Spatial Sense · SHS1 Additional Mathematics

Lesson notes

Learning Objectives

Indicator: 1.2.1.LI.3 - Apply the knowledge of ratio to divide a line segment in a given ratio either internally or externally.

By the end of the lesson, learners can:

Explain the meaning of internal and external division of a line segment in a given ratio using their own words and diagrams.
Use the algebraic formula (x, y) = ((m x₂ + n x₁)/(m + n), (m y₂ + n y₁)/(m + n)) to find the coordinates of a point that divides a line segment internally in the ratio m : n.
Use the algebraic formula (x, y) = ((m x₂ - n x₁)/(m - n), (m y₂ - n y₁)/(m - n)) to find the coordinates of a point that divides a line segment externally in the ratio m : n.
Construct geometrically the point that divides a line segment in a given ratio using a ruler, compass and set square.
Solve word problems involving division of line segments in given ratios, set in practical contexts.

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Curriculum details

Strand: Geometric Reasoning and Measurement (Strand 2)
Sub-strand: Spatial Sense (2.1)
Content standard: 1.2.1.CS.1 - Demonstrate knowledge and understanding of spatial sense in relation to lines and angles between intersecting lines. 1.2.1.LO.1 Describe the properties of lines, including parallel, perpendicular and midpoints. 1.2.1.LO.2 Derive the equation of a line in various forms, find the shortest distance between a point and a line and the perpendicular distance from an external point to a line. 1.2.1.LO.3 Solve problems on acute angles between two intersecting lines. 1.2.1.LO.4 Perform algebraic manipulations of Vectors and resolve vectors using the triangle, parallelogram and polygon laws of addition.
Indicator: 1.2.1.LI.3 - Apply the knowledge of ratio to divide a line segment in a given ratio either internally or externally.
Suggested placement: Semester 1, Week 20 (Week 20 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
Source document note: The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

exemplars - p.142: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page
Curriculum reference: NaCCA curriculum document, p. 142

Exemplars (from the NaCCA curriculum)

Learning Experience: Learners use the Jigsaw method to investigate the division of line segments in a given ratio. Leader to discuss findings with group members.
Activity 1:
- Use collaborative learning to link the concepts of division by a given ratio algebraically to line segments.
- Group leaders to initiate Talk for Learning by discussing division by a ratio generally.
Example 1: Dividing GHC 50 in the ratio 2: 3.
Solution: The group leader to build on what others will say to establish that, suppose you are asked to divide a line segment 𝐴 in the ratio 2: 3. It means that you to find a point 𝐶, such that 𝐴: 𝐶 = 2: 3 𝐴: 𝐶 = 2: 3
RS ) Group members to discover that 𝐴: 𝐶 = 2: 3 is the same as TS = "

A geometric division construction on a triangle, labelled with repeated points and ratio AC to CB.

Activity 2: Use collaborative Learning to discuss dividing a line segment in a given ratio internally geometrically. Group leaders to lead discussions on how to geometrically locate point C.
Example 1: Divide a given line segment 𝐴 in the ratio 2: 3. Solution: Learners to experiment together in groups by drawing a line segment of any length
- Draw any ray 𝐴 inclined at an acute angle to AB, and mark 2 + 3 = 5 equal intervals on 𝐴 with a compass.
- Name the divisions 𝐴 ! , 𝐴 ) , 𝐴 " , 𝐴 , , and 𝐴 . .
- Join 𝐴 . 𝑡 𝐵.
- Through 𝐴 ) draw a line parallel to 𝐴 . 𝐵. RR RS
- Learners in groups establish that ! = RT . R ! R *
Activity 3: The group leader initiates talk on dividing a line segment in a given ratio internally algebraically.
- Leaders of the jigsaw groups to initiate talks on the division of a line segment in a given ratio algebraically. #0 ! $%0 $ #A ! $%A $
- Learners in their groups establish the formula (𝑥, 𝑦) = i #$% , #$% j
Example 1: Find the coordinates of the point that divides the line segment 𝐴(3,1) and 𝐵(3, −5) in the ratio 2: 1 internally.
2(3) + 1(3) 2(−5) + 1(1) (𝑥, 𝑦) = u , v 2+1 2+1 = (3, −3)
Learners in groups solve direct and indirect questions on internal division and present and solve examples individually.
Activity 4: Group leaders initiate talk on the division of a line segment in a given ratio externally.
- Leaders of the jigsaw group to initiate discussion in their various groups.
- Learners in their groups establish the formula (𝑥, 𝑦) = i ! $ , ! $ j. #0 *%0 #A *%A #*% #*%
Example 1: Find the coordinates of the point that divides the line segment 𝐴(3,1) and 𝐵(3, −5) in the ratio 2: 1 externally.
2(3) − 1(3) 2(−5) − 1(1) (𝑥, 𝑦) = u , v 2−1 2−1 = (3, −11) Activity 5: Learners in collaborative groups solve direct and indirect questions.
Teaching and Learning Resources:
- SHS Curriculum
- Mathematical set calculators.
Assessment (1.2.1.AS.3). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding; Level 3 Strategic reasoning.