SHS3 Additional Mathematics · Semester 1, Week 4
Applications of Algebra
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Curriculum details
- Strand
- Modelling with Algebra (Strand 1)
- Sub-strand
- Applications of Algebra (1.2)
- Content standard
- 3.1.2.CS.2 - Demonstrate the ability to use and apply knowledge of matrices in linear transformations and apply a linear transformation to solve problems in context. 3.1.2.LO.1 Construct compound statements and truth tables using connectives. 3.1.2.LO.2 Apply linear transformation in: finding images of points and object points. finding reflections and rotations of points and plane figures.
- Indicator
- 3.1.2.LI.1 - Find the equation of the image of a line under a linear transformation.
- Suggested placement
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Semester 1, Week 4
(Week 4 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
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The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:
- exemplars - p.450: 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.450: 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. 450
Exemplars (from the NaCCA curriculum)
Think-pair-share, Talk for Learning, and Building on what others say. Learning Experience: Learners in pairs discuss the concept of matrices in transformations. Activity 1 Concept of matrices in transformation Learners in convenient groups discuss the following ideas. The image of a point under transformation is generally denoted by 𝐴 ! , expressed as 𝐴 → 𝐴 ! . The transformation 𝑇 can also be represented by using the functional notation 𝑇(𝐴) = 𝐴 ! . Given the linear transformation, 𝑥 ! = 𝑎 + 𝑏 and 𝑦 ! = 𝑐 + 𝑑, the matrix of the linear transformation is expressed as: 𝑎 𝑏 𝑀 = É Ê 𝑐 𝑑 Example 1 Find the matrix of the linear transformation, x 1 = 3x + y and y 1 = 2x - 3y Solution 𝑥 ! = 3𝑥 + 𝑦 𝑦 ! = 2𝑥- 3𝑦 𝑥 ! 3 1 𝑥 ö ! ÷ = É ÊÉ Ê 2 −3 𝑦 𝑦 3 1 The matrix transformation is 𝑀 = É Ê 2 −3 Activity 2: Equation of image of a line under a given transformation. Learners in small groups discuss and determine the equation of the image of a line under a given transformation. Linear Transformation of a Line: To find the equation of the image of a line under a translation, reflection, rotation, or enlargement: - Find the coordinates of any two points on the line. - Find the coordinates of the images of the two points from step 1. - Use the two image points from step 2 to find the slope of the image line. - Use one of the image points, the slope from step 3, and the equation of a line to find the equation of the image line. Example: Find the equation of the image of the line in part (a) under a 90û anticlockwise rotation about the origin. Solution: To find the equation of the image line, we first start by finding two points that lie in the original line. Let us choose (0,6) and (2,10). Then, we find the image of these two points under the transformation (a 90û anticlockwise rotation). The image point of (0,6) under the rotation is (6,0), and the image point of (2,10) under the rotation is (10,2). We can now use the two image points to find the slope of our image line. Then, we can use the slope and one point to find the equation of the image line. Let's use (6, 0). The equation of a line is 𝒚 = 𝒎 + 𝒃, where m is the slope and b is the y-intercept 𝑦 = 𝑚 + 𝑏 ! 𝑦 = ) 𝑥 + 𝑏 put in (6,0) and solve for b ! 0 = ) (6) + 𝑏 0 = 3 + 𝑏 𝑏 = 3. 𝟏 𝟏 So, the equation of the image line is 𝒚 = 𝟐 𝒙 + 𝟑 or 𝒚 − 𝟐 𝒙 − 𝟑 = 𝟎 Teaching and Learning Resources: - SHS Curriculum, Graph boards, mathematical set, ICT tools Assessment (3.1.2.AS.1). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding; Level 3 Strategic reasoning.