mei.org.uk Issue 54 July/August 2016 Curriculum update...
Transcript of mei.org.uk Issue 54 July/August 2016 Curriculum update...
J u l y / A u g u s t 2 0 1 6 m e i . o r g . u k I s s u e 5 4
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Maths Item of the Month
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Curriculum update
GCSE
A summary of general GCSE changes (all subjects) is available on the gov.uk website.
This Ofqual document sets out the re-sit opportunities exam boards must provide for GCSEs (graded A* to G).
We’ve created a summary of the GCSE legacy re-sit information, which you will find in the Useful Links on our Forthcoming Mathematics Curriculum Change web page.
A level
This Ofqual document sets out re-sit opportunities exam boards must provide after the last scheduled sitting of legacy AS and A levels.
We’ve created a summary table of the new A level re-sit information, which you will find in the Useful Links on our Forthcoming Mathematics Curriculum Change web page.
M4 is edited by Sue
Owen, MEI’s Marketing Manager. We’d love your
Conference update
We have a different format of M4
magazine this month as we know that
the end of the term is nigh!
We recently held our annual conference
and wanted to share some of it with
you, especially if you were unable to
attend, so on the next page you’ll find a
few photos from the three-day event,
and an adapted version of the Delegate
Challenge for you to use with students
(or to try yourself!). This was devised by
Carol Knights, MEI’s Extension and
Enrichment Coordinator, and sponsored
by Oxford University Press. The original
activity, targeted at teachers, included
QR codes for participants to scan to
find more information related to the
questions. The QR codes (and
resources) were available on each of
the exhibition stands on the Friday of
the conference.
Overall winner of the Delegate
Challenge was Alaric Stephen from
Hereford Sixth Form College. Alaric
was presented with a prize of an Oxford
University Press goody bag and choice
of mathematics books, also a binary
wall clock. The photograph (top right)
shows Alaric with Carol Knights.
We awarded other prizes: some
delegates spotted a gold label on a
leaflet in their delegate bag, which won
them a prize from that organisation.
We celebrated ten years
of MEI’s Maths Item of
the Month with a
display around the
Chancellors’ Building of
problems and challenges
published each month on
our home page. A list of Maths Items of
the Month categorised by GCSE/A level
topics can be seen at: Maths Items of
the Month Curriculum mapping.
Conference session resources will be
uploaded over the summer to our
conference archive.
#MEIConf2016
@MEIConference
In this issue
Curriculum Update: GCSE
changes, GCSE legacy resits,
new A level resits
July/August focus: Conference
update
KS3-KS5 Teaching Resource:
Summer Challenge 2016
University of Bath 30 June - 2 July
Top to bottom:
Jonny Griffith, our
Conference after-
dinner speaker; Tom
Button, one of our
quizmasters; Sue de
Pomerai, our other
quizmaster and
occasional Ada
Lovelace
impersonator...
MEI Conference 2016
End of term
resource
The final teaching and
learning resource of
the academic year is
a replica of the
delegate challenge
from MEI’s Annual
Conference.
You should find some
questions are suitable
for KS3 students
whilst others will
challenge A level
students. The order of
difficulty is hard to
ascertain since the
mathematical content
of one question might
be judged as simpler
than the content of
another, but the
amount of problem
solving or ‘juggling
with maths’ required
makes it more
challenging.
However, we have
attempted to present
them in approximate
order of difficulty, for
you and your students
to enjoy.
The PowerPoint
version of the
resource can be
downloaded from the
M4 web page -
remove the teacher
notes and answers
before sharing it with
your students! Enjoy! M4 Magazine will return next term, with a September/October edition.
The after-dinner quiz
that took place on
Thursday evening was a
popular event. Hodder
provided the wine with
dinner, and OCR
sponsored the quiz.
Pictured left are Neil
Ogden from OCR with
the winning quiz team
and their prize boxes of
chocolates.
Friday’s exhibition enabled delegates to
find out about the support offered by
various maths-related organisations.
Rob Eastaway and Ben Sparks kindly
provided some magical maths
entertainment for the lunch queue!
Challenge 1
Using only the numerals
1, 3, 5, 7 and 9, plus a
decimal point if required,
find 3 numbers whose
sum is 30.
Challenge 2
How many prime numbers
less than 10 000 have a
digit sum of 2?
Challenge 3
What digit should replace
the question mark?
2
67
8?6
5
4
7
9
Challenge 4
How many different sets of consecutive positive
integers sum to 105?
Challenge 5
Starting with a large cube, and leaving it in situ,
can you use 6 cuts to divide the cube into 27
identical smaller cubes?
If you are allowed to
rearrange the various
slices between cutting,
what’s the least number of
cuts needed?
Challenge 6
What is the largest even
integer that cannot be
written as the sum of 2 odd
composite* positive
integers?
* A whole number with more than 2
factors…i.e. not 1 and not prime.
Challenge 7
Can you use each of the digits 1 to 9, exactly
once each to make a set of 6 prime numbers?
What’s the largest prime
number that can appear in
a set?
Challenge 8
How many factors does 2016 have?
(Including 1 and itself)
Challenge 9
What number does ‘BOB’ represent in this
division where letters take the place of digits?
BOB= .TALKTALKTALKTALK…..
DID
Challenge 10
In a non-special quadrilateral the angle bisectors
are constructed.
If the blue angle is 125°, what’s the yellow angle?
125°
Challenge 11
The radius of
the blue circle
is 20cm and
the radius of
the purple
circle is 12cm.
What’s the radius
of the yellow
circle?
Challenge 12
side length of
small pentagon÷
side length of
large pentagon?
What is:
Challenge 13
A rectangle and equilateral triangle are inscribed
into circles of radii 2cm.
They have the same area as each other.
What are the dimensions of the rectangle?
*Leave answer in surd form
Challenge 14
Two boats are crossing a river from opposite sides.
When they first meet, they are 720 metres from the
near shore.
When they reach the opposite shores, they stop for
10 minutes and cross the river again, but this time
they meet 400 metres from the far shore.
How wide is the river?
Challenge 15
What is the diameter of
the largest semi-circle
which can be inscribed
in a square of side
length 1cm? *Leave answer in surd form
Teacher notes: Summer Challenge 2016The final edition of the academic year is a replica of the delegate
challenge from MEI’s Annual Conference.
You should find some questions are suitable for KS3 students whilst
others will challenge A level students.
The order of difficulty is hard to ascertain since the mathematical
content of one question might be judged as simpler than the content of
another, but the amount of problem solving or ‘juggling with maths’
required makes it more challenging. However, we have attempted to
present them in approximate order of difficulty, for you and your
students to enjoy.
All answers are given in the teacher notes below.
Enjoy!
Teacher notes: Maths required for solutions
Q Maths Q Maths Q Maths
1 Adding only 6 Composite numbers 11 Similar triangles,
simple algebra.
2 Prime numbers 7 Prime numbers 12 Angles in polygons,
trigonometry.
3 Multiplication 8 Factors - prime
factorisation helps
13 Pythagoras, solving
quadratics (although
it’s a quartic)
4 Consecutive
number sums
9 Division, recurring
decimals
14 Rates of change,
solving equations
5 Understanding of
a cube (shape)
10 Angles (triangles,
vertically opposite,
quadrilaterals)
15 Tangents, 1:1:√2
triangle, solving
equations
Teacher notes: Just answers
Q Maths Q Maths Q Maths
1 5.3+5.7+19=30
Many other
solutions
6 38 11 7.2cm
2 3 (2, 11, 101) 7 641 12 Cos 36° (0.809017)
3 0 8 36 13 √(8+√37) by √(8-√37)
4 7 9 242 14 1760m
5 6 10 55° 15 4-2√2 or equivalent
Teacher notes: Hints, ideas and solutionsChallenge 1: What whole numbers can you make?
Challenge 2:
Can you find the set of possible numbers are we should check?
For a digit sum of 2, there must be just a 2 with some zeros or two 1s
with some zeros, so the contenders are: 2 20 200 2000 11 101
110 1001 1010 1100.
Anything that ends in a zero is divisible by 10 and therefore not prime;
removing these leaves: 2 11 101 1001
1001 = 7 x 11 x 13, the other 3 are prime.
Challenge 3: this is “simply” spotting that each row is the row above
multiplied by 8. 9 x 8 = 72, 72 x 8 = 576, 576 x 8 = 4608
Teacher notes: Hints, ideas and solutionsChallenge 4:
One way to approach this is to look at the divisibility of 105.
If it is exactly divisible by an odd number or gives an answer ending
with a ‘.5’ when divided by an even number there is a set with that
amount of numbers in it. (but not 21 numbers with 5 in the middle as
this goes into negative values)
For example: 5 x 21 is 105 which means there is a set of 5 numbers
with 21 as the middle number.
For example: 105÷6 = 17.5 which means there is a set of 6 numbers
with 17 18 as the middle pair of numbers.
This leads to the following 7 sets:(2) 52 53
(3) 34 35 36
(4) 19 20 21 22 23
(5) 15 16 17 18 19 20
(6) 12 13 14 15 16 17 18
(7) 6 7 …….15
(14) 1 2 …….14
Teacher notes: Hints, ideas and solutionsChallenge 5: Consider the central cube. It has 6 sides that need to be
individually cut from the larger cube, therefore no matter how you
rearrange the slices, you’ll still need 6 cuts.
Challenge 6:
Can you write down some odd composite numbers?
9 15 21 25 27 33 35 39 45…
What do you notice about these? They are all multiples of 3 and/or 5.
So if we can find a run of 14 even numbers which can all be written as
the sum of a pair of odd composites, all we’d have to do is add 30 to
one of the composites to get the next ones. (Why would this work?)
40 to 68 are all possible… e.g. 40 = 25+15 so 70 = 55 + 15.
List all the pairs of odd numbers which sum to 38 and you will find that
at least 1 of them is prime each time.
Teacher notes: Hints, ideas and solutionsChallenge 7: A list of prime numbers will really help… see next slide.
We have 9 digits to play with and need 6 prime numbers.
This means that we could potentially have:
A. 3 single digit numbers and 3 double digit numbers
B. 4 single digit numbers, 1 double digit numbers and 1 triple digit number
C. 5 single digit numbers and 1 four digit number.
There are only four prime single digits (2 3 5 7), which rules out option C.
For option B we would need all of the single digits as primes and would
then need to make a 2 digit and a 3 digit from 1 4 6 8 9. Clearly the
numbers would have to end with a 1 and a 9, so possible pairings to check
are:
(41 689) (41 869) (61 489) (61 849) (81 469) (81 649)
(49 681) (49 861) (69 481) (69 841) (89 461) (89 641)
Bold numbers are prime, so 641 is the largest prime that could appear.
Teacher notes: Hints, ideas and solutionsChallenge 7: Prime numbers to 1000
2 3 5 7 11 13 17 19 23 29
31 37 41 43 47 53 59 61 67 71
73 79 83 89 97 101 103 107 109 113
127 131 137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223 227 229
233 239 241 251 257 263 269 271 277 281
283 293 307 311 313 317 331 337 347 349
353 359 367 373 379 383 389 397 401 409
419 421 431 433 439 443 449 457 461 463
467 479 487 491 499 503 509 521 523 541
547 557 563 569 571 577 587 593 599 601
607 613 617 619 631 641 643 647 653 659
661 673 677 683 691 701 709 719 727 733
739 743 751 757 761 769 773 787 797 809
811 821 823 827 829 839 853 857 859 863
877 881 883 887 907 911 919 929 937 941
947 953 967 971 977 983 991 997
Teacher notes: Hints, ideas and solutionsChallenge 8: Find the prime factorisation of 2016
Challenge 9: http://mathforum.org/library/drmath/view/56803.html for a
very similar problem
Challenge 10: In the diagram, 2a+2b+2c+2d=360°, so a+b+c+d=180°
180°-(a+b) =125°
a+b = 55°
c+d =125°
180°-125°=55°=yellow angle
180°-(c+d)
180°-(a+b)
d
d
cc
b
b
aa
125°
Teacher notes: Hints, ideas and solutionsChallenge 11:
Construct lines from the circle centres which are
perpendicular to the sides of the ‘cone’. These form
similar triangles.
If the radius of the yellow circle is ‘r’, what other
lengths can you write?
Adding some other lines in to form
more right angled triangles, what
lengths do you know?
r
r
Teacher notes: Hints, ideas and solutionsChallenge 11:
Working with this diagram you
should see some similar
triangles.
(12+r)/ (12-r) = 32/8 = 4
12+r = 48-4r
5r=36
r= 7.2cm
Teacher notes: Hints, ideas and solutionsChallenge 12:
sin36°=small/x
small= xsin36°
tan36°=large/x
large=xtan36°
small/large = (xsin36°)/(xtan36°)
small/large= cos36° x
large
small
36°
Teacher notes: Hints, ideas and solutionsChallenge 13:
What are the dimensions of the triangle?
What’s the area of the triangle?
3√3
What do we know about the rectangle?
x2+y2=16 (then y2=16-x2)
Also, xy=3√3, so x2y2=27
3
3
21
2
y
x4
Substituting for y2 we get x2(16-x2)=27 x4-16x2+27=0
Use the quadratic formula to obtain x2=(8+√37) and x2=(8-√37)
Since y2=16-x2 when x2=(8+√37), y2=(8-√37) and vice versa, so the
dimensions of the rectangle are √(8+√37) and √(8-√37)
Teacher notes: Hints, ideas and solutionsChallenge 14:
This is an old problem, origin unknown, usually set in yards so that the
answer of 1760 yards is a mile.
One method for solving it is given here:
http://www.braingle.com/palm/teaser.php?op=2&id=19622&comm=1
Teacher notes: Hints, ideas and solutionsChallenge 15:
Think about a semi-circle whose diameter
joins the midpoints of the sides of the square
and touches the top.
It can rotate within the square getting larger
(and smaller at times, depending on which bits
move).
When it reaches 45° it has reached a
maximum so the diagram for the largest semi-
circle becomes as shown.
(Think about what happens if it isn’t at 45°)
Teacher notes: Hints, ideas and solutionsChallenge 15:
The diagonal line is a line of symmetry.
Drawing some other lines on the diagram.
Since the semi-circle just touches the square,
the sides become tangents, so drawing the
normal OT creates an isosceles right-angled
triangle (OVT).
Since we have a diameter and a square on it,
P must also be on the circle.
Add in a perpendicular line through O.
Assume the radius is r.
Find the “side length” of the yellow triangle
(which is also right-angled and isosceles).
Then “side length” + r =1.
VT
O
P
r
P
O
T V
r