mathematics reasoning - activities for developing thinking skills

7/27/2019 Mathematics Reasoning - Activities for Developing Thinking Skills

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Activities for developing thinking skills

SMILEM AT HE M AT I CS



C o n t e n t s

I n t r o d u c t i o n

Introduction 2Activities referenced to the Frame work for Teaching Ma them atics 3Activities referenced by National Curriculum level 4

Year 7

Largest Product 6Mdbius Band 8Shapes and Numbers 10Shapes from Squares 12Some Sums 14Symm etry Investigation 16Up the Stairs 18Wigg ly Tessellations 20Without Looking 22

Year 8

Dots 24Half Time Scores 26Odd and Even Chains 28Origami Boxes 30Panorama 32Polygons 34Quarters 36Shapes from Triangles 38Tessellating Pentom inoes 40

Year 9

4-in-a-Line 42Boxing Areas 44Drawing Integer Triangles 46Follow the Path 48Frogs Puzzle 50How many Squares 52Number Pyramids 54Painted Cubes 56Tower of Hanoi 58

Answers

Year 7 60Year 8 63Year 9 66

Additional Support Materials

Suggested mental and oral starters 70Other publications from Smile mathem atics 71



I n t r o d u c t i o n

Reasoning conta ins 27 activities, fully referenced to the Year 7, 8 and 9 teachingprogrammes f rom the Framework for Teaching M athemat ics 2 00 1. Each act iv i ty hasbeen ass igned to a particular year group to help departments integrate Using andApplying Mathemat ics and Thinking Skills into theirKey Stage 3 S c h e m e of Work .

To allow more flexibility, each activityhas also been referenced to the appropr ia teNational Curriculum Assessment levels . Each activity is accompan ied by teacher ' snotes which include:

• information on topic area, level , duration and resources• learning objectives including those taken from the relevant sections of the

Framework for Teaching Mathemat ics 2001• a description of how the activity can be used within the classroom, rais ing

issues for differentiat ion.

A photocopiable Resource Sheet accompanies each activity. This can be used asan OHT to introduce the activity to the whole class or photocopied so that each student has a copy to work f rom.

Compat ib le ora l and mental s tar ters have been suggested and are taken fromSMILE Mathemat ics publ icat ions Imaginings , Mathsin Your Head and Reckonings .

"There are f ive categories of thinking skil ls embedded in the National Curr iculum:information processing skil ls , enquiry skil ls , creative thinking skil ls , reasoning skil lsand evaluation skills." (Framework for Teaching Mathematics 2001)

© RBKC Smile Mathemat ics 2002



A c t i v i t y R e f e r e n c e : N a t i o n a l C u r r i c u l u m L e v e l

Na t io n a l C u r r i cu lu m Lev e l sYe a r 7 3 4 5 6 7 8

Largest Product

Mobius Band

Shapes and Numbers

Shapes from Squares

Some Sums

Symmetry Investigation

Up the Stairs

Wiggly Tessellations

Without Looking

Ye a r 8 3 4 5 6 7 8Dots

Half Time Scores

Odd and Even Chains

Origami Boxes

Panorama

Polygons

Quarters

Shapes from Triangles

Tessellating Pentominoes

Ye a r 9 3 4 5 6 7 84-in-a-Line

Boxing

Drawing Integer Triangles

Follow the Path

Frogs Puzzle

How many Squares ?

Number Pyramids

Painted Cubes

Tower of Hanoi



L a r g e s t P r o d u c t

TopicWritten methods

Level5-7

Duration1 lesson

Learning Objec t ivesMul t ip ly and divide three-digit by two digit whole numbers (Y7 p. 104).Check results by considering whether it is of the right order of magnitude(Y7 p. 110).

Recognise that some permutations w i l l give equal products because of theirsymmetry.

ResourcesNumber cards 1 - 9, x and =. Calculators or spreadsheet.

DescriptionWith students working in pairs, use the 1, 2, and 3 cards and the x and = cards toexplore what products can be made. Agree as a class the six differentmultiplications, 1 x 23, 1 x 32, 2 x 13, 2 x 31, 3 x 12, 3 x 21. Then identify which ofthe multiplications gives the largest product.

Students then work in pairs, using cards 1, 2, 3, 4, x and =, finding out the numberof different products, the largest product and the smallest product.

Possible ExtensionExtend to include 5 digits. Unless using a spreadsheet, it may be best to concentrateon finding the largest product only. This can then be extended to 6, 7, 8 and 9 digits.

Possible O u t c o m e sStudents generate results for 4 digits and identify the largest product (Level 5).Students generate the 36 results for 4 digits and use their understanding to predictthe largest product for 5 digits (Level 6).

Students systematically work out the total number of products for 6 digits andpredict the largest product (Level 7).

Associated Vocabularydigit, largest, multiplication, product



Reasoning R e s o u r c e s h e e t


Different produ cts can be m ade using the digits 1 to 4:

•\ r

1

\ r

2 3 4

For example:

2 x 1 3 4 = 2681 x 2 x 3 x 4 = 24

2 4 x 1 3 = 312

Find as many products as you can using these four digi ts .

• W ha t is the largest prod uct?

• W hat is the sm allest product?

Chal l engeWhat is the largest possible product using the digits 1,2,3, 4, 5, 6, 7, 8 and 9?



M o b i u s B a n d

TopicGeometrical reasoning: lines, angles and shape

Level4-7

Duration1-2 lessons

Learning Objec t ivesVisualise and identify 3D shapes.Investigate the effect upon Mobius Bands of various folds and cuts.

U se correctly the vocabulary, notation and labelling conventions for lines, anglesand shapes (Y7 p. 178).

ResourcesStrips of paper about 3cm by 30cm. Scissors, glue or sellotape and coloured pencilsor pens.

DescriptionGive out the Resource Sheet. Ask students to make several Mobius Bands from thestrips of paper by giving the strip a half turn and then joining the two ends to make aclosed loop. Students should then investigate what happens when they:

• attempt to colour one side red and the other side blue

• draw a line along the centre of the Mobius Band and then cut along it

• cut the Mobius Band into thirds (it may help to start with a wider strip of paper)

• cut a slit in the strip of paper and push one end through before joining, then extendthe slit right around the Mobius Band

Possible ExtensionsWhat happens with more than one half-turn?

Possible O u t c o m e sStudents successfully complete the four tasks and begin to describe their results(Level 4).

Students use appropriate language to present their results (Level 5).

Students begin to justify their results and attempt other variations (Level 6).

Students justify their results and correctly predict what they think wi l l happen forother variations before actually trying them (Level 7).

Associated Vocabularyedge, face, Mobius Band, prediction, topology.



To draw this pattern

• Shade one square.

• Using a different colour, shade every square that touc hes it along an edge.Record the number of squares you shade.

• W ith a third colour, shad e every squa re that touch es anothe r squa re along an edge .Record the number of squares you shade.

• With a fourth colour, shade every square that touche s another square along an edge.Record the number of squares you shade.

12

4

8

Continue the pattern of colours and sequence of numbers.

C h a l l e n g e |

Use isometric paper.

Start with an equilateral triangle and follow a similar rule.Write the sequence of numbers .

U s e o t h e r s t a r t i n g s h a p e s t o g e n e r a t e d i f f e r e n t n u m b e r s e q u e n c e s .



S h a p e s f r o m S q u a r e s

TopicTransformations

Level3-5

Duration1 lesson

Learning Objec t ivesBegin to identify and use angle, side and symmetry properties of triangles and

quadrilaterals (Y7 p. 184).Recognise and visualise the transformations of a 2-D shape (Y7 p.206).

ResourcesCut-out squares, O H P and centimetre squared paper.

Descr ipt ionU s i n g an OHR explore and discuss why there is only one possible shape made from2 squares and only 2 shapes from 3 squares. Then investigate how many different

shapes are possible using 4 squares, 5 squares, .. .

Encourage students to discuss congruency, reflections and rotations in order to

determine whether their shapes are unique. This can be done as a useful plenary

session.

Possible ExtensionStudents can go on to look at 6 squares and identify which of the shapes are nets of

a cube.

Poss ib le OutcomesStudents recognise congruent shapes ( L e v e l 3).Students are able to identify all 12 shapes using 5 squares ( L e v e l 4).Students are able to identify all the shapes using 6 squares that w i l l produce a net of

a cube ( L e v e l 5).

Assoc ia ted Vocabula rycongruent, prediction, reflection, rotation, shape.




2 s q u a r e s - ^

3 s q u a r e s " ^

1 s h a p e

2 s h a p e s

4 s q u a r e s

? s h a p e s

5 s q u a r e s ^

? s h a p e s



S o m e S u m s

TopicIntergers, powers and roots

Levels :3-6

Duration1 lesson

Learning Objec t ivesRecognise and use multiples and factors (divisors) in simple cases (Y7 p.52).

Recognise the first few triangle numbers (Y7 p.56).

Generate sequences from practical contexts and describe the general term in simplecases (Y7 p. 156)

ResourcesNone

Description

Brainstorm a definition of 'consecutive numbers' (keep to positive integers).A sk students for some more examples of numbers which can be made by adding 2consecutive numbers. Take a couple of examples then ask students to find andrecord their own solutions and identify any patterns. Y o u may wish to encourageweaker students to stick to numbers between 1 and 20. Extend the problem bylooking at 3 consecutive numbers, 4 consecutive numbers, 5 consecutive numbers,etc.

Possible ExtensionsW h i c h numbers cannot be made?

What if you include 0?

What if you look at consecutive odd numbers, consecutive even numbers?

Possible O u t co mesStudents find which numbers between 1 and 20 can be written as the sum of 2,3 and 4 consecutive numbers (Level 3).

Students find sequences of numbers that can be written as the sum of consecutivenumbers (Level 4).

Students f ind a rule for generating the next term in each sequence (Level 5).

Students f ind a rule for generating the nth term in each sequence (Level 6).

Associated Vocabularyconsecutive, expression, sequence, sum, term.



S o m e S u m s

C o n s e c u t i v e n u m b e r s are integers which are next to each other on thenumber l ine:

0 1 2 3 4 5 6 7 8 9 10 11

H 1 1 1 1 1 1 1 1 1 1 h

For example:2 , 3 a n d 4 are c o n s e c u t i v e8 and 9 are c o n s e c u t i v e5 and 7 are n o t c o n s e c u t i v e

13 can be writ ten as the sum of 2 consecutive numbers:

6 + 7 = 1 3

18 can be writ ten as the sum of 3 consecutive numbers:

5 + 6 + 7 = 1 3

Investigate which numbers can be writ ten as the sum of. . ... . 2 cons ecutive num bers. . . 3 consecutive numbers.. . 4 cons ecutive num bers... etc.

W hich num bers betwee n 1 and 20 cannot be written as the sum of conse cutive num bers?




Level3 - 5

Duration1 - 2 lessons

Learning Objec t ivesUnderstand and use the language and notation associated with reflections,translations and rotations (Y7 p.202).Recognise and visualise the transformation and symmetry of a 2-D shape: reflectionin given mirror lines, and line symmetry; rotation about a given point, and rotationalsymmetry; (Y7 p.204, 206).

ResourcesCentimetre squared paper (2cm squared paper is more suitable for weaker students),scissors, display paper and coloured pencils, mirror and tracing paper. An O H P and acopy of each shape either cut out from squared paper or from coloured O H T.

DescriptionA s k students to make copies of the three shapes on squared paper and cut them out.Using the O H P, invite students to place the three shapes so that they create asymmetrical shape and then identify the lines of symmetry.With students working in pairs, get them to recreate the shape, check the lines ofsymmetry with a mirror and record their results, using a dotted line to record thelines of symmetry.This activity is a rich source of discussion and pupils should be encouraged todiscuss the type of symmetry their shapes have, i.e. reflective symmetry, number of

lines of symmetry, rotational symmetry and order of rotational symmetry.Teachers may want to specify rules in advance, however for many groupsdetermining their own rules is a very useful activity and can involve discussions oncongruency. This activity is a rich source of display material.

Possible O u t c o m e sStudents generate symmetrical shapes using the three pieces (Level 3).Students identify and record the lines of symmetry (Level 4).Students are able to identify shapes with rotational symmetry (Level 5).

Associated Vocabularycongruent, diagonal, horizontal, line of symmetry, order of rotational symmetry,reflective symmetry, rotational symmetry, symmetry, vertical.



S y m m e t r y I n v e s t i g a t i o n

Copy these 3 shapes o n to squared paperand cut them out .

Use the p ieces to make sym metr ica l sha pes .

Record your results , showing any l ines of symmetry.



U p t h e S t a i r s

TopicSequences, functions and graphs

Level5-7

Duration3-4 lessons

Learning Objec t ivesGenerate and describe simple integer sequences (Y7 p. 144).

Generate sequences from practical contexts and describe the general term in simplecases (Y7 p. 154).

ResourcesO H T of Resource Sheet

DescriptionBrainstorm with the class how many different ways you can climb a 6 step staircase.Record the possible ways that the class finds. You may wish to begin with norestrictions on the size of steps and introduce these at a later stage, for examplecombinations of 1 and 2 steps only. Discuss whether 5 steps followed by 1 step isthe same as 1 step followed by 5. Encourage students to move on from drawingsteps and f ind a way to efficiently code their work to include all possible moves.

E.g . With 5 steps, 4, 1 or 1, 4 or 3, 2 or 2, 3 etc.

3, 1, 1 or 1, 3, 1 or 1, 1, 3 etc.

Possible O u t c o m e sStudents are able to organise their results and systematically record or code their

data (Level 5).

Students generalise the number of ways of climbing an 'n' staircase (Level 6).

Students generalise the effects of step restrictions on results, leading into theFibonacci sequence (Level 7).

Associated Vocabularycode, Fibonacci sequence.



TopicGeometric reasoning: lines, angles and shapes.

Level3-5

Duration1 lesson

Learning Objec t ivesConstruct tessellating shapes.

Recognise and visualise transformations and symmetries of shapes (Y7 p.202-212).

Understand and use the language and notation associated with reflections,translations and rotations (Y7 p.202).

Resources4cm squares, scissors, glue, card and A4 paper. Copies of Resource Sheet or an O HTo f the Resource Sheet.

DescriptionBrainstorm a definition of 'tessellation' including the word 'congruent'. Findexamples of shapes which tessellate and shapes that do not. L o o k at a hexagon,triangle, circle and the wiggly tessellation from the resource sheet and discuss with

students whether each shape w i l l tessellate and why.

Give students copies of the Resource Sheet and go through the steps for creating awiggly tessellation as a whole class (use the O H P or a large model to demonstrate).Stress the use of appropriate vocabulary, such as 't ranslation'. Students then use 4cmsquares to create their own wiggly tessellations on A4 paper.

Possible ExtensionTry starting with a different shape, for example an equilateral triangle or a regularhexagon.

Possible O u t c o m e sStudents make a template and create a tessellation from the given instructions

Students tessellate several shapes to an appropriate level of accuracy and begin toexplain why the method works (Level 4).

Students extend their understanding of wiggly tessellations to create tessellationsstarting from hexagons and triangles (Level 5).

Associated Vocabularycongruent, tessellation, translation.

(Level 3).




W i g g l y Te s s e l l a t i o n s

Here is how to make a wiggly tessellation . . .

A Start with a square' piece of paper.

Draw a wiggly linefrom one side to theother.

Draw anotherwiggly line from thetop to the bo ttom.

Cut along the two wiggly lines.

Take the bottomtwo pieces andtranslate themup to the top.

Now take two ofthe side piecesand translate themto the other side.

J V

7 Stick your new shape onto card to make a template and cut it out.• Che ck that your shape tesse llates by tracing round the tem plate on A4 paper.

T r y m a k i n g u p s o m e w i g g l y t e s s e l l a t i o n s o f y o u r o w n .



W i t h o u t L o o k i n g

TopicGeometrical reasoning: lines, angles and shape

Level4-7

Duration1 lesson

Learning Objec t ives

U se 2-D representations to visualise 3-D shapes and deduce some of their properties( Y 7 p.200).Develop confidence with isometric drawing.

ResourcesMu l t i l i n k or Centicubes and isometric paper.

DescriptionMake sure students are able to use isometric paper for simple 3D drawings beforeattempting this activity. Div ide the class into pairs. Gi v e each student 6 cubes and

ask them to make a so l id without their partner seeing it. Each pair then swaps theirsolids under the table. Without looking, each student then draws the so l id they havebeen given on isometric paper. When students have finished their drawings theyshould compare them with the actual solids to determine whether they are correct.They should then try the activity again for different solids. Y ou may wish toencourage weaker students to use less cubes or to stick to simpler solids, e.g. solidswith one layer only, solids made by starting with a 3 by 2 by 1 cuboid and movingone cube only.

Possible ExtensionsUsing more cubes w i l l make the activity significantly more difficult.

Y ou could vary the rules of the activity by giving students a short period to studyeach so l id before it is hidden and they attempt to draw it from memory.

Possible O u t c o m e sStudents draw simple solids made from cubes using isometric paper (Level 4).Students draw more complex solids made from cubes using isometric paper(Level 5).Students identify key features of solids by touch and sight (Level 6).Students draw complex solids on isometric paper from memory (Level 7).

Associated Vocabularycube, cuboid, edge, face, isometric, rotation, symmetrical, vertex.




Each person:

• Make a solid using 6 cubes w i t h o u t y o u r p a r t n e r s e e i n g i t .

• Swap your solid with your partner's solid under the table.

• Draw the solid you have been given on isometric paper w i t h o u t l o o k i n g .

• Now look at the solid and check that you have drawn it correctly.

• Try it again for other solids.



D o t s


Level

5-7Duration1-2 lessons

Learning Objec t ives

Generate and describe integer sequences (Y8 p. 147).B e g i n to use linear expressions to describe the nth term of an arithmetic sequence,justifying its form by referring to the activity or practical context from which it wasgenerated (Y8 p. 155).

ResourcesSquare dotty paper.

Descr ipt ionEmphasise to students that this investigation involves squares and rectangles drawnat 45° to the paper. Show students the examples on the resource sheet. Make surethat they understand why they are described as 3 by 3 and 5 by 2. Discuss with

students the importance of adopting a systematic approach using the suggestion onthe resource sheet. Ask students to investigate the number of dots in differentsquares and rectangles. Encourage them to tabulate their results, describe patternsand look for algebraic generalisations.

Poss ib le OutcomesStudents generate results systematically and describe any patterns (Leve l 5).Students describe their rules algebraically and attempt to justify them in words(Leve l 6).Students f ind an algebraic generalisation for the number of dots inside any rectangle(Leve l 7).

Assoc ia ted Vocabula ryalgebraic, generalisation, rectangle, sequence, square



D o t s

This inve stigation is abou t squa res an d rectan gles wh ich are draw n at 45 ° to the paper.

Invest igate the number of dots in different squares and rectangles which are drawn at45 ° to the pap er.

It will be easier to find a rule if you start with rectangles with width 2.

Th is squ are is 2 by 2. Th is recta ngle is 2 by 3. This rectangle is 2 by 4.How many do ts? How many do ts? How many do ts?

What about rectangles with width 3 . . . with width 4 . . . any width?



H a l f - T i m e S c o r e s

TopicIntegers, powers and roots

Level3-6

Duration1-2 lessons

Learning Objec t ivesRecognise and use multiples and factors (divisors) (Y8 p.53).Begin to use linear expressions to describe the nth term of an arithmetic sequence,justifying its form by referring to the activity or practical context from which it wasgenerated (Y8 p. 155).

DescriptionG o through the example on the resource sheet with the whole class, recording all theresults on the board. Y o u may wish to use a topical score from a recent footballgame, e.g. Sunderland 1 Manchester United 4. Or you could choose any other sportwhich has two halves and scores in increments of 1, such as netball or hockey.

Discuss with students how they can be sure they have found all the half-time scoresand how they can generate these systematically.Div ide the class into small groups and ask each group to investigate how manypossible half-time scores there are for different full time scores. When they havesufficient results, encourage students to describe any patterns or rules they havefound.Y o u may wish to collate the class' results in a two-way table during a plenarysession. This may help students to find the generalisation for any f ina l score.

Possible O u t c o m e s

Students f ind all possible half-time scores for a given f ina l score (Level 3).Students f ind a systematic way of generating their results, tabulate their results andfind simple patterns such as 'i t goes up in 2' (Level 4).Students f ind linear expressions to describe their sequences of results (Level 5).Students find an algebraic generalisation for the number of possible half-time scoresfo r any f ina l score (Level 6).

Associated Vocabularyfactor, generalisation, multiple, sequence.



H a l f - T i m e S c o r e s

T h e f i n a l score of the ga m e is 3 - 2.What do you think the h a l f - t i m e score was?

W rite dow n all the po ssible h alf-time sco res for a final score of 3 - 2.How many are there?

Investigate the possible half-time scores for different final scores.



O d d a n d E v e n C h a i n s


Level3-6

Duration1-2 lessons

Learning Objec t ivesGenerate and describe integer sequences (Y8 p. 145).A d d , subtract, multiply and divide integers (Y8 p.49).

ResourcesCentimetre squared paper.

DescriptionArrange the class into similar ability groups or pairs. Expla in to the whole class thatyo u are going to create a number chain. This is a sequence of numbers generatedusing the following rules to f ind successive terms in the sequence:

• if a number is even, halve it

• if the number is odd, add 1

Think of a number to start from and demonstrate how to generate a number chainusing these rules (e.g. 9 —» 10—> 5 —» 6 —> 3 —> 4 —> 2 —> 1). Discuss why it is notnecessary to go any further than 1. Then ask students to investigate number chains.Each group should be allocated different starting numbers. Results from the wholeclass can be presented together in one diagram (see Answers).More able students should be encouraged to look at numbers up to 32 and tabulatethe starting number and the number of terms in each chain. They should look at

which numbers have the longest chains and which have the shortest. They couldthen draw a graph to illustrate their results.

Possible O u t c o m e sStudents recognise that all chains end in 1 and attempt to explain why (Level 3).Students collate the sequences on a summary diagram and make general statementsabout their results such as "The longest chain starts with 17" (Level 4).Students analyse their results, e.g. "The shortest chains start with a power of 2", andattempt to justify their analysis (Level 5).

Students give algebraic generalisations, e.g. "The longest chains start with 2n+l"

(Level 6).

Associated Vocabularyeven, halve, odd, powers, sequence.





Think of a number.

If it is even - halve it.

If it is odd - add one.

Continue along the same rules.

o

Record your sequence.

T r y s t a r t i n g w i t h d i f f e r e n t n u m b e r s .

W h a t h a p p e n s ?

How long are your number chains?

What is the longest chain?

What is the shortest chain?

Do larger numbers always give longer chains?



TopicMeasures and mensuration

Level5-7

Duration2 lessons

Learning Objec t ivesK n o w and use the formula for the volume of a cuboid (Y8 p.239).Calculate volumes and surface areas of cuboids (Y8 p.241).

ResourcesA n OHT of the Resource sheet. A4 coloured paper, square coloured paper. Pencils,rulers, scissors. Lentils or rice. A couple of boxes already made up.

DescriptionRun through the step by step instructions using the Resource sheet as an O H T. After

making their first box, students can explore the difference in volumes, using

lentils/rice if necessary. Tr ia l and improvement and/or calculations may make furtherprogress possible. Students might want to compare results from changing theorientation of the paper.

Possible O u t c o m e sStudents generate boxes and are able to work out the volumes (Level 5).Students explore the relationship between changes in dimension and resultantchanges in volume (Level 6).Students systematically work out the dimensions with the greatest volume and therelationship between the volume and surface area (Level 7).

Associated Vocabularydimension, maximum, minimum, orientation, surface area, trial and improvement,volume.



I O r i g a m i B o x e s

Take an A4 sheet of paper.

Fold in half and open out.Fold in half the other way andopen out .

N-

Fold the left and rightedges to meet along thecentre .

Do not open out.

J

Fold back along the dottedline.

A

V

The corners must touch thecrease l ines.

Fold the top and bottomedges to meet along thecentre .

Then open ou t .

Fold all 4 corners.Do not open out .

/\ s/ \

/ Ni \

\ *\ J

\iss

The corners must touch thecrease l ines.

6 Gent ly pu l l apar t . . .

Pull

. . . to form an open box.

J

• What are the dimensions of the basic box?

• What is the volume of the box?

• What is the largest volume box you can make out of a sheet of A4 paper?



P a n o r a m a


Level5-6

Duration1 lesson

Learning Objec t ivesMake simple scale drawings (Y8 p.217).Use bearings to specify directions (Y8 p.233).

ResourcesAngle indicator. One copy of a tourist map for each pair of students. Maps areusually available free from local Tourist Information offices showing points ofinterest. Alternatively, use a London Tourist map and suggest that students plan avisit.

DescriptionIn pairs, students look at the map and suggest 10 places that they would visit. Lis tthese places on the board. Ask the students to f ind and mark these places on theirmap. The class should then agree a place where these can be viewed, e.g. a highchurch tower, The London Eye, etc. This point should also be marked on the map.From this point ask the students to draw the North line on the map. Using the angleindicator, measure the angle of the first place and record this as a three digit bearingin a table.

Students continue to find the bearings of a ll the places listed. They then use these todraw a panorama. Students can discuss the advantages and disadvantages of using apanorama. Students select another central viewing point, f ind the bearings of the

same places and draw a new panorama.

Possible ExtensionStudents could exchange their panorama with someone else and challenge them towork out the central viewing point. Alternatively students could construct alongitudinal panorama from their panoramic view.

Possible O u t co mesStudents measure and record bearings and use these to draw a panorama (Level 5).Students are able to work out back bearings (Level 6).

Associated Vocabularyangle, bearings, clockwise, panorama, reflex angle, three digit.



P a n o r a m a

Houses of Par l iamentand Big Ben

The diagram above shows a panorama of some buildings that you mightsee from the top of the London Eye.

• St Paul's Cathedral is on a bearing of 055°.• The Tower of London is on a bearing of 072°.

Check these bearings on the tourist map.

1. Use the panorama and an angle indicator to find the bearings of:a) Houses of Parl iament and Big Benb) Nelson's ColumnCheck your answers on the tourist map.

2. There are other s i tes from which you can view London, such as:a) Top of Parliament Hillb) The Monumentc) Waterloo Bridge. . .

Draw an accurate plan of the panorama from your chosen si te.

3. W hat information do pan oram as show ?What information do they n o t show?

R e m e m b e rBearings are always measured clockwise, s tart ing from North.



P o l y g o n s


Level4-7

Duration1-3 lessons

Learn ing Objec t ivesDeduce and use formulae for the area of a triangle; calculate areas of compoundshapes made from rectangles and triangles (Y8 p.237).

ResourcesSquare dotty paper, O H P. Pinboards and elastic bands (optional).

Descr ipt ionShow students a copy of the resource sheet on the O H P. Make sure studentsunderstand the terms and vocabulary used. Discuss how to f ind the area of thepolygons in the examples. Now give out the resource sheets and ask students toexplore other polygons drawn on centimetre square dotty paper. Encourage studentsto look for a relationship between the number of dots on the perimeter, the numbero f dots inside and the area of the polygon. Y o u may wish to direct them towardskeeping one of the three variables constant.

Poss ib le OutcomesStudents f ind accurately the area and number of dots for several polygons and recordtheir results (Leve l 4).Students work systematically, keeping one variable constant. They tabulate theirresults and comment on patterns that they have found ( L e v e l 5).

Students express rules using formulae and attempt to justify them ( L e v e l 6).Students look for the relationship between their rules and f ind a generalisation forany polygon (Leve l 7).

Assoc ia ted Vocabula ryarea, generalisation, perimeter, polygon, relationship, variable.



P o l y g o n s

This polygon is drawn on centimetre square dotty paper :• • • • • • • •

• • • • • • • •

It has 9 dots on the perimeter.

It has 2 dots inside

• • •

• • • a

• • • a

\• a

•

It has an area of 5^ cm 2 .

• •

• *

/ • . >

This polygon is a lso drawn on centimetre square dotty paper :• • • • • • • •

• • •

• • •

» at • • •

' A How ma ny dots does it have/ \ on the perim eter?

/ • \ • • •

\ How ma ny dots does it have^ • • inside?

• • • <

What is its area?

» • • •

• • • a • • • •

Invest igate other polygons drawn on centimetre square dotty paper.See if you can f ind any relat ionship between:

• the number of dots on the per imeter

• the number of dots inside• the area of the polygo n




Level4-6

Duration1 lesson

Learning Objec t ivesK n o w that if two 2-D shapes are congruent, corresponding sides and angles areequal (Y8p.l91).

Transform 2-D shapes by simple combinations of rotations, reflections andtranslations (Y8 p 205).Explore equivalent fractions.

ResourcesO H P and transparency of square dotted paper, square dotted paper, mirrors, scissors.

Description

Discuss with the class ways of splitting a 4 x 4 grid into four equal parts using anO H P and a transparency of square dots. Students may in i t ia l ly suggest the obviouscross and diagonal cuts. They should be encouraged to explore some more unusualcuts. Some students may find it difficult to cut their square up into more complexparts because of difficulties in replicating the first cut. They should be encouraged touse reflection, rotation and symmetry to aid them, finding how many lines ofsymmetry exist in each of their designs. Students should progress to drawingtessellations of the quarters they have discovered and as a further extension couldlook for ways of cutting up a larger square or the original square into other fractionalparts.

The class should then conclude that there are many ways of quartering a 4 x 4square. Some of the tessellations produced could be used for display work. Be awareof students not working on the correct square size. A brief discussion of square sizesthat can be exactly quartered is useful.

Possible O u t c o m e sStudents create congruent shapes (Level 5).

Students explore how some quarters w i l l tessellate in only one way and others inmany ways (Level 6).

Students explain why some quarters w i l l tessellate in only one way and others in

many ways (Level 7).

Associated Vocabularycongruent, fractions, quarters, reflective symmetry, rotational symmetry, squarenumbers, tessellation.



Q u a r t e r s

Find as many ways as you can to divide a 4 x 4 grid into4 congruent par ts .

Explore the different ways to tessellate each quarter.



T e s s e l l a t i n g P e n t o m i n o e s

A pentomino is created from five squares touching along one edge.

• • DThese f ive squares

• • can make this pentomino.

The pen tom ino will tessellate in only one way.

The pento mino will tessellate in more than one way.

Investigate the tessellat ions of al l the 12 pentominoes.

• W hich pen tom ino will tessellate in f ive different ways?

• W hich tessellat ions can be colou red using only two colours?

• Wh at proper t ies of pentom inoes determ ine how many differentways it will tessellate?



Reasoning Ye a r 9

B o x i n g A r e a s


Level5-8

Duration1-2 lessons

Learn ing Objec t ivesGenerate sequences from practical contexts and write an expression to describe thenth term of an arithmetic sequence (Y9 p. 155).

Deduce properties of the sequence of square numbers from spatial patterns (Y9 ext.p. 159).

Resources1/2 centimetre squared paper.

Descrip t ionCheck students are familiar with square numbers before you start. L o o k at the

maximum and minimum boxed areas for 4 squares with the class as a whole. Askstudents to find the maximum boxed area for different numbers of squares up to 8.After 10-15 minutes, pool the class's results and draw out through discussion therule for the maximum boxed area (n2). Now ask students to investigate theminimum boxed area for any given number of squares. After some time (15-20minutes) two rules may w e l l have become apparent (2n and 2n-l). At this pointdraw the class together to explore which numbers fo l low each of the two rules.Highlight the role of counter example in disproving a rule.

Then ask students to investigate further and to determine a rule for which numbersfo l low each of the two rules above.

Poss ib le OutcomesStudents determine the minimum boxed area for any given number of squares. Theyidentify one of the sequences of numbers which fo l low the rule 2n - l , e.g. 5, 8, 11,14, .. . (Level 5).

Students identify the sequence of sequences which fo l low the rule 2n - l , i.e. 5, 8, 11,... add 3, 13, 18, 23, ... add 5, etc. (Level 6).

Students articulate the pattern of summing consecutive square numbers to f ind thefirst number in each sequence and attempt to generalise in words or symbols(Level 7).

Students produce and justi fy a sophisticated generalisation for which numbersfo l low each of the two rules (Level 8).

Assoc ia ted Vocabu larycounter-example, generalisation, maximum, minimum, sequence.



Reasoning 1 Ye a r 9

D r a w i n g I n t e g e r T r i a n g l e s

TopicGeometrical reasoning: lines angles and shapes

Level5-7

Duration3 lessons

Learning Object ivesU se straight edge and compasses to construct a triangle (Y9 p.222).Develop and apply coding to simplify a situation.

ResourcesCompasses and rulers.

Descrip t ionBrainstorm a method for constructing triangles using compasses and a ruler. Using

this method, ask the class to explore how many different triangles they can constructwith integer length sides. L o o k at triangles with longest side: 3cm, 4cm, 5cm, . ..

Useful discussion may arise focusing on a suitable method of describing triangles,e.g. (3, 2, 2), and on combinations that w i l l not form triangles, e.g. (3, 1, 2) and (3,I, 1)

Poss ib le OutcomesStudents are able to construct a triangle using ruler and compasses (Level 5).Students are able to generalise which dimensions w i l l generate different types oftriangles (Level 6).Students justify their results (Level 7).

Assoc ia ted Vocabu larydimension, generalisation, rectangle, square.



F o l l o w t h e P a t h

TopicConstruction and loci

Level6-8

Duration1 lesson

Learning Objec t ivesF i n d the locus of a point that moves according to a simple rule (Y9 p.225).K n o w and use the formulae for the circumference and area of a circle (Y9 p.235).

ResourcesRulers, pencils, compasses and angle indicators.T h i n strips (straight edged) of card, squares and other cut-out shapes.Large square for demonstration, blu-tac.

DescriptionA s k students to visualise the locus of a vertex of a square as it is rotated

successively about each of its vertices along a straight line. Then use a large squarewith a piece of blu-tac on one corner (or a smaller square on an OHP) todemonstrate this to the whole class. Discuss the shape of the locus (including thenumber of arcs in one cycle) using appropriate vocabulary.

Get students, working in pairs, to stick a strip of card onto paper to use as a base forrotating their square. Ask them to practise 'walking' their square along the strip ofcard by doing successive rotations before attempting to draw the locus of one vertexas the square moves. This is more easily done if students use compasses to securethe centres of rotation. Encourage students to describe the locus, identifying thecentres of each arc and calculating the length of, and the area underneath, the locus

(one cycle only). Ask them to recreate the locus accurately using ruler andcompasses before going on to investigate the loci for other points and shapes (e.g.equilateral triangle, rectangle, right-angles and isosceles triangles).

Possible O u t c o m e sStudents recreate the locus accurately using ruler and compasses (Level 6).Students identify the centres of each arc and calculate the length of, and the areaunderneath, the locus. They look at the loci of other shapes (Level 7).Students f ind algebraic expressions for the length of, and area underneath, the loci ofa range of shapes in terms of the length of sides (Level 8).

Associated Vocabularyarc, circumference, cycle, loc i , locus, path, radius, rotation, vertex.




Start with a square cut out from card and a long strip of card.Stick your strip of card onto a large sheet of paper.Rotate the square along the edge of the strip as shown below.Draw in the path followed by a vertex of the square as you go.

/Rotate about this point...

/.. . then rotate about this point

• W ha t will the locus of the vertex look like if yo u continue to rotate the squa re a longa straight line? Describe it as accurately as you can.

• W hat abou t the locus of the other vertices?• W ha t abo ut the locus of the cen tre of the squa re?• Try rotating different sha pes along a straight line.



F r o g s P u z z l e


Level4-7

Duration1-4 lessons

Learning Objec t ivesGenerate terms of a linear sequence using term-to-term and position-to-termdefinitions of the sequence (Y9 p. 147).

Generate sequences from practical contexts and write an expression to describe thenth term of an arithmetic sequence (Y9 p. 155).

ResourcesCounters in two different colours. The 'Frogs' program from the M i c r o S M I L E'Mathematical Puzzles' C D (optional).

DescriptionThere are several ways of introducing this activity to the whole class. Y o u could use7 chairs at the front of the class and ask 6 students to represent the frogs. The rest ofthe class could direct the students on how to move whilst you keep a record of themoves. Alternatively you could use the M i c r o S M I L E program 'Frogs'. Discuss withstudents how they should record their moves in a way which would enable them torecreate or refine their solution. Encourage them to record the number of hops andslides as w e l l as the total moves.

Having solved the ini t ial problem, students should go on to investigate for differentnumbers of grey and white frogs. Suggest that they start off with less frogs, an equalnumber of grey and white frogs and the number of l i l ies one more than the total

number of frogs. Encourage students to record their results in a table, to analysetheir results and to find a generalisation.

Possible O u t c o m e sStudents solve the in i t ia l problem (Level 4).

Students select an appropriate diagrammatic representation of their moves (Level 5).Students describe patterns, make hypotheses and test them (Level 6).Students find an algebraic generalisation for the number of moves, hops and slidesfor any number of grey and white frogs (Level 7).

Associated Vocabularyalgebraic, generalisation, hop, hypothesis, model, move, slide.




There are 7 lilies on a pond.There are 3 grey frogs and 3 white frogs.

The aim is to swap the grey frogs with the white frogs.

Rules:

• Grey frogs can only mo ve to the right.• W hite frogs can only mo ve to the left.• A frog can s l i d e onto an empty lily or h o p over a frog of a d i f f e r e n t colour.

How many moves are needed to swap the 3 grey frogs with the 3 white frogs?How many hops?

How many slides?

You could set up a model of the puzzle by using two different coloured counters and agrid of 7 squares. The starting positions of the counters are shown below.

Try changing the number of grey and white frogs.Can you f ind a rule for the num ber of moves needed ?What about for the number of hops and slides?



Reasoning1 Ye a r 9

H o w M a n y S q u a r e s

TopicGeometrical reasoning: lines, angles and shapes

Leve l5-7

Duration1-2 lessons

Learn ing Objec t ivesUnderstand congruence (Y9 p. 190).Generate sequences from practical contexts and write an expression to describe thenth term of an arithmetic sequence ( Y 9 K p.155).

ResourcesCentimetre square paper and a chess board.

Descr ipt ionA s k the class how many squares there are on the chess board but tel l them that theanswer you are looking for is not 64. Gi v e the class 5 minutes to think about the

problem and then invite students to explain how they might start answering thequestion. Some students may need support to develop a systematic approach. It mayhelp to start with a simpler example, e.g. ' How many squares in a 3 x 3 grid?'

Possible ExtensionsExtend to f ind how many squares are in an n x m rectangular board and then toexplore how many a x b rectangles are in an n x m rectangular board.

Poss ib le OutcomesStudents are able to generate a system to identify 204 squares on a chess board

(Leve l 5).Students generalise their method and are able to work out the number of squares in a15x15 square board ( L e v e l 6).

Students generalise a method for working out the number of squares in an n x mrectangular board ( L e v e l 7).

Assoc ia ted Vocabula rydimension, generalisation, rectangular, square.



N u m b e r P y r a m i d s

TopicEquations, formulae and identities

Level3-8

Duration1-2 lessons

Learning Objec t ivesU se index notation for integer powers and simple instances of the index laws( Y 9 p.59).Derive and use more complex formulae (Y9 ext. p. 139).

DescriptionDemonstrate to the whole class how the number pyramid given on the resource sheetis built. Then ask students to investigate other number pyramids starting in i t ia l lywith three consecutive numbers at the base. You may need to guide students on theirchoice of variables, e.g. choose a for the first number in the base and p for the peaknumber. Encourage students to f ind an algebraic generalisation for any three

consecutive numbers at the base and then attempt to justify it. Get students to go onto investigate number pyramids starting with four consecutive numbers, five

consecutive numbers, etc. and look for an overall generalisation. They w i l l need tointroduce a third variable, e.g. n, for the number of consecutive numbers at the base.Fo r a more rigorous algebraic justification, students may wish to use variables a, b,c, . .. for the numbers in the base and to use the binomial coefficients from theChinese Triangle (also known as Pascal's Triangle).

Possible O u t c o m e sStudents generate several number pyramids correctly (Level 3).

Students make general comments about their results (Level 4).Students generate results systematically and describe any patterns (Level 5).Students find an algebraic generalisation for three consecutive numbers (Level 6).Students find an algebraic generalisation for the peak number for any startingnumber and any number of consecutive numbers at the base (Level 7).Students justify their rule using proof by induction or binomial coefficients(Level 8).

Associated Vocabularyadjacent, algebraic, binomial coefficient, consecutive, generalisation, justify, proof

by induction.



P a i n t e d C u b e s

TopicGeometrical reasoning: lines, angles and shapes

Level4-8

Durat ion1-2 lessons

Learn ing Objec t ivesSimpl i fy or transform algebraic expressions (Y9 p. l 19).Generate terms of a sequence using term-to-term and position-to-term definitions ofthe sequence (Y9 p. 153).Identify and use the properties of area and volume.

ResourcesMu l t i - l i n k cubes or plain cubes and sticky paper ( S M I L E 1322 A Blue Cube andS M I L E 1323 A Red Cube optional)

Descr ipt ionShow the class a 3 x 3 x 3 cube with stickers on each outside face. Ask the class:H o w many small cubes?H o w many cubes have stickers on 1 face?Where are they positioned?H o w many cubes have stickers on ... 2 ... 3 ... 0 faces?Where are they positioned?

Then ask the class to explore a 4 x 4 x 4 cube, a 5 x 5 x 5 cube, etc. Weaker studentsmay want to bui ld a 2 x 2 x 2 cube and a 3 x 3 x 3 cube for themselves, and usestickers to answer the above questions.

Possible ExtensionsWhat about painted cuboids?

Poss ib le OutcomesStudents recreate the 3 x 3 x 3 cube, finding and recording their results ( L e v e l 4).Students record correct results for other cubes and identify patterns ( L e v e l 5).Students are able to generalise for an n x n x n cube (Leve l 6).Students record correct results for some cuboids ( L e v e l 7).Students are able to generalise for any cuboid ( L e v e l 8).

Assoc ia ted Vocabula rycube, edge, face, vertex, volume.



T o w e r of H a n o i

TopicIntegers, powers and roots

Level5-8

Duration1-3 lessons

Learning Objec t ivesU se index notation for integer powers and simple instances of the index laws(Y 9 p.59).Derive and use more complex formulae (Y9 ext. p. 139).

ResourcesTower of Hanoi (if this is not available, students can make their own puzzle usingdifferent sized pieces of card and a grid of 3 squares). The 'Hanoi' program from theM i c r o S M I L E 'Mathematical Puzzles' CD (optional).

Description

Start off with a Tower of Hanoi with 4 discs. Make sure students understand therules for moving the discs and demonstrate the first few moves if necessary. Discusswith students how they might record their moves in a way which would enable themto recreate or refine their solution. You may l ike to discuss why the largest disc onlyneeds to move once.

Div ide the class into similar ability pairs. Ask each pair to solve the ini t ial problemand, when they are sure they have found the minimum number of moves, investigatefo r different numbers of discs. The number of moves increases very quickly soencourage them to start with 2 discs, 3 discs, etc.

Possible O u t c o m e sStudents find and record the minimum number of moves for 4 discs (Level 5).Students generate a table of results for various numbers of discs and describe anypatterns they see (Level 6).

Students find a generalisation for the minimum number of moves for any number ofdiscs (Level 7).

Students present and justify their rules algebraically (Level 8).

Associated Vocabularyalgebraic, generalisation, index, minimum, power.



T o w e r o f H a n o i

Tower of Hanoi is a logic puzzle from Vietnam.There are three poles and a tower of different sized discs on one of the poles.

The aim of the puzzle is to move all of the discs onto either of the other two poles.

Rules :

• Only one disc can be mo ved at a t ime.• A disc can neve r be plac ed on top of a sm aller disc.

Can you solve the puzzle?W hat is the minimu m nu mb er of mo ves nee ded to mov e the discs?

Investigate the minimum number of moves needed for 2 discs , 3 discs , . . .Can you find a rule for any number of discs?




For 4 digits:N o. of different w ay s: 36Larges t produc t: 1312Sm allest product: 234

M o b i u s B a n d

A) Imp ossible as, top olo gic al^ , the Mobius Band has only one face (and one

edge) .B) A s ing le Mobius Band wi th two comple te tu rns .C) Tw o inter locking M obius B and s, one with a half turn and the other with two

turns .D) A single Mob ius Band with two com plete turns.

S h a p e s a n d N u m b e r s

Sta rting with a triangle the se qu en ce s is: 3, 6, 9, 12 . . .Starting with a sq uar e the se qu en ce s is: 4, 8, 12, 16 . . .Star t ing with a hex ago n the seq uen ces is: 6 , 12, 18, 24 . . .


Num ber of squa res used Nu mb er of different sha pes ma de

S o m e S u m s

Numbers which can be wr i t ten as the sum of consecu t ive numbers :2 consecutive numbers - 3 , 5 , 7 , . . .3 consecutive numbers - 6 , 9 , 12, . . .4 con sec utive num bers - 10, 14, 18, . ..c consecu t ive numbers - c t h tr iangle number, c t h t r iang le numberThe n t h term of the above sequence is:

12345

1125

12

c(c + 1)

2+ ( n - 1 ) c

Powers o f 2 cannot be made .



If you include 0, the nt h term of the sequence is:c ( ° - 1 ) / *^ — - ^ + ( n - 1 ) c

For consecut ive odd numbers , the n t h term of the sequence is:

c 2 + 2c(n -1 )

(powers of 2 become possible)

For consecut ive even numbers , the n t h term of the sequence is:

c(c + 1) + 2 c ( n - 1 )

S y m m e t r y I n v e s t i g a t i o n

The number of different shapes found wil l depend upon shapes al lowed, e.g.whether a shape could be added by touching at a vertex, or whether rotationalsymmetry is used.

U p t h e S t a i r sThere are 8 different ways of climbing a 5 step staircase by going up in one or twosteps at a t ime.With no step size restrictionsNum ber of s teps Way s of cl imbing

3 44 85 16i i

n 2n-1

With largest step size 2Num ber of s teps Wa ys of cl imbing

3 34 55 86 137 21I t

Resul ts give a Fibonacci-type sequence

An average s tai rcase in a 2 s torey house has 13 s teps. There are 377 ways ofclimbing up 13 steps in one or two steps at a t ime.



W i g g l y Te s s e l l a t i o n s

If the steps are followed correctly, the resultant shape will tessellate.The resultant shape will always tessellate if the starting shape itself tessellates.


No answers required .



D o t sStudents may obtain the fo l lowing general isat ions:

For width 2 dots, the nu mb er of dots inside is 3L - 1 (whe re L is the le ngth).For width 3 dots, the num ber of dots inside = 5L - 2.For width 4 dots, the nu mb er of dots inside =7 L - 3.For width W dots , the number of dots ins ide = (2W - 1 ) L - (W - 1) .Looking at the problem using geometr ical reasoning givesnum ber of dots ins ide = LW + ( L - 1)(W - 1) , which is equal to the abo ve.

H a lf T i m e S c o r e sPupils may find patterns when recording their results in a 2 way table such as:

For the final sc ore n, 0, the rule is 'add T .For the final sco re n, 1, the rule is 'add 1 and dou ble' .For a final sc ore n, m, the rule is '(n + 1) x (m + 1)'.


For s tar t ing num bers between 1 and 32, the resul ts can be summ arised as fo l lows:

• 1 8 -7 — *1 9 — 2 0

2 1 — - 2 2

2 3 — - 2 4

31 — - 3 2

2 9 — - 3 0

2 7 — - 2 8

2 5 — 2 6

9\•10-

11

12-

16-t

15

14-

t13

8

3

4 2 — 1

Number of Starting Numbersterms to 1

0 11 22 43 3, 84 6, 7, 165 5, 12, 14, 15, 326 10, 11 , 13, 24, 28, 30, 317 9, 20, 22, 23, 26, 27, 298 18, 19, 21, 25

9 17

O r i g a m i B o x e s

The dimensions of the bas ic box are:height = 5 .3cm, length = 15cm, and width = 10.4cmVolume of bas ic box is 826.8cm3

By varying the fold l ine in Stage 3 i t is possible to create boxes with a greaterv o l u me .

P a n o r a m a

The bearing of the Houses of Parl iament and Big Ben from the London Eye is 235°.The b earing of Nelson 's Colu mn in Trafalgar S quare from the L ondon Eye is 312 °.



P o l y g o n s

If A = area, P = no. of dots on perimeter, N = no. of dots inside, then thegeneralisation is given by:P = 2A - 2N + 2 ~

Q u a r t e r s

Possible results :

S h a p e s f r o m T r i a n g le s

No. ofTriangles

Investigation 1No. of shapes

Investigation 2No. of nets

Investigation 3No. of tessellating shape s

1 1 0 1

2 1 0 1

3 1 0 1

4 3 2 3

5 4 0 3

6 12 5 12

Te s s e l l a t i n g P e n t o m i n o e s

Here are the 12 pentominoes



Reasoning Ye a r 9

A n s w e r s

4 - i n - a - l i n e

The generalisations for 4-in-a-l ine on an m by n board are:Num ber of row wins = m(n - 3) wh en m > 4Num ber of colum n wins = n(m - 3) wh en n > 4Nu mb er of diagon al wins = 2(m - 3)(n - 3) wh en m > 4 and n > 4Total number of winning l ines = 4mn -9(m + n) + 18 when m > 4 and n > 4

The generalisations for x-in-a-l ine on an m by n board are:Nu m be r of row win s = m(n - (x - 1)) wh en m > xNu m be r of co lum n win s = n(m - (x - 1)) wh en n > xNu m be r of diag ona l wins = 2(m - (x - 1))(n - (x - 1)) wh en m > x and n > xTotal nu m ber of win nin g lines = 4m n - 3(x - 1)(m + n) + 2 (x2 - 2x + 1) when m> x and n > x

B o x i n g A r e a s

An y num ber wh ich can be writ ten as (x - 1 )2 + x 2 + a(2x - 1), for so m e integ ervalue of a, has a minimum boxed area of 2n - 1.

Any other number has a minimum boxed area of 2n.

D r a w i n g I n t e g e r T r i a n g l e s

Max. edge length No. of tr iangles

3cm 44cm 65cm 96cm 12

e.g: 5cm(5 , 5, 5) (5, 5, 4) (5, 5, 3) (5, 5, 2) (5, 5, 1)(5, 4, 4) (5, 4, 3) (5, 4, 2)(5, 3, 3)5 + 3 + 1 = 9 t ri angles

General isa t ions :Odd numbers

n - > n + ( n - 2 ) + ( n - 4 ) + ... + 1 = ( n - 2 )2

Even numbersn -> n + (n - 2 ) + (n - 4) + ... + 2 = (n - 2 )(n - 1)




The locus of the vertex of a square after successive rotat ions about each of i tsvert ices along a straight l ine (one cycle only) is as follows:

If the square has s ides R cm, the locus could be descr ibed as :A quarter circle with radius R cm and centre A, followed by a quarter circle withradius V2R cm and centre B, followed by a quarter circle with radius R cm andcen t re C .

The length of the locus (for one cycle only) is:

22TIR 2TW2R 2TIR

+ + = 7 l R4 4 4

The area underneath the locus (for one cycle only) is:

TtR R 2

— + — +4 2

TI (V 2R) 2 R 2 TTR 2

+ — +2 4

(TU + 1)RS


The number of moves for 3 grey and 3 white frogs is 15 (6 sl ides and 9 hops).Table of results for equal numbers of white/grey frogs and one extra l i ly:

Grey/white frogs Slides Hops Moves

1 2 1 3

2 4 4 8

3 6 9 15

4 8 16 24

100 20 0 10 000 10 20 0n 2n n 2 n(2 + n)



Reasoning Ye a r 9

A n s w e r s

H o w M a n y S q u a r e s ?

There a re 204 squares on a s t andard 8 x 8 ches s board .Able s tudents may generalise for an n x m rectangular board given m>nm n + (m - 1 )(n - 1) + (m - 2)(n - 2) + ... + (m - n + 1) x 1

N u m b e r P y r a m i d s

If p = peak num ber, a = f irs t numb er in base , n = numb er of conse cutive num bersin base, then the overall generalisation is given by:

p = 2 ( n - 1 ) a + 2 ( n - 1 ) ( n - 1 )

P a i n t e d C u b e s

For a 3 x 3 x 3 cube

No. of stickers 0 1 2 3 Total No. of cubes

No. of cubes 1 6 12 8 27

Gen eralisation for any cub e: (x - 2 )3 + 6(x - 2)2 + 12 (x - 2) + 8 = x3

To w e r o f H a n o i

One possible way of coding the moves:

• Give each disc a number from 1 (for the smallest) to 4 (for the largest).• Give each pole a letter from A (the starting pole for the discs) to C.• The solution for 4 discs is then given by:

1 -> B, 2 -> C, 1 -> C, 3 -> B, 1 -» A , 2 -> B, 1 -> B, 4 -> C, 1 -> C, 2 -> A ,1 -> A, 3 -> C, 1 -> B, 2 -> C, 1 -> C

The minimum number of moves is given by:

Number of discs Minimum No. of moves

2 3

3 7

4 15

5 31

n 2"-1



S u g g e s t e d o r a l a n d m e n t a l s t a r t e r s

Year 7Largest Product Beat the ClOCk (Maths in your Head page 4)

Mobius Band Right-Handed GlOVe (Imaginings page 5)

Shapes and Numbers Multiple Grids(Maths in yourHead page 18)

Shapes f rom Squares Square (Imaginings page 26)

Some Sums Who is Left Standing? (Reckonings page 12)

Symmetry Investigation Introducing Pentominoes (imaginings page 36)Up the Stairs Sequences (Reckonings page 2)

Wiggly Tessellations Cuts (Imaginings page 11)

Without Looking Cube (Imaginings page 12)

Year 8Dots Rectangle (Imaginings page 6)

Half Time Scores Thinking and Listening (Reckonings page 8)

Odd and Even Chains FiZZ-BuZZ(Reckonings page 2)

Origami Boxes Isosceles Right-An gled Triangle (imaginings page 10)

Panorama ROUndabOUt (Imaginings page 34)

Polygons Equilateral Triangle and Circles (imaginings page 9)

Quarters Fraction Dice (Maths in your Head page 14)

Shapes from Triangles Large Equilateral Triangle (imaginings page 7)

Tessellating Pentominoes Large Red Square (imaginings page 8)

Year 94-in-a-Line Think Of a Number (Imaginings page 32)

Boxing Areas Factor Chains (Reckonings page 12)

Drawing Integer Triangles Straight Line (Imaginings page 3)

Follow the Path Geometric Rotations (Imaginings page 25)

Frogs Puzzle YeS Or NO (Reckonings page 13)

How Many Squares? Cutting C orners (Imaginings page 2)

Number Pyramids In Your M ind (Imaginings page 23)

Painted Cubes Wet Cubes . . . (Imaginings page 17)

Tower of Hanoi HOW Many Different ...? (Maths in your Head page 15)



O t h e r p u b l i c a t i o n s f r o m S m i l e M a t h e m a t i c s

M a k e s S e n s e S e r i e s• Mult ipl icat ion Makes Sense

• Fract ions, Decimals and Percentages M ake Sense 1 ( levels 2 -4 )

• Fract ions, Decimals and Percentages Make Sense 2 ( levels 5-7)

• Ratio Makes Sense

M e n t a l M a t h e m a t i c s• Maths in your Head

• Reckonings

• Imaginings

W h o l e C l a s s Te a c h i n g

• Whole Class Mathematics Projects• Nice Ideas in On e Place - Volu m es 1 and 2

For detai ls of these publicat ions and many other resources to enhance any schemeof work contact us for a catalogue.

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mathematics reasoning - activities for developing thinking skills

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