Linear Algebra Map Narrative Orientation Chris Olley

Linear Algebra Map (getting your head around the maths) Narrative (organising the story you are going to tell) Orientation (getting into the ideas) We will plan a sequence of lessons, design activities and teach some of it.

Linear Algebra: The Map 1 One minute silent thinking (What is linear algebra? What does it consist of? How does it fit together? How does it fit in?) Share elements with your group (max 3 words before someone else speaks) Groups go round sharing (max 3 word) elements ONLY when you have run out can you use your phone to check Wikipedia or Wolfram Mathworld

Linear Algebra: The Map 2 You must do this entirely on your own. DO NOT look at what anyone else is doing. Guard your work!! Create a structural diagram in which you put together all of the elements of linear algebra. DO NOT do it ‘mind map’ style with the title in the middle and cloud shapes randomly placed! Impose organising principles e.g. – use the dimensions of the page to show the breadth/scope of the topic and depth/complexity of the elements Include precedents and antecedents i.e. what comes before and what comes after the elements in your map.

Linear Algebra: The Map 3 Share your map with your group and explain the organising principles that you have used. Choose one map from your group and rework it onto A1 size paper. – Maximise coherence and structure – Maximise diachronicity (multiple interconnectedness) Stick on the wall and review

Linear Algebra: The Narrative 1.Equations are statements involving unknowns and an equality relationship, where the value(s) of the unknown(s) are fixed by the equality. We need to develop mechanisms for finding these values. 2.Objects can be transformed in position, orientation and size in a plane. We need to develop mechanisms for describing these transformations. These two come together at: 3.Matrix arithmetic can be used to describe transformations and combinations of them and also to find solutions to systems of equations.

Developing the Narrative 1.Equations are statements involving unknowns and an equality relationship, where the value(s) of the unknown(s) are fixed by the equality. We need to develop mechanisms for finding these values. 2.Objects can be transformed in position, orientation and size in a plane. We need to develop mechanisms for describing these transformations. 3.Matrix arithmetic can be used to describe transformations and combinations of them and also to find solutions to systems of equations. In pairs (or three if in a group of 5).... choose one of the narrative strands (1,2 or 3) Construct a more detailed narrative for this strand using the map. What will student develop first then next, until they have covered the whole of the description in the narrative strand. Write this as list of things to be covered. Share your detailed narrative with the full group.

Equations 1. We can solve equations informally posed in the form “I'm thinking of a number...”. 2. We can develop systematic methods for solving equations using do/undo and algebraic manipulation. 3. There are relationships between algebraic solutions and graphical solutions 4. With more than one variable, solutions can be found with more than one equation. (a) Develop algebraic solutions (b) Develop graphical solutions.

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

Map: The Sequence Plan Equations 1. We can solve equations informally posed in the form “I'm thinking of a number...”. 2. We can develop systematic methods for solving equations using do/undo and algebraic manipulation. 3. There are relationships between algebraic solutions and graphical solutions 4. With more than one variable, solutions can be found with more than one equation. (a) Develop algebraic solutions (b) Develop graphical solutions. What is the mathematical content needed to engage with this? Brainstorm in your full group and share elements for your map... Now in pairs/threes... Think of a suitably descriptive but sufficiently snappy, title. Give a more detailed overview of the content. What do you think students will find hard to do here? What will they need particular practice doing? What will is conceptually tricky or counterintuitive? Detail specific technical vocabulary and notations that you will need students to use. Now fill in the first page of the sequence plan document...

Narrative: The Sequence Plan... In our pairs/threes... You have six lessons to work on this. From your map, make a more detailed narrative: What will student develop first then next, until they have covered the whole of the description in the detailed narrative strand. Add detail to your narrative until it is clear what specific mathematics will be dealt with. Take this narrative and divide it into six parts. Give each part a descriptive title (the lesson name). Add detail to your narrative until it is clear what mathematics will be dealt with. Fill in the first three columns of the Sequence Plan table 4. With more than one variable, solutions can be found with more than one equation. (a) Develop algebraic solutions (b) Develop graphical solutions.