Development of number through the history of mathematics. Logarithms

Transcription

1 Development of number through the history of mathematics

2 Development of number through the history of mathematics Topic: Tables of numbers Resource content Teaching Resource description Teacher comment Background to development of the history of mathematics resources 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: Four-figure logarithm tables page 1 Activity sheet 2: Four-figure logarithm tables page 2 Supporting historical information (Activity sheets 1 and 2) Activity sheet 3: Four-figure logarithm tables both pages Activity sheet 4: Three-figure logarithm tables page 1 Activity sheet 5: Three-figure logarithm tables page 2 Activity sheet 6: Three-figure logarithm tables both pages Activity sheet 7: A page of an exercise book from 1964 using four-figure tables Activity sheet 8: A page of an exercise book from 1964 using three-figure tables Activity sheet 9: explained Resource description Pages of four-figure (or three-figure) logarithms are provided together with pages from a learner s exercise book of Learners work in pairs/groups to work out how logarithms were used to calculate multiplications in order that they might better understand the importance of the role of logarithms in the development of mathematics (and for some to provide a better background when they meet them again at A level). Teacher comment This resource was initially put together to support the subject knowledge of secondary mathematics teacher trainees and then extended to be used by teachers on a Mathematics Development Programme for Teachers. Although all A level mathematics learners have to study logarithms as part of their course it appears that logarithms are poorly understood by many of these learners even after they have successfully completed the A level course. This resource offers a solution. 1

3 Mathematical goals To help learners to: develop a better understanding of how to use two way tables (three- or four figure) 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 compare and contrast different ways in which calculations might be performed realise that logarithms are based on powers (here we use powers of 10) and that they are used to convert a multiplication into an addition understand that before calculators existed people had worked out ways to calculate tables of numbers to high degrees of accuracy recognise that some methods of completing calculations do not give exact answers realise that approximations are important when working out calculations using logarithms know how to create tables of figures using a formula on a spreadsheet recognise that calculators and spreadsheet have been pre-programmed with mathematical functions in order to allow them to work properly Starting points: An ability to read figures from columns of a table. Although not essential it may be useful if learners have completed the Tables of Numbers resource from this set of resources. This module is aimed at KS3 and KS4 learners. It may help if learners have done some work on indices knowing, for example, that 2 2 x 2 4 = 2 6 and have some background work on estimations and rounding. Materials required: For each pair of learners you will need a mini-whiteboard and: Either the four-figure tables Activity sheets 1 and 2, ideally on A3 paper: first two pages of four-figure square tables OR Activity sheet 3 which contains both pages of the four-figure logarithm tables Activity sheet 7: A page of an exercise book from 1964 using four-figure tables Or the three-figure tables Activity sheets 4 and 5, ideally on A3 paper: first two pages of three-figure square tables OR Activity sheet 6 which contains both pages of the three-figure logarithm tables Activity sheet 8: A page of an exercise book from 1964 using three-figure tables Interactive whiteboard and projection resources You may find it easier to project the Activity sheets, using a data projector, a visualiser 2

4 or an overhead projector with a transparency. Alternatively you might want to the 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. Time needed: At least one hour. What I did: Beginning the session In this resource learners work as archaeologists (or mathematical detectives) to examine the table of logarithms, translate what they see and then interpret them and suggest meanings. This would match, to some extent, the process that archaeologists and mathematicians have gone through when working with the original materials. Whole group discussion (1) Interpreting the artefacts and moving onto logarithms I set the scene by providing historical information see video (: Historical information). Here are some points for the learners: Tables of logarithms (log tables) allow you to replace multiplications with additions. They were used in mathematics classrooms until the early 1990s having been discovered in the early 1600s. The use of logarithms made multiplication much quicker than working it out long-hand. Results were always approximate (but this was always good enough). I give out Activity sheets 1 and 2 (4 and 5 if you want to use three-figure tables) and ask learners to look at them and see if they can see how to use them, knowing that they are used in some way to give the answer to a multiplication but by using addition. When noone has any ideas I then ask them to look at the entries for 2, 3 and 6; I then suggest 2 and 4; then 2, 4 and 8; then 2, 5 and 10. I extend keeping it so that multiplications are easy and come to an answer under 10. When pupils have noticed that adding the numbers in the tables for the first two figures gives the number in the tables for the other figure I ask them to try more examples of their own to see if it always happens. This appears to create a problem when the answer comes to more than ten - for example 5 and 4 does not appear to give 20 ( or 1.301) because 20 appears in the table as 3010 (or 301). Working in groups (2) Finding out more I now hand out Activity sheet 7 (8 for three-figure tables). Learners should now discuss the work of the 13 year old and work out how the tables were used. Here is the example for four-figure logs (abbreviation of logarithms), with the example on Activity sheet 8 (three-figure versions) following. 3

5 In the example above: 1.74 is read from the 17 row 4 column to give the is read from row 28, column 2 with mean difference from the 3-column added on to give , so (which is 10 times bigger) has logarithm (this is the hard bit) is converted back by looking for 6912 in the middle of the table and the closest available in 6911 (at row 49 column 1) and the 6912 can be made exactly by adding the mean difference of 1 from the 1-column so giving as the solution to the number with logarithm is 10 times bigger than the number with logarithm , so the answer is (which rounds to 49.1 to 3 significant figures) alternatively using the approximation of the sum to 2 x 30 = 60 the answer is 49.1 (rather than 4.91 or 491) The three-figure tables work in the same way for the example below but without the extra part for the mean difference column. Examples 1 and 3 from the 13-year old s follow the same idea. Learners should make up some more for themselves and check their work by using calculators, either to check the sum directly or to read off the logarithms - they need to use the log button, so log 1.74 gives and log gives giving a total of then to take this back to a number they need the 10 x button (on the same key as the log button) to get the answer of etc. To keep things easier you could set the accuracy of the computer to be 4 or 3 figures according to the tables being used. 4

6 Division follows using subtraction. For ease I make sure that divisions always give a result that is bigger than 1 so that the calculator and the written method are the same. Note the following: logarithms do not exist for 0 or negative numbers the logarithm of 1 is zero (since 10 0 = 1) logarithms of numbers between 0 and 1 are negative, (e.g. 0-1 = 0.1), but the old method does not appear to show this the old written method of using logs would have 0.2 as whereas on a calculator it appears as (which equals ), this is the meaning of the notation. the old written method allowed for a consistent use of the log tables, but created problems for learners also learners using logs would usually have in the same book a set of antilogarithms these would convert a logarithm back into a four (or threefigure) number (you read the numbers from the outside rather than reversing the process of finding the number in the middle of the log table. Whole group discussion (3) What the learners have found out A video provides an explanation of how to work out multiplications using logarithms. For other sources and websites see the Supporting historical information (Activity sheets 1 and 2) that follows. The key feature is that Henry Briggs (in 1617), using the work of John Napier (1614) discovered logarithms in base 10 which in effect allowed every number to be written as a power of 10, so for four-figures tables in base 10 the number 2 = The log of each number was found in the tables so allowing multiplication of a pair of numbers became conversion to logs (using the tables) adding the logs and then converting the log back to a normal number however the price was that answers would not only be approximate since the log of a number is not usually exact. The same could be done in any positive base number and the only one in use in books of tables was for the constant e (2.178 ). Reflection This resource was initially put together to support the subject What learners might do next: You could ask learners to find out about: the slide rule Napier s bones the gelosia method of multiplication 5

7 Russian peasant multiplication other methods of multiplication how logarithms are worked out 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: Relevant information is found on the page Supporting historical information (Activity sheets 1 and 2). 6

8 Activity sheet 1 Four-figure logarithm tables page 1 These are given to the first four figures with no account being given to place value. 7

9 Activity sheet 2 Four-figure logarithm tables page 2 These are given to the first four figures with no account being given to place value. 8

10 Supporting historical information (Activity sheets 1 and 2) A number of sites offer information about the discovery or invention of logarithms and the chronology of their use. Basically John Napier worked out logarithms of sines and Henry Briggs then adapted them into base 10 logarithms a few years later (however they were not complete). The complete table of logarithms for numbers from 1 to 100,000 was published in The MacTutor History of Mathematics archive gives information about Napier and his discovery of logarithms in 1614 with the publication of A Description of the Admirable Table of. The title page of the original can be seen. The text of this is available at a site dedicated to John Napier where the translation into English by Edward Wright can also be found. Further information on Napier and logarithms can also be found at the History of Computing site where it also details how Henry Briggs published Logarithmorum Chilias Prima the first set of tables of base 10 logarithms. A translation of his book Arithmetica Logarithmica is available and more about Briggs is found at the MacTutor History of Mathematics archive. Jo Edkin s site offers interactive examples of how to use four figure log tables. Cleave books offer a booklet on A collection of some methods for doing multiplication that have been devised and used in previous times. This provides an overview of using logarithms (page 4), instructions of how to use three-figure tables (pages 10 and 11) and a table of logarithms on page 15. There are two short videos found at MathsTube under the heading : one a brief history, the other showing how logarithms work. 9

11 Activity sheet 3 Four-figure logarithm tables, both pages These are given to the first four figures with no account being given to place value. 10

12 Activity sheet 4 Three-figure logarithm tables page 1 These are given to the first four figures with no account being given to place value. 11

13 Activity sheet 5 Three-figure logarithm tables page 2 These are given to the first three figures with no account being given to place value. 12

14 Activity sheet 6 Three-figure logarithm tables, both pages These are given to the first three figures with no account being given to place value. 13

15 Activity sheet 7 A page of an exercise book from 1964 using four-figure tables Here is a page of a 13 year old s exercise book from 1964 where four-figure logs have been used. 14

16 Activity sheet 8 A page of an exercise book from 1964 using three-figure tables Here is a page of a 13 year old s exercise book from 1964 where three-figure logs have been used. 15

17 Activity sheet 9 explained Taken, with permission, from Jo Edkin's website. 16