Jignashi

20 February, 2023

 


Times Tables

 

by

 

Chandrakant Doshi

 

Email:  chandra@agnani.co.uk

 


 

Abstract :

 

This note outlines a novel approach to help the children learn the Times Tables.

     

 

One topic in mathematics that turns children off the subject is the learning of   times or  multiplication tables. Many approaches have been devised to help children in memorising these tables. As they progress through the school, recalling multiplication of two numbers is required for manytopics in maths. 

 

Another approach to help the children learn these times tables is presented here. Parents and grandparents can play an important part here. Children love to play games and they definitely love to show off. What better way then to invite a child/grandchild to test their parents/grandparents knowledge of times table? This  becomes a game and there is plenty of opportunity to show off. Incentive to learn the tables comes automatically with the game.

 

Multiplication tables run from 2 to 10 or 12. Generally they are printed as columns but they can also be printed in a grid format. The latter are for use by the older children. To help children learn their times tables with greater interest a partially filled grid of the times table could also be utilised. This should engage their minds in working out the missing steps.

 

Once a child has learnt the times table then applying that knowledge involves the child running through, in her/his mind, that particular table starting from one. They carry on till the required multiplier is reached. 

 

Here the times 5 table provides some advantage. Multiplication of any two numbers, where at least one number is greater than 5, can be performed with increasing ease than the usual method of running through, in their mind or by the use of fingers, the times table of one of the number, starting from 1. 

 

Children could be shown how to utilise their knowledge of times 5 table to run through the multipliers higher than 5. With right sort of examples their knowledge of times tables can be reinforced.

 

Let us take a specific example. We wish to know the result of multiplying 7 x 3. We could run through the times table for 3, starting from 1 till we arrive at 7. 

Another way is to go straight to the times 5 table. We have 3 times 5 = 15, then running forward we have 3 times 6 = 18 and finally 3 times 7 = 21. 

 

In both cases the result is 7 x 3 = 21. 

 

This requires mastering times 5 table thoroughly. Children learn times 10 table very quickly, once they realise that all they need to do is put 0 at the end of the number being multiplied. Times 5 table has a similar ease of obtaining the result: multiplication by 5 results in a number ending in either 5 or 0. This should prove a great help to mastering the times 5 table. Children will quickly pick up the fact that multiplying 5 by an even number always results in the number ending in 0. Even 0, odd 5 becomes the rule. This proves an easy introduction to odd or even numbers.

 

Times tables can also be printed in a grid format (see Table A1). Familiarising its use helps a child to see the multiplication in reverse order, ie, 3*7=7*3.  A challenge for a child is to start using a partially filled multiplication table so the child gets to reinforce his knowledge of the table in small step.  Table A2 is an example of a partially filled multiplication table. Similar tables can be prepared as necessary. 

 

Children, and perhaps even the parents, may need to be taught how to use these Tables. One way to proceed is to let the child ask a few questions using the table. Then gradually permit the child to pick some questions from the tables and encourage the child to ask more questions from memory only. Proceeding this way the child should be able to come up with more questions from memory to test their parent/grandparent. 

 

Parents are busy people and not all may find time to practice such exercises with their children. Grandparents, if living near, could prove helpful if they are prepared to undertake the task. Some parents may need to learn their times tables but their experience and maturity should make this task fairly easy. 

 

 

 

 


 

 









08 February, 2023

 

Papers on Academia

 


Email: chandra@agnani.co.uk


21st. September, 2024


Link to Asokan Chhatra added.


8th. February, 2023                          

 

Some of my early papers and notes are to be found on this blog but the later works are uploaded to Academia. 

 

Below is the list of my work on Academia, including the links to access them. 

 

This list will be updated as new papers are added to Academia.

 


Asokan Chhatra


https://www.academia.edu/123896197/Asokan_Chhatra


 

Harappan Linear Measures

 

https://academia.edu/resource/work/91385667

 

 

Accuracy of Ancient Weights

 

https://academia.edu/resource/work/55221784

 

 

Ancient Weights: Two Cultures

 

https://academia.edu/resource/work/42941631

 

 

Comparison Ancient Weight Systems

 

https://academia.edu/resource/work/30143220

 

 

Ancient Egyptian Weight Systems

 

https://academia.edu/resource/work/4846620

 

An Externally Pivoted Balance

 

https://academia.edu/resource/work/8066648

 

Ancient Weights from South Asian Civilization

 

https://academia.edu/resource/work/4846613

 

 

27 January, 2021

Ancient Weights:2 cultures

Ancient Weights: 2 Cultures

by

Chandrakant Doshi


Key words:  weights, ancient, Harappan, Indus, Mesopotamian, Ur, Nippur, Mohenjo Daro, Dholavira, stone, chert, hematite.



Mesopotamian and Harappan civilizations both had their own weight systems. This note examines these systems, recognising that the recovered weights from each site are a mix of two grades of weights (details below), one accurate and one less so and demonstrates that the  accurate grades from these bronze age cultures have similar accuracy.  These two grades will be referred to here as Class A and Class B weights, specimens of Class A being the more accurate of the two Classes. 

For this analysis weights from Mohenjo Daro1, Dholavira2, Nippur3 and Urare used. In all these weight collections the details of the materials of weights are available. This is important, as it will become clear further down.

Harappan weights were first analysed by A. S. Hemmy in late 1920s and he noted the difference in accuracy of weights of different materials when analysing Mohenjo Daro specimens. 
“The weights not made of chert are on the whole not so accurate as those that are”, he wrote in Further Excavations at Mohenjo Daro1, (MjD FEM), page 605.

A. S. Hemmy5 also analysed weight systems of the Harappan and Mesopotamian civilizations and concluded that the Harappan weights were by far the more accurate of the two. For comparison, Hemmy used Mohenjo Daro weights on the one hand and a collection of weights from Ur and Susa on the other. No references were given for the sources of these data. But he did observe that
“In the Indus system, the great majority conform to one standard (13.625 gm or 210.2 gr.), and may be divided into two grades, one very accurate, one much less so.”

W. B. Hafford6 also noted two different grades in the weight collection from Nippur:

“Were the well-made hematite and bronze pieces the official or royal weights used by an upper echelon of merchants and bureaucrats, while the crude types of less dense stone and less regular form were those of the more local people?”

When a wide range of values need to be displayed on a chart then converting each number that represents the weight of a specimen to its logarithm makes it possible to display it on a standard size chart. Such a chart is referred to as a log chart. Logarithm or log of a number is expressed with respect to another number, called the base. Commonly used base is 10. Charts used here are all to base 2, which has the advantage that values which are halves or doubles of another value have log values that differ by 1. Since most weight systems have values which are halves or doubles of previous values, specially at the lower end, then the choice of 2 is an advantage. With suitably scaled axis, this is easily seen. Divergence from the binary scale also shows clearly. And the use of 2 as a base provides a better resolution.

To prepare this special chart, the weight list is sorted by the material of the specimens and for each material, by the value of the specimens, lowest value first. The numerical value of each weight is converted to log to base 2. These log values are then plotted on a chart. As can be seen in the accompanying charts, vertical axis represents log(W) while the horizontal axis represents the item number on the list. The vertical axis is suitably adjusted to have the value of the unit weight fall on a grid line.

Sorting the weights by material and then value results in the weights on the chart successively moving up as well as to the right. Once the weights of the first material are exhausted then the plotted point moves down and to the right. So, jumps from top to bottom are due to the change of material. This way, weights of different materials can be identified on the chart. 

An examination of the chart reveals that for one material the display is a staircase ( implying many specimens are close together in value) while for the rest of the materials, the display comprises of mostly scattered points, with no systematic structure evident. A full sequence of specimens of this single material, from the lowest value to the highest, is present in the collection. This is not the case for the rest of the materials.

Chart D below displays these weights from Dholavira arranged in alphabetical order by material and within each material by numerical order, lowest first. Weights made of chert are over-plotted in red and shows the typical staircase representing a weight system. The flats are made by a large number of specimens close in value. Some of the other materials also show a few specimens on a flat but generally there are few specimens that make up the flat and the range of nominal values covered by the staircase is rather small. So, there are indeed two grades of weights present in the recovered collection from Dholavira. 



A similar situation is observed with weights recovered from Mohenjo Daro by Ernest Mackay1. Chart A displays these weights.





There are two papers by W. B. Hafford available online and which include a list of weights from the Mesopotamian sites of Ur2 and Nippur3. Each list includes, besides the weight of the specimen, the material it is made of, the shape, ratio of the specimen and the derived unit of the system. Charts for each of these two weight lists, along similar line to Chart D reveal two grades of weights in each. Hematite, a hard stone, is the material for one of the grades of weights and is over-plotted in red in the charts below. This can be seen in Chart 1 for Ur and Chart 2 for Nippur.

Mesopotamian weights are scattered across various museums and institutions and Hafford has put considerable effort to collect them together. As he notes, there are many more of these in the museums in Iraq which were not accessible to him.








Looking at the four charts above it is clear that weight collection from each site can be separated into two categories. The criteria that partitions these weights is the type of stone used in manufacturing them: chert for Harappan weights and hematite for Mesopotamian weights. Both of these are hard stones and the weights made of these materials are referred to as Class A weights. To establish that these Class A weights are the more accurate of the collection, we need to examine their frequency distribution charts. 

A frequency distribution chart displays the number of specimens in a given interval of weight. Easiest way to achieve this is to normalise the whole weight list first.  Each specimen allocated to a ratio or nominal weight is divided by that nominal weight. A specimen weighing exactly the nominal weight will return a value of 1.00, while values below or above the nominal value will return a value less than or greater than 1.00, respectively. 

The range of normalised values is divided into intervals of 0.01, starting with the interval at the nominal value. This is symmetrically arranged about it, so the interval covers the range 0.995 to 1.005 of the normalised value. The entire range of the normalised values is similarly covered. 

The frequency distribution is the number of weight specimens occurring within any given interval. A plot of these numbers is the frequency distribution of that weight system. These distributions are based on a unit weight of 13.625 grams for the Harappan system and 8.40 grams for the Mesopotamian system. 

The plots display the distribution of Class A and Class B weights on the same chart for each of the four sites. The Harappan weights, Chart B and Chart E show the peak frequency for Class A at the normalised value of 1.00. Very few of Class B specimens are present in Chart B whereas very large number are present in Chart E. 

Among the Mesopotamian weights, the peak of Class A weights at Ur (Chart 3) occurs at the normalised value of 0.97 with the frequency at normalised value of 1.00 is only slightly lower and is the second highest peak in the distribution.  For Nippur the peak for Class A weights does occur at the normalised value of 1.00.















Frequency distribution charts of the Class A weights for both the cultures are mostly symmetrical about the nominal weights. This confirms that the weights of chert and hematite are more accurate than the rest of the collection, as already noted by Hemmy1,5 and Hafford3. Since a specimen of hard stone is less prone to chipping, it is likely that these weights were issued by some central authority. Class B weights cannot have been made by the same authority because, if they were, they could easily have manufactured them using the hard stones. It is likely that they were made by some regional authority which could handle the softer materials used in their manufacture. It is also possible that they were made by individuals for their specific use.

An examination of the frequency distribution of Class B weights from the four sites show that they differ widely. This would support the view that many of these were made by individuals for the activity they were engaged in, such as mixing spices, medicinal products or alloys like bronze.   

The frequency distribution of Class B weights from MjD FEM is hardly distinguishable from that of Class A material (Chart B). This was the only Harappan weight collection with material data available to Hemmy when he first analysed weights from this culture. It is no surprise therefore at the conclusion he reached, knowing the chert weights were more accurate than other materials. None of the Harappan weights he subsequently analysed had material information recorded. 

Frequency distribution presents a qualitative picture of the weight system. Standard deviation would represent a quantitative measure of a weight system. A normalised value of a weight specimen represents its deviation from its nominal weight, which has a normalised value of 1.00. The whole list of normalised values of a system can be used to calculate the standard deviation of that collection. 

In Table 1 below are listed standard deviations of weights from 4 sites under 2 headings. Class (A + B) specimens are a mix of all the materials whereas Class A specimens are made of only a specific stone, different for each culture. 





Standard deviations of MjD FEM, Nippur and Ur are close in value for Class A collection of weights. Compare this with the same for Class (A + B). This clearly demonstrates that Class A specimens, made of hard stone, are the officially authorised weights. All the others should be considered unauthorised specimens,  made of different materials and may not have been closely calibrated to the authorised specimens. A mixture of these two sources of weights do not correctly reflect the accuracy of weight systems of each culture. 

Dholavira weights present a unique picture. First the whole collection needs to be separated into 3 different standards R. S. Bisht has identified. Only then the weights of Harappan standard can be examined. As it is, the Class A weights have their peak at the normalised value of 1.00 but the distribution is not symmetrical about it.

More weight data from new sites in both cultures need to be examined to see if they support the view that the Class A weights have similar accuracy in both these cultures. In that respect Dholavira weights need to be re-examined. W. B. Hafford has collected new Mesopotamian weight material (email, 15 August, 2019) that he is examining, and it would be interesting to see how the result of this new material fits with the scheme described above. 

The analysis presented here demonstrates that both Harappan and Mesopotamian civilisations had achieved similar accuracy with their weight systems. It was the mix of two grades of weights that masked the true accuracy of the Mesopotamian weight system. It may be that this mix of two grades of weights was a general practice among these two as well as other cultures. 






References:

1/       Further Excavations at Mohenjo Daro, Ernest Mackay (1938), pp 603

2/      Dholavira Excavation Report - R. S. Bisht (2015). This is available online and can be accessed here: http://asi.nic.in/pdf_d…/dholavira_excavation_report_new.pdf 

3/      Balance Pan Weights from Nippur by W. B. Hafford

4/      The Balance Pan Weights from Ur by William B. Hafford, in Akkadica 133 (2012)

5/         The Statistical Treatment of Ancient Weights by A. S. Hemmy, Ancient Egypt, December 1935 pp 83

6/      Balance Pan Weights from Nippur by W. B. Hafford: page 34


© 2020 Chandrakant Doshi


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30 May, 2019

Dholavira Weights



Dholavira weights

by


Chandrakant Doshi





Key words:   Harappa, Indus, Dholavira, Gujarat, ancient weights



Weights have been recovered from most major sites of the Harappan civilization, including Dholavira, a Harappan site in Kutch, Gujarat. It was excavated from 1989 by R. S. Bisht and in the excavation report Bisht lists 996 weight1specimens. Dholavira flourished in third millennium BCE and has revealed a few unique features not identified elsewhere in the Harappan Civilization, including the heaviest weight recovered from any site. Bisht has identified 3 different weight systems. One of these is the standard Harappan system with the unit weight of 13.625 grams. This note examines the weights that conform to the standard Harappan system.

Weight collection at Dholavira consists of 20 different shapes and 44 different materials. Many of the shapes and materials have not been seen among the recovered weights from earlier excavated sites at Harappa2, Mohenjo Daro2and Chanhu Daro2. These shapes and materials are listed in tables below. 

One useful way of studying the relations between the weights and highlight the systematic nature of the system is by an appropriate chart. A weight system would show the characteristic staircase where the flats or rungs are produced by specimens with values in close proximity to each other. A log chart is an appropriate way of displaying this systematic nature as it can accommodate the wide range of values on a single chart. 

A chart of all the 996 specimens is shown in Image 1. The weights are all sorted in ascending order and converted to log2, as this provideshigh resolution and highlights the binary steps of the weight system. The converted values are plotted on the chart. The specimens within 5% of the Harappan weight system are over plotted in red on the chart. 



Table 1 lists the weight ratios and the nominal weights of the Harappan system as well as the frequencies, within 5% of the nominal weights, of the Dholavira collection. 






Table 2 lists all the shapes and their frequencies. The largest number of weights are listed as cuboids (387), followed by spheroids (124). 






The chart below, Image 2, is a plot of all the 996 weights. For this chart, the weights are all arranged in alphabetical order and then, for each shape, the weights are sorted in ascending order. This way all the shapes can be identified with the plot of each shape starting at the bottom of the chart and reaching towards the top. The plot of the cuboid specimens, which distinctly shows the characteristic staircase of a weight system, is over plotted in red for easy identification. 



Weights are made of 44 different materials. These are listed in Table 3.




A few of the weights with sufficient number of specimens are plotted in the chart below, Image 3. First, the weights are separated by material and, for each material, the weight list is sorted in ascending order and log of each weight to base 2 is plotted on the chart below.The weights of 6 different materials are displayed on the chart. 




The vertical scale of the chart is adjusted to show the binary nature of the system at lower weight values. 3.77 on the vertical axis corresponds to 13.625 grams. Weights of chert, agate and sandstone show distinct staircase, indicating many specimens at the nominal values of the weight system. That is not the case with the other three materials. 


Another way of characterising a weight system is by plotting its frequency distribution. This is shown in Image 4. As most of the weight specimens follow a binary scale, the range of the distribution is from 0.75 to 1.50 normalised values. Height of the columns in the chart represent the frequency. The columns in blue comprise all the specimens within a given range, irrespective of their condition of preservation. Red columns represent only perfect, meaning undamaged, specimens.



The peak frequency is at the nominal values, i.e. 1.00 on the chart. R. S. Bisht has identified 3 different weight systems, which all contribute to the distribution. Frequency distribution of only the chert specimens gives a better representation of the Harappan weight system. This is shown in Image 5.





A. S. Hemmy analysed the Harappan weight system and established the unit of the system at 13.6253grams. Weights below the unit formed a binary sequence of successive halves. Heavier weights followed the sequence of 1, 2, 4, 10 for each decade starting from the unit weight. At the top end the sequence was 100, 200, 400, 800 as no weights at ratio 1000 were recovered at that time. Only two damaged specimens support Hemmy’s top ratio of 800. 

M. S. Vats carried out excavation at Harappa in the 1930s, although the excavation report was not published until 1940. In the report Vats mentions a heavy weight of 26535.6 grams4, though this is not included in the official weight5list. No reason is given for this omission. 

Among the Dholavira weights is one perfect specimen of limestone, weighing 13720 grams, which is close to the ratio 1000. (Acc No. 54404). The heaviest weight is also a limestone specimen weighing 35700 grams, (Acc No. 54550). This is 31% higher than the nearest attainable nominal value of 27250 grams. The condition of preservation of this last piece is not recorded.

Two additional weight systems Bisht reports have not been analysed in this note. This will be undertaken at a later date. Any weight specimens conforming to these additional systems recovered from other sites will have to be examined at the same time. 

The analysis of the weights that conform to the standard Harappan weight system confirms the remarkable accuracy of these weights as first observed by A. S. Hemmy6and Dholavira weights demonstrate that this standard was prevalent throughout the Harappans civilization.



Acknowledgements


I would like to thank Harappa.com for putting the link to the Dholavira Excavation Report on Tweeter (@Ancient_Indus). This note would not have been possible without access to the material found at Dholavira.

The link to the report:   fb.me/8z9Nkaucl




References:

1/.      Weights and Measures, page 334 onwards, Dholavira Excavation Report, available Online:  http://asi.nic.in/pdf_d…/dholavira_excavation_report_new.pdf
         The link was provided by Harappa.com@Ancient_Indus


2/.      Weights from these sites were analysed in a previous paper, which can be viewed here: https://jignashi.blogspot.com/2010/01/

3/.      Chanhu Daro Excavations, 1943 pp 236


4/.      Excavations at Harappa, M. S. Vats, 1940, p. 57, item 8 

5/.      Excavations at Harappa, op cit, pages 363-365

6/.      Further Excavations at Mohenjo Daro, E. J. H. Mackay,  
          1938, p. 603



© 2019 Chandrakant Doshi




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