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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