problems with patterns and numbers: masters for photocopying

Problems with Patterns an'dNumbers

Masters forPhotocopying

Joint Matriculation BoardShell Centre for Mathematical Education

Teaching Strategic Skills - Publications List

Problems with Patterns and Numbers - the "blue box" materials

• School Pack - Problems with Patterns and Numbers 165 page teachers'book and a pack· of 60 photocopying masters.

• Software Pack - Teaching software and accompanying teaching notes. Thedisc includes SNOOK, PIRATES, the SMILE programs CIRCLE, ROSEand TADPOLES, and four new programs* KAYLES, SWAP, LASER andFIRST . Available for BBC B & 128, Nimbus, Archimedes and Apple IT(* the Apple disc only includes the five original programs).

• Video Pack - A VHS videotape with notes.

The Language of Functions and Graphs - the "red' box~' materials

• School Pack - The Language of Functions and Graphs 240 page teachers'book, a pack of 100 photocopying masters and an additional booklet Traffic:An Approach to Distance-Time Graphs.

• Software"Pack - Teaching software :and accompanying teaching not.es."The;disc includes TRAFFIC, BOTILES, SUNFLOWER and BR~GES.Available for BBC B & 128, Nimbus, Archimedes and PC.

• Video Pack - A VHS videotape with notes.

For current prices and further information please write to: PublicationsDepartment, Shell Centre for Mathematical Education, University of Nottingham,Nottingham NG7 2RD, England.

©Shell Centre for Mathematical Education, University of Nottingham, 1984.

Note: We welcome the duplication of the materials in "this package for useexclusively within the purchasing school or other institution.

Masters for Photocopying

CONTENTS

Examination Questions The Climbing Game 4 (12)*Skeleton Tower 5 (18)Stepping Stones 6 (22)Factors 7 (28)Reverses 8 (34)

Classroom MaterialsUnit A Introductory Problems 10 (45)

Al Organising Problemst 11 (46)A2 Trying Different Approachest 13 (50)A3 Solving a Whole Problemt 15 (54)

Unit B B1 Pond Borders 17 (72)Pupil's Checklist 18 (73)

B2 The First to 100 Game 19 (76)Pupil's Checklist 20 (77)

B3 Sorting 21 (80)Pupil's Checklist 22 (81)

B4 Paper Folding 23 (88)Pupil's Checklist 24 (89)

Unit C C1 LaserWars 25 (96)C2 Kayles 26 (98)C3 Consecutive Sums 27 (100)

A Problem Collection The Painted Cube 28 (106)Score Draws 29 (108)Cupboards 30 (110)Networks 31 (112)Frogs 32 (114)Dots 33 (116)Diagonals 34 (118)The Chessboard 35 (120)

1 Ili'~llllli~imlN15819

A Problem Collection (cont'd) The Spiral GameNimFirst One HomePin Them DownThe "Hot Fat Tune" GameDomino SquareThe Treasure Hunt

Support Material

2. Experiencing Problem Solving: A Treasure Hunt Problem

36 (124)37 (126)38 (128)39 (130)40 (132)41 (134)42 (136)

44 (146)

5. Assessing Problem Solving: Skeleton Tower ProblemSkeleton Tower Marking Scheme6 Unmarked Scripts6 Marked ScriptsNotes on Marked ScriptsMarking Record Form

45 (18)46 (19)47 (163)55 (163)63 (163)64 (164)

* The numbers in brackets refer to the corresponding pages in the Module book.

t The masters for AI, A2 and A3 should be used to form four page booklets, byphotocopying back to back onto paper and folding this paper in half.

2

SpecimenExamination

Questions

3

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THE CLIMBING GAME

Finish

This game is for two players. •A counter is placed .on the dot labelled '"start" and the • •players take it- in turns to slide this counter up the dotted •grid according to the following rules: • •At each turn, the counter can only be moved to an adjacentdot higher than its current position. •Each movement can therefore only take place in one of • •three directions: ••

.",' . • •./ •

• •The first player to slide the counter to the point labelled •'''finish'' wins the game.

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(i) This diagram shows the start of one game, playedbetween Sarah and Paul.Sarah's moves are indicated by solid arrows ( .--)Paul's moves are indicated by dotted arrows (- - -+)It is Sarah's turn. She has two possible moves.Show that from one of these moves Sarah can ensurethat she wins, but from the other Paul can ensure thathe wins.

(ii) If the game is played from the beginning and Sarahhas the first move, then she can always win the gameif she plays correctly.Explain how Sarah should play in order to be sure ofwinning.

Start

©Shell Centre for Mathematical Education, University of Nottingham. 1984.

4 (12)

SKELETON TOWER

(i) How many cubes are needed to build this tower?

(ii) How many cubes are needed to build a tower like this, but 12 cubes high?

(iii) Explain how you worked out your answer to part (ii).

(iv) How would you calculate, the number of cubes needed for a tower n cubeshigh?


5 (18)

STEPPING STONES

A ring of "stepping stones" has 14 stones in it, as shown in the diagram.

start

A girl hops round the ring, stopping to change feet every time she has made 3hops. She notices that when she has been round the ring three times, she hasstopped to change feet on each one of the 14 stones.

(i) The girl now hops round the ring, stopping to change feet every time shehas made 4 hops. Explain why in this case she will not stop on each one ofthe 14 stones no matter how long she continues hopping round the ring.

(ii) The girl stops to change feet every time she has made n hops. For whichvalues of n will she stop on each one of the 14 stones to change feet?

(iii) Find a general rule for the values of n when the ring contains more (or less)than 14 stones.

©Shell Centre for Mathematical Education, University of Nottingham, 1984.6 (22)

FACTORS. .

The number 12 hC;lssix factors: 1,2,3,4, 6 and 12.

Four of these are even (2,4, 6 and 12)

and two are odd (1 and 3).

(i) Find some numbers which have all their factors, except 1, even.

Describe the sequence of numbers that has this property.

(ii) Find some numbers which have exactly half their f~ctors even. Again

describe the sequence of numbers that has this property.

Explain in both part (i) and part (ii) why your result is true.

@Shell Centre for Mathematical Education, Uniyersity of-Nottingham, 1984.7 (28)

REVERSES

Here is a row of numbers: 2, 5, 1, 4, 3.

They are to be put in ascending order by a sequence of moves which reversechosen blocks of numbers, always starting at the beginning of the row.

Example:

2, 5, 1, 4, 3 reversing the first 4 numbers gives 4, 1, 5, 2, 3

4, 1, 5, 2, 3 reversing the first 3 numbers gives 5, 1, 4, ") 3..,5, 1, 4, 2, 3 reversing all 5 numbers gives 3, 2, 4, 1, 5

1, 2, 3, 4, 5

(i) Find a sequence of moves to put the following rows of numbers in ascendingorder

(a) 2, 3, 1(b) 4, 2, 3, 1(c) 7, 2, 6, 5, 4, 3, 1

(ii) Find some rules for the moves which will put any row of numbers inascending order.

©Shell Centre for Mathematical Education, University of Nottingham, "19s4.8 (34)

ClassroomMaterials

9

INTRODUCTORY PROBLEMSThese are different kinds of problem to those you are probably used to. They donot have just one right answer and there are many useful ways to tackle each ofthem. Your teacher is interested in seeing how well you can tackle theseproblems on your. own. The methods you use are as important as the answersyou get, so please write down everything you do, even if you are not sure it isright.

1 TargetOn a calculator you are only allowed to use the keys

You can press them as often as you like.You are asked to find a sequence of key presses that produce a given number inthe display. For example, 6 can be produced by

3x4-3-3=(a) Find a way of producing each of the numbers from 1to 10.You must "clear"your calculator before each new sequence.(b) Find a second way of producing the number 10. Give reasons why one waymight be preferred to the other.

2 Discs

Here are two circular cardboard discs. A number is written on the top of eachdisc. There is another number written on the reverse side of each disc.By tossing the two discs in the air and then adding together the numbers whichland uppermost, I can produce anyone of the following four totals:

11, 12, 16, 17.

(a) Work out what numbers are written on the reverse side of each disc.

(b) Try to find a different solution to this problem.

3 Leagues

A top division has 22 teams. Each team plays all the other teams twice-once athome, and once away. Games are usually played on Saturdays, but sometimeson Wednesdays too. The season lasts about 35 weeks.There is a proposal to expand this top division to 30teams.How many matches in all would be played, and how'~any matches would eachteam play? What would the effect be on the length offue playing season?

II

©Shell Centre for Mathematical Education, Uniyersity of Nottingham, 1984.

10 (45)

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POND BORDERS

Joe works in a garden centre that sells square ponds and paving slabs to surroundthem. The paving slabs used are aliI foot square.The customers tell Joe the dimensions of the pond, and Joe h~s to work out howmany paving slabs they need.* How many slabs are needed in order to surround a pond 115feet by 115feet?

* Find a rule that Joe can use to work out the correct number of slabs for anysquare pond.

* Suppose the garden centre now decides to stock rectangular ponds.Try to find a rule now.

* Some customers want Joe to supply slabs to surround irregular ponds like theone below:-

T4 ft

...1k--3ft~

(This pond needs 18 slabs. Check that you agree).

Try to find a ~ule for finding the number of slabsneeded when you are given the overall dimensions(in this case 3 feet by 4 feet).

Explain why your rule works.


17 (72)

POND BORDERS ... PUPIL'S CHECKLIST

Try some simple cases * Try finding the number of slabs needed for somesmall ponds.

Be systematic * Don't just try ponds at random!

Make a table

Spot patterns

Find a rule

Check your rule

* This should show the number of slabs needed fordifferent ponds. (It may need to be a two-waytable for rectangular and irregular ponds).

* Write down any patterns you find in your table.(Can you explain why they occur?)

* Use these patterns to extend the table.* Check that you were right.

* Either use your patterns, or look at a picture of thesituation to find a rule that applies to any sizepond.

* Test your rule on small and large ponds.* Explain why your rule always works.

©Shell Centre for Mathematical Education, University ofNottingham, 1984.

18 (73)

THE "FIRST TO 100" GAME

This is a game for two players.Players take turns to choose any whole number from 1to 10.They keep a running total of all the chosen numbers.The first player to make this total reach exactly 100wins.

Sample Game:

Player 1's choice Player 2's choice Running Total

10 105 15

8 238 31

2 339 42

9 519 60

8 689 77

9 8610 96

4 100

So Player 1wins!

Play the game a few times with your neighbour.Can you find a winning strategy?

* Try to modify the game in some way, e.g.:- suppose the first to 100 loses and overshooting is not allowed.- suppose you can only choose a number between 5 and 10.


19 (76)

THE "FIRST TO 100" GAME ... PUPIL'S CHECKLIST

Try some simple cases

Be systematic

* Simplify the game in some way:e.g.:- play "First to 20"e.g.:- choose numbers from 1 to 5e.g.:- just play the end of a game.

* Don't just play randomly!* Are there good or bad choices? Why?

Spot patterns *

*

Are there any positions from which you canalways win?Are there other positions from which youcan always reach these winning positions?

Find a rule

Check your rule

* Write down a description of "how to alwayswin th~s game". Explain why you are sure itworks.

* Extend your rule so that it applies to the"First to 100" version.

* Try to beat somebody who is playingaccording to your rule.

* Can you convince them that it always works?

Change the game in some way * Can you adapt your rule for playing a newgame where:- the first to 100 loses, (overshooting is not

allowed)- you can only choose numbers between 5

and 10.


20 (77)

SORTING

50 red and 50 blue counters are placed alternately in a line across the floor:RBRBRBR ... RB

By swapping adjacent counters (see arrows) they have to be sorted into 2groups, with all the reds at one end and all the blu~s at the other:RRR ... RRRBBB ... BBB

* What is the least number of moves needed to do this?How many moves are needed for n red and n blue counters?

* What happens when the counters are placed in different starting formations:For example RRBBRRBBRRBB RRBBor RBBRRBBRRBB RBBR

* What happens when there are red, blue and green counters arrangedRBGRBG ... RBG .What happens with 4 colours?What happens with m colours?

* Invent and explore your own arrangement of counters.Write about your findings.


21 (80)

SORTING ... PUPIL'S CHECKLIST

Try some simpie cases * Try finding the number of moves needed forjust a few counters.

Be systematic * Try swapping counters systematically.

Find a helpful representation * If you are unable to use real counters, canyou find a simple substitute?

* Can you use the simple cases you havealready solved, to generate further cases byadding extra pairs of counters rather thanstarting from the beginning each time?

Ma'ke a table * Make a table to show the ,relationshipbetween the number of counters and thenumber of swaps needed.

Spot patterns * Write about any patterns you find in yourtable.(Can you explain why they occur?)

* Use these patterns to extend the table.* Check that you were right.

Find a rule * Use your patterns, or your representation,to find a rule that applies to any number ofcounters.

Check your rule * Test your rule on small and large numbers ofcounters.

* Try to explain why your rule must alwayswork.


22 (81)

PAPER FOLDING

For this investigation, you will need a scrap of paper.Fold it in half, and then in half again. In both cases you should fold left overright. Open it out and look at the folded creases:

first fold

second fold

now unfold:

1/.-=: l~

/J~ I]

You should see 3 creases - one "up" and two "down".

* Now suppose you were able to fold your paper strip in half, left over right, 6times, and then unfold it completely.Predict the total number of creases you would get.How many of these are "up" creases and how many are "down"?What order would these creases come in?

* Explain hoW you can predict the number and order of creases for 7,8, ... folds.

* Try folding the paper in a different way and explore the patterns in thepositioning and number of your creases. Write about your findings.For example, here is a tricky two-step case ...

Left to right thenBottom to top . . .

tI

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Any patterns?

andagain ...

andunfold ... (gasp!)

I II IIt t, ,

----I I - --•, ,I ,, ,.

• -, --- ----• ,I tt t1---- I I --- -I ,I ,! !


23 (88)

PAPER FOLDING ... PUPIL'S CHECKLIST

Try some simple cases

Be systematic

Find a helpful representation

Make a table

Spot patterns

Find a rule

*

*

:I:

*

*

*

**

*

It is very difficult to fold a normal sheet ofpaper in half 6 times. (J ust think how thick itwill be!), so try just a few folds first.

Make sure that you always fold from left toright - don't turn your paper over inbetween folds!

Invent symbols for "up" and --down"creases.

Use your symbols to record your results.

Make a table to show the relationshipbetween the number of times the paper isfolded and the number of upward anddownward creases, and also the order inwhich these creases occur.

Write about any patter~s you find in yourtable. Can you explain why they occur?

Use these patterns to extend the table.

Check th~lt you were right.

Use your patterns to find rules that apply toany number of folds.

*Check your rule

Extend the problem

Test your rules on large and small numbersof creases.

* Try to explain why they work.

* Invent your own system of folding.

* Try to predict what will harpen, then checkto see if you were right.

©Shell Centre for Mathematical Education, University of Nottingham, 1984_

24 (89)

LASER-WARS

¢- and. represent twotanks armed with laserbeams that annihilateanything which lies to theNorth, South, East or Westof them. They movealternately. At each move atank can move any distanceNorth, South, East or Westbut cannot move across orinto the path of theopponent's laser beam. Aplayer loses when he isunable to move on his turn.

......•..... .. ....... ....... .... ... .......

....... ....... ....... ....... .¢-. ....... ...

( . .............. laser beams)

*

*

Play the game on the board below, using two objects to represent the tanks.Try to find a winning strategy, which works wherever the tanks are placed tostart with.

Now try to change the game in some way ...


25 (96)

KAYLES

This is like an old 14th century game for 2 players, in which a ball is thrown at a

number of wooden pins standing side- by side:

The size of the ball is such that it can knock down either a single pin or two pins

standing next to each- other. Players alternately roll a ball and the person who

knocks over the last pin (or pair of pins) wins.

Try to find a winning strategy. (Assume that you can always hit the pin or pins

that you aim for, and that no one is ever allowed to miss).

Now try changing the rules ...


26 (98)

CONSECUTIVE SUMS

The number 15 can be written as the sum ofconsecutive whole numbers in three differentways:

15=7+815= I +2+3+4+515=4+5+6

The number 9 can be written as the sum ofconsecutive whole numbers in two ways:

9=2+3+49=4+5

Look at numbers other than 9 and 15 and findout all you can about writing them as sums ofconsecutive whole numbers.

Some questions you may decide to explore . • .

Spaces for your own questionswhen you think of any.

Write about your discoveries.Try to explain why they occur.


27 (100)

*

THE PAINTED CUBE

Imagine that the six outside surfaces of a large cube are. painted black. Thislarge cube is then cut up into 4,913 small cubes. (4,913=17x 17x 17).

How many of the small cubes have:o black faces?1 black face?2 black faces?3 black faces?4 black faces?5 black faces?6 black faces?

* Now suppose that you cut the cube into n3 small cubes ...


28 (106)

SCORE DRAWS

"At the final whistle, the score was 2-2"

What was the halt time score? Well, there are nine possibilities:

0--0; 1-0; 0-1; 2-0; 1-1; 2-1; 2-2; 1-2; 0-2

* Now explore the relationship between other draw~ matches, and thenumber of possible half-time scores.

There are six possible ways of reaching a final score of 2-2:

l. '0--0, 1-0, 2-0, 2-1, 2-22. 0--0, 1-0, 1-1, 2-1, 2-23. 0--0, 1-0, 1-1, 1-2, 2-24. G--O, 0-1, 1-1, 2-1, 2-25. 0--0, 0-1, 1-1, 1-2, 2-26. 0--0, 0-1, 0-2, 1-2, 2-2

*

*

How many possible ways are there of reaching other drawn matches?

Finally, consider what happens when thefinal score is not a draw.


29 (l08)

CUPBOARDS

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A factary sells cupbaards in twa standard widths: 5 dm and 7 dm.(Note: 1 dm=1 decimetre=10 centimetres).By placing cambinatians af these cupbaards end to.end, they can be fitted into.raams af varia us sizes.

Far example, two.5 dm and three 7 dm cupbaards can be fitted into.a raam 31 dmlang.

31

k_7_~_cupboards ~

*

*

*

*

Haw can yau fit a raam 32 dm long?

Explare raams with different lengths. Which anes can be fitted exactly withcupbaards. Which cannat?

Suppase the factary decides to. manufacture cupbaards in 4 dm and 7 dmwidths. Which raams cannat be fitted naw?

Investigate the situatian far ather pairs af cupbaard sizes.Can yau predict which raams can ar cannat be fitted?


30 (110)

NETWORKS

A network is a set of lines (or "arcs"), junctions (or "nodes") and spaces(or "regions") which compose a shape.The network shown above is composed of 12 arcs, 7 nodes (marked with blobs)and 7 regions (these are numbered-notice that we have included the outside asa region).

Networks can be of two kinds:Connected, like this . . . or disconnected like this . . .

o

Draw your own connected networks. Find a rule connecting the number of arcs,nodes and regions. Try to explain why your ~uleworks.

Can you adapt your rule t9 work for disconnected networks?

A cube has 6 faces, 8 comers (or vertices) and 12 edges.Explore the relationship between the number of faces, vertices and edges forother solid shapes.

Can you find any exceptional cases?

"@ShellCentre for Mathematical Education, University of Nottingham, 1984.

31 (112)

FROGS

These two frogs can change places in three moves

Move 1

Move 2

Move 3

l_t~I"11"1G?1_11"1 Tdll

Rules

* A frog can either hop onto an adjacentsquare, or jump over one other frog tothe vacant square immediately beyondit.

* The white frogs can only move fromleft to right the black frogs can onlymove from right to left.

The frogs shown below can be interchanged in 15moves. E~plain how.

How many moves would be needed to interchange 20 white and 20 black frogs?- n white and n black frogs?

Now suppose that there are an unequal number of black and white frogs.These frogs can be interchanged in 11 moves. Explain how.

How many moves are needed to interchange 15white and 20 black frogs?- n white and m black frogs?


32 (114)

DOTS

You will need a supply ofdotty paper.

The quadrilateral shown inthis diagram has an area of161f2 square units.

The perimeter of thequadrilateral passes through 9dots.

13 dots are contained withinthe quadrilateral.

Now draw your own shapes and try to find a relationship between the area, thenumber of dots on the perimeter and the number of dots inside each shape.

Try to find a similar result for a triangular dot lattice.(You will of course have to redefine your unit of area).


33 (116)

DIAGONALS

K ......e •• e.... ~ ..::.'.. .• .

.. ~ ..........:..... )k:... - ... : . .• ....•. . .-. .": ....

'::':::. :-:~

:i·.:·<:·. - ... ....... :..'": ::: ........ -. - .... ". ".

'X " ........ .• -....... (':.:..... •.: -.- :•.... :.... " :"•.": ...>f(.: - ..... ... .• ..."a •• _•• •.:..:.." .•... ..... :

A diagonal of this 5 x 7 rectangle passes through 11 squares.

These have been shaded in the diagram.

* Can you find a way of forecasting the number of squares passed through ifyou know the dimensions of the rectangle?

* How many squares will the diagonal of a 1000 x 800 rectangle pass through?

©Shell Centre for Mathematical Education, UnIversity of Nottingham, 1984.

34 (118)

THE CHESSBOA~D

*

*

*

*

*

How many squares are thereon an 8x8 chessboard?(Three possible squares areshown by dotted lines).

How many rectangles arethere on the chessboard?

Ca~ you generalise yourresults for an nXn square?

How many triangles arethere on this 8 x 8 grid?How many point upwards?How many point down-wards?

Look for other shapes in thisgrid and count them.

III

~ --- --,I ,

~ ", II, ,II~I

,I I

I


35 (120)

~• •

THE SPIRAL GAME

•

•

This is a game for two players. Place a counter on the dot marked H! H. Now takeit in turns to move the counter between 1 and 6 dots along the spiral, alwaysinwards. The first player to reach the dot marked HtH wins.

Try to find a winning strategy.

Change the rule for moving in some way and investigate winning strategies.

@Shell Centre for Mathematical Education, University of Nottingham ~1984.

36 (124)

NIM

This is a game for 2 players.Arrange a pile of counters arbitrarily into 2 heaps.Each player in turn can remove as many counters as he likes from one of theheaps. He·can, if he wishes, remove all the counters in a heap, but he must takeat least one.The winner is the player who takes the last counter.

Try to find a winning strategy.Now change the game in some way and analyse your own version.


37 (126)

"FIRST ONE HOME"

..•~-- -- •/./ I

/ I/ I

JI III,. _.

FIt'II5M

This game is for two players. You will need· to draw a large grid like the oneshown, for a playing area.

Place a counter on any square of your grid.

Now take it in turns to slide the counter any number of squares due West, Southor Southwest, (as shown by the dotted arrows).

The first player to reach the square marked "·Finish" is the winner.


38 (128)

"- ..: :

PIN THEM DOWN!

WALLA game for 2 players.Each player puts counters of his colour in an endrow of the board. The players take it in turns to slideone of their counters up or down the board anynumber of spaces.No jumping is allowed. The aim is.to prevent youropponent from being able to move by pinning himto the wall.

•••••0 0 0 0 0

WALL

WALL

WALL

Can you find a winning strategy?


39 (130)

THE "HOT FAT TUNE" GAME

This is a game for two players.

Take it in turns to remove anyone of the nine cards shown above.

The first player to hold three cards which contain the same letter is the winner.

Try to find a winning strategy.


40 (132)

DOMINO SQUARE

This is a game for 2 players.You will need a supply of 8 dominoes or 8 paper rectangles.Each player, in tum, places a domino on the square grid, so that it covers twohorizontally or vertically adjacent squares.After a domino has been placed, it cannot be moved.The last player to be able to place a domino on the grid wins the game.

For example, this board shows the first five moves in one game:

(It is player 2's turn. Howcan he win with his nextmove?)

Try to find a winning strategy .


41 (134)

THE TREASURE HUNT

This is a game for two players.You will need a sheet of graph paper on which a grid has been drawn, like theone below. This grid represents a desert island.

1000 _r--- r-r-r--r--r------,

500 ~-+--_+___+___+-+--+--t-__+___+___tw

s

E

o 500 1000

One .player "buries" treasure on this island by secretly writing down a pair ofcoordinates which describes its position.For example, he could bury the treasure at (810,620).

The second player must now try to discover the exact location of the treasure by"qigging holes", at various positions.For example, she may say "I dig a hole at (200,200)".

The first player must now try to direct her to the treasure by giving clues, whichcan only take the form:"Go North", "Go South", "Go East", "Go West", or "Go South and East" etc.In our example, the first player would say "Go North and East".

***

*

Take it in turns to hide the treasure.

Play several games and decide.who is the best "treasure hunter".

How should the second player organise her "hole digging" in order todiscover the treasure as quickly as possible?

What is the least number of holes that need to be dug in order to be sure offinding the treasure, wherever it is hidden?


N

~ ~

'""' -~ ~~'""' -c= ~

~ 0

~N

~ ~

'""' -~ ~~'""' -c= ~

~ 0

~N

N ~

'""' -~ ~~~ -= ~

.~ 0

~N

~ ~~ -~ ~~~ -= ~

~ 0

~

••••• roQ.c E ~ Q M.~ M • ...-.4 a) ~~ E ro Q

~ ~ ::J~

~ro 0 a) Cd

00 ~ 1'00-4 U ~ ~

<t: CO u Q ~ ~

64 (164)

NOTES ON MARKED SCRlPfS

ScriptAEmma In part (iii) Emma was awarded only 1 mark out of 2 since her answer did

not explain clearly that she had added the numbers from 1 to 11.In part (iv) she was given 1mark out of2 as her answer showed evidence of asystematic approach although it was incomplete.

ScriptBMark In part (i) Mark's answer was correct and although no working was shown

he was given both marks.Although Mark's diagram for part (ii) is correct, there are three errors inhis solution. He shoul4 have had 66 cubesx4+12 and, in addition, hiscalculation of 45x4+ 11 is incorrect. He was given 1 mark out of 4.

Script CIan Ian has misunderstood the question and assumed the tower to have' a

hollow middle.In part (i) his answer is therefore wrong and he gets no marks.In part (ii) he has made two errors: he assumed the tower has a hollowmiddle and has 13 layers. He was therefore given 2 marks out of 4.In part (iii), his explanation of his calculation is not complete and so hescores 1 mark out of2.In part (iv) his answer is not correct and scores no marks.

ScriptDColin In part (ii) Colin has made two errors in multiplication for h=11 and h=12.

Since each answer has been worked out independently using c=h x w onlythe error in h=12 need be penalised. So Colin scores 3 marks out of 4.In part (iii) he scored both marks for a clear, complete and correctexplanation of his method.In part (iv) the three formulae on the left hand side are correct andsufficient to solve the problem, although they are not organisedsystematically. He was therefore awarded 1 mark out of2.

ScriptEPeter

ScriptFPaul

In part (ii) there is some doubt as to how Peter has worked out his answer. Itmay be that he has attempted to build onto the original tower andcalculated the number of extra cubes ne~ded but has forgotten to add on the66. We are giving him the benefit of the doubt by taking this view althoughthis may mean a slightly inflated mark. He was awarded 3 marks out of 4 for

(".)part ,11 •

In part (iii) his explanation of his method is not very clear and he wasawarded 1 mark out of 2.

Paul's answer is of a very high standard. He was awarded 10marks out of 10despite the algebraic error in the last part.

63 (163)

SCRIPT F PAUL (continued)

~ I "H,\hpl~ -this /)vm~ ~ 4.

(i)~4~~

~ "d.d twe.lue. ::

4. TiY ~ ib~ 1) cu~ h'9he

~ <II.'1f:l 1J.4. mi:ldk 11 -bkd:s i !fro all WI- k.di -4 bloc/cs 0+ ()-1 k~h ~Ik& ~Ia.~ eq~' "~ ':t:'-t 'Z. : (n:!.l;-+ (0-1 L 1. ; 4#" .! 1""-, I

2 '

1m eW.t i) it> tk -tow'.~ _I):l + (4,,-f}

2


62 (163)

42

2

SCRIPTF PAUL

11~1 d~sI~'-rower ~s 5 ~~ each~~ ..Jd~ in fWT1'4Yl(cU tmk~! J-2-5.(\.5:~ 1mn\Jc.-1rr i\UIWZ',(~ f\~ -~ 6Y{)qf ex rftCU'Julw N.n1~rs I~

.:r2 +:x.2..'

1, 1fr~ ~o-f 4~ ~bAtrie ik DVQ'chkd bqse,~4k ~u\ct !t5Y lr~~~U~.

1. 1.~ .•.'::(. ~ c; + S : 30 :. {52 2. L

~ rtMtlpfy th~ ~ 4 (~~)

2 d3 10 bu, td & ker D- cubEs h11

~ (l.l).X1l.j'-H..e frladk ~IJQ \,1m so y~ ~ r& ~d~ 1m ele~Y\ bla:k\

2

continued


61 (163)

SCRIPT E PETER

(1) (f'f.-6 -==- ~44-i7 - d..8;-

4- {. g - ~~-lr 'I '=t - *36.".- lfo'-f- 'I- lO "4' 1- Ii

...... 4-f,r-+-t + 6 3~lo


60 (163)

SCRIPT D COLIN

a..rr\ 0t.U'\\- o~ c.u'oQo f\QQc\ Q.d l-o be.A, \e1 tluL\otJ:Jer ore 66 2(c.) ! t.(),d'h

(~) ls)! c, 'ND o~ bare ~r °a ~:1eP...L-

66 \\ 5I ~ ql l~ 6~ ~;{(]) IS 7q ! \S?> \1 9

10 \q (V Iq q\\ '1.2\ 1..\ d~

\1 "-G 2.?> \ \ 3-a:>

t\rnOUl\~ ol- steps : t,Q.lCO ~~ M,nu.~ one...L0 ,d~ ~ b~ '::' OMau...t'\~C! s¥ rt\u It\p\,~ ~ h..cc Qdd <me..nN'\Ou.t\~.o~ c.uOQ.~ :: h~Cjht ml.<l\\,p\;ed b~ U)d~~ oj- ~tlt2. 2So 0. 'rouxzr o! "'e,~hl Il. LZeUld na-Qd l..S6 eubQ~ to be..

t:Ot\s;\rw:J~J o~ 1- Gr.

~= H-\

Cos=- ~ -Ic:

\,.:) ;::"""if

H= s. IH-~

-"'" 1


59 (163)

®

SCRIPTC IAN


58 (163)

1

o.

SCRIPT B MARK

6) (')(2)

~ 66 Gv lre~.cc:::.:rZ c:'..,b c':S

2

1

®


SCRIPT A EMMA (continued)

Q I tr h'gh:-1OUJe.r

Elli - t, 1<4- -r ~ ;:. 2~-

\' ( ( H9h: -

~

IS 1- Y. - i:JO ~6 "::. 66-

It II ~ hi9h:- ~10 l(\(. -: 4.0 -t <) :. ~S·.

,. ( . 3 h~h fu sx'+ :: 11 -+ 3 ~ IS

It ,t a I ~\(. '=. If T 2 -= b

\ ' " DX4-- .. 0 l' I ~

~jXJHert1Sa 1cJ~(£ wl1f ht{p me to spt>t-

1hz eti(fr:rt~ fDttun ll{- iYlf dtffo-tr.cq fPtw" = If. 1haJgh~of tow(r tlmeSI h£ightofhu:tr Jno of btOd(SCJSal ': no. cf b\oc1s used [s Q JX2Cttrril

@©Shell Centre for Mathematical Education, University of Nottingham, 1984.

56 (163)

SCRIPT A EMMA

(::J) .,..6 ::- )f:,6 i 1hi1is +ht= n~. ef cubes 'f)(.(.dcd

,he S" ~p ob~vt, ~ ;nt ta "t'ntf CDliotmn .

~each arm wo~cl C6Ylsis}- ot:-

2

-to- =='"

-10-I- 1-0

I

1IIIII

4:lbtr + )"1- .::. 1m) ~~. ~Id tqwoJ C( llwJU whivh is J2

~ h'~.3> I jllSl- ,~warlc.td our the. 110· of cubes ~ 1 arm ~ s~~

ctr n UAht.s h-'9h and d!crecu OJ drMJr1 fi> 1. I o,\AJhplitc( -this ~ 4-ltS ~-e. o..r"e ~ a.rrYlS I I tntn a.dded 1Ylt trfctJ ~JVtf" of tm

rower g"(\ h1 1h6 ~u-. 1'f} " fu, e.a~ <lTYY1 .siQrfS 1. C1(bt" d(fVJn I w(fU1d frr.!H~ wn~t

~ -. -:: -t .---.~J----: ; ,~.: -_!--"J!::...-' .-- '-

) mvt dtci~ec\ 10 iry Scn1t SlfT)phr eY11rnpll.S 10 Se{. ;{- Ic.oJ'\ spcr- ~ ~rT\S

continued


55 (163)

~ 11 •. ~RcR a: gu,c~ ~ TO M~t~ A,SJ('EUfltJ'J TO~ oF n BUl r:.<;.fil4H.

SCRIPT F PAUL (continued)

~ I h1dhp'~ +hh numb- ~ 4.

t6~4~~

. ~ "tld ~ue :

len add 11 ~ tk iniAl.~ _ J) J. ~ (4,,-1)

~


54 (163)

·SCRIPT F PAUL

ll-thllll'r I-o.~5 slb~ ea en .cJirB ~~ in NJlT'On wi rmJo.r -J~ J - 2- s.4 .s:tk fmn~ Irr nUfi""I(~ ~~ "in crrrJof or fy'ClIJulCif 1UVl~~ I~

:r2 ~:x.2-"

1, 1ir each ~o-f f~ ~brie f~ nwbhhd bqse,u;eik ~ula hrr lr~~f\\;~.

, 1. .••..;:t. -;.::x.. ~ S' -+- S : 30 :. 15

:2 1. .:L

IhN ftrv\hply H1tS 6., 4 (I-k ~ )

2 ~3 10 hUI'd & ~eY Q cubEs h11

Tok ~.~ Madk ~~ blah so '1l1J. ~ ~ft W\t~ fa, e\e~Y1 bla:kzl

continued


53 (163)

SCRIPT E PETER


SCRIPT D COLIN

S PR., \ Q..tc(\ Tot.CQ..('""~.

.~\\\e.. a..tn0l.U'\\- o~ c:..ub~ f\~Q.d rn. ee) I

66I ~ q\~ ~:Z0q } \'03

l<~ \q CV

"1..2\

\1- 2SG

\\

\o(My ore 66ls)

~r o~ep-L-567g

qd~\ \

~f1\OUi\~ ol- s\eps :: l-,C21CQ h~ M,nu.:). one...

U) ,d\in ~ bor.:e -: OMm.u'\} ok s\r:fs fY\u fhp\,~ \o~ h.oo Qdd Ot'e.nN:\Ou.t\-to ..~ e.uOQ.~ = h<i~h~ m\04\Lp\;ed b~ U)\d~~ oj- ~t:R.

~ 0. \otcQr of "'e\~h~ J 1... czx:u\d nQ.Qd l..S6 e.ubQ,:\ Q, be.t:o'\<s;\C'\AC.tQ.J 04 t Gf-

to= \-4-\

CoS::.~-I


51 (163)

SCRIPTC IAN

) 'ls-Z'?f7'\5 ~T' 'czl:'" §2 eJ:b )€s. 4').,.c-+\6~ \l.-~<2:>+""-'t-\ '::6 \2) ),: i ~ Co, •• 't£ c-..c> "\+\r1-~'\"2-~~~"2c>-+"Z.~ ..• '""2.~'"2.","~ "\.0 T~\~~

- :)\~ <::::.u-~. ~5 \,~ \1>5 ,.,,--z..\\ Z.S5 "'-c:x&~5

~ ') \ i\ 9,..~ '"' ~ 1. ~. ,So C>- \;'~....... ~\-.,J.-~C Qf, "--~ •...(\

\",~ ~<;\\s\ \..a...\ {~ \, "2'1'\~::' '""" , ~<d. ~'bo..~~o ~, 50 \~~\-. ~~ ~\. ~ \ "'~ '\.. ~ ~V () \: \"t.

~.1>,) Clt.'o.aS -= \..x...=<~0 ~ iC \ -. \ ~I: X~ ,..•••

~ q,..~(' 0~ ~ ~~~- 'aQ~ '2. C ~"~~5

~~~ -'"~'6 ~ ~ \" ~"'~~ "5 ~~ I o..'t-~ ~0fJ.. ~o.Jfi.. ~~'k O~ cu.,\,~ - ~J' ~ ~C)\\~ \~.

@Shel1 Centre for Mathematical Education, University of Nottingham, 1984.

50 -(163)

SCRIPT B MARK

6i:I 66 ~'"be~.jqz ~~bc'S


49 (163)

SCRIPT A EMMA (continued)

a towe.r tr hcgh: -

Ifu - b xtJ,. 1- 4, ;. 2l{-

" c ( ~9 'n ~-

hn 's 1- '+ -:. bO ~ E> '::: bG

I( It )h9h

:- ~10 (\(. ~ t{O 1" <) ::. tI-S •

" ( . 3 h~h fu '3 ~ 't ':: n..1" 3 ~ )~

..••. 4- "1" 2 btt " a ( ;( \(. ~ ~

l • " D)c4-- -:.0 1" I -.: I"

~fXJHert1 s .a -tabf<r WJ11 h{fp me to ~t-

1hl c..tiff~ J=Otttrn tf -lhf dr{famcq ftt:ut" = \J..

ha'9h~ 0 f tow(rtlmfS, h6ght of h;u.t.r Jno of bto&$ tISld ':: no. ef b\ods usr.d fs Q PJtf:trril


48 (163)

SCRIPT A EMMA

{;,D .,..6 = ) f:,6 } 1hjsis +hc n~. af CLtbIS '0«.dcd

,he S- up abt.'1t, ~ int tan)'nlJ CDh.•mn ."i)

each Q1Y\'\ wou.ld cmsis}- ot:--

r-=IooI--~1-- ~

tr,HI

:2fltr + Jl- :;. (m)- }j~. WMid fqwal Q.. -mwu whic.h is 12~ hi~.3) ) j.ur i.~ warktd our tnt no. of cubcs ~ 1 arm ~ .sttntCn~

Q.t- n tJAbt.s h'9f1 and dtcrecuYlj dtMlfl fz> 1. I thulhplitet -jhis ~ 4-45" ~ ore 4- a.rMS I I ihtn a.dded iht ~J heJ9Y1F of tm

ftJ'Wt r §Yl h1 1h'6 rtSW)J-·

.,} r fu. mck arm sfarts 1. ,"bt" dr:flAJn I w(fU1d f,r.!H..1j Wr)~t

:---: : '-t .---..! .. .:-!---! .':-=-=.1-1

1 mVt dtciaed 10 ~ Sa11t si~ p lor e'Itlrnpl!.S 10 Se!- ;r J

c.a.n spcr ~ ptt7trns

continued

©Shell Centre for Mathematical Education, 'University of Nottingham, 1984.

47 (163)

SKELETON TOWER ••• MARKING SCHEME

(i) Showing an understanding of the problem by dealing correctly with a simplecase.Answer: 66~ marks for a correct answer (with or without working).

Part mark: Give 1mark if a correct method is used but there is an arithmeticalerror.

(ii) Showing. a sy~~ematlcattack In the extension to a more difficult case.Answer: 2764 marks if a correct method is used and the correct answer is obtained.

Part marks: Give 3 marks if a correct method is used but the work contains anarithmetical error or shows a misunderstanding (e.g. 13 cubes inthe centre column).Give 2 marks if a correct method is used but the work containstwo arithmetical errors/misunderstandings.Give 1 mark if the candidate has made some progress but thework contains more than two arithmetical errorslmisunderstandings.

(iii) Describing the methods used.

2 marks for a correct, clear, complete description of what has been donepr~viding more than one step is involved.Part mark: Give 1 mark if the description is in~omplete or unclear but

apparently correct.

(iv) Formulating a general rule verbally or algebraically.

2 marks for a correct, clear, complete description of method.Accept "number of cubes=n(2n-l)" or equivalent for 2 marks. Ignore anyerrors in algebra if the description is otherwise correct, clear and complete.

Part mark: Give 1 mark if the description is incomplete or unclear but showsthat the candidate has some idea how to obtain the result for any·given value of n.


46 (19)

SKELETON TOWER

(i) How many cubes are needed to build this tower?

(ii) How many cubes are needed to build a tower like this, but 12 cubes high?

(iii) Explain how you worked out your answer to part (ii).

(iv) How would you calculate the number of cubes needed for a tower n cubeshigh?


45 (18)

A TREASURE HUNT PROBLEM

This is a game for two players.

The diagram below,represents an island, and each dot represents a possiblelocation for some buried treasure. (In this case there are 30 possible hidingplaces).

321

1 2 3 4 5 6 7 8 9 10

One player has to guess the location of the treasure, and the other has to providea Hclue" after each guess, which can only be of the following kind:

After the first guess, the clue is either "warm" or "cold" according to whetherthe treasure is located at a neighbouring point or not.

After each succeeding guess, the clue is either "warmer", "colder", or "sametemperature", depending on whether the guess is closer to, further awayfrom or the saIne distance from the treasure as the previous guess.

The-aim is to discover the treasure with as few guesses as possible.

* In the sample game shown below, the first guess, Gl, was (8,3). The cluegiven was "cold", so the treasure is not on any neighbouring points (shownwith a0)'

3GI

0 X 02 00 01 G::!

X

1 2 3 4 5 6 7 8 9 10

*

*

The second guess, G2, was (8,1) ...Show that, wherever it is buried, the treasure can always be located witha total of 5 guesses (including Gland G2). Is this the minimum number?

Now try to find the minimum number of guesses needed for a differentgrid ...

What is the best "guessing" strategy?

©Shell Centre for Mathematical Education, University ofNottingbam~ 1984.

44 (146)

SupportMaterials

43

problems with patterns and numbers: masters for photocopying

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