Development of number through the history of mathematics ... · The Rhind mathematical papyrus I use my IWB version to display Activity sheet 1 (the Rhind mathematical papyrus image)

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History of Mathematics Multiplication

Development of number

through the

history of mathematics

Multiplication

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History of Mathematics Multiplication

Development of number through the history of mathematics Topic: Tables of numbers

Resource content Teaching

Resource description

Teacher comment

Mathematical goals

Starting points

Materials required

Time needed

What I did

Reflection

What learners might do next

Further ideas

Artefacts and resources Activity sheets and supporting historical information

Activity sheet 1: The Rhind mathematical papyrus

Supporting historical information (Activity sheet 1)

Activity sheet 2: The piece of papyrus

Supporting historical information (Activity sheet 2)

Activity sheet 3: Another piece of papyrus

Supporting historical information (Activity sheet 3)

Activity sheet 4: Bamboo pieces

Supporting historical information (Activity sheet 4)

Activity sheet 5: A small piece of bamboo

Supporting historical information (Activity sheet 5)

Activity sheet 6: Egyptian numbers

Activity sheet 7: Arrangements of Egyptian numbers

Activity sheet 8: Chinese numbers

Activity sheet 9: Egyptian and Chinese numbers

Activity sheet 10: Which scripts?

Resource description A selection of artefacts representing papyri and bamboo fragments. Learners work in groups as teams of archaeologists or mathematical detectives to examine, interpret and then give meanings to mathematical artefacts. Once done the mathematics that underpins the artefacts is explored (place value and multiplication methods).

Teacher comment

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The resource is set within the context of the history of mathematics and how mathematics is discovered and/or invented. Considerable importance is attached to the use of „real‟ artefacts and historical accuracy – though interpretation is more open to debate. Options are available to link to real artefacts. This is a lesson that has been used over a number of years with beginning teachers and with a cohort of teachers on the Mathematics Development Programme (a 40 day course for secondary non-specialists teachers of mathematics). These teachers have then used this lesson with their learners. This version is a considerable adaptation of earlier versions having been supplemented with more resources and a greater variety of mathematical resources.

Mathematical goals To help learners to:

understand how an archaeologist might interpret „evidence‟ collected from an artefact

develop a better understanding of the importance and advantage of using a place value system

realise that sometimes different mathematical developments occurred in different cultures

understand that similar developments in mathematics often occurred in many parts of the world

place mathematical development in a historical and geographical context

become more familiar with different methods of multiplication

compare and contrast different methods of multiplication

Starting points An ability to multiply two numbers. This module is aimed at KS3 learners.

Materials required For each pair of learners you will need:

Activity sheet 2: A piece of papyrus

Activity sheet 3: Another piece of papyrus

Activity sheet 4: Bamboo pieces

Activity sheet 5: A small piece of bamboo

Materials to create posters

Internet for some/all learners (optional) Interactive whiteboard and projection resources You may find it easier to project the Activity sheets, using a data projector, a visualiser or an overhead projector with a transparency. Alternatively you might want to use the Promethean ActivStudio and Smart Notebook IWB versions of the activities. Wherever items for display are subject to copyright restrictions direct links are provided for them.

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Activity sheets 1 to 9

„Supporting historical information‟ Activity sheets 1 to 5

Time needed At least one hour and up to three hours. If time is short, fewer artefacts may be considered and the use of the resource adapted.

What I did: Beginning the session I told the learners that they will work as archaeologists (or mathematical detectives) to examine, interpret and then give a meaning to mathematical artefacts – in effect they need to „translate‟ what they see, interpret them and suggest meanings. I let them know that they are matching, to some extent, the process that archaeologists and mathematicians have gone through when working with this or similar materials, and this would include hiding results from other groups. I use the IWB versions, but if you cannot do that you can use the Activity sheets in the most appropriate way for your classroom. Whole group discussion (1a) How has mathematics developed? First, I set the scene, and ask some questions (answers are included to help you):

How do you think we know about mathematics from years ago? Mostly from written records and artefacts (usually marked or written on).These records/artefacts are then interpreted by mathematicians, archaeologists or specialists – they do not always agree (especially about the older artefacts).

Was mathematics invented or discovered? There is a debate about this with some people arguing it is an invention, others that things are discovered and others believe it is a bit of both. There is perhaps no correct answer. A view from the Ask a scientist website is as below. “Before deciding whether mathematics was invented or discovered, we must clarify terminology.

Discovered: The thing always existed. Someone found it. Invented: The thing did not previously exist. Someone created it.

Then we must clarify what mathematics is. Mathematics is a tool, a model. It is something that we can use to describe or model parts of reality, or any other system based on quantifiable things. Mathematics can be used to model finance, logic, even colour. Mathematics itself did not exist before the first mathematicians. What was discovered was how to mould the model to fit reality. What was invented was the model itself. Mathematics was invented. How to use mathematics was discovered.”

Which people were involved in mathematical discovery/invention and where did they come from? All over the world though more tends to be known about people from the western world. Different aspects of mathematics became important at different times, and often mathematics has developed in different ways at different speeds in different parts of the world.

How long ago did people invent or make discoveries about mathematics? There is considerable debate about this. The Wikipedia Timeline of mathematics suggests 70,000 BCE and the process still continues.

Is mathematics an area where people are still finding out things? Yes. How it is

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used can be seen, for example in a booklet called Where the Maths you learn is used. Learners may mention some of the following comments with varying levels of detail (often very little) –make a note of what they say since you may want to follow up on these things later. Lists are here also for you to give them a broad idea of where mathematics has come from.

Arab explorers – algebra; geometry; number system; tessellations

Babylonians – „cuneiform‟ writing; clay tablets; 360 degrees in a circle

Chinese – sticks/rods for numbers;

Egyptians – hieroglyphics; rope measurements; papyrus; pyramids

Greeks – Pythagoras; geometry; pi (i.e. , and circles

Indians – numbers, trigonometry For a list of key sources of information see the bibliography section. Here are three different timelines that provide information related to the history of mathematics. The MacTutor timeline is in parts, starting at 800 BCE (Before the Common Era, the same as AD/BC notation) offering a timeline of mathematicians. The Wikipedia timeline goes back to 70,000 BCE but there is discussion, in particular, about what is missing. Whole group discussion (1b) The Rhind mathematical papyrus I use my IWB version to display Activity sheet 1 (the Rhind mathematical papyrus image) and make use of the items found on the page Supporting historical information (Activity sheet 1). I then set the challenge: “This is a piece of papyrus. What do you think it was used for and where do you think that it comes from? “ I take in comments and suggestions. See, in particular the transcript of the BBC podcast for answers [since original plan was used the BBC have taken down the resource noted here]– but basically the Rhind mathematics papyrus is a set of questions and worked solutions of how to solve 84 problems (paragraphs 13 and 14 of the transcript) with problem titles on the papyrus in red and solutions in black. Working in groups (1) The mathematical detective/archaeologist: examining the artefacts I then have ready for each team of archaeologists or mathematical detectives the artefacts which represent

a piece of papyrus (Activity sheet 2)

another piece of papyrus (Activity sheet 3)

bamboo pieces (Activity sheet 4) - in the form of a jigsaw

a small piece of bamboo (Activity sheet 5)

Which scripts? (Activity sheet 10) - optional

I give out Activity sheets 2 and 4 first to alternate groups/pairs of learners. The first time that you use Activity sheet 4 learners will have to cut out the jigsaw pieces (store them for later use in envelopes) – it helps if each set is a different colour. I tell them to keep their work hidden from the competition (the others). I usually say something like:

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“Examine the artefact you are given, translate it and then interpret what it shows. Once you have done this let me have your ideas.” I circulate and listen to what the learners are saying. I do not tell them anything but instead use questions if they get appear to be stuck such as: “What have you tried?”, “What do you think this means?”, “Is there a link between …?”, “Do you think that this is important?”, “What could …?” etc. Once they have completed their activity I give out the other one to them. When a group has completed both of these I give out Activity sheets 3 and 5 as necessary. For those who finish quickly you can use the optional Activity sheet 10 „Which scripts?‟. Alternatively I sometimes ask learners to make up their own Egyptian multiplications using either of the methods or translate numbers into and out of Egyptian and/or Chinese. If you have any learners who know Chinese then they may be able to tell you how to write numbers in Chinese (where a zero now does exist). Whole group discussion (2) Interpreting the artefacts I project the images and ask learners for the translations and their interpretations. I let them offer suggestions before and try to get something from most groups (using what I have heard to decide the order of responses). Often I am surprised by what some learners say and some will do much better than I expect while others do not do this as well as I expect. Working in groups (2) Finding out more I now offer learners one of a number of options with, for example a poster as the outcome and each group talking about what they have found out:

compare the Egyptian and Chinese number systems and look at the advantages and disadvantages of each

consider the two different ways to complete the Egyptian multiplication and look to see when one method is quicker than the other

research more multiplication methods

make a complete nine-nines rhyme resource

write numbers in both scripts to see how they differ

compare Egyptian multiplication with the different methods modern methods that they have been taught (what are the advantages of one system over the other)

compare either number system with that used now and show the advantages and disadvantages of each system

Whole group discussion (3) What the learners have found out I then have a discussion on the posters or other results that learners have come up with. Key feature: neither (Egyptian or Chinese systems) had a zero, so place value (as we use it now) is not possible. The advantages of place value systems are that:

they only need 10 different digits

you do not have to keep inventing more symbols as numbers get bigger

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multiplication, some might argue, is easier

the system easily extends to decimals (once the decimal point is „invented‟)

numbers between 0 and 1 can be easily expressed

Reflection Learners at all stages, even beginning mathematics teachers and non-subject specialists, often do not know why and where mathematics came from or why we do things in the way that we do. They do not know of the chronology or the geography of mathematics, yet mathematics is an important part of everything around us (see Where the Maths you learn is used). Most learners, when involved, become interested in finding out about the history of mathematics – and this lesson has been a trigger for them to find out more.

What learners might do next: I suggest that learners look to find out more about:

Napier‟s bones

the gelosia method of multiplication

Russian peasant multiplication

other methods of multiplication

Further ideas Other modules that use a similar approach are: found at the History of Mathematics Mathemapedia entry at the NCETM portal.

Artefacts and resources: The Rhind mathematics papyrus No longer in place – but someone may have copies of these. Note that the British museum website links direct to the 15 minute podcast while the BBC History of the World Episode 17: Rhind Mathematical Papyrus provides eight interactive images, a link to both play and download the podcast as well as the transcript of the podcast. Egyptian mathematics sites for teachers MacTutor History of Mathematics has a section on Egyptian mathematics. Egyptian mathematics sites for learners and teachers Jo Edkin provides a page on Egyptian numbers. Multiplication is done slightly differently to what is shown here. Jo Edkin also offers a page on many number systems, including Egyptian. Papyrus paper with copyright clearance at Wikipedia. Mark Millimore‟s website offers many things for teaching about the Egyptians including an ancient Egyptian calculator, photos and other products. The Eyelid website also offers some Egyptian problems translating numbers into symbols. They offer the symbols below (and you can use these “I have no objection to people using the material on this site for Educational, non-profit purposes provided I'm

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credited with a link back to this site”.)

The British Museum also provides more on Egypt including information about the pyramids. The Great Scott! Site provides a hieroglyph translator and also translates numbers into Egyptian. Sites about Chinese mathematics for teachers The MacTutor site at St Andrews offers an overview of Chinese mathematics for teachers and information about Chinese numerals. Victor Katz offers a reading list with books on Chinese mathematics in the section headed Ancient Mathematics. David Joyce offers a view of the history of Mathematics in China, though most links do not work, but it does offer a timeline.

Sites about Chinese mathematics for teachers and learners Jo Edkin provides a page on Chinese numbers. Multiplication is done slightly differently to what is shown here. Jo Edkin also offers a page on many number systems, including Chinese. The site also shows the formal way to write these numbers for financial reasons. You can convert numbers into Chinese. Chinese character for zero. Toshuo.com is a personal blog that converts Chinese numbers into Arabic numbers, but can be used to see the symbols for over 10,000.

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Development of number through the history of mathematics Activity sheet 1 The Rhind mathematical papyrus This is a piece of papyrus. What do you think it was used for and where do you think it comes from?

The diagram above, available at Wikipedia, shows the Rhind mathematical papyrus. We would suggest that you instead use the information on the next page entitled Supporting historical information (Activity sheet 1). Promethean ActivStudio and Smart Notebook IWB versions are available.

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Supporting historical information (Activity sheet 1) For copyright reasons the following can only be listed as links. We suggest that you go to The Rhind mathematical papyrus: British Museum (the main page at the British Museum for this item) and the Rhind mathematical papyrus image: use instead of the supplied Activity sheet 1. The MacTutor history of mathematics archive contains more information about the mathematics on the Rhind papyrus. The MacTutor history of mathematics archive provides an overview of Egyptian mathematics. The Wikipedia article on the Rhind mathematical papyrus also provides some information about the discussion over the Rhind papyrus (also referred to elsewhere as Ahmes papyrus). The Rhind mathematical papyrus: the BBC podcast (does not always work on all browsers).

These below are no longer in place but you may find someone who has these. Or they may be broadcast again.

The Rhind mathematical papyrus: BBC History of the World (the main page at the BBC, has an interactive section that includes several images and a video. You can also link to the 15 minute podcast and read the transcript of the podcast. You can also find here description of a few of the problems on the papyrus and an overview of what it was (basically, its a revision guide for anyone wanting to enter the Egyptian 'civil service' of the time - about 1550BCE. Transcript of the BBC podcast (audio stream of the BBC podcast.)

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Development of number through the history of mathematics Activity sheet 2 The piece of papyrus This is a piece of papyrus. Translate and then interpret the papyrus. It is thought that is contains a calculation.

The blank papyrus paper has been overlaid with „hieroglyphs‟.

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Supporting historical information (Activity sheet 2) The translation of the papyrus is as below.

The calculation is 25 x 73 = 1825. The Egyptian system is one of „juxtaposition‟ where symbols for numbers are written next to each other. The Egyptians did not have place value or the zero. The number system is based around 10, but the powers of two are used for multiplication. Roman numerals also used a form of „juxtaposition‟ but there were other differences. Each power of ten has a different symbol (see Activity sheet 6 for some suggestions of what the symbols mean). A multiplication such as 25 x 73 was worked out by taking one unit lot of the 7), then continuing by doubling to get two lots of 73 (146), then doubling again to four lots of 73 (292) until the next doubling process took you beyond the 25 lots needed: in effect you have used the powers of 2. (In the example above it stops at 16 since double 16 is 32.) Then you need to take just those powers of two that combine to make the multiplier of 73, here 25 is made up of 1 + 8 + 16. The other powers of two are struck out. The final total is found by adding the corresponding multiples of 73.

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Development of number through the history of mathematics Activity sheet 3 Another piece of papyrus Translate and interpret this piece of papyrus. How is it the same and how is it different?

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Supporting historical information (Activity sheet 3) The translation of the papyrus is as below.

Later the Egyptian system of multiplication became „more efficient‟. This method uses the understanding that multiplying by ten (or by 100 or 1000) just changes the symbols but keeps the „shape‟ of the number the same. In this way you only ever have to get the powers of two up to 8, then you can use multiplying these by ten (or 100 etc) to help with the multiplication. The rest follows in a similar way.

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Development of number through the history of mathematics Activity sheet 4 Bamboo pieces Here is a jumble of bamboo pieces. They fit together to make mathematical sense. Original symbols taken, with permission, from Jo Edkin's website. Idea adapted from Smile resource.

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Supporting historical information (Activity sheet 4) Here is the correct arrangement.

Here the only nine symbols are needed to make numbers 1 to 9 and a tenth then allows numbers up to 99, with the next symbol not needed until 1000. So a number like 345 is written as 3 then symbol for 100, 4 then symbol for 10 then 5. The system as first used did not have a zero but modern Chinese does. Note that before this system was in place an earlier system of vertical and horizontal rods was used.

Key source: Li Yan and Du Shiran, Chinese Mathematics: A Concise History, translated by John N. Crossley and Anthony W. C. Lun (Oxford: Clarendon Press, 1987)

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Development of number through the history of mathematics Activity sheet 5 A small piece of bamboo Here is a piece of bamboo that was found in China. What is it?

Adapted from diagram in Li Yan and Du Shiran (page 14) where they report that “… many bamboo and wood strips with traces of Chinese characters have frequently been excavated in North-West China). Most date from the Hàn Dynasty (206 BC-220 AD). They are usually known as „Hàn strips‟. Some of these record the Nine-nines rhyme.” According to the same source (page 13) “In ancient times this rhyme was different from the present one which all Chinese school children know by heart. It started with „Nine nines makes eighty-one‟. So it was called the „Nine-nines rhyme‟.” Key source: Li Yan and Du Shiran, Chinese Mathematics: A Concise History, translated by John N. Crossley and Anthony W. C. Lun (Oxford: Clarendon Press, 1987).

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Supporting historical information (Activity sheet 5) Here is the translation of a piece of bamboo that was found in China.

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Development of number through the history of mathematics Activity sheet 6 Egyptian numbers

1 The symbol for one may come from a finger. Everyone starts off counting on their fingers!

10 The symbols get more complicated as the numbers get bigger. The symbol for ten is a piece of rope.

100 The symbol for a hundred is a coil of rope.

1,000 The symbol for a thousand is the lotus or water lily. It shows the leaf, stem and rhizome or root. It seems odd not to show the flower, but you can eat the root.

10,000 The symbol for ten thousand is a single, large finger. Perhaps it is a finger ten thousand times as big as the symbol for one!

100,000

The symbol for a hundred thousand is a tadpole. It seems to be nearly turning into a frog. If you want to know why this is the symbol for such a large number, imagine a pool full of frog spawn all turning into tiny frogs.

1,000,000

The symbol for a million is a god called Heh. It also means just a very large number, like 'squillion'. I think it looks like a fisherman describing how big was the fish that got away - "It was enormous!"

Taken, with permission, from Jo Edkin's website.

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Development of number through the history of mathematics Activity sheet 7 Arrangements of Egyptian numbers

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Development of number through the history of mathematics Activity sheet 8 Chinese numbers

Taken, with permission, from Jo Edkin's website. The symbols stop at 10,000 so 1 million is considered as 100 lots of 10,000.

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Development of number through the history of mathematics Activity sheet 9 Egyptian and Chinese numbers

Taken and adapted, with permission, from Jo Edkin's website. Taken with permission, from the Eyelid website.

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Development of number through the history of mathematics Activity sheet 10 Which scripts? The „Which scripts?‟ „optional activity from Smile is found on page 37 of the booklet. In this six numbers 2, 25, 58, 85, 13 and 100 are written in five scripts including Chinese. Cut out the 30 numbers and match the numbers and scripts. For copyright reasons the diagrams are not included here but can easily be downloaded.