Linear Algebra II Orientation using ICT Chris Olley

Narrative: The Sequence Plan 1.Find two numbers which satisfy a pair of equations in two unknowns. Amusing mock practical settings. 2.Develop systematic algebraic methods with whole number coefficients and whole number solutions. 3.Represent the same sets of equations graphically and generate the same results. 4.Use this method with any coefficients/solutions but (i) it is only approximate and (ii) it sometimes doesn't work. 4. With more than one variable, solutions can be found with more than one equation. (a) Develop algebraic solutions (b) Develop graphical solutions. 5.Use the algebraic method in all cases and spot when it will not work. 6.Extend the method to non-linear cases.

1. Transformations in a coordinate grid. The relationship between object points and image points. A mapping from an object to an image Reflection – Line of reflection Rotation – Centre of rotation – Angle of rotation (direction = anticlock) Enlargement – Centre of enlargement – Scale factor Translation – Vector Also: Glide reflection Stretch Sheer

2. Develop systematic algebraic methods with whole number coefficients and whole number solutions. View/CAS Solve your sets of equations systematically. Write a report with examples.

Apps/Linear Solver Find pairs of equations with whole number solutions. (Check you agree the solutions). Write them down (with the solutions) Apps/Function Take each pair, rearrange to Y= Press Plot Read the solutions (use + or – to zoom) 3. Represent the same sets of equations graphically and generate the same results. Write a report of what you have found with examples

Pedagogic Task (Orientation) Discourse Pedagogy DS+DS- Reserved Situation/ Composition Components/ Assembly Ostensive Exposition/ Problem Demonstration/ Drill

Orientation What would be needed for you to work in this way with students? What materials needed to be produced or equipment used? How were you organised to engage with the idea? By what mechanism did the activity orient the learner (you?) to the mathematics

The Session 1.Planning (Total = 1hr) a.Sketch out a local map b.Design a pedagogic task with maximum 5 mins teacher input and minimum 2 mins participant activity. 2.Teaching (70 mins) a.Timekeeper ( 2 = 7) b.Videographer (good sound + easy sharing) 3.Reviewing (Immediately after write notes on...) a.Practical issues (could they be heard, were the instructions clear, did they engage with you,...) b.Maths issues (was the maths accurate, did this deepen your understanding of the idea) c.Pedagogic task (identify the nature of task(s) from the grid) Discourse Pedagogy DS+DS- Reserved Situation/ Composition Components/ Assembly Ostensive Exposition/ Problem Demonstration/ Drill

Narrative 1.Find two numbers which satisfy a pair of equations in two unknowns. Amusing mock practical settings. 2.Develop systematic algebraic methods with whole number coefficients and whole number solutions. 3.Represent the same sets of equations graphically and generate the same results. 4.Use this method with any coefficients/solutions but (i) it is only approximate and (ii) it sometimes doesn't work. 5.With more than one variable, solutions can be found with more than one equation. (a) Develop algebraic solutions (b) Develop graphical solutions. 6.Use the algebraic method in all cases and spot when it will not work. 7.Extend the method to non-linear cases. Transformational Geometry 1. Transformations of the plane without coordinates. (a) Parameters for different transformations. (b) The relationship between object and image. (c) Classify combinations of transformations. 2. Transformations in a coordinate grid. The relationship between object points and image points. Matrices and Matrix Arithmetic 1. Matrices 2. Matrix arithmetic. Transformational Geometry (Reprise) 3. Matrix arithmetic transforms object to image coordinates for the transformations identified. 4. Matrices can be found for each transformation. 5. Combinations and Inverses of transformations and their matrices. Equations (Reprise) 6. Matrices and their inverses solve systems of linear equations