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