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transformation
a + 2b + 3c = 10
3a + b + 2c = 18
2a + 3b + c = 14
two lengths: 10cm and 8cm
cut a length off one so that it is the mean of the
other two
[find two solutions]
y = x2 – 10x + 24
translate this quadratic (parabola) 5 to the left
what is the new equation?
David Wells
what are a, b and c?
use the diagram to show that
(a – b)2 = a2 + b2 – 2ab
b
a
transformation
cut these into two congruent
(identical) shapes with one
cut – which is connected,
dot to dot
show how to transform a regular pentagon into
an isosceles triangle (of equal area)
AB and CD are parallel chords, 6cm apart
C
10cm
prove that a number plus its reciprocal is
always greater than or equal to 2
D
x +
x –
A
14cm
B
1
x
?
how far is the line AB
away from the centre?
1
x
generalising
draw four shapes with an area of 10 squares
and a perimeter of 14
a=
3
4
b=
1
7
find: ab + a + b
find fractions for ‘a’ and ‘b’ so that ab + a + b = 1
what do they have in common?
can you find a general rule?
a triangle’s angles are divided in the ratio of 3
consecutive integers
four consecutive numbers are multiples of
2, 3, 4 and 5 (in this order)
what always happens?
what could they be?
in general?
generalising
(9n – 8) –
(7n – 25)
is always a multiple of 13
what are the missing numbers?
22 + 21 – 20 =
in general?
23 + 22 – 21 =
24 + 23 – 22 =
25 + 24 – 23 =
a, b, c and d are any four consecutive numbers
what is (a2 + d2) – (b2 + c2)?
find digits A, B, C and D, all different
so that AB + CD = DC + BA
e.g. 97 + 24 = 42 + 79
a general rule?
reversing the question
five numbers
have a mean = 4
mode = 3
range = 9
what could they be?
a two-digit number is
divided by the sum of
the digits
what is the smallest
result?
[find two solutions]
AB
= smallest?
A+B
1
1
1
+
=
a
b
6
x
y
4
0
2
7
12
14
0
5
35
90
[find five solutions]
what’s the rule?
reversing the question
n=3
726 = 462 + 264
a 3-digit number + the
reverse of it = 726
n=4
find two other solutions
draw matchstick patterns for an nth term rule:
6n
766 = PQR + RQP
find three solutions
6n + 2
6n + 3
6n + 5
example
5
10n + 5
2
2n – 1
4n – 2
4n + 2
10n – 5
8n + 20
18n – 36
12n – 24
12n + 30
two answers
see Michael Fenton’s Desmos tasks
2n + 1
find a quadratic that passes through (6, 0) and
(0, 6) with (4, – 2) as a lowest point (vertex)
find a quadratic that passes through (– 2, 0) and
(8, 10) with (2, – 8) as a lowest point (vertex)
questions on boring topics
find the volume and surface areas of
the two cuboids:
2×2×6
1½ × 4 × 4
1.
× 1.
2.
× 1.
= 2. 1
1.
× 1.
= 2. 1
1.
use the digits 1 to 6, once only
.
+
.
to make:
• largest possible
• smallest possible
• 7.05
• 10.02
• 11.91
• 13.44
• 11.82
× 1.
= 2. 1
= 2. 1
×
+
use the digits 2 to 7 (once only) in the circles
multiply to get the products in the table
add these nine products together
how do you place the six digits to get the
highest possible total?
questions on boring topics
M(8)
use any of the digits: 1 , 2 , 3 , 4, 5 , 6 , 7 , 8
but you can’t use a digit twice (or more) in:
% of
M(13)
M(13)
to get as close as you can to:
(a)
(b)
(c)
(d)
(e)
M(7)
400
700
100
500
300
3 solutions possible
M(7) is a 2-digit multiple of 7, one digit in each box
use the four numbers 1 , – 2 , 5 and – 7
once only in a sum:
(
+
) – (
to make the numbers:
(a) + 1
(b) – 1
(c) + 9
(d) – 9
(e) + 15
(f) – 15
+
) =
ab – ba = 1
ab – ba = 7
5a – 2b = 32
7a – 10b = 36
2a – 2b = 64