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

Click here for the MEI

Maths Item of the Month

Disclaimer: This magazine provides links to other Internet sites for the convenience of users. MEI is not responsible for the availability or content of these

external sites, nor does MEI endorse or guarantee the products, services, or information described or offered at these other Internet sites.

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