Active Learning in FP1 Aims To identify three topics which would benefit from a new approach because – Students find them hard (Proof by Induction) – Students find them strange to start with (Matrices) The Activities Algebra-Free Proof by Induction Defining Matrices through transformation Competitive Team Graph Sketching – Students lack discipline in them (Graph Sketching) To try out three activities that actively engage the learners Proof by Induction Students struggle with two aspects of this: – the concept – the algebra Let’s separate these aspects and nail the concept without the algebra Three problems with no or minimal algebra… Proof by toppling dominoes (aka Proof by Induction) In fact, we normally reverse steps 1 and 2, which would be no good for the dominoes, but makes for a neater proof… Proof by induction Step 1 Check it is true for the first one Step 2 Show that if the result is true for any individual number, then it is also true for the next one Result It is true for all integers bigger than the first one Proof by toppling dominoes (aka Proof by Induction) Toppling dominoes Step 1 Set up the dominoes so that each domino will knock the next one over Step 2 Knock the first domino Result All the dominoes fall over Proof by induction Step 1 Show that if the result is true for any individual number, then it is also true for the next one Step 2 Check it is true for the first one Result It is true for all integers bigger than the first one Algebra-free proof by induction Prove that every 2n x 2n grid can be covered with Lshaped 3 tiles so that 1 square is left uncovered and this can be in any desired position. Every road in a particular country is one-way. Every pair of cities is connected by exactly 1 direct road. Prove by induction that there exists a city which can be reached from every other city either directly or via at most one other city. Let’s start with the main application Introducing Matrices Most courses introduce matrices as a way of recording information. They begin with addition, subtraction and multiplication by a scalar Multiplication of matrices is often contrived, or worse, simply defined. We define matrices as a way of recording a linear transformation – perhaps referring to the need to be able to program a computer to do it. We define matrix multiplication to be that required for transformations to work (the MEI text does this bit). Initial work involves lots of drawing and sketching Students’ first reaction to matrices is generally ‘What’s the point?’ Reflection in the y-axis We can see that x → -x & y → y. or x’ = -x y’ = y So we can now write these equations as a pair of simultaneous equations as multiples of x and y. x’ = -1x + 0y y’ = 0x + 1y Finally we can summarise the equations coefficients by using matrix notation. Reflection in the x-axis What happens to x and y? Write this as a pair of simultaneous equations. (2,-1) (-2,1) −1 0 0 1 Try some more transformations Reflection in the line y = x Reflection in the line y = -x Enlargement, scale factor 2, centre O Rotation 90°anticlockwise (2,1) (2,1) Now write this information in a matrix A different way of thinking about transformation matrices Look at our general simultaneous equations: x‘ = ax + cy y‘ = bx + dy with matrix: a c b d What happens if we transform the points (1,0) and (0,1)? Where next? Try this worksheet with students in pairs or threes. Still thinking in terms of So if we find the images of the points (1,0) and (0,1), we get the matrix of the transformation straight away. either writing the transformations as linear transformations and using the matrix as a shorthand, or considering what happens to (1, 0) and (0, 1) And then? Introduce multiplication in terms of applying the transformation to a set of points. Move on to combinations of transformations. And finally formalise algebra of matrices. Competitive Team Graph Sketching Students are good at identifying vertical asymptotes and intersections with axes. They tend to have a sloppy approach to completing the rest of the sketch, however carefully we teach them a step-by-step method. Competitive Team Graph Sketching Competitive Graph Sketching requires each team member to have responsibility for one part of the sketch. Rotating the roles will give everyone experience of each requirement. All members of the group are fully involved. Weaker students can be helped to complete their task by stronger ones, but the team will only improve as a whole as each member gets better at his/her task, so that they can all work at the same time. Create mixed ability teams to make the competition fair. The Roles Points: Vertical asymptotes and intersections Above: Behaviour approaching vertical asymptotes from above Below: Behaviour approaching vertical asymptotes from below Infinity: Behaviour tending towards positive and negative infinity The Competition Each member of the team does their bit on paper – aim to look at limits algebraically, but allow weaker students to work numerically. (All students should check on a calculator.) The Points Person draws and labels the axes after he/she has worked out the asympotes, then draws in the asymptotes and marks and labels the intersections. The rest of the team add in their bits, check it works and join it all up. The first team to finish get a small prize.