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SHS3 Additional Mathematics · Semester 1, Week 5

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.2 - Use linear transformation to determine image and object points.
Suggested placement
Semester 1, Week 5 (Week 5 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.452: 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.452: 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. 452

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 using linear transformation to determine image and object points.
Activity 1: Image of a point under linear transformation Learners in convenient groups discuss and determine the Image of a point under a given Linear transformation.
To find the image of a point under a given linear transformation; 𝑥
- Identify the given point or the position vector, É 𝑦Ê.
𝑎 𝑏
- Identify the mapping represented by the 2×2 matrix É Ê. 𝑐 𝑑
- Multiply the position vector on the left by the mapping to obtain the image of the point. 𝑎 𝑏 𝑥 𝑎 + 𝑏 É Ê É 𝑦Ê = ö ÷. 𝑐 + 𝑑 𝑐 𝑑
Example 1 1 0 Find the image of A(1, 0), B(2, 0) and C(2, 2) under the mapping represented by the matrix É Ê. 2 1
Solution: A(1, 0), B(2, 0) and C(2, 2). 1 0 Mapping É Ê 2 1 Image of A (1, 0);
Worked matrix multiplication mapping points A, B and C to image coordinates.
Worked matrix multiplication mapping points A, B and C to image coordinates. 
Example 2: Find the image of the matrix where the point is (-2, 3) and the matrix 𝐵(𝑥, 𝑦) → (3𝑥 + 5𝑦, 2𝑥 + 𝑦).
A worked matrix-times-object calculation producing an image vector.
A worked matrix-times-object calculation producing an image vector. 
Activity 2: Transformation given a matrix Learners in small groups find the transformation given the Matrix.
1 0 Note: Given the matrix É Ê, the linear transformation 𝑇, represented by the matrix, is 𝑇(𝑥, 𝑦) → 2 1 (𝑎 + 𝑏, 𝑐 + 𝑑)
−1 2 Write down the linear transformation defined by the matrix É Ê 3 −4 −1 2 Let the matrix be 𝑀 = É Ê 3 −4 The linear transformation, 𝑇, defined by 𝑀, is 𝑇:(𝑥, 𝑦) → (−𝑥 + 2𝑦, 3𝑥- 4𝑦).
Activity 3 Geometric transformation Learners in small groups describe geometrically a plane transformation.
1 1 Example 1: Let 𝑀 = É Ê. Describe the linear transformation 𝑇 geometrically. 0 1
Solution: Reading the columns of 𝑇 tells us that 𝑇(1, 0) = (1, 0), 𝑇(1,0) = (1,0), and 𝑇(0,1) = (1,1); the transformation 𝑇 thus turns the unit square into a parallelogram with base 1 and height 1 as shown.
Three coordinate panels showing a square transformed by translation along diagonal guide lines.
Three coordinate panels showing a square transformed by translation along diagonal guide lines. 
1 0 Example 2: Let 𝑀 = É Ê. Describe the linear transformation 𝑇 geometrically. 0 −1
Solution: Reading the columns of 𝑀, we have 𝑇(1,0) = (1,0) and 𝑇(0,1) = (0, −1). So 𝑇 fixes (1,0) and "flips" (0,1) to its negative (0, −1). Thus, vertical directions are "flipped" so that (𝑥, 𝑦) is sent to (𝑥, −𝑦), and 𝑇 is reflection in the x-axis.
Coordinate panels showing a square transformed about the origin to new axes.
Coordinate panels showing a square transformed about the origin to new axes. 
Teaching and Learning Resources:
- SHS Curriculum, Graph boards, mathematical set, ICT tools
Assessment (3.1.2.AS.2). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding; Level 3 Strategic reasoning.
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