Good tasks, good questions, good teaching, good learning …. Anne Watson Leeds PGCE Feb 2007
Decimals! 10% of % of 23.23
Teaching context All learners generalise all the time It is the teacher’s role to organise experience It is the learners’ role to make sense of experience
(a)P = (1, -1) (b)P = (-2, -4) (c) P = (-1, -3) (d) P = (0, -2) (e) P = (½, -1½ ) (f) P = (-1½, -3½) (g) P = (0, 0) (h) P= (-2, 2) Taxicab distances Let A =(-2, -1)
Phrases we are not going to use Today we are going to do page 93 … Then they did the exercise … I gave them a worksheet … They practised …
Gradient exercise 1: (4, 3) & (8, 12) (-2, -1) & (-10, 1) (7, 4) & (-4, 8) (8, -7) & (11, -1) (6, -4) & (6, 7) (-5, 2) & (10, 6) (-5, 2) & (-3, -9) (-6, -9) & (-6, -8) (8, 9) & (2, -9) (7, -8) & (-7, 5) (-9, -7) & (1, 4) (-4, -3) & (4, -2) (2, -5) & (-3, -7) (1, 6) & (-1, -3) (-1, 0) & (5, -1) (-3, 5) & (-3, 2)
Gradient exercise 2: (i) (4, 3) & (8, 12)(ii) (-2, -3) & (4, 6) (iii) (5, 6) & (10, 2)(iv) (-3, 4) & (8, -6) (v) (-5, 3) & (2, 3)(vi) (2, 1) & (2, 9) (vii) (p, q) & (r, s)(viii) (0, a) & (a, 0) (ix) (0, 0) & (a, b)
Gradient exercise 3: (4, 3) & (8, 12)(4, 3) & (4, 12) (4, 3) & (7, 12)(4, 3) & (3, 12) (4, 3) & (6, 12)(4, 3) & (2, 12) (4, 3) & (5, 12)(4, 3) & (1, 12)
What do you see?
Use of controlled variation 4 pens plus 5 pencils cost £ pens plus 2 pencils cost £ oranges plus 3 apples cost £ oranges plus 1 apple cost £ stamps plus 5 envelopes cost £ stamps plus 4 envelopes cost £3.60
Controlling variation and using layout to show structure sin 2 x + cos 2 x = 1 2 sin 2 x + 2 cos 2 x= 2 3 sin 2 x + 3 cos 2 x = 3 4 sin 2 x + 4 cos 2 x = 4 e x sin 2 x + e x cos 2 x = e x cosx sin 2 x + cos 3 x = cosx
Giving choice; learners’ examples Multiply each of the terms in the top row by each of the terms in the bottom row in pairs: x – 1x + 1x + 2 x + 3 Add some more options of your own
Answers worth comparing Simplify these: 6/10 18/20 6/8 14/16 Now simplify these: 15/25 45/50 15/2035/40 Compare the answers
Sorting 2x + 13x – 32x – 5 x + 1-x – 5x – 3 3x + 33x – 1-2x + 1 -x + 2x + 2x - 2
Sorting processes Sort into two groups – not necessarily equal in size Describe the two groups Now sort the biggest pile into two groups Describe these two groups Make a new example for the smallest groups Choose one to get rid of which would make the sorting task different
Sorting grids +ve y-intercept -ve y-intercept Goes through origin +ve gradient -ve gradient
Sorting trees What is the x-coefficient? 123
Comparing In what ways are these pairs the same, and in what ways are they different? 4x + 8 and 4(x + 2) Rectangles and parallelograms Which is bigger? 5/6 or 7/9 A 4 centimetre square or 4 square centimetres
Ordering Put these in increasing order: 6√2 4√3 2√8 2√9 9 4√4
Put these in order of …… x √2 e x/2 3 √ x 2 2 x x -2/3 x √2 x 3/2 3 √ x 2 x 2sin x x -2/3
Arguing about … Anne says that when a percentage goes down, the actual number goes down - Is this always, sometimes or never true? John says that when you square a number, the result is always bigger than the number you started with - Is this always, sometimes or never true?
Characterising Which multiples of 3 are also square numbers? Which quadratic curves go through (0,0)?
Needing harder methods Find a number half-way between: 28 and and and and and and.0064
Needing harder methods Find a number half-way between: and
Using numbers as placeholders 1 x 7 1 x 7 1 x 7 … x 7 3 x 14 3 x 21… x 14 … 21
Varying order … 2x – 3 x + 4(5x + 2)/2
Varying order …. adding 1 dividing by 1 subtracting 1 multiplying by 1 substitute n for 1 and find values for n which change the order
… and another Find a quadratic whose roots have a difference of three … find another
Purposeful textbook tasks
Summary of key ideas Exercise design: expectation, surprise, practice Control variation: a lot, a little, what? Interplay of examples and generalisation Visual impact Complexifying Choice Making up examples Comparing answers Sorting Ordering Arguing about … Characterising Leading into harder methods Numbers as placeholders … and another