some (quick) ways to (probably) make consolidation
tasks at least a bit more interesting
•
•
•
•
reverse the question
greater generality
seek (exhaust) all possibilities
look at/for alternative methods
solve 3x – 4 = 20
make up some equations
with a solution of x = 8:
make them as complicated
as you can
what integer solutions can
you find to 3x – 4y = 20?
what numbers must ‘n’ be for 3x – n = 20
to have an integer solution?
in trying to solve 3x – 4 = 20, what happens if
you divide everything by 3 first?
what are the
coordinates of the
mid-point between
(2 , 1) and (6, 9)?
which pairs of points have
(4, 5) as a mid-point?
if (4, 5) is one third of the way
along a line what could the two
end-points be?
what other integer points lie on the
line joining (2, 1) to (6, 9)?
can you find two (or more) ways to find the
mid-point between two points?
what is 86 – 79?
find several two, 2-digit subtraction
sums with an answer of 7
what general properties do the digits
in a two, 2-digit subtraction sum,
AB – CD have if the result is 7?
find all the (positive) options for the result of a
subtraction sum
using the digits 6, 7, 8 and 9 (without repeats)
how can you see that the result of the sum 86 – 79
must be 7 by relating the numbers to multiples of 7?
what is the mean
average of 0 , 3 and 15?
give sets of three numbers
with a mean of 6
can the mean be smaller than
the median for a set of three
integers?
find all integer sets of 4 numbers
with a mean of 6 and a range of 6
can you think of (another) way to find the
mean of 138, 142, 135, 148 and 137?
the rectangle is reflected and
ends up at (1, – 2), (5, – 2),
(1, – 4) and (5, – 4) what is
the mirror line’s equation?
what happens to the
coordinates of a shape when
you reflect it in y = x?
reflect the
rectangle in the
line y = x
what happens to the four corner
coordinates when the shape is
reflected in y = x + 1?
can you use a rotation followed by a reflection that
has the same effect as reflecting in y = x?
Simpsons
Simpsons
200
Bart
180
160
Homer
Maggie
height cm
Marge
140
120
100
80
60
Lisa
Abraham
Mr Burns
40
20
0
0
10
20
30
40
50
60
70
80
age years
Thanks to the Shell Centre
a large bag of flour weighs 24kg
it costs £21.50
a sponge cake uses
150g of this flour
what are the questions that these calculations find out?
(a) 24000
150
(b)
24
21.50
(c) 2150
24
(d)
21.50
× 150
24000
play your cards right
the numbers 0 to 9 are arranged randomly
0 1 2
3 4 5
6 7 8
9
0 1 2
3 4 5
6 7 8
9
?
probability of
H:
L:
H:
L:
H:
2
6
L:
4
6
H:
3
5
L:
2
5
H = higher
L = lower
can you suggest
an experiment where the probability is ¾ ?
some subtraction sums with an answer – 5 ?
two coordinate pairs with a gradient of ½ between them ?
a quadrilateral whose diagonals cut at 90o ?
two numbers with a highest common factor of 15 ?
a hexagon with rotational but no line symmetry ?
numbers that might round to 350 ?
numbers with exactly five factors ?
the dimensions of a cuboid with a volume of 24 cm3 ?
a ratio question with an answer £180 and £240?
Egyptian fractions
the Ancient Egyptians only used
unit numerator fractions
they turned other fractions into
sums of two or more fractions,
all with a numerator of 1
(apart from ⅔)
they used fractions with
different denominators
1
1
5
=
+
12 18
36
1
1
9
=
+
16 20
80
Egyptian
fractions
Ancient Egyptians
turned fractions into a
sum of fractions with
unit numerators
can you find other
ways to represent
these two fractions in
this ‘Egyptian’ way?
Egyptian
fractions
one way to do this
is to split up the
numerator into a sum
so that the two numbers
are both factors of the
denominator
so that they cancel…
5
36
4
1
5
=
+
36 36
36
1
1
5
=
+
9 36
36
Egyptian
fractions
sometimes you cannot do this
straight away
so, look at an
equivalent fraction
until you can find two (or more)
numbers that are both factors
of the denominator
which cancel…
2
7
remember that the
denominators must
be different
7
1
8
=
+
28 28
28
1
1
2
=
+
4 28
7
5
12
try to make this out of two unit fractions
5
12
and another way
11
12
Egyptian
fractions
try to make this out of two unit fractions
7
12
and another way
7
12
why does this need three unit fractions?
try to find two ways to do this
Egyptian fractions
try to write these as the sum of two fractions with numerator 1
and different denominators, as the Ancient Egyptians did
(1)
4
9
(8)
7
13
(15)
find two ways for
3
8
(9)
2
5
(16)
find two ways for
3
10
(17)
find three ways to write
as the sum of two
unitary fractions, with
different denominators
1
8
(18)
find three ways to write
as the sum of two
unitary fractions, with
different denominators
1
10
(19)
find four ways to write
as the sum of two
unitary fractions, with
different denominators
2
15
(20)
find three ways to write
as the sum of three
unitary fractions, with
different denominators
4
5
(2)
7
20
(3)
10
21
(10)
2
9
(4)
11
18
(11)
5
14
(5)
11
24
(12)
4
49
(6)
5
9
(7)
2
15
(13)
(14)
6
23
9
50
harder
Egyptian
fractions
2
7
try to write these
as the sum of
two unit fractions
they can all
be done
4
19
8
23
5
21
2
13
answers
1
4
1
5
1
3
1
7
1
6
+
+
+
+
+
1
28
1
95
1
69
1
91
1
14
=
=
=
=
=
2
7
4
19
8
23
2
13
5
21
1
1
2
=
+
6 ??
11
1
1
?
=
+
3 33
11
1
1
3
=
+
? ??
11
1
1
6
=
+
? ??
11
1
1
1
=
+
12 ???
11
Egyptian
fractions
some of the
‘elevenths’
family
why/how do
these work in
this way?
1
1
1
=
+
5 20
?
1
1
1
=
+
8 56
?
1
1
1
=
+
? ??
3
1
1
1
=
+
? ??
10
1
1
1
=
+
?
??
6
Egyptian
fractions
Egyptian
fractions sum to
another Egyptian
fraction
what is a general
rule for these?
1
1
1
=
+
n
n(n – 1)
?
substitute some
numbers for ‘n’
what happens?
try to prove that this
will always work
2
2
2
=
+
n+1
n(n + 1)
n
substitute some odd
numbers for ‘n’
what happens?
2
4
2
=
+
n+2
n(n + 2)
n
substitute some even
numbers for ‘n’
what happens?
what do these last two statements establish?