Development of number
through the
history of mathematics
Tables of numbers
Development of number through the history of mathematics
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: Pairs of numbers
 Activity sheet 2: Two sets of four-figure Mathematical Tables: one from 1905 the
other from the second part of the 20th century
 Activity sheet 3: Tables of numbers: four-figure tables, page 1
 Activity sheet 4: Tables of numbers: four-figure tables, page 2
 Activity sheet 5: Tables of numbers: four-figure tables, both pages
 Activity sheet 6: Tables of numbers: three-figure tables, page 1
 Activity sheet 7: Tables of numbers: three-figure tables, page 2
 Activity sheet 8: Tables of numbers: three-figure tables, both pages
 Activity sheet 9: Earlier examples of tables of numbers
Resource description
Tables of squares numbers are provided in both four-figure and three-figure format.
Learners are not told that these are tables of squares, though the starter activity
involves looking at the squares of the numbers from 1 to 40.
Learners work in pairs/groups or as a class to see how tables of numbers were used to
help with calculations in pre-calculator times.
Two other resources are also available one on tables of logarithms and their use, the
other on the slide rule.
Teacher comment
Learners take calculators for granted without ever thinking about how devices were
used to help make calculations easier before the calculator.
This also provides an opportunity for learners to read and use two-way tables.
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Tables of numbers
Development of number through the history of mathematics
Mathematical goals
To help learners to:
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develop a better understanding of how to use two way tables (three- or four –
figure)
be able to use the tables to read squares
recognise the difficulty in reading square roots from the table of squares
find an approximation for the square root of a number
realise that over time methods of calculation have changed
place mathematical development in a historical and geographical context
become more familiar with different methods of calculation
know how to create tables of figures using a formula on a spreadsheet
Starting points:
An ability to read figures from columns of a table. Although not essential it may be
useful if learners have completed the Multiplication resource from this set of resources.
It will also help if learners are familiar with Pythagoras’ theorem.
This module is aimed at KS3 learners.
Materials required:
For each pair of learners you will need:
Either The four-figure sets of tables
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Activity sheets 3 and 4: first two pages of four-figure square tables OR Activity
sheet 5 instead
Or The three-figure sets of tables
 Activity sheets 6 and 7: first two pages of three-figure square tables OR Activity
sheet 8 instead
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 and 2
Activity sheets used with learners (four-figure tables Activity sheets 3 and 4, or
Activity sheet 5 OR three-figure tables Activity sheets 6 and 7, or Activity sheet
8)
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Tables of numbers
Development of number through the history of mathematics
Time needed:
About one hour.
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)
Pairs of numbers starter activity
First I display Activity sheet 1: Pairs of numbers and ask learners to look at the pairs. I
have calculators available for learners. Here are some sample comments
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Second number is the square of the first.
As you go down the list the differences are 3, 5, 7, 9 etc.
These differences get bigger each time by 2.
The end number of the square numbers cycle in a pattern 1, 4, 9, 6, 5, 6, 9, 4, 1, 0 then
start again.
2
2
2
2
2
2
2
For example 11 and 19 end in the same digit as do 12 and 18 , 13 and 17 , and 14
2
and 16 .
No square numbers end in 2, 3, 7 or 8.
th
th
The difference between e.g. the 9 and 10 square numbers is equal to 9 + 10.
Difference in bottom row squares goes up by 200 each time.
2
2
40 is 100 times 4 and so on.
Working in groups (1)
The mathematical detective/archaeologist: examining the artefacts
Now I set the scene: a typical classroom in England in the 1950s – early 1980s, (i.e.
before calculators were in schools and when their grandparents might have been at
school), and learners have these books of four-figure tables on their desks (three-figure
tables in the early mid 1970s - 1980s). I then show the cover for the four-figure tables
(Activity sheet 2) and then Activity sheets 3 and 4 stating that these were pages that a
typical 12 -1 5 year old would use in their mathematics lessons. I then pose the
question: “What do you think the tables were used for?”
Alternatively you could use the three-figure versions - Activity sheets 6 and 7.
Learners may not link this to the earlier discussion (i.e. that the tables are tables of
squares). Tables show squares of numbers where those before 31.6 do not show the
decimal point in the correct place (so 112 is shown as 1210). Comments below refer to
the four-figure tables, though similar things apply for the three-figure tables.
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Tables of numbers
Development of number through the history of mathematics
At some point learners need to be aware that the break in row 31 comes as the figures
in the body of the table ‘start again’ from 1000.
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2
Tables allow learners to read squares of four-figure numbers, so 10.67 is found from the
table in the 10 row in the column headed 6 (giving figures, without a decimal point of
1124) with a mean difference (of 15) that is found in the 10 row column headed 7 making
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the total 1124 + 15, which is 1139, then knowing that 10 is about 100, the answer is
given as 113.9
Learners who used these books had to know how to do approximations to put the
decimal point into the correct place.
The answer on the calculator is 113.8489 so the answer is incorrect in the fourth figure.
Answers using the tables were then usually rounded to three significant figures (this can
be ignored if you want)
These can be used to find square roots by reversing the process but you have to be very
careful because each number ‘appears’ twice in the list (e.g. 4410 appears in the body of
2
2
the table twice at both 21 and 66.51 ). So the square root of 4.41, 441, 44100, all get
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linked to 21 and 0.441, 44.1, 4410 all get linked to 66.51 . You might not wish to follow
this part through.
Without the calculator you needed to be able to read the tables, make an approximation
to the answer to get the correct value from the figures given and then round it to three
significant figures (you may wish to ignore the rounding at this point). For three-figure
tables you would round to two significant figures.
Whole group discussion (2)
Interpreting the artefacts
We then discuss the tables of squares, how to use them and look at mathematical
problems they might help solve. This usually leads onto the suggestion of use in
Pythagoras-type questions. I usually ask learners to use them to solve a few
Pythagoras-type questions that they have to make up and solve (and a friend will then
check their working with a calculator).
Finally learners are asked to compare solving the same problem in three different ways,
looking in particular at what skills are needed to work in each of these three situations
where:
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no calculation tools are available
square tables are available
calculators are available.
Learners should be aware that square root tables also existed for square roots from 1 to
100.
Once all this is completed I then take them back to the original activity on the squares of
the first 40 numbers – looking at how 4 and 40 squared produce squares that are 100
times apart (402 = 1600 is 100 times 42 = 16 and a number between 12 and 13 squared
will be 160).
I have since found early versions of tables of numbers and would now show these to
learners. The links for these are shown on Activity sheet 9.
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Tables of numbers
Development of number through the history of mathematics
Reflection
Once they have realised how the tables work learners find the square root tables quite
straightforward. They have much more difficulty with finding square roots – in particular
knowing which one to use. I find that the part they have most difficulty with is
approximating to a square root.
What learners might do next:
You could ask learners to find out about:
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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:
These are found at the History of Mathematics Mathemapedia entry at the NCETM
portal.
Artefacts and resources:
A chronology of tables
The Schoyen Collection contain ’11 of the earliest examples form Sumer and Babylonia
where mathematics was invented’ including details about MS 3047 which is a
multiplication table for numbers in the Sumerian base 60 system – considered by some
to be the oldest known evidence of mathematics (from the 27th century BCE).
VAT 12593 is the catalogue number of what is believed to be the ‘world’s oldest datable
table’ from about 2600 BCE (Campbell-Kelly, M.). It is found on page numbered 30. The
chapter is also on the author’s site (Eleanor Robson) and is found on the 13th page of
the file, numbered page 30). A ‘translation’ of this is found in the work of Friberg on
page 149.
An old Babylonian clay tablet referred to as Plimpton 322 offers a table of Pythagorean
triples. An exhibition called Before Pythagoras: The Culture of Old Babylonian
Mathematics offers photographs of the tablet and further information.
The Alphonsine Tables provide info about planetary positions and movements, and a
page of them is given on Wikipedia. There is more information about them on the
History of Science website.
Barlow’s tables of squares, cubes, square roots, cube roots and reciprocals of all
integer numbers up to 10000 were printed in 1839 with the following in the forward:
I cannot ascertain that any tables of square roots, cube roots, or reciprocals,
comparable in extent to those of Mr. Barlow, were ever printed before his. The
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Tables of numbers
Development of number through the history of mathematics
tables of squares and cubes up to 10,000 were printed by Guldinus in 1635, by J.
P. Buckner (according to Murhard) in 1701, and by Seguin in 1801. The cubes to
100003 ,and the squares up to 254002, by Dr. Hutton, were published by the
Board of Longitude in 1781.
Barlow's tables can be downloaded directly as a pdf file.
Bibliography
Campbell-Kelly, M., Croarken, M., Flood R.G. & Robson, E. (eds.), The History of
Mathematical Tables from Sumer to Spreadsheets, Oxford: Oxford University Press
2003.
Campbell-Kelly, M. The history of mathematical tables: from Sumer to spreadsheets
Friberg, J. A remarkable collection of Babylonian mathematical texts
Robson, E. Tables and tabular formatting in Sumer, Babylonia, and Assyria, 2500 BCE
– 50 CE
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Tables of numbers
Development of number through the history of mathematics
Activity sheet 1
Pairs of numbers
Look at the pairs of numbers below and write down as
many things as you can about them or about any
patterns you see. Also look for things that are the
same at different places in the list.
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Tables of numbers
Development of number through the history of mathematics
Activity sheet 2
Two sets of four-figure Mathematical Tables: on the left from 1905 and on the
right from the second part of the 20th century
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Tables of numbers
Development of number through the history of mathematics
Activity sheet 3
Tables of numbers: four-figure tables, page 1
These are given to the first four figures with no account being given to place value.
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Tables of numbers
Development of number through the history of mathematics
Activity sheet 4
Tables of numbers: four-figure tables, page 2
These are given to the first four figures with no account being given to place value.
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Tables of numbers
Development of number through the history of mathematics
Activity sheet 5
Tables of numbers: four-figure tables, both pages
These are given to the first four figures with no account being given to place value.
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Tables of numbers
Development of number through the history of mathematics
Activity sheet 6
Tables of numbers: three-figure tables, page 1
These are given to the first three figures with no account being given to place value.
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Tables of numbers
Development of number through the history of mathematics
Activity sheet 7
Tables of numbers: three-figure tables, page 2
These are given to the first three figures with no account being given to place value.
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Tables of numbers
Development of number through the history of mathematics
Activity sheet 8
Tables of numbers: three-figure tables, both pages
These are given to the first three figures with no account being given to place value.
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Tables of numbers
Development of number through the history of mathematics
Activity sheet 9
Earlier examples of tables of numbers
Some of these are not included for copyright reasons, but could be shown.
MS 3047 - a multiplication table for numbers in the Sumerian base 60 system –
considered by some to be the oldest known evidence of mathematics (from the 27th
century BCE).
VAT 12593 - what is believed to be the ‘world’s oldest datable table’ from about 2600
BCE (see 13th page of the file, numbered page 30). This file also contains a map of the
Middle East. A translation of the table is available.
The Alphonsine Tables (below) of 1483 were among the earliest printed mathematical
tables.
Barlow’s tables of squares, cubes, square roots, cube roots and reciprocals of all integer numbers
up to 10000 were printed in 1839.
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Tables of numbers
Development of number through the history of mathematics
Supporting historical information (Activity sheet 9)
Earlier examples of tables of numbers
Babylonian cuneiform writing (Plimpton 322)
Source: Wikipedia
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Tables of numbers