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Have you
decoded my
message yet?
Warwick
2 / 7 / 2013
David Crawford
Leicester Grammar
School
Using UKMT
materials in the
classroom
• UK JUNIOR MATHEMATICAL CHALLENGE
• THURSDAY 26th APRIL 2012
Organised by the United Kingdom Mathematics Trust
from the School of Mathematics, University of Leeds
RULES AND GUIDELINES (to be read before starting)
1. Do not open the paper until the Invigilator tells you to do
so.
2. Time allowed: 1 hour.
No answers, or personal details, may be entered after the
allowed hour is over.
3. The use of rough paper is allowed; calculators and
measuring instruments are forbidden.
• 2008 JMC Question 14
• 2009 IMC Question 9
• 2010 SMC Question 1
Answers
JMC
½
IMC
20:31
SMC
4
Classroom use?
Classroom use?
Use linked questions to
create an activity
http://www.ukmt-resources.org.uk/
In the diagram PQ = PR = RS. What is the size of
angle x?
Q
R
36°
x
P
J
54°
K
72°
N
S
L
90°
144°
M
108°
Classroom use?
Use questions to create a
mini competition
The Millfield Team Competition
(with thanks to Ceri Fides)
I collect about 15 questions on a particular topic using the testbase software (or you
could even just use a whole IMC or SMC paper) and then print out sets of questions
on coloured paper. Each team gets one set of questions. Each question has three
boxes at the bottom for answers (for this reason I normally remove the multiple
choice option and just look for numerical answers. The questions are each worth
different amounts of points and the team to answer a question first gets double
points. The points for each question and current leader board are displayed
throughout the lesson. The teacher just sits at the desk and students bring questions
up. If they are correct hold onto the slips and if they are incorrect cross out one of
the boxes and hand the slip back. When the teacher gets a chance they can record
the scores on the score board (this is where the colours are invaluable as you can
see which team handed in what and I keep all the slips in a pile so I can see which
came in first).
Question 6 ( 4 points)
Gladys plants 60 gladioli bulbs. When they flower, she notes
that half are yellow; one third of those which are not yellow
are red; and one quarter of those which are neither yellow
nor red are pink. The remainder are white.
What fraction of the gladioli are white?
Qu
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Pts
3
3
3
3
4
4
5
5
5
5
5
6
8
9
Total
A
6
3
3
6
8
8
10
10
12
16
18
104
B
3
3
3
3
4
9
25
C
3
3
6
D
3
6
TEAM
The first team to complete a question will get double points
6
10
18
9
28
• Write down three different one
digit numbers
• Using two of these at a time,
form six different two digit
numbers
• Add up the six two digit numbers
• Divide by the sum of the original
one digit numbers
• And (hopefully) you all get ...
•22
• Intermediate Mathematical Olympiad and Kangaroo
• (IMOK)
• Olympiad Cayley Paper
• Thursday 15th March 2012
All candidates must be in School Year 9 or below (England and Wales), S2 or below
(Scotland), or School Year 10 or below (Northern Ireland).
READ THESE INSTRUCTIONS CAREFULLY BEFORE STARTING
1. Time allowed: 2 hours.
2. The use of calculators, protractors and squared paper is forbidden.
Rulers and compasses may be used.
3. Solutions must be written neatly on A4 paper. Sheets must be STAPLED together in the top left corner
with the Cover Sheet on top.
4. Start each question on a fresh A4 sheet. You may wish to work in rough first, then set out your final
solution with clear explanations and proofs. Do not hand in rough work.
5. Answers must be FULLY SIMPLIFIED, and EXACT. They may contain symbols such as ,fractions, or square
roots, if appropriate, but NOT decimal approximations.
6. Give full written solutions, including mathematical reasons as to why your method is correct.
Just stating an answer, even a correct one, will earn you very few marks; also, incomplete or
poorly presented solutions will not receive full marks.
7. These problems are meant to be challenging! The earlier questions tend to be easier; the last two
questions are the most demanding.
Do not hurry, but spend time working carefully on one question before attempting another.
Try to finish whole questions even if you cannot do many: you will have done well if you hand
in full solutions to two or more questions.
What do these two questions have in common?
•Two different rectangles are placed together, edge to edge, to form a large
rectangle. The length of the perimeter of the large rectangle is ⅔
of the total perimeter of the original two rectangles.
Prove that the final rectangle is in fact a square.
•Teams A, B, C and D competed against each other once. The results
table was as follows.
Team
Win
Draw
Loss
For
Against
A
3
0
0
5
1
B
1
1
1
2
2
C
0
2
1
5
6
D
0
1
2
3
6
(a) Find (with proof) which team won in each of the six matches.
(b) Find (with proof) the scores in each of the six matches.
Answer 1
Both are questions from the Year 9
Olympiad paper from 2009
Answer 2
Both are questions chosen by the Welsh
government for their publication
“Developing higher order mathematical skills”
as exemplars for what KS3 pupils can achieve and
to support preparation for the 2012 GCSE where
applications of mathematics and solving multistep problems will be important
One for you
to try!
• B5 Calum and his friend cycle from A to C,
passing through B.
• During the trip he asks his friend how far they
have cycled.
• His friend replies “one third as far as it is from
here to B”.
• Ten miles later Calum asks him how far they have
to cycle to reach C.
• His friend replies again “one third as far as it is
from here to B”.
• How far from A will Calum have cycled when he
reaches C?
• Solution
• Let the points at which Calum asked each question be P
and Q.
• Let the distances AP and QC be x miles and y miles
respectively.
• Then the distance from P to B is 3x miles and the point
P must lie between A and B.
• Similarly the distance from B to Q is 3y miles.
• Thus .
A____P____________B_______________Q_____C
x
3x
3y
y
• Since they have cycled 10 miles between P and Q, we
know that 3x + 3y = 10, and so
• AC = 4x + 4y = 4/3 (3x + 3y) = 4/3 × 10 = 13⅓ miles .
??
• Another good place to find problems is the
site of our Canadian counterparts
•www.cemc.uwaterloo.ca
Team Maths Challenges
For Sixth Form – Senior Team Maths
Team Maths Challenges
For Sixth Form – Senior Team Maths
Challenge
For Years 8 + 9 – Team Maths Challenge
Team Maths Challenges
For Sixth Form – Senior Team Maths
Challenge
For Years 8 + 9 – Team Maths Challenge
NEW!!! (Resources only)
Primary Team Maths Challenge
Classroom use?
Classroom use?
Absolutely ideal as they are
designed for pupils working
collaboratively
1.
2.
3.
4.
Crossnumber
Relay
Mini-relay
Group round
1. Crossnumber
3. Mini-Relay
B1
In the Mathstown greengrocer’s shop:
9 apples cost the same as 6 bananas, and
4 bananas cost the same as 3 coconuts.
At these same prices, A apples cost the
same as 8 coconuts.
Pass on the value of A.
B2
T is the number that you will receive.
Five numbers (in no particular order) are:
10, 4, 20, 15, T
Their mean is a and their median is b.
Pass on the value of 2a – b.
B3
T is the number that you will receive.
A regular T-sided polygon has all of its
interior diagonals drawn in.
The polygon has D different lengths of
interior diagonal.
Pass on the value of D.
B4
T is the number that you will receive.
Sophie cycles the first 150 metres of her
race in 8T seconds, then cycles the
remaining 250 metres in 12T seconds.
Calculate her average speed for the whole
400 metre race, in metres per second.
Answers
B1
B2
B3
B4
16
11
4
5
Two for you to try !
One from the Year 8/9 Final last
year
One from a Senior Regional
round
Answers
A1
7
A2
36
A3
72
A4
24
C1
C2
C3
C4
12
9
7
15
Head to Head
Head to Head
2
7
12
9
5
42
150 999
5
15
25
19
11
85
301 1999
f(n) = 2n + 1
Head to Head
2
5
10
4
7
28
103
19
7
30
0.5
200
52 903 3.25 40 003
f(n) = n2 + 3
Head to Head
4
6
14
8
15
11
98
128
2
3
7
2
5
11
7
2
f(n) = largest prime factor of n
Head to Head
8
12
16
24
40
6
99
5000
2
3
4
4
6
2
9
70
f(n) = integer part of square root of n
Head to Head
6
12
16
1
3
8
11
20
3
6
7
3
5
5
6
6
Six
Twelve
Sixteen One
Three
Eight Eleven Twenty
f(n) = number of letters in the word n
• Write down a sequence of six
numbers with a constant
difference between the numbers
• Use these six numbers (in order)
to write down two simultaneous
equations
• Ax + By = C
• Dx + Ey = F
• Solve these equations
• And (hopefully) you all get
•x = -1
•y = 2
The end
Thank you for listening
and joining in
Bonus
Starter questions