Good tasks, good questions, good teaching, good learning ….

Anne WatsonLeeds PGCEFeb 2007

Decimals!

10% of 232.3

20% 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)

a

a

a

What do you see?

Use of controlled variation

4 pens plus 5 pencils cost £2.60 4 pens plus 2 pencils cost £2.00

5 oranges plus 3 apples cost £2.36 5 oranges plus 1 apple cost £2.12

8 stamps plus 5 envelopes cost £3.908 stamps plus 4 envelopes cost £3.60

Controlling variation and using layout to show structure

sin2x + cos2x = 1 2 sin2x + 2 cos2x = 2 3 sin2x + 3 cos2x = 3 4 sin2x + 4 cos2x = 4 exsin2x + excos2x = ex cosx sin2x + cos3x = 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 – 1 x + 1 x + 2 x + 3x – 1 x + 1 x + 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 15/20 35/40

Compare the answers

Sorting

2x + 1 3x – 3 2x – 5

x + 1 -x – 5 x – 3

3x + 3 3x – 1 -2x + 1

-x + 2 x + 2 x - 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

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 342.8 and 3.4 38 and 44

-34 and -28 9028 and 9034 .0058 and .0064

Needing harder methods

Find a number half-way between:

41

and 21

83 and

43

52 and

74

ba and y

x

Using numbers as placeholders

1 x 7 1 x 7 1 x 7 … 7 2 7 3 7 4

3 x 7 3 x 14 3 x 21…7 8 7 15 7 22

9 x 14 …21 21

Varying order …

2x – 3 x + 4 (5x + 2)/2

Varying order ….

adding 1 dividing by 1subtracting 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 … 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