<p> Some History of Mathematics</p><p>1 June 2009 HM/JZ Who invented the numbers on the right? When?</p><p>How did they get to Europe? Who brought them to Europe? Why?</p><p>How was the use of these numbers ‘popularised’?</p><p>2 June 2009 HM/JZ Magic Squares</p><p>This is a MAGIC SQUARE 8 1 6 Each row adds up to ...... ? Each column adds up to ...... ? 3 5 7 Each diagonal adds up to ...... ?</p><p>What is it that makes a square 4 9 2 magic?</p><p>What patterns can you find in this magic square?</p><p>Colour in the odd numbers. What do you notice?</p><p>Can you explain why ‘5’ has to go in the middle square?</p><p>Some more magic squares</p><p>3 June 2009 HM/JZ Find the magic number first to help you to fill in the blank squares 1 8 15 5 7 14 4 6 13 22 10 19 3 11 18 2</p><p>In each of the 3 magic squares you 30 39 1 10 19 have now completed there is something special about one of the 7 9 27 29 diagonals? What is it?</p><p>Where is the number 1 in each 46 6 8 26 37 magic square? 5 14 16 25 34 36 What is the connection between the size of the square, the middle number and the magic number? 13 15 24 33 42 4 21 23 43 3 12 31 40 49 2 11 20</p><p>Magic Squares again</p><p>In 1612, Claude Bâchet found out how to make a 3 by 3 Magic Square from this arrangement</p><p>1 4 2 7 5 3 8 64 June 2009 HM/JZ 9 Can you finish Bâchet’s magic square? All the lines must add up to ...... ?</p><p>What is the method? </p><p>Make a Bâchet 5 x 5 magic square. Start like this:-</p><p>1 6 2 11 7 3 16 12 8 4 21 17 13 9 5 22 18 14 10 23 19 15 24 20 25</p><p>Now make a 7 x 7 Bâchet magic square.</p><p>Look for patterns.</p><p>Can you predict?</p><p>What are the rules before you start the square?</p><p>What is the biggest Bâchet magic square you can make on your A4 paper?</p><p>Explain carefully what you have to do.</p><p>5 June 2009 HM/JZ INVESTIGATE Join the middle of each square in order.....1 -> 2 -> 3 etc. What do you notice? Can you predict?</p><p>Use two colours to shade in the odd and even numbers. What do you see? Can you make any predictions? How many evens? How many odds?</p><p>Compare the size of the square and the middle number? What is the link?</p><p>RESEARCH</p><p>Find out some different ways to construct magic squares.</p><p>Is it possible to make a magic square with an even number along the sides?</p><p>Benjamin Franklin was very interested in Magic Squares. Find some of the ones he ‘invented’. What else was he famous for?</p><p>In 1514, Albrecht Dürer, an artist, used a magic square in one of his paintings. Can you make a copy of his magic square?</p><p>Find out about the Lo-Shu Magic Square from Ancient China from thousands of years ago </p><p>What can you find out about Claude Gaspar Bâchet (1581-1638)? What can you find out about the maths he discovered?</p><p>Different ways with multiplication</p><p>Look carefully at these different methods and then try to work out how to use them. Practise each method until you can do it.</p><p>6 June 2009 HM/JZ 1 Gelosia multiplication Arab.....over 800 years old</p><p>How do you find the numbers to go into the triangles?</p><p>How do you find the numbers in italics?</p><p>From this GELOSIA I have found that 264 x 53 = 13 992</p><p>Can you understand how I have worked this out?</p><p>Use this method to work out these multiplications.</p><p>The first one is started for you.</p><p>3 5 1 35 x 23 2 2 124 x 16 3 3 32 x 243</p><p>4 215 x 346</p><p>5 367 x 431</p><p>6 1298 x 532</p><p>Which part of the Gelosia represents the UNITS the TENS the HUNDREDS the THOUSANDS and so on? Colour the units section in one colour, the tens in another colour and so on. 2 Russian multiplication ...... doubling and halving</p><p>This method is hundreds of years old.</p><p>I am going to use the method to do 27 x 35. Numbers in left column are halved...... ignore remainders. Follow the method carefully. Numbers in right column are 27 35 doubled.</p><p>Cross out rows with even 7 number on left. June 2009 HM/JZ</p><p>Add up the right hand column. 13 70 6 140 3 280 1 560 945</p><p> so 27 x 35 = 945</p><p>Check this on your calculator.</p><p>Use Russian multiplication to work out</p><p>1 42 x 56 2 65 x 9 3 36 x 125</p><p>4 35 x 27 5 256 x 32 6 248 x 347</p><p>8 June 2009 HM/JZ 3 Egyptian multiplication</p><p>This method is over 2000 years old. It was found on an ancient Egyptian document called the Rhind Papyrus I want to find 25 x 38 First I work out 1 x 38 = 38 2 x 38 = 76 4 x 38 = 152 8 x 38 = 304 16 x 38 = 608</p><p>(How do I do this quickly and easily?)</p><p>Then I work out how to make 25 with these tables</p><p>25 = 1 + 8 + 16</p><p> so 25 x 38 = 38 How do I know what 304 numbers to add + 608 together? 950</p><p>Look at this set of numbers and carry it on until you have 10 numbers</p><p>1 2 4 8 16 ...... </p><p>How is the sequence built up?</p><p>Make these numbers by adding together numbers from the sequence. a 14 b 28 c 35 d 129 e 56 f 100 g 1000 h 1999</p><p>Can you find a number that cannot be made by adding numbers from the sequence?</p><p>Use the Egyptian multiplication method to work out</p><p>1 56 x 25 2 33 x 42 3 75 x 42 4 23 x 43 5 135 x 48 6 279 x 83</p><p>Think about the ways of multiplying you have discovered on the last 3 pages. Which did you like best? Which did you find hardest? Practise the method you liked best so you can use it whenever you multiply.</p><p>9 June 2009 HM/JZ Count It!</p><p>We count in TENS...... probably because we have 10 fingers. But we could count in sets of 5, or 20, or 2 or ...... </p><p>Our counting system is called Hindu-Arabic because it originated in India and came to Europe via the Arab countries of North Africa.</p><p>Some peoples from other parts of the world developed different number systems.</p><p>Look carefully at the MAYAN system from Mexico about 500 BC to 600 AD.</p><p>Our numbers are arranged in columns, headed .... Th (10 x 10 x 10) H (10 x 10) T (10) U (units) ...... Mayan numbers are in rows. </p><p>Mayan row value Mayan numbers are read from the bottom up. 8000 (twenty x twenty x twenty) 400 (twenty xtwenty) 20 (twenty) 1 (units) Complete the following table...look carefully at the symbols used.</p><p>Babylonian Number System</p><p>10 June 2009 HM/JZ This system uses just two symbols. Their value depends on which column they are in. Can you fill in the missing numerals in the Babylonian system?</p><p>Copy and complete the following:-</p><p>The Babylonian system uses sets of ...... </p><p>We still use some of the Babylonian system.</p><p>We use SIXTIES when we are working with ...... </p><p>60 seconds = 1...... and 60 minutes = 1 ...... </p><p>360 (6x60) degrees = ...... </p><p>Find out anything else you can about the Babylonians</p><p>11 June 2009 HM/JZ Egyptian Number System</p><p>The Egyptian system has more symbols than the Babylonian system</p><p>Use Egyptian numbers to write down your age the number of your house the number of this month your height in centimetres today’s date...day:month:year (in full) the answer to 4 567 x 1 328</p><p>12 June 2009 HM/JZ Roman Numbers </p><p>Like the Egyptians, the Romans used many symbols.</p><p>1 I 5 V 10 X 50 L</p><p>100 C 500 D 1000 M</p><p>C stands for centum which is Latin for 100, and M is for mille..Latin for 1000</p><p>IV is 4...... one before 5 VI is 6...... one after 5</p><p>1 2 3 4 5 6 7 8 9 10 11 12 I II III IV V VI VII VIII IX X XI XII</p><p>40 50 60 90 100 110 400 500 600 900 1100 XL L LX XC C CX CD D DC CM MC</p><p>3 237 is MMM CC XXX VII in the Roman number system</p><p>Write the numbers in the previous list in Roman numerals. Write the same numbers in the Babylonian system. Which system is easiest to use? Which system gives the longest ‘number’ for the year? Which system do you prefer? Why?</p><p>13 June 2009 HM/JZ Pa-Kua … Ancient Symbols of the Orient</p><p>This is the South Korean flag? Can you use compasses and ruler and construct it?</p><p>14 June 2009 HM/JZ Fu His Hexagrams … From Ancient China</p><p>A hexagram is a symbol in 6 parts.</p><p>The Fu Hsi hexagram is made up of a yin (broken line) that stands for 0 and a yang (solid line) that stands for 1. </p><p>The position of the solid line is important.</p><p>Complete the table number Fu Hsi number Fu Hsi number Fu Hsi 0 8 16 1 9 17 2 10 18 3 11 19 4 12 20 5 13 21 6 14 22 7 15 23 Can you get to 63? What happens next?</p><p>15 June 2009 HM/JZ Binary System</p><p>What does binary mean? Why is it important? (Hint – you have a mobile phone – I have a laptop)</p><p>Imagine only being able to count in sets of 2 and only having the symbols 0 and 1 to use.</p><p>You would need to think about the way the numbers were grouped and the column headings you used. </p><p>Look at this table fours twos units 1 1 2 1 0 Can you see how I 3 1 1 am working out my 4 1 0 0 binary numbers? 5 1 0 1 6 1 1 0 7 1 1 1 8 what do I do now? </p><p>Can you continue the table to 64? Think carefully about the number of columns you will need and the headings you will use.</p><p>Find out why the binary system is important to us today and where it is used.</p><p>How can you change the binary number 100111001 back into a decimal number ? (decimal system uses sets of 10.....our usual system!)</p><p>256 128 64 32 16 8 4 2 1 1 0 0 1 1 1 0 0 1</p><p>First put it in a table with the binary system’s column headings</p><p>Now you can see that you have one 256, one 32 ,one 16,one 8 and one 1</p><p>256 + 32 + 16 + 8 + 1 = 313 so 1001110012 = 31310</p><p>Convert these binary numbers into decimal numbers</p><p>1 11101 2 101101 3 1001001 4 11001100</p><p>16 June 2009 HM/JZ The Golden Rectangle</p><p>Copy this pattern and carry it on for 2 more </p><p>Write down the length of the side of each new square you add - the shaded square side length</p><p>1 1 2 3 5 ...... </p><p>Investigate the number pattern ...can you carry it on for 20 terms?</p><p>Find out who it is called after.</p><p>For each rectangle you have drawn work out long side short side (long side  short side)</p><p>What happens?</p><p>This picture shows the Parthenon in Greece. Measure the length and the </p><p> width of the rectangle. Work out length  width. What do you notice?</p><p>Rectangles with this proportion are called Golden Rectangles. They are supposed to be the most pleasing and popular rectangles.</p><p>17 June 2009 HM/JZ Can you find any Golden Rectangles around school...check windows, tables, doors and so on. </p><p>Construct a Golden Rectangle First draw a square, halve it and put in the diagonal of one half.</p><p>B Open compasses to length AB. With centre A draw an arc. Where the arc meets the baseline, C, is the corner of the Golden Rectangle. Complete the rectangle. A C Check that it is Golden </p><p>Draw a regular pentagon (each angle is 108o if that helps) Join each vertex to every other vertex (a vertex is a ‘corner’) – join the alternate corners, this is Measure all the different lines in the design. Can you find the Golden Ratio by dividing one line by another? How many ways can you find it?</p><p>Research ..... Find out where Leonardo da Vinci, a sixteenth century Italian artist, found the Golden Ratio.</p><p>Your own research</p><p>Think about something in the History of Mathematics that you would like to find out about.</p><p>18 June 2009 HM/JZ Maybe an ancient counting system like Chinese Rod numerals</p><p> or the story of weights and measures,</p><p> or the story of calendars,</p><p> or a particular topic..such as prime numbers magic squares Pascal’s triangle or...... ?</p><p> or a famous mathematician like Hypatia Pythagoras Sophie Germain Euler Ada Lovelace or ...... ? Think about your presentation Video Powerpoint Podcast Poster</p><p> http://www-groups.dcs.st-and.ac.uk/~history/Indexes/HistoryTopics.html http://groups.csail.mit.edu/medg/people/doyle/gallery/mathematicians/ http://www-history.mcs.st-and.ac.uk/history/Indexes/Women.html http://www.mcs.surrey.ac.uk/Personal/R.Knott/ http://www.agnesscott.edu/lriddle/women/chronol.htm ac means UK university website edu means USA uni</p><p>Here are some useful website addresses If you find some other useful websites make a note of the addresses.</p><p>19 June 2009 HM/JZ</p>