good tasks, good questions, good teaching, good learning …. anne watson leeds pgce feb 2007
DESCRIPTION
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 experienceTRANSCRIPT
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