Development of number

through the

history of mathematics

Multiplication

Development of number through the history of mathematics

1Multiplication

Development of number through the history of mathematicsTopic: Multiplication

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 ingroups as teams of archaeologists or mathematical detectives to examine, interpret andthen give meanings to mathematical artefacts. Once done the mathematics thatunderpins the artefacts (place value and multiplication methods). is explored

Teacher comment

The resource is set within the context of the history of mathematics and howmathematics is discovered and/or invented. Considerable importance is attached to the

Development of number through the history of mathematics

2Multiplication

use of ‘real’ artefacts and historical accuracy – though interpretation is more open todebate. Options are available to link to real artefacts. This is a lesson that had beenused over a number of years with beginning teachers and with a cohort of teachers onthe Mathematics Development Programme (a 40 day course for secondary non-specialists teachers of mathematics). These teachers have then used this lesson withtheir learners. This version is a considerable adaptation of earlier versions having beensupplemented with a greater variety of mathematical resources.

Mathematical goals

To help learners to:

understand how an archaeologist might interpret ‘evidence’ collected from anartefact

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

realise that sometimes different mathematical developments occurred in differentcultures

understand that similar developments in mathematics often occurred in manyparts 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 visualiseror an overhead projector with a transparency. Alternatively you might want to use thePromethean ActivStudio and Smart Notebook IWB versions of the activities.Wherever items for display are subject to copyright restrictions direct links are providedfor them.

Development of number through the history of mathematics

3Multiplication

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 beconsidered 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) toexamine, interpret and then give a meaning to mathematical artefacts – in effect theyneed to ‘translate’ what they see, interpret them and suggest meanings. I let them knowthat they are matching, to some extent, the process that archaeologists andmathematicians have gone through when working with this or similar materials, and thiswould include hiding results from other groups.

I use the IWB versions, but if you cannot do that you can use the Activity sheets in themost appropriate way for your classroom.

Whole group discussion (1a)How had mathematics developed?

First, I set the scene, and asked some questions (answers are included to help you):

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

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

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 issomething that we can use to describe or model parts of reality, or any other systembased on quantifiable things. Mathematics can be used to model finance, logic, evencolour. Mathematics itself did not exist before the first mathematicians. What wasdiscovered was how to mould the model to fit reality. What was invented was the modelitself. Mathematics was invented. How to use mathematics was discovered.”

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

How long ago did people invent or make discoveries about mathematics? Thereis considerable debate about this. The Wikipedia Timeline of mathematics suggests70,000 BCE and the process still continues.

Development of number through the history of mathematics

4Multiplication

Is mathematics an area where people are still finding out things? Yes. How it isused can be seen, for example in a booklet called Where the Maths you learn is used.

Learners usually 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 onthese things later. Lists are here also for you to give them a broad idea of wheremathematics had 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 ofmathematics. The MacTutor timeline is in parts, starting at 800 BCE (Before theCommon Era, the same as AD/BC notation) offering a timeline of mathematicians. TheWikipedia timeline goes back to 70,000 BCE but there is discussion, in particular, aboutwhat is missing.

Whole group discussion (1b)The Rhind mathematical papyrus

I used my IWB version to display Activity sheet 1 (the Rhind mathematical papyrusimage) and made use of the items found on the page Supporting historicalinformation (Activity sheet 1).

I then set the challenge: “This is a piece of papyrus. What do you think it was used forand where do you think that it comes from? “

I took in comments and suggestions. See, in particular the transcript of the BBC podcastfor answers – but basically the Rhind mathematics papyrus is a set of questions andworked 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 had ready for each team of archaeologists or mathematical detectives theartefacts 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 gave out Activity sheets 2 and 4 first to alternate groups/pairs of learners. The firsttime that you use Activity sheet 4 learners will have to cut out the jigsaw pieces (storethem for later use in envelopes) – it helps if each set is a different colour. I told them to

Development of number through the history of mathematics

5Multiplication

keep their work hidden from the competition (the others). I usually say something like:

“Examine the artefact you are given, translate it and then interpret what it shows. Onceyou have done this let me have your ideas.”

I circulated and listened to what the learners said. I did not tell them anything butinstead used questions when they appeared to be stuck such as: “What have youtried?”, “What do you think this means?”, “Is there a link between …?”, “Do you thinkthat this is important?”, “What could …?” etc.

Once they had completed their activity I gave them the other. When a group hadcompleted both of these I gave out Activity sheets 3 and 5 as necessary. For thosewho finished quickly I used the optional Activity sheet 10 ‘Which scripts?’. Alternatively Isometimes asked learners to make up their own Egyptian multiplications using either ofthe 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 towrite numbers in Chinese (where a zero now does exist). There are different symbolsfor 1 to 10,000: so 100,000 is ten lots of 10,000.

Whole group discussion (2)Interpreting the artefacts

I projected the images and asked learners for their translations and interpretations. I letthem offer suggestions and tried to get something from most groups (using what I hadheard to decide the order of responses). Often I was surprised by what some learnerssaid and some did much better than I expected while others did not do this as well as Iexpected.

Working in groups (2)Finding out more

I then offered learners one of a number of options with, for example a poster as theoutcome and each group then had to present their results (in another lesson):

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

consider the two different ways to complete the Egyptian multiplication and lookto 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 modern methods that you 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

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:

Development of number through the history of mathematics

6Multiplication

they only need 10 different digits you do not have to keep inventing more symbols as numbers get bigger 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-subjectspecialists, often do not know why and where mathematics came from or why we dothings in the way that we do. They do not know of the chronology or the geography ofmathematics, yet mathematics is an important part of everything around us (see Wherethe Maths you learn is used).

Most learners, when involved, become interested in finding out about the history ofmathematics – and this lesson had 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 papyrusThe British museum website links direct to the 15 minute podcast while the BBC Historyof the World Episode 17: Rhind Mathematical Papyrus provides eight interactiveimages, a link to both play and download the podcast as well as the transcript of thepodcast.

Egyptian mathematics sites for teachersMacTutor History of Mathematics had a section on Egyptian mathematics.

Egyptian mathematics sites for learners and teachersJo Edkin provides a page on Egyptian numbers. Multiplication is done slightly differentlyto what is shown here.Jo Edkin also offers a page on many number systems, including Egyptian.Papyrus paper with copyright clearance at Wikipedia.

Development of number through the history of mathematics

7Multiplication

Mark Millimore’s website offers many things for teaching about the Egyptians includingan ancient Egyptian calculator, photos and other products.The Eyelid website also offers some Egyptian problems translating numbers intosymbols. They offer the symbols below (and you can use these “I have no objection topeople using the material on this site for Educational, non-profit purposes provided I'mcredited with a link back to this site”.)

The British Museum also provides more on Egypt including information about thepyramids.The Great Scott! Site provides a hieroglyph translator and also translates numbers intoEgyptian.

Sites about Chinese mathematics for teachersThe MacTutor site at St Andrews offers an overview of Chinese mathematics forteachers and information about Chinese numerals.Victor Katz offers a reading list with books on Chinese mathematics in the sectionheaded Ancient Mathematics.David Joyce offers a view of the history of Mathematics in China, though most links donot work, but it does offer a timeline.

Sites about Chinese mathematics for teachers and learnersJo Edkin provides a page on Chinese numbers. Multiplication is done slightly differentlyto what is shown here.Jo Edkin also offers a page on many number systems, including Chinese. The site alsoshows 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, butcan be used to see the symbols for over 10,000.

Development of number through the history of mathematics

8Multiplication

Activity sheet 1The Rhind mathematical papyrus

This is a piece of papyrus. What do you think it was used for and where do you think it comesfrom?

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 Supportinghistorical information (Activity sheet 1).

Promethean ActivStudio and Smart Notebook IWB versions are available.

Development of number through the history of mathematics

9Multiplication

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 theBritish Museum for this item) and the Rhind mathematical papyrus image: use instead of thesupplied Activity sheet 1.

The MacTutor history of mathematics archive contains more information about the mathematics onthe 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 aboutthe 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).

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

Transcript of the BBC podcast (audio stream of the BBC podcast.)

Development of number through the history of mathematics

10Multiplication

Activity sheet 2The piece of papyrus

This is a piece of papyrus. Translate and then interpret the papyrus. It is thought that is contains acalculation.

The blank papyrus paper had been overlaid with ‘hieroglyphs’.

Development of number through the history of mathematics

11Multiplication

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 fornumbers are written next to each other. The Egyptians did not have place value or the zero. Thenumber 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 had a different symbol (see Activity sheet 6 for some suggestions of what thesymbols 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 thepowers of 2. (In the example above it stops at 16 since double 16 is 32.) Then you need to takejust 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 correspondingmultiples of 73.

Development of number through the history of mathematics

12Multiplication

Activity sheet 3Another piece of papyrus

Translate and interpret this piece of papyrus. How is it the same and how is it different?

Development of number through the history of mathematics

13Multiplication

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 theunderstanding 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 theseby ten (or 100 etc) to help with the multiplication. The rest follows in a similar way.

Development of number through the history of mathematics

14Multiplication

Activity sheet 4Bamboo piecesHere 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 Smileresource.

Development of number through the history of mathematics

15Multiplication

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 numbersup to 99, with the next symbol not needed until 1000. So a number like 345 is written as 3 thensymbol for 100, 4 then symbol for 10 then 5. The system as first used did not have a zero butmodern Chinese does. Note that before this system was in place an earlier system of vertical andhorizontal rods was used.

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

Development of number through the history of mathematics

16Multiplication

Activity sheet 5A 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 bambooand wood strips with traces of Chinese characters have frequently been excavated in North-WestChina). 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 presentone 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 JohnN. Crossley and Anthony W. C. Lun (Oxford: Clarendon Press, 1987).

Development of number through the history of mathematics

17Multiplication

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

Development of number through the history of mathematics

18Multiplication

Activity sheet 6Egyptian numbers

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

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

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

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

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

100,000

The symbol for a hundred thousand is a tadpole. Itseems to be nearly turning into a frog. If you want toknow why this is the symbol for such a large number,imagine a pool full of frog spawn all turning into tinyfrogs.

1,000,000

The symbol for a million is a god called Heh. It alsomeans just a very large number, like 'squillion'. I think itlooks like a fisherman describing how big was the fishthat got away - "It was enormous!"

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

Development of number through the history of mathematics

19Multiplication

Activity sheet 7Arrangements of Egyptian numbers

Development of number through the history of mathematics

20Multiplication

Activity sheet 8Chinese 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.

Development of number through the history of mathematics

21Multiplication

Activity sheet 9Egyptian and Chinese numbers

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

Taken with permission, from the Eyelid website.

Development of number through the history of mathematics

22Multiplication

Activity sheet 10Which scripts?

The ‘Which scripts?’ ‘optional activity from Smile is found on page 37 of the booklet. In this sixnumbers 2, 25, 58, 85, 13 and 100 are written in five scripts including Chinese. Cut out the 30numbers and match the numbers and scripts. For copyright reasons the diagrams are not includedhere but can easily be downloaded.