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									                                                Maths Year 2000 Scotland
                                             Resource Sheet PA26-1
                                                            Level A

Teacher’s Planning Sheets
                 THE PARK (LEVEL A)




  The Resource Sheets PA26-2, PA26-3, PA26-4 show
  the planning for one topic. They illustrate how
  mathematics can be developed in context.



  Pupil materials are provided on Resource Sheets
  PA27 and PA28.
KEY               ACTIVITIES                                                       RESOURCES             STRAND          ASSESSMENT
QUESTIONS

What do you see Walk to the park stopping at identifiable features, e.g. zigzag     Paper &              Position –      Do they point out
on the journey to lines, shops, phone box, bollards, stone wall, scout hut. Get the Clipboards           Movement        appropriate
the park?         children to draw each feature.                                                         Time –          landmarks?
                     Back at school, paint each feature and use as sequencing       Paint & paper        Sequence        Can they sequence
                     game                                                                                                them appropriately?
                     Draw picture maps of the journey, based on the features                                             Are maps recognisable
                                                                                                                         as the journey?

                 On walk discuss shapes seen on pavement, make rubbings of         Thin paper and        Range of shapes:
                 iron covers and names shapes in pattern.                          wax crayons           squares,         Can they name
                    Display the shapes in class                                    Backing card          triangles,       shapes?
                    In class, use small pieces of card, stick into patterns, and   (e.g. from cereal     rectangles,
                    make rubbings                                                  box), small           circles
                                                                                   pieces of card
                                                                                   cut into squares,
                                                                                   circles, etc., glue
                 At park discuss how many steps they think they have taken                               Measure in non- Are estimates
                   Count on the way back (if possible!)                                                  standard unit   appropriate?
                   Back at school cut out feet and display number of steps taken Paper, scissors         Range of        Can they count steps?
                   to park (may take too much room!)                                                     numbers (level
                                                                                                         B)
What is at the   Free play on all items
park?            Each group play on one item thinking of words behind, in front                                          Can they describe
                 of, above, under, on top, through. Then the class watches each                          Positions and   movement
                 group separately giving sentences using these words.                                    movement        appropriately?
                 Back at school, paintings illustrating children at park involved
                 in each description.
                 Children create sequences using park pictures                    Resource Sheet         Patterns &
                                                                                  PA27: Patterns         sequences
                                                                                  in the Park
                                                                                                                                                 Resource Sheet PA26-2
                                                                                                                                                                         Maths Year 2000 Scotland
KEY                ACTIVITIES                                                         RESOURCES       STRAND             ASSESSMENT
QUESTIONS

If you were        Count how many play things and compare areas of each play                          Range of           Can they count tiles?
going to replace   thing                                                                              numbers            Can they suggest
the soft-tiles     What size does each have to be? – discuss ways of measuring                        Measure in         appropriate units of
under each play                                                                                       convenient non-    measure?
thing, how         Each child uses own non-standard unit (length, breadth of each                     standard units
would you order    tile and of each “patch”)                                                          length - area
what you need?
I want to create  Count railings in each section – how many sections? How tall                       Range of            Are railings drawn or
railings back at  are railings?                                                                      numbers             made a representation
school               Measure in hands, etc. What shape do they form at the top?                      Measure in          of size and number in
                     Draw the railings                                                               convenient non-     section at park?
                     Back at school, paint or make 3D railings from rolls of paper,   Large roll of  standard units of
                     using the information discovered                                 paper          length
                                                                                                                                                 Level A Theme: The Park continued




How many          Estimate, then try and count                                        Resource Sheet Range of           Can they estimate and
people can sit on Is it enough for the whole class?                                   PA28 How       numbers            count?
each seat?        How many seats do we need? Estimate, try, then count                many?          Add                Can they measure in
                                                                                                                        non- standard units?
How many           Discuss suitable non-standard units                                                Measure in        Can they suggest
different ways     Estimate                                                                           convenient non- reason/s for the
can you find of    Try variety of hands, feet, thumbs, etc.                                           standard units of different answers?
measuring          Compare differing answers and discuss why different number of                      length and
length and         hands depending on child                                                           estimate
height of seat?
Draw a plan of     Discuss what information they need to remember                                     Range of           Can they record info?
the park back at   Count number of swings, rungs of climbing frame, seesaws etc.                      numbers            Is model accurate
school             and record in a table                                                              Info handling      representation of
                   Make 3D model of playpark in junk (or own invented one)                            Collecting -       numbers collected?
                                                                                                      display
                                                                                                                                                                      Level A
                                                                                                                                                                                     Maths Year 2000 Scotland
                                                                                                                                                       Resource Sheet PA26-3
KEY               ACTIVITIES                                                     RESOURCES         STRAND           ASSESSMENT
QUESTIONS

How many          Guess number of children who could all be occupied at a time                     Range of
children is the   Go round dropping children off at each item                                      numbers
play park         Count round – do the whole class become occupied? How
designed for?     many spare places or places needed are there?                                    Add / Subtract   Can they estimate and
                                                                                                                    count?
How long does a Count the number of swings equal to one game of hopscotch,                         Time -           Can they compare
game of          count the number of seesaws equal to one game of hopscotch,                       measuring in     times of activities?
hopscotch last? continue with other play park items, e.g. How many seesaws = HBJ Year 2 p.12       non- standard
                 10 swings etc.                                                   Textbook         unit
                    Back at school invent timer to give everybody the same time                                     Can they make some
                    on each activity (e.g. swinging beanbag)                                                        kind of timer?
                    If possible, try timer out at park
What is the most Draw favourite item                                                               Information
popular play     Display the drawings: (1) in sets; (2) in columns above name of                   handling
item in park?    item, (3) by threading bead onto appropriate string; (4) connect                  Collect -
                 child’s name to object name with line                                             organise -       Can they interpret
                 Oral work: counting, making sentences, comparing                                  display -        results?
                                                                                                   interpret
Can you invent a Look at Pip and see how big he is – how far can he move and     Pip               Position and
park plan on     paint large park plans in groups.                                                 movement
floor for Pip?   Give directions for a friend to walk round                                                         Can they move Pip
                 Use Pip to visit various points                                 CDROM:                             around?
                 Use “Pathway to Bearings” ‘Playpark’ CDROM, or similar          Pathway to                         Do they achieve
                                                                                 Bearing Disc 03                    success with program?
                                                                                                                                            Resource Sheet PA26-4
                                                                                                                                                                    Maths Year 2000 Scotland
                                                          Maths Year 2000 Scotland
                                                             Resource Sheet
Name: ____________________                                            PA27
                                                                    Level A

                        Patterns in the Park


Draw the next object in the line to make a pattern. Add an object
to each end of the last line.




Add an object to the start and to the end of this line:




Make up two of your own and give them to a friend to try.
                                                     Maths Year 2000 Scotland
                                                         Resource Sheet
Name: ____________________                                        PA28
                                                                Level A

                          How many?


Draw the number of things in each box.
Write the number

                               Picture           Number


   swings


   seesaw


   hopscotch


   climbing frame


   slide


   animal rockers



            How many play things are there altogether?
                                                Maths Year 2000 Scotland
                                             Resource Sheet PA29-1
                                                            Level B

Teacher’s Planning Sheets
                THE STREET (LEVEL B)




  The Resource Sheets PA29-2, PA29-3, PA29-4 show
  the planning for one topic. They illustrate how
  mathematics can be developed in context.



  Pupil materials are provided on Resource Sheets
  PA30–PA36.
Key: [M]: Mental work; [W]: Written work; [C]: Calculator work

 KEY                 ACTIVITIES                                                      RESOURCES         STRAND              ASSESSMENT
 QUESTIONS

 How many            Oral work estimating how many vehicles and what types of                          Information
 vehicles pass the   vehicles                                                                          Handling
 school in ten       Carry out traffic survey outside school and at corner of another Clipboards       Collect -           Can children record
 minutes?            road near the school                                                              Organise -          information and
 Do you think the                                                                                      Display -           discuss results?
 same number/        Plot results in table and graph form                            Small & large     Interpret
 more/less will      Discuss results                                                 square graph
 pass another        Survey at different times of the day                            paper for results.
                                                                                                                                                    Level B Theme: The Street




 road in the same                                                                    Graphing           Add and subtract
 time?                                                                               computer
 Do you think it     Discuss results                                                 program
 will vary           Number work finding total number of cars etc, difference
 according to the    between cars, lorries, etc. [W]
 time of day?
 How many            Note down number plates from cars parked outside school         Resource        Range & type of
 different number    Order and sort numbers                                          Sheets:PA30 and number                Do they take a logical
 plates can you      Use digits to generate other number using (+, -, x ) [M] [C]    PA31: Number Add, subtract            approach?
 find?               Solve clues to make number plates                               Plates 1 and 2  and multiply
 The teachers        Measure length and breadth of a car parked.                                     Measure and
 need a car park.    Survey teachers to find out how many car slots are needed       cm. tape        estimate
 Can you draw        Discuss how much space is needed for manoeuvring                metre stick                           Are their estimates
 out the car park    Mark out concrete to show slots correct size                    chalk           Range of shapes       reasonable?
 on concrete?                                                                                                              Is concrete marked
                                                                                                                           appropriately?
 How many            Go for a walk and sketch all the different shapes you see       Resource Sheet:
 different shapes    In class, make display of the signs                             PA32: Signs
 can you find in     Sort them into categories depending on corners and sides                          Angle
 street signs?       Use template to check for right angles.
                     Use a mirror and see which signs have line of symmetry                            Symmetry
                     Try other signs from the Highway Code.                          Highway Code
                                                                                                                                                                        Resource Sheet PA29-2
                                                                                                                                                                                                Maths Year 2000 Scotland
KEY              ACTIVITIES                                                       RESOURCES        STRAND        ASSESSMENT
QUESTIONS

What do you see Look at lampposts on both sides of a road and note down the       Resource Sheet Patterns and
on lampposts? letters and numbers                                                 PA33: Lamppost sequences
                  Discuss pattern formed                                          Patterns 1 and 2               Can they spot a
                  Resource Sheet Lamppost Patterns 2 [M]                                                         pattern?
What numbers Collect a selection of numbers from hydrant plates near the          Clipboards
do you see on     school
hydrant posts? Oral work guessing what numbers mean                                                Measure and
                  (the top number is the diameter of the water pipe in mm, the                     estimate
                  bottom number the distance from the hydrant plate to the pipe   Metre stick or
                  outlet in metres)                                               Trundle wheel
                  Measure distance to water cover to check.                                                      Can they measure?
What time does Use either knowledge gathered when children were standing          Clocks         Time            Can they apply time?
                                                                                                                                        Level B Theme: The Street continued




the bus stop      doing car survey or information from timetable                  Resource Sheet
outside school? Stand and check bus passes at scheduled time                      PA34: The Bus Money
How much does Discuss early/late?
it cost to travel Class work on clocks, and timetables                                             Add,
into town/        Oral work using money to pay fares                                               Multiply      Can they apply
another           Using knowledge gather information, and discuss how to                                         money?
destination?      display it
Can you make a
timetable and
display fare
information?
                                                                                                                                                                              Maths Year 2000 Scotland
                                                                                                                                        Resource Sheet PA29-3
                                                                                                                                                       Level B
KEY               ACTIVITIES                                                       RESOURCES       STRAND            ASSESSMENT
QUESTIONS

How are house    Look at house numbers on three different streets. Choose          Resource Sheet Range and type
numbers          streets where at least one has even and odd numbers on each       PA35: House       of numbers
arranged?        side of the street, and another where the houses are numbered     Numbers is
                 consecutively on each side of the street.                         suitable as a
                 Discuss number patterns: next to each other, odds and evens       template: fill in
                 etc.. Discuss road breaks, are numbers always consecutive after   the names of the Patterns and     Can they discuss
                 road break?                                                       streets you have sequences        patterns and generate
                 Make a frieze of front doors with house numbers. Have a           chosen. The                       own?
                 variety of sequences displayed [M] [W]                            street for part 1
                                                                                   should have       Add, subtract
                                                                                   consecutive       and divide
                                                                                   numbers, and for
                                                                                   part 2, odd and
                                                                                   even.
What numbers Look at a post-box. Copy down all the information on label.           Resource Sheet Time
are displayed on Draw digital time on clocks.                                      PA36: The Post
a post box?      Be at a post-box when letters are collected.                      Box                               Do they make
                 Estimate the number of letters collected, then ask.                                                 reasonable estimates?
                 Calculate number of letters collected at that post-box for the                    Add, Multiply     Do they get reasonable
                 day, the week, etc.    [C]                                                                          answers?
                                                                                                                                              Resource Sheet PA29-4
                                                                                                                                                             Level B
                                                                                                                                                                       Maths Year 2000 Scotland
                                                         Maths Year 2000 Scotland
                                                     Resource Sheet PA30
Name:_____________________                                       Level B


                           Number Plates 1


   Write down the number plates of TEN cars, like this:
        G 637 GOA             E757 JPJ       H 832 TFS
   Use the numbers from each plate and write them in a list:
        e.g.      637
                  757
                  832
                  ...
   1. Organise your list so that the largest number is
      first and all the other numbers are in order of size
      below that number.
   2. Put your numbers into two lists. One list for odd
      numbers. One list for even numbers. Are there
      more odd numbers or even numbers?
   3.   Make three lists of numbers under headings:
               smaller than 200   200 – 800 greater than 800
   4. Write each number in words.
                                                   Maths Year 2000 Scotland
                                                Resource Sheet PA31
Name:_____________________                                 Level B/C


                              Number Plates 2


   1. Choose one number plate
              e.g.      G 637 GOA
       Write down the number
              e.g.      637
       How many numbers can you make using the digits
       and + and – ?
              e.g.      6 = 3 + 7 = 16
                        6+3–7 = 2


   2. Use a calculator. How many numbers can you
      make using the digits and + and – and x ?
       e.g.          (6 x 3) + 7 = 25
                     (6 + 3) x 7 = 54


   3. Choose another number plate and try these
      questions again. Did you make the same amount
      of new numbers as you did before?
                                                         Maths Year 2000 Scotland
                                                      Resource Sheet PA32
Name:_____________________                                       Level B/C


                             Signs


   1. Write the name of each shape next to the sign.




   2. Use a mirror. If a sign has a line of symmetry,
      draw the line of symmetry over the sign.
           Red circles tell you not to do something
           Blue circles tell you what you must do
           Triangles give warnings
           Rectangles give information
   3. Design three signs of your own.
       They must each be a different shape.
   4. Explain what they are for.
                                                     Maths Year 2000 Scotland
                                                Resource Sheet PA33-1
Name:_____________________                                  Level B/C


                    Lamp Post Patterns 1


   1. Look at four lamp posts in a row
       Write down the code for each one


   2. Cross the road
       Write down the code for four more
       What do you notice?


   3. Go to another road
       Look at the codes.
       Write down four from each side.


       What do you notice?




   4. Write down the names of four streets near where
      you live.
   5. Invent a code for four lamp posts on each of these
      streets.
                                                           Maths Year 2000 Scotland
                                                      Resource Sheet PA33-2
Name:_____________________                                        Level B/C


                       Lamp Post Patterns 2


                                                            L
                                                            A
     L
                                                            5
     A
                                                            9
     5
                                                  L
     3
                                                  A
                                                  5
                                                  8


   1. Fill in the missing codes on the lampposts above.
   2. These codes are all from one side of the street.
      Fill in the empty spaces.

         B      B       B      B      B       B
         R      R       R      R      R       R
                        3
                        2



   3. Complete these patterns:
         102,   104,    ____, ____, ____, ____, ____
         86,    84,     82,    ____, ____, ____, ____
         35,    37,     ____, ____, ____, ____, ____
         191,   189,    ____, ____, ____, ____, ____
         ____, ____, ____, 50,        52,     ____, ____
         ____, ____, 27,       ____, 23,      ____, ____
         ____, ____, ____, ____, ____, 103,               105
                                                                     Maths Year 2000 Scotland
                                                                  Resource Sheet PA34
Name:_____________________                                                    Level B


                                 The Bus


   1. When does the __________ bus pass the school?
      Draw the MINUTE HAND for each clock
               11 12 1             11 12 1             11 12 1
             10        2         10        2         10        2
             9          3        9          3        9          3
             8         4         8         4         8         4
               7 6 5               7 6 5               7 6 5



   2. Go to school gates at four times when the bus is
      due.
       Is it always on time? Is it early? Is it late?
       How many people are on the bus?
       Why do you think this is?
       Where is the last stop of the bus?
       When does the bus get there? Draw the minute hand.
               11 12 1             11 12 1             11 12 1
             10        2         10        2         10        2
             9          3        9          3        9          3
             8         4         8         4         8         4
               7 6 5               7 6 5               7 6 5



       The _____________ bus leaves _______________
       at ________________________ and arrives at
       school at ________________________.
       Draw digital times:
           Leaves:           :            Arrives:            :
                                                         Maths Year 2000 Scotland
                                                    Resource Sheet PA35
Name:_____________________                                     Level B/C


                        House Numbers
   1. Write down six house numbers next to each other
      in each of these streets:
           Street 1: ______________________
           Street 2: ______________________
   2. Use the house numbers from Street 1:
               Choose any two next door numbers
               Add them together
               Add 19
               Divide by 2
               Subtract the first number
       What is your answer?
   3. Try with another pair of numbers, next door to
      each other. What happens? If you try again with
      another pair, does it happen again?
   4. Use house numbers from Street 2:
               Choose any two next door numbers
               Add them together
               Add 19
               Divide by 2
               Subtract the first number
       What problem do you have?
       What instruction can you add in to sort it out?
                                                              Maths Year 2000 Scotland
                                                           Resource Sheet PA36
Name:_____________________                                            Level B/C


                           The Post Box


   1. Copy the information about
      collection times on the post box
      into this square:
   2. Draw collection times for week
      days.
              11 12 1          11 12 1
            10        2      10        2
            9          3     9          3
            8         4      8         4
              7 6 5            7 6 5




              11 12 1          11 12 1        11 12 1
            10        2      10        2    10        2
            9          3     9          3   9          3
            8         4      8         4    8         4
              7 6 5            7 6 5          7 6 5



   3. Go to the box when one of the collections is going
      to be made.
   4. Estimate how many letters are emptied from the
      box.


   5. Ask the postman if he knows roughly how many
      letters are emptied in each collection.
   6. Calculate how many letters are collected in one
      day. (Round each collection to nearest ten.)
   7. Calculate how many letters are collected in one
      week.
                                                                                 Maths Year 2000 Scotland
                                                                              Resource Sheet PA37
N a m e : ____________________                                                            Level B


                                    Quadrilaterals
Draw differently-shaped QUADRILATERALS by joining dots on the 9-dot square.

Two have already been drawn.

Find another fourteen.
                                                                                       Maths Year 2000 Scotland
                                                                                   Resource Sheet PA38
N a m e : ____________________                                                                 Level B


                                         Pentominoes
Pentominoes are made by arranging five squares together so that each square touches at least one other
square along the length of a side.

There are twelve different pentomino shapes in total. Two have been drawn for you . Draw the other ten.
                                                                                        Maths Year 2000 Scotland
                                                                                    Resource Sheet PA39
Name:______________________                                                                     Level C


                                               Targets
The Target is the number in the circle. Use the other four numbers in any order and plus, minus,
multiplication and division signs to make the Target number.




      Make up some Targets of your own
                                                                                       Maths Year 2000 Scotland
                                                                                   Resource Sheet PA40
N a m e : ____________________

                                  Make a Pop-up Card!

How to make a basic mechanism


     Take a sheet of card

     Fold in half

     Cut along two parallel lines, which are perpendicular to the folded edge

     Score, then fold along the dotted line shown

     Pop up!




Decorating the card
     Decorate the card by sticking on shapes as shown

     Place another sheet of paper behind so that you can’t see the hole when the card is shut


Alternative pop-up mechanism
     Take a sheet of card

     Fold in half

     Cut a single line perpendicular to the folded edge

     Score, then fold along the dotted lines as shown

     Pop up!


Challenges
     Make a pop-up Christmas tree

     Make pop-up letters
                                                                                            Maths Year 2000 Scotland
                                                                                     Resource Sheet PA41-1



                                     Structures & Straws
In this activity we explore the things you can build with triangles whose sides are all the same length.

Making shapes with straws
First we’ll make a triangle and a square from straws and compare their properties.
To make a triangle or square:
1. Take a bendy straw, and bend it. Either side of the bend are two straight lengths of straw: the short
   end, and the long end.
2. Take a pair of scissors, and cut down the length of the short end up to the bend. Don’t cut through
   the bend itself.
     If possible, make the cut by inserting one blade of the scissors into the straw, and cut up to the bend.
3. You can now squish the short end so that it is thinner than the long end of another straw. Squish the
   short end of the straw, and push it into the long end of another bendy straw.
4. When you have three or four straws joined together in a line, join the short end of the last straw to the
   long end of the first straw, to create a triangle or a square.
Once you have a triangle and a square made up, try distorting them. Can you change the square into
another kind of shape? - what about the triangle?
You should have found that the square is easily changed into a diamond shape (a rhombus) just by changing
the angles between the sides. However the only way to change the triangle is to change the length of at
least one side. To do this you have to break it. It is this property that makes them strong.

Building structures
1. Make lots of triangles to experiment with. What do you notice about the triangles you have made?
2. Join the triangles together to form a variety of 3D shapes. Line up the edges of two triangles and bind
   them with a bit of sticky tape round the middle. Add other triangles to the free edges and then fold
   the outer ones up to meet each other. Try to make symmetrical shapes.
3. Which 3D shape has the fewest triangles?
4. Once you have built a collection of pyramids try building towers from them. Arrange 3 on the table so
   that you can balance corners of the base of a 4th one on their points. This will make a bigger
   pyramid. Tape these 4 pyramids together. Try to arrange more to balance this pyramid on top. Make
   more layers to fit underneath. Remember to tape them in place as you go along. Try to make each
   layer with as few pyramids as possible to make a tower. How high can you go?
                                                                    Maths Year 2000 Scotland
                                                                Resource Sheet PA41-2



                  How to make a triangle from straws
You will need:

       3 straws                             Scissors


What to do:

       Bend each straw




       Take each straw, and cut the short end
       up to the bend with the scissors




       Squash the short end of the short straw and
       push it inside the long end of the second straw




       Now, squash the short end of the second straw,
       and push it inside the third straw




       Finally, squash the short end of the third straw,
       and push it inside the long end of the first straw, to
       make a triangle




Well done! You only need four of these and a square made from straws to
make a pyramid!
                                                                                           Maths Year 2000 Scotland
                                                                                       Resource Sheet PA42
                                                                                                 Level C/D

                                        Mirror Alphabet




           A B C D E F G
           H I J K L M N
           O PQ R S T U
            VWX Y Z
1. Stand a mirror on the paper along the top of these letters so you can see the reflection of the letters in
   the mirror. There are 9 that look the same in the mirror as on the paper. Which ones are they?
2. Try making words from these letters. Write them in block capitals and see if you can read them in the
   mirror.
3. Find the 11 letters that you can still read if you put the mirror through the middle of the letter from top
   to bottom. Which ones are they?
4. Make words from those letters that you can read when you put the mirror down the middle of the
   middle letter. Words that are spelt the same forwards and backwards are called palindromes. Are all
   palindromes symmetrical?
5. Try making up a sentence from these letters. Now, write it down back to front. Where do you put the
   mirror so that you can read it?
                                                                                         Maths Year 2000 Scotland
                                                                                  Resource Sheet PA43-1



                                       Patchwork Tiles

1. There are three templates for patchwork jigsaw puzzles with this sheet. Cut out the squares and
   triangles. Do not to mix up the tiles for different patterns. Old Windmills has 16 small right angled
   triangles; Maple Leaf has 8 large right angled triangles and 5 squares; Ohio Star has 16 small right
   angled triangles and 5 squares.

    To make them more difficult you can cut the squares in the Maple Leaf and Ohio Star into the same
    size of triangles as the rest of the puzzle.

2. Arrange the patchwork quilt pieces into a pattern.

            •     Try to create one of the traditional square patchwork quilt designs. (The Ohio Star, Old
                  Windmills or the Maple Leaf).

            •     Do any of the patterns look symmetrical?

            •     Do you have to use some of the smaller shapes to create larger shapes?

            •     Try to create a new patchwork quilt design.

    These traditional designs are mathematical as they have either reflectional symmetry (as with a mirror)
    or rotational symmetry (the item has been moved around a point before being repeated) - some have
    both. All these designs are square (either 2x2 or 3x3) and most of the pieces are either squares or
    right-angled isosceles triangles.

    Patchwork quilts (both square and hexagonal) are a form of what mathematicians call a tiling or
    tessellation. This is a way of covering an area of any size with a few tiles without any gaps or
    overlaps. Mathematicians study tilings because they are interesting and have many practical
    applications from the study of crystals to tiling a floor.

3. Look out for man-made and natural tile patterns.
                      Maths Year 2000 Scotland
                  Resource Sheet PA43-2



Patchwork Tiles

  Maple Leaf
                      Maths Year 2000 Scotland
                  Resource Sheet PA43-3



Patchwork Tiles

Old Windmills
                      Maths Year 2000 Scotland
                  Resource Sheet PA43-4




Patchwork Tiles

  Ohio Star
                                                                                           Maths Year 2000 Scotland
                                                                                   Resource Sheet PA44-1



                                          Möbius Strips

What do I do?
1. Take a rectangular piece of paper. I’ll refer to
   the corners as A, B, C, and D, as shown. How
   many sides does this piece of paper have?
2. What shape would you create if you were to
   stick the short edges together, matching A to C,
   and B to D? (Try it, if you like.) Does the paper
   still have two sides?
3. If you stuck the paper together into a ring, make it into a rectangle again (carefully cut with the
   scissors if you need to).
4. Now make another ring, but this time give one end a half twist, so you match A with D and B with C.
   Join up the short ends so you have a single ring, as shown. This is called
   a Möbius Strip or Möbius Band.
5. How many sides does a Möbius Strip have? Take a pen and start
   midway between the ‘edges’ of the strip. Draw along the length of the
   strip, down the centre. What happens?
6. Take a brush pen, and hold the edge of the Möbius Strip against the pen
   nib. Colour the edge of the Möbius Strip by holding the highlighter still
   and rotate the Möbius Strip around (this is quite tricky). Are there any
   edges left that are not coloured in? How many edges does the Möbius
   Strip have?
7. Carefully take a pair of scissors and cut down the central line (ask for
   help with this part if you need it). What shape do you have now? How many sides does it have?
8. If the shape you have created has a thick enough band, you can use it. Otherwise, create a new
   band, but this time give one end two half twists before sticking the short ends together matching A to
   C and B to D, but with a whole twist in the paper. How many sides does this have? Draw a line in
   the middle of the strip again. Carefully cut down this line. What have you created?
                                                                                         Maths Year 2000 Scotland
                                                                                   Resource Sheet PA44-2


Möbius Strips continued...




Is this maths?
Yes! The Möbius Strip or Möbius Band was discovered by August Ferdinand Möbius, a nineteenth
century German mathematician and astronomer. Möbius strips are unusual as they have only one side and
one edge, but aren’t infinite objects. Möbius was one of many mathematicians who pioneered topology –
the mathematical study of knots, stretchy surfaces, and objects that get flexed or bent about. Topology is
useful to people studying everything from molecules of DNA to the shape of the universe.

Where can I find Möbius strips?
The B. F. Goodrich Company patented a conveyor belt in the form of a
Möbius strip which lasts twice as long as conventional belts, as the wear across
the belt was completely even.
Some continuous-loop recording tapes were shaped as Möbius Strips to
double the playing time. In the 1960s Sandia Laboratories used Möbius Strips
in the design of versatile electronic resistors.
Möbius Strips also feature in the work of artists, including, most famously,
M.C. Escher.
One form of the ‘chasing arrows’ recycling symbol is based on the Möbius
band, another is the Möbius band with three twists.

What else can I try?
1. Try altering the number of twists you make before sticking the short ends together. What happens
   when you put in 3, 4 or 5 half twists? What happens when you cut these down the centre?
2. Try repeatedly cutting the Möbius strip down the centre line, and then repeating with each subsequent
   band you get. What happens? Which bands are connected to which other bands? Do the bands
   cycle from one type of band to another?
3. Find out about Klein bottles: a three-dimensional bottle with only one surface.
                                                                                             Maths Year 2000 Scotland
                                                                                     Resource Sheet PA45-1



                                            Penrose Tiles

Pupil investigation
1. Cut out tiles from the sheet
2. There are two shapes of tiles here – kites and darts. Kites are, well, kite-shaped, and darts are
   shaped like arrowheads.
3. You have to follow the matching rule in order to get the correct effect: Each tile has two lines: a solid
   line and a dotted line. The dotted lines must join to dotted lines, and the solid lines must join to the
   solid lines. Also the whole length of the tiles’ sides must line up.
4. Start with a single tile, and add tiles according to the matching rule. Try to cover the table – it is
   harder than it looks! You may get stuck and have to remove some tiles and try again. There are
   seven different ways of arranging the tiles round a point – can you find any of them?

Is this maths?
Yes! Tilings of all shapes and sizes are studied in maths. Tiling is a way of covering an infinitely large area
using copies of a few different tiles, or shapes. Tiles can be any shape which can completely cover a flat
surface. Squares and regular hexagons – e.g. floor tiles or honeycombs are simple examples.
Penrose tiles, named after the British mathematician Sir Roger Penrose, who discovered them in 1974, are
very unusual. The pattern of tiles created with Penrose tiles is never periodic – it does not repeat in a
regular fashion, like wallpaper does. But, the pattern you see in any small section will be found nearby –
and, eventually, in any other pattern of Penrose tiles.

Where can I find Penrose Tiles and other tiles?
Why do mathematicians study tilings? They are interesting in themselves, and also have many practical
applications. One such application is in crystallography – the study of how chemical atoms fit together.
Some chemical substances will form crystals in a manner similar in layout to the Penrose tiles. A French
company has recently found a very practical application for substances that form these quasi-crystals: they
make excellent non-scratch coating for frying pans.
Look out for tile patterns around you, both man-made and natural. How often do they repeat? What
shape, or shapes, are the tiles?
                                                                                       Maths Year 2000 Scotland
                                                                                Resource Sheet PA45-2



                                         Penrose Tiles


                                                                         Kite




                                Dart




These are the seven “legal” ways of arranging tiles about a single point. The names were suggested by
John Conway, another mathematician who has worked with Penrose tiles.
                                                              Maths Year 2000 Scotland
                                                          Resource Sheet PA46-1



                                         Pyramid Puzzle


Make these shapes from building bricks




Now use them to make a pyramid like this
                                                                                  Maths Year 2000 Scotland
                                                                          Resource Sheet PA46-2



                                      Questions


Some sum!

Find numbers whose sum is 12 and whose product is as large as possible.

For example, 2 + 5 + 5 = 12 and 2 x 5 x 5 = 50. You should be able to do better than that.




The river puzzle

The river Puzzlement is 50m wide. It is spanned by a new bridge, of which one quarter lies on the
West bank and one eighth lies on the East bank. How long is the bridge?




Magic TEN

Place a straw on each line of the diagram. How many straws are there altogether?




Now remove six to leave ten

								
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