SHS1 Additional Mathematics · Semester 1 Week 20: Spatial Sense

Lesson notes

Learning Objectives

Indicator: 1.2.1.LI.2 - Predict the midpoint of a line segment given two points and find the generalisation of the midpoint of a line segment.

By the end of the lesson, learners can:

State the definition of a midpoint and explain why it divides a line segment into two equal parts.
Predict the coordinates of the midpoint of a line segment when given the coordinates of its two endpoints.
Derive the generalised midpoint formula, M = ((x₁ + x₂)/2, (y₁ + y₂)/2), from the idea of finding the average of the x-coordinates and the average of the y-coordinates.
Apply the midpoint formula to solve problems involving horizontal, vertical and diagonal line segments.
Use the midpoint formula in reverse to find one endpoint of a segment when the midpoint and the other endpoint are given.

This lesson builds directly on Week 19 (1.2.1.LI.1) where learners described the properties of lines, including the idea that a midpoint is a point on a segment that divides it into two equal halves. This week, learners move from describing midpoints to actually calculating them using coordinates. In Week 20’s next slot (1.2.1.LI.3), learners will extend this idea to divide a line segment in a given ratio, which relies on understanding the midpoint as the special case of the ratio 1:1. Do not introduce ratio division in this lesson.

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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.2 - Predict the midpoint of a line segment given two points and find the generalisation of the midpoint of a line segment.
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.
Curriculum reference: NaCCA curriculum document, p. 141

Exemplars (from the NaCCA curriculum)

Learning Experience: Learners in pairs will perform class activity and use their information to describe how the midpoint of a line can be obtained from a given line segment and present answers.
Activity 1: Initiate Talk for Learning to discuss how the midpoint of a line can be obtained from a given line segment and present answers.
Example: From line segment AC shown below, AB is halfway from AC; hence, point B is the midpoint.

Figure from the shs1 additional mathematics curriculum, printed page 142

Teaching and Learning Resources:
- SHS Curriculum
- Mathematical set calculators.
Assessment (1.2.1.AS.2). The document marks no depth-of-knowledge level for this indicator.