SHS1 Additional Mathematics · Semester 2, Week 4
Spatial Sense
Lesson notes
Learning Objectives
Indicator: 1.2.1.LI.1 - Recognise and explain various forms of vectors and apply the knowledge to find unit vectors.
By the end of the lesson, learners can:
- Define a vector and distinguish it from a scalar, giving at least three examples of each from everyday situations.
- Express a given vector in column form, in terms of the unit vectors i and j, and in magnitude-direction form (r, θ).
- Calculate the magnitude of a vector given in column form or in terms of i and j.
- Identify parallel vectors from a given set of vectors by expressing one as a scalar multiple of another.
- Find the unit vector in the direction of a given vector by dividing the vector by its magnitude.
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Sign in with phone numberCurriculum 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.1 - Recognise and explain various forms of vectors and apply the knowledge to find unit vectors.
- 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.
- 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.157: 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
- exemplars - p.160: a stacked fraction, matrix, vector or other two-dimensional construct is flattened by the text layer; all visible parts require comparison with the rendered page
- Curriculum reference
- NaCCA curriculum document, p. 157
Exemplars (from the NaCCA curriculum)
Learning Experience: Learners in a mixed-ability group conceptualise the idea of vectors and, perform algebraic manipulations on vectors, and interpret it geometrically. Activity 1: Using Think-Pair-Share, learners recollect the definition of vector, vector notation, types and examples of vectors. Activity 2: - Use collaborative learning to express vectors in a Cartesian plane, express vectors in magnitude and direction form, and calculate the magnitude of a vector. «««««⃗ and «««««⃗ - Learners in groups express vectors. 𝐴 𝐵 from 𝐴(2,4) and 𝐵(−3,7) as column (𝑥 𝑦 ), or 𝑥 L + 𝑦 L . «««««⃗ = (2 4 ) and similarly 𝑂 «««««⃗ = (−3 7 ). Example: 𝐴(2,4) will become 𝑂 - Learners in groups brainstorm to express a vector in magnitude and direction form. Example: Aku's house is 40 metres away from her school, and the bearing is 150°. Write the location of Aku's house in vector form. Answer: (40𝑚, 150°) - Learners in groups estimate distances of their sitting positions and the directions in relation to their teacher's table and write in vector forms. Example: - Esi's position is (5𝑚, 060°) - Edem's position is (8𝑚, 120°) etc. Activity 3: In collaborative groups, Learners identify parallel vectors from given vectors and make generalisations of parallel vectors. Example 1: Identify parallel vectors from the following given vectors 𝑎 = (4 3 ), 𝑏 = (6 4 ), 𝑐 = (2 − 3 ), 𝑑 = (8 − 12 ),𝑒 = (3 2 ) Expected Responses: 𝑑 = 4𝑐 and 𝑏 = 2𝑒. Therefore, vectors 𝑑 and 𝑐 are parallel, and vectors 𝑏 is parallel to vector 𝑑 Generalisation: Two vectors are parallel when one vector can be expressed as a scalar multiple of the other. Learning Experience: Through collaborative learning and activities, learners determine the resultant vectors using the polygon laws of addition. Activity 1: - In collaborative groups, learners discuss triangle laws of addition. - The group leader initiates Talk for Learning on the triangle law of addition, and Learners in groups represent the triangle law of addition with a diagram. Addition of vectors
- Learners in groups apply the triangle law of addition to subtraction of vectors and represent it with a diagram. Subtraction
- Learners represent multiplication of a vector by a scalar diagrammatically. Scalar multiplication
- Learners in collaborative groups perform algebraic manipulations on vectors. Example 1: ! ! Given the 𝑚 = (3 − 4 ) and = (5 0 ), 𝑝 = , (𝑚 + 𝑛) and 𝑞 = ) (𝑚 − 𝑛) Show that |𝑝 + 𝑞| = |𝑝 − 𝑞|. Solution 1 𝑝 = [(3 − 4 ) + (5 0 )] 4 𝑝 = (2 − 1 ) 1 𝑞 = [(3 − 4 ) − (5 0 )] 2 = (−1 − 2 ) (𝑝 + 𝑞) = (1 − 3 ) and (𝑝 − 𝑞) = (3 1 ) |𝑝 + 𝑞| = √10 and |𝑝 − 𝑞| = √10 ∴ |𝑝 + 𝑞| = |𝑝 − 𝑞| Activity 2: - In collaborative groups, learners discuss parallelogram laws of addition and represent it in a diagram. - The group leader initiates discussion on parallelogram and polygon laws of addition and represents it in a diagram.
- Through brainstorming, learners in groups establish that the triangle and parallelogram laws of addition give equal resultants. Teaching and Learning Resources: - Graph board or paper - Compasses - Mathematical tools - Dynamic tools like GeoGebra - Video clips Assessment (1.2.1.AS.1). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 2 Skills of conceptual understanding; Level 3 Strategic reasoning.