) , 𝑦 ) ), choose any arbitrary point within the line, say 𝑃(𝑥, 𝑦).

- If 𝑃 lies on the same line, then the gradient of the line 𝐴 is equal to the gradient of the line 𝐴, that is 𝑀 RT = 𝑀 RU . A *A A*A $
- If the relationship is expressed algebraically, we will have ! $ = . 0 ! *0 $ 0*0 $
- If 𝑦 − 𝑦 ! is made the subject, the formula becomes A *A 𝑦 − 𝑦 ! = (𝑥 − 𝑥 ! ) ∙ ! $ 0 ! *0 $
- This can be rewritten as 𝑦 − 𝑦 ! = 𝑚(𝑥 − 𝑥 ! )
Example 1: Find the equation of the line that passes through the points (−2,4) and (4,8).
Steps:
- In mixed-ability groups, learners use the formula A *A A *A 1*, ) 𝑚 = ! $ 𝑜 $ ! to find the gradient of the line as 𝑚 = = " . 0 ! *0 $ 0 $ *0 ! ,*(*))
- Group leaders in the various groups lead their members to substitute 𝑚 and either (𝑥 ! , 𝑦 ! ) of (𝑥 ) , 𝑦 ) ) into the formula 𝑦 − 𝑦 ! = 𝑚(𝑥 − 𝑥 ! ) or 𝑦 − 𝑦 ) = 𝑚(𝑥 − 𝑥 ) ). Choosing (−2,4), The substitution results to 2 𝑦 − 4 = (𝑥 + 2) 3
- Use the correct algebraic manipulation and make 𝑦 the subject. 2 16 𝑦 = 𝑥 + 3 3
Activity 3: Use collaborative learning to find the Equation of a line given a gradient and one point.
Learners in mixed-ability group use their previous knowledge of finding the equation of a line to find the equation of a line given a gradient and one point.
Learners in groups deduce that when given a gradient and one point, 𝑚 and (𝑥 ! , 𝑦 ! ) will be substituted in the formula 𝑦 − 𝑦 ! = 𝑚(𝑥 − 𝑥 ! ) or 𝑦 − 𝑦 ) = 𝑚(𝑥 − 𝑥 ) )
𝑦 is made the subject of the equation or equate the expression to 0.
- ! Example 1: Find the equation of a line passing through the point (6, −2) and has a gradient , .
Solution: 𝑦 − 𝑦 ! = 𝑚(𝑥 − 𝑥 ! ) −1 𝑦 + 2 = (𝑥 − 6) 4 4𝑦 + 𝑥 + 2 = 0 or 𝑦 = , − ,
- 0 !
Activity 4: In mixed-ability groups learners discuss equation of a line (Intercept Form).
Learners in the mixed-ability group recollect how to find 𝑥 and 𝑦 intercepts.
Example 1: Find the 𝑥 and 𝑦 intercepts of the equation 4𝑦 + 𝑥 + 2 = 0
Solution: The 𝑦-intercept is found by substituting 𝑥 = 0 −1 ∴ 𝑦 = 2 𝑥-intercept is found by substituting 𝑦 = 0 ∴ 𝑥 = −2
Learners in their groups apply the knowledge of intercepts and the equation of a line, that is 𝑦 = 𝑚 + 𝑐, to deduce the equation of a given gradient and an intercept.
Learners in groups investigate the conditions that make it impossible to use the intercept form to find the equation of a line. Learners discover that the conditions are:
- If the line is parallel to an axis.
- If the line passes through the origin.
Activity 4: General Equation of a line (Standard Form)
Learners in mixed-ability equate various examples of equations of a line to 0 and establish that the general equation of a line is given by 𝑎 + 𝑏 + 𝑐 = 0.
Learners in groups investigate what happens to the general equation of a line when 𝑎 or 𝑏 is zero. Learners discover that:
-
- ; When 𝑎 = 0, 𝑏 + 𝑐 = 0 ∴ 𝑦 = ' , these lines are horizontal and parallel to the 𝑥 − 𝑎

- ;
- When 𝑏 = 0, 𝑎 + 𝑐 = 0 ∴ 𝑥 = & , these lines are vertical and parallel to the 𝑦 − 𝑎

Learners in groups confirm that the equation of a vertical line cannot be written in the form 𝑦 = 𝑚 + 𝑐. The equation 𝑎 + 𝑏 + 𝑐 = 0 is the most general equation for a straight line and can be used where other forms of equation are not suitable.
Teaching and Learning Resources:
- SHS Curriculum
- Mathematical set calculators.
Assessment (1.2.1.AS.4). The document marks these depth-of-knowledge levels for this indicator: Level 3 Strategic reasoning.
Extended Critical thinking and reasoning