Spatial Sense · SHS1 Additional Mathematics

Lesson notes

Learning Objectives

Indicator: 1.2.1.LI.2 - Perform algebraic and graphical operations (addition, subtraction, scalar multiplication) and their geometrical interpretation.

By the end of the lesson, learners can:

Define a vector in two dimensions and represent it in both column form (x/y) and component form (xi + yj), distinguishing it from a scalar quantity.
Add two or more vectors algebraically by adding corresponding components, and interpret the sum as the resultant vector.
Subtract one vector from another algebraically by subtracting corresponding components, and interpret the difference geometrically as the vector joining the heads of the two vectors.
Multiply a vector by a scalar algebraically, and describe the effect on the vector’s magnitude and direction (same direction if scalar is positive, opposite direction if negative).
Perform graphical operations of vector addition (triangle and parallelogram laws) and scalar multiplication on graph paper, and verify that the graphical result matches the algebraic computation.

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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.2 - Demonstrate knowledge and understanding of spatial sense relating to vectors in two dimensions and perform algebraic operations on vectors and their geometrical interpretations. 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 - Perform algebraic and graphical operations (addition, subtraction, scalar multiplication) and their geometrical interpretation.
Suggested placement: Semester 2, Week 4 (Week 24 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. 161

Exemplars (from the NaCCA curriculum)

Collaborative Learning, Talk for Learning Approaches, Project-based Learning, Experiential Learning:
Learners in a mixed-ability group discuss the concept of vectors, perform algebraic manipulations on vectors and make geometrical interpretations.
Activity 1: Learners will work in convenient groups (ability, mixed-ability, mixed gender, etc.) to perform algebraic and graphical operations (addition, subtraction, scalar multiplication) and their geometrical interpretation.
Activity 2: Learners will collaboratively embark on a project on the applicability of vectors in real life and make presentations. Examples:
- Calculate the angle between two vectors;
- Determine a vector normal to a plane;
- Calculate the moment of a force about a point; and
- Calculate the moment of a force about a line.
Teaching and Learning Resources:
- Graph board or paper
- Compasses
- Mathematical tools
- Dynamic tools like GeoGebra
- Video clips
Assessment (1.2.1.AS.2). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 2 Skills of conceptual understanding; Level 3 Strategic reasoning.