Read this: “ ‘Twas brillig, and the slithy toves
Did gyre and gimbal in the wabe;
All mimsy were the borogoves,
And the mome raths outgabe.”
(from Jabberwocky by Lewis Carroll)
Now answer these questions:
1. What were the slithy toves doing in the wabe?
2. How would you describe the state of the
borogroves?
3. What can you say about the mome raths?
Read this: “ ‘Twas brillig, and the slithy toves
Did gyre and gimbal in the wabe;
All mimsy were the borogoves,
And the mome raths outgabe.
And now answer these questions:
4. Did you need to understand the text in order to
answer questions 1 to 3?
5. Why were the borogroves mimsy?
6. How effective was the mome raths’ strategy?
Find the mean, mode, median and range of the
numbers:
3, 7, 9, 1, 3, 14, 7, 3, 4, 9
1. Answer the question.
2. If I change a 3 for a 4 what happens to the measures?
3. Can you change one number so that the mean changes
but none of the other measures change?
4. Can you change one number so that the median
changes but none of the other measures change?
5. If an extra 3 is added what will change?
6. Can you change one number so that the range is
reduced but none of the other measures change?
7. How can you change just one number so that two of
the measures change?
Question: The monthly charge for a mobile phone is £25.
This includes 300 minutes free each week. After that there is
a charge of 5p per minute. Calculate the cost of using the
phone for 540 minutes, 600 minutes, 310 minutes and 450
minutes for each of the 4 weeks in one month.
1. What other questions could you ask about this situation?
2. What would happen to the bill if only 100 minutes
were used in the third week?
3. What difference is it likely to make to the bill if calls after
6.00 pm are only 3p per minute?
4. Is it good value?
5. What other information would you want to know about
the charges for this phone before you agreed to have it?
6. Can you make the question more realistic?
Question: Tim buys a pack of 12 cans of cola for £4.80.
He sells the cans for 50p each. He sells all of the cans. Work
out his percentage profit.
1. Change some or all of the numbers so that the percentage
profit is still the same.
2. If he can get 24 cans for £9.20 should he buy them?
3. He wants to make at least 10% profit. What should he
do?
4. The price has gone up to £5.20 for 12 cans, what should
he do to maintain his percentage profit.
5. What else could you ask about this situation?
Question: Rebekah has 35 sweets. She shares then in
the ratio 4 : 3 with her brother Daniel. Rebekah keeps the
larger share. How many sweets does she keep?
1. What happens if Rebekah has miscounted and there are
only 33 sweets?
2. What other totals will ‘work’ with the given ratio?
3. What other ratios will ‘work’ with the given total?
4. Change the question so that Rebekah gets 21 sweets
and Daniel gets 15.
5. Change the question so that Rebekah gets 3 more than
Daniel.
6. What if there are 3 children instead?
Question:
In a sequence u1 = 3, u2 = 9 and u3 = 15.
Find u6 and a formula for un.
1. Is 90 a term in the sequence?
2. Are all terms a multiple of 3? How do you know?
3. Can you find other sequences with the same u3 and u6?
Describe their patterns.
4. How could you change the question so that 100 is a
term of the sequence?
5. What would you have to change to get u6 = 40?
6. What difference would it make if u1 = 4?
7. What could the next part of the question be?
Question: Express x 2  8x  3 in the form ( x  a) 2  b
where a and b are integers to be determined. Hence write
2
down the transformation that sends y  x to the graph
y  x 2  8x  3
1. What difference would it make to the answers if
the 8 was changed to a 10?
2. What difference would it make to the answers if
the 3 was changed to a -3?
3. Change the question so that the answer to the
second part is a translation of
  3
 
5 
4. How could you make the question harder?
5. How could you make the question easier?
Question:
Express
x  6 x  y  4 y  3 in the form ( x  a)  ( y  b)  r
2
2
2
Hence write down (a) the centre and (b) the radius of the
circle.
1. Answer the question.
2. What difference will it make to the answer if the 6 is
changed to an 8?
3. What would you have to change to get a radius of 5?
4. How could you change the question so that the
centre changed but the radius stayed the same?
5. What else could you ask?
6. How could you make the question harder?
2
2
Alternatively give one of these:
Question:
Express
2
2
2
(
x

a
)

(
y

b
)

r
in the form
Hence write down (a) the centre and (b) the radius of the circle.
OR
Question:
Express
in the form
Hence write down (a) the centre and (b) the radius of the circle.
OR
Question:
Express
Hence write down (a) the
in the form
and (b) the
of the circle.
Prompts to open up closed questions:
1. What happens when …..?
2. What happens if I change …..?
3. What difference would it make if …..?
4. What would the next part of the question be?
5. What happens next ….?
6. How could you make it true that …..?
7. What could you change so that …..?
8. How could you make the question easier …..?
9. How could you make the question harder …..?
1. A young person’s railcard gives one third off the normal
price. Jenny uses her railcard to buy a ticket. The normal
price of the ticket is £36.45. Work out how much she pays
for the ticket.
2. A text book costs £12.99. Work out the cost of 20 books.
A teacher can afford to buy 12 of the books.
Write 12 out of 20 as a percentage.
3. Andy spends one third of his pocket money on a computer
game and one quarter on a ticket to a football match. Work
out the fraction of his pocket money that he had left.
4. Bronze is made from copper and tin.
The ratio of the weight of copper to the weight of tin is 3 : 1.
(a)
What weight of bronze contains 36 grams of copper?
(b)
Work out the weight of copper and the weight of tin in
120 grams of bronze.
5. (a) Change 10 kilograms to pounds.
(b) Change 7 pints to litres.
6.
8cm
2cm
4cm
10cm
The diagram shows 3 small
rectangles inside a large
rectangle. The large
rectangle is 10cm by 8 cm.
Each of the small
rectangles is 4cm by 2cm.
Work out the area of the
region shaded in the
diagram.
A baker offers a 3 tiered wedding cake.
Each layer is cylindrical. The bottom layer
has diameter 29cm, the middle layer has
diameter 23cm and the top layer has
diameter 17cm.
What other mathematics could be linked to this
question?
Write questions that use as many different
aspects of mathematics as you can think of.
What extra information would you need in order
to answer your questions?
Percentage of male and female
who had drunk alcohol on 5 days
or more in the week prior to
interview.
Mean consumption of
children aged 11 – 15 who
drank in the last week.
What questions could you ask?
Can you write a hard question, a medium
question and an easy question all of which
can be answered from the graph or the
chart above?
What other questions could you ask?
10 cm
30º
What questions could you ask?
x
(3, 10)
x
(1, 4)
What questions could you ask?
y  x  3x  x  5
3
2
What questions could you ask?
Distance
from home
Time
What questions could you ask?
What information would you need in order
to answer each question?
Trapezoidal tables are put together in pairs so that they will seat
6 students.
What questions could you ask?
What information would you need in order
to answer each question?
Type 1
Type 2
x cm
x cm
8cm
13cm
6cm
7cm
Type 3
Type 4
13cm
12cm
11cm
8cm
x cm
x cm
Type 5
8cm
Type 6
15cm
Find the length of the
diagonal of the rectangle.
18
24
Find x and y.
8 cm and 10cm are the lengths of two sides of a right angled
triangle. Find possible triangles and use Pythagoras to justify
that they are indeed right angles triangles.
The hypotenuse of a right angled triangle is 10cm. Find
possible triangles and use Pythagoras to justify that they
are indeed right angles triangles.
One of the sides of a right angled triangle that is adjacent
to the right angle is 10cm. Find possible triangles and use
Pythagoras to justify that they are indeed right angles
triangles.
One side of a right angled triangle is 10cm. Find possible
triangles and use Pythagoras to justify that they are indeed
right angles triangles.
Instead of :
How about:
Is this diagram
possible? Justify
your answer.
Find x and y.
18
18
24
24
Or this one:
Put possible lengths on the
diagram so that the right
angles work. Justify your
decisions. Can you
generalise?
Not to scale.
Creating more thinking problems
All problems should make you think but the thinking problems
are problems that are being focussed on here:
• make you stop and think and keep you thinking.
• explore understanding.
• are usually open with more than one answer.
• are easily accessible but can be extended very naturally into
something quite complex.
• require some problem solving and reasoning skills.
• cannot be solved just by trial and error or rote learned rules
Original Problem:
Plot the points A (1, 1), B (2, 3), C (4, 4) and
D (3, 2). Join them up to create the shape ABCD
and name this shape.
Thinking Problem:
The coordinates of one vertex of a rhombus
are (3, 4). Give possible coordinates for the
other vertices.
Original Problem:
Plot the points A (1, 1), B (2, 3), C (4, 4) and
D (3, 2). Join them up to create the shape ABCD
and name this shape.
Thinking Problem:
The coordinates of one vertex of a rhombus
are (3, 4). Give possible coordinates for the
other vertices and justify that your shape is a
rhombus in as many different ways as
possible.
Original Problem:
Find the volume of this triangular
prism.
State the units of your answer.
8 cm
9cm
6cm
Area of the sloping
face is 48cm²
Thinking Problem:
Find possible dimensions for the prism. Which give the maximum
volume? Does this prism have the largest surface area?
Original Problem:
Find the area of each shape.
2
6
3
8
4
5
5
6
10
Thinking Problem:
Find the area of each shape in as many different ways
as you can and show that each gives the same answer.
2
x
3
3x
x
5
5
x
10
Original Problem:
Construct two box plots from the stem and leaf:
3
4
8
7
3
5
5
7
3
6
8
6
2
4
4
5
7
6
8
8
7
7
1
3
3
3
1
1
0
5
5
5
2
8
9
5
9
6
3
1
4/5 represents 4.5 cm
1
Thinking Problem:
Construct a possible back to back stem and leaf that is
represented by these box plots.
Original Problem:
A driver covers 110 km in 3 hours and then travels
at 70 km h-1 for a further 2 hours. What is his
average speed?
Thinking Problem:
The average speed of a journey is 60 km h-1.
Construct a possible travel graph.
Justify that its average speed is 60 km h-1 and
describe the journey.
Find a possible equation of the line. Justify your answer.
Can you generalise your answer?
x (2, 6)
x (6, 3)
Find a possible equation for the other line. Justify your answer.
Can you generalise your answer?
2y + 3 x = 5
Give a possible equation for the circle. Justify your answer.
y=x