writing
down multiplication sums
You will recall that multiplication is the adding of a collection of objects several times.
Multiplication sums are laid out as a sort of shorthand
so, for instance, adding a collection of two objects together
three times can be written as 2 x 3 = 6. In words, three
lots of two objects added together makes six objects,
or two times three equals six.
The sum can also be written as below:

T 
U 


2 
x 

3 



6 

When the total of items added together is more than nine,
they are made into a collection of ten items and any left
over are kept in a collection of ‘units’.
In the sum written out below, there are no units left
over, as two times five makes ten, which is written as
one ten and no units: 10.

T 
U 


5 
x 

2 


1 
0 


^{1} 

The next multiplication sum is adding together two lots
of 12: 12x2. Although the learner may have made a cross
table up to twelve (we have only shown a
crosstable up to ten) or may remember their two times
table up to twelve (or their twelve times table), by laying
the multiplication sum out as below, they will have a
method that will work for any multiplication made with
a collection smaller than ten.

T 
U 

1 
2 
x 

2 


2 
4 




The answer for adding two collections of two goes below
in the units column and, likewise, the one collection
of ten times two goes in the tens column.
Where the continuous adding of collections smaller than
ten  the units  results in a total greater than nine,
the group of ten items is carried forward to the tens
column, to be added to the result of the tens column multiplication.

T 
U 

1 
4 
x 

3 


4 
2 


^{1} 

‘Long multiplication’ or extended multiplication
is similar to the simple multiplications above, but the
multiplications for each part  units, tens....  is calculated
and recorded separately, then added together for a final
answer.
Take, for example, the long multiplication of 83 times
27. This is broken into two parts:
 multiplying 83 with tens  two lots of ten (20), this
subsum being 83x2  and
 multiplying 83 with units  seven units (7), this
subsum being 83x7.
These subsum results are then added together.
To write the multiplication with tens, in this case two
tens (twenty), a holding digit  zero  is put in the units column. This holding digit ensures
that the multiplication with tens is automatically placed
in the correct columns for the final addiing up (addition).
Note that
 I have added more column headings, T for Thousands
and H for Hundreds, to help keep the calculation organised
 carrying figures are included as needed;

T 
H 
T 
U 




8 
3 
x 


2 
7 


1 
6 
6 
0 
multiplication with two
tens 


5 
8^{2} 
1 
multplication with seven
units 



2 
2 
4 
1 
addition of the two subsums 



^{1} 
^{1} 



Here is another long multiplication, this time multiplying
123 by 123. This one is broken into three parts:
 multiplying 123 with hundreds  one lot of one hundred
(100), this subsum being 123x100  and
 multiplying 123 with tens  two lots of ten (20),
this subsum being 123x2  and
 multiplying 123 with units  three units (3), this
subsum being 123x7.
These subsum results are then added together.
To write the multiplication with hundreds, in this case
one hundred (100), holding digits  zeroes  are put in the tens and the units columns.
To write the multiplication with tens, in this case two
tens (twenty), a holding digit  zero  is put in the units column.
These marker digits ensure that the multiplication with
hundreds or with tens are automatically placed in the
correct columns for the final addiing up (addition).


T 
H 
T 
U 




1 
2 
3 
x 


1 
2 
3 


1 
2 
3 
0 
0 
multiplication with one
hundred 


2 
4 
6 
0 
multiplication with two
tens 



3 
6 
9 
multplication with three
units 



1 
5 
1 
2 
9 
addition of the three subsums 




^{1} 
^{1} 



writing
down division sums
Writing down division sums is done somewhat differently from addition, subtraction
and multiplication sums. Here, we will start by describing
division into less than ten parts. Later, we will do some
‘long (but not very long) division’.
Division sums are often written in the ‘shorthand’
form of, say, 14÷2=7 (or 14/2=7).
In this
division sum, two is being constantly subtracted from
fourteen  in fact, it is subtracted seven times. The
notation ÷ or / can be described
(or interpreted as) ‘constantly subtracted by’,
or ‘divided by’.
Although for a relatively simple division sum, it is
not necessary to lay out the division sum as done for
dividing bigger numbers, it can be regarded as a good
habit, as well as providing a consistent method. The more
complicated the sum, the more it is helpful to write it
so the steps in the division can be followed, and so much
of the working is incorporated.
So, for writing down and providing clarity of working, 14÷2=7 is writtten thus:
The number being divided is put
inside the Lshape, the number doing the dividing is put
on the left of the vertical line, and the answer is put
above the horizontal line.
For larger sums, involving hundreds or even thousands,
as well as tens and units, it is essential to hold in
mind that each column, from left to right, holds ten times
as many objects as the column to its immediate right 
hundred is ten lots of ten, and ten is ten lots of units
(one).
Thus, in the division sum below, three lots of a hundred,
and five lots of ten, and two units are to be divided
(or shared) into eight parts.
The next division sum is 352÷8.
The working out the sum (calculation) in four steps is
followed by an explanation, the numbers being used in
the sum written in bold.
 Three, being smaller than eight,
cannot have eight subtracted from it
even once
(eight does not divide into three).
With the three being a collection
of ten times as many objects as the five to its right, the three and five together behave rather
like thirtyfive.
 This ‘35’
is divided by eight, or
eight is subtracted until there is nothing or less than
eight left.
In this case, eight can be subtracted four times (can be divided into 35
four times) with three left over 
a remainder of three (in pink to distinguish it from the division sum as
originally written).
The answer of four is written above
the horizontal line, in the tens column (the division
was of 35 tens).
 The remainder of three (remember that is three lots of ten) is added to the two units, being written as a small
three just before the two.
The last part of this division is three tens plus two units (32) divided by eight (or eight subtracted, in this
case, four times).
 The answer of four is written above
the horizontal line, in the units column (the division
was of 32 units).
Now this many seem rather a laborious explanation to
those who have done division so many times, and who have
properly learnt their times tables. For them, these steps
now happen pretty well automatically (except perhaps remembering
some of the times table and working out the reminder to
carry forward). But for a beginner, it is a great help
to be aided by explanations of each step that do not glide
over a step so simple that it just assumed to be done
without instruction.
The next division sum is 5293÷7.
The working out the sum (calculation) is shown in four
steps, the numbers carried being shown in pink.
The final subsum of 43÷7 gives
a reminder of of one, or one left over. This is recorded
at the end of the answer as r.1  r. being an abbreviation of remainder.
calculation
‘shortcuts’
There a number of calculation ‘shortcuts’
to help with divisions. However, some are more useful
than others, and some are practically no use at all, other
than as an amusement and diversion. They should not be
used as a sunstitute for understanding how division works,
but regarded as possibly useful tools that may speed up
a calculation.
 All numbers divisible by two end
in an even number (2,4,6,8) or in zero (0).
 For all numbers divisible by three,
the sum of the digits is also divisible by three, but
that sum will be a much smaller, more recognisable number.
For instance, 483 ‘adds up’ like this: 4+8+3=15,
and 15 is divisible by three (15÷3=5),
so 483 must be divisible by three.
Here is a longer example, 123456789. The digits add
1+2+3+4+5+6+7+8+9=45. 4 and 5 make 9, so 123456789 can
be divided by three.
 If the last two digits form a number divisible by four, then the complete number is divisible
by four.
Example: 569828. The last two digits are 28. 28÷4=7,
therefore 569828 is divisible by four.
 All numbers ending in either five (5) or in zero (0)
can be divided by five.
 Numbers divisible by six must be
divisible by both two and three. That
is, the last digit will be even or zero, and the number’s
digits add up to three or a multiple of three.
 Determining whether a number is divisible by seven is a bit more complicated. There are several methods,
but I shall just explain one method here.
The idea is, double the last digit and subtract that
doubling from the rest of the digits. If the result
can be divided by seven, then so can
the complete number.
For example using the number 434, doubling 4 gives 8:
438=35. 35 is divisible by 7 (five times), so 434 is
divisible by seven. Note that as the number being examined
becomes larger, so this method become ever more laborious.
Thus it can take nearly as much effort to check for
divisibility as it does to do the division by seven.
 To see whether a number can be divided by eight,
check its last three numbers.
If the first number is even, the number is divisible
by eight if the last two numbers are divisible by eight.
So 648 can be divided by 8.
If the first number is odd, subtract 4 from the last
two numbers. If that result is divisible by eight, then
the whole number will be as well. Thus, for 1352 subtract
4 from 52: 524=48. 48 can be divided by by 8, so 1352 is divisible by 8.
 A number is divisible by nine if
the sum of its digits can be divided by nine. For example,
the digits of the number 261 add up to nine (2+6+1=9).
Thus 261 is dismissible by nine.
 All numbers ending in zero can be divided by ten.
writing
down long division sums
Long division, dividing a number bigger than ten into
another number, can be seen as doing several, consecutive,
smaller division sums.
For larger sums, involving hundreds or even thousands,
as well as tens and units, it is essential to hold in
mind that each column, from left to right, holds ten times
as many objects as the column to its immediate right 
hundred is ten lots of ten, and ten is ten lots of units
(one).
So a long division sum such as 952÷17,
nine lots of hundred, and five lots of ten, and two units
(or ones) are to be divided (or shared) into seventeen
parts. The smaller, subsums are organised thus:
 the hundreds divided by seventeen,
 next the tens, plus any hundreds that are a reminder
from the first subdivision sum, divided by seventeen,
 then the units, plus any tens that are a reminder
from the second subdivision sum, divided by seventeen.
The number being
divided is put inside the Lshape, the number doing
the dividing is put on the left of the vertical line,
and the answer is put above the horizontal line.
The working out the sum (calculation) in four steps below
is followed by an explanation, the numbers being used
in the sum written in bold.
 17 is too big to divide into 9, 17 does not ‘go into’ 9.
Remember, you are seeing how many times 17 can be subtracted, in this case from 9.
If it is helpful, you can put a zero above the nine,
on the answer line, to show 17 ‘goes
into’ 9 no times, but this zero will not be included
in the final answer. (We have ‘written’
zero lightly to indicate that it is optional.)
 The remainder  9  from the first subsum is put
as its result. This is written similarly to the subtraction
sums described above.
Now we bring down the 5 tens to create
the next number for the next subsum.
 17 ‘goes into’ 95 five
times  you can subtract 17 from 95
five times.
The 5 is written as part of the answer at the top of
the division sum.
The remainder  10  from this second subsum is put
as its result.
 Now we bring down the 2 units (ones)
to create the next number for the next subsum. 17 can be subtracted six times from 102  17 ‘goes in’ 102 six times.
The 6 is written as part of the answer at the top of
the division sum.
There is no reminder.
A bigger long division sum, say with thousands as well,
or with a larger divisor,
is performed in a similar fashion.
In doing long and complex calculations, some learners
can take delight (and satisfaction) in doing the multiplications
and subtractions needed to complete the subsums. Other
learners may become frustrated or overwhelmed. It is normal for humans to make regular errors when transcribing in
detailed calculations, just as you will find misspellings
even in the most rigorously edited books.
“Let him [the abbot] so temper all things that
the strong may have something to strive for and the
weak have nothing to dismay them.”
“IF A BROTHER IS COMMANDED TO DO THE IMPOSSIBLE
If it happens that orders are given to a brother which
are too heavy or impossible, let him receive the order
of his superior with perfect gentleness and obedience.
But if he finds that the weight of the burden is altogether
beyond his strength to fulfil, then let him explain
to his superior the reasons why he cannot do
it, patiently at a suitable time, without showing
any pride or resistance or contradiction. Then, after
his representations, if the superior remains firm in
requiring what he has ordered, let the subject realise
that it is better so, and out of charity, trusting
in the help of God, let him obey.”
[The rule
of Saint Benedict for monasteries]
Good teaching is not simple.
writing
down longer long division sums
As described just
previously, long division sums, which involve dividing
a number bigger than ten into another number, can be seen
as doing several, consecutive, smaller division sums.
For these larger sums, it is essential to hold in mind
that each column, from left to right, holds ten times
as many objects as the column to its immediate right 
 hundred thousand is ten lots of ten thousand
 ten thousand is ten lots of a thousand
 thousand is ten lots of a hundred
 hundred is ten lots of ten
 and ten is ten lots of units (one).
So with a longer long division sum such as 987,654,321÷145,
nine lots of hundred million,
and eight lots of ten million,
and seven lots of a million,
and six lots of hundred thousand,
and five lots of ten thousand,
and four lots of thousand,
and three lots of hundred,
and two lots of ten,
and one unit (or one) are to be divided (or shared) into
one hundred and fortyfive parts.
The smaller, subsums are organised thus:
 the hundred millions divided by a hundred and fortyfive,
 next the ten millions, plus any hundred millions that
are a remainder from the first subdivision, divided
by a hundred and fortyfive
 then the millions, plus any ten millions that are
a remainder from the second subdivision, divided by
a hundred and fortyfive
In this long division sums, the first
calculable subdivision, that is one that starts with
a number greater than the divisor, 145, is the next
subdivision.
 next the hundred thousands, plus any millions that
are a remainder from the third (first calculable) subdivision,
divided by a hundred and fortyfive
 then the ten thousands, plus any hundred thousands
that are a remainder from the fourth (second calculable)subdivision,
divided by a hundred and fortyfive
 next the thousands, plus any ten thousands that are
a remainder from the fifth (third calculable) subdivision,
divided by a hundred and fortyfive
 then the hundreds, plus any thousands that are a remainder
from the sixth (fourth calculable) subdivision, divided
by a hundred and fortyfive,
 next the tens, plus any hundreds that are a reminder
from the seventh (fifth calculable) subdivision sum,
are divided by a hundred and fortyfive,
 lastly the units, plus any tens that are a reminder
from the fourth (sixth calculable) subdivision sum,
divided by a hundred and fortyfive divided by seventeen.
The number being
divided is put inside the Lshape, the number doing
the dividing is put on the left of the vertical line,
and the answer is put above the horizontal line.
The working out the sum (calculation) in six steps below
is followed by an explanation, the numbers being used
in the sum written in bold. Numbers carried
forward as part of multiplications [going to the left]
are shown in pink, while numbers
borrowed as part of subtractions and taken to the next
column [to the right] are shown in blue.
Here follows a worked example for the longer longer division
sum 987,654,321÷145. I have ‘cheated’
a bit with the choice of divisor, so that the multiplications
needed to caculate the necessary multiples of the divisor
are not too time consuming. Of course, you can construct
ever more interesting longer long division sums for yourself.
How about 1,023,456,789÷777?
 145 is too big to go into either 9 or 98, 145 does not ‘go into’ either 9 or 98. Remember, you are seeing how
many times 145 can be subtracted.
If it is helpful, you can put a zero above the 9 and the 8, on the answer line, to show 145 ‘goes into’ 9 and into
98 no times, but these zeros will not be included in
the final answer. (We have ‘written’ zero
lightly to indicate that it is optional.)
So, the first subdivision sum is 987÷145.
145 ‘goes into’ 987 six
times  you can subtract 145 from 987
six times.
[Mental arithmetic tip: consider
that 145 is 1505. Two lots of 150 is 300, six lots
of 150 is 900, and six lots of 145 will be less (six
times 5 less) than 900. So 145 can be subtracted six
times from 987.]
The 6 is written as part of the answer at the top of
the division sum.
The remainder  117  from this first subsum is put
as its result. This is written similarly to the subtraction
sums described above.
 Next we bring down the 6 hundred
thousands to create the next number for the next subsum.
145 ‘goes into’ 1176 eight
times  you can subtract 145 from 1176
eight times.
[Mental arithmetic tip: consider
that 145 is 1505. Two lots of 150 is 300, eight lots
of 150 is 1200, and eight lots of 145 will be less (eight
times 5 less) than 1200. So 145 can be subtracted eight
times from 1176.]
The 8 is written as part of the answer at
the top of the division sum.
The remainder  16  from this second subsum is put
as its result.
 Now we bring down the 5 ten thousands
to create the next number for the next subsum.
145 ‘goes into’ 165 once
 you can subtract 145 from 165 one
time.
The 1 is written as part of the answer at the top of
the division sum.
The remainder  20  from this third subsum is put
as its result.
 Next we bring down the 4 thousands
to create the next number for the next subsum.
145 ‘goes into’ 204 once
 you can subtract 145 from 204 one
time.
The 1 is written as part of the answer at the top of
the division sum.
The remainder  59  from this fourth subsum is put
as its result.
 Now we bring down the 3 hundreds
to create the next number for the next subsum.
145 ‘goes into’ 593 four
times  you can subtract 145 from 593
four times.
The 4 is written as part of the answer at the top of
the division sum.
The remainder  13  from this fifth subsum is put
as its result.
 Next we bring down the 2 tens to
create the next number for the next subsum. But 145 is bigger than 132, so 145 can be subtracted
from 132 no times  145 goes into 132
no times. So a zero is written as part of the answer
at the top of the division sum.
We now bring down the 1 unit to create
the number for the final subsum.
145 can be subtracted nine times from
1321  145 ‘goes into’
1321 nine times.
[Mental arithmetic tip: consider
that 145 is 1505. Two lots of 150 is 300, nine lots
of 150 is 1500, and nine lots of 145 will be less (nine
times 5 less) than 1500. So 145 can be subtracted nine
times from 1321.]
9 is written as part of the answer at the top of the
division sum.
There is a reminder of 16.
end notes
 Tom Lehrer, author
of many sharp, if also hilarious, songs, wrote one on
doing sums  New Math, involving the sum 342173.
New Math
Introduction
Some of you who have small children may have perhaps
been put in the embarrassing position of being unable
to do your child's arithmetic homework because of
the current revolution in mathematics teaching known
as the New Math. So as a public service here tonight
I thought I would offer a brief lesson in the New
Math. Tonight we're going to cover subtraction.
This is the first room I've worked for a while that
didn't have a blackboard so we will have to make
due with more primitive visual aids, as they say
in the "ed biz." Consider the following
subtraction problem, which I will put up here: 342
 173.
Now remember how we used to do that. three from
two is nine; carry the one, and if you're under
35 or went to a private school you say seven from
three is six, but if you're over 35 and went to
a public school you say eight from four is six;
carry the one so we have 169, but in the new approach,
as you know, the important thing is to understand
what you're doing rather than to get the right answer.
Here's how they do it now.
You can't take three from two,
Two is less than three,
So you look at the four in the tens place.
Now that's really four tens,
So you make it three tens,
Regroup, and you change a ten to ten ones,
And you add them to the two and get twelve,
And you take away three, that's nine.
Is that clear?
Now instead of four in the tens place
You've got three,
'Cause you added one,
That is to say, ten, to the two,
But you can't take seven from three,
So you look in the hundreds place.
From the three you then use one
To make ten ones...
(And you know why four plus minus one
Plus ten is fourteen minus one?
'Cause addition is commutative, right.)
And so you have thirteen tens,
And you take away seven,
And that leaves five...
Well, six actually.
But the idea is the important thing.
Now go back to the hundreds place,
And you're left with two.
And you take away one from two,
And that leaves...?
Everybody get one?
Not bad for the first day!
Hooray for new math,
Newhoohoomath,
It won't do you a bit of good to review math.
It's so simple,
So very simple,
That only a child can do it!
Now that actually is not the answer that I had in
mind, because the book that I
got this problem out of wants you to do it in base
eight. But don't panic. Base
eight is just like base ten really  if you're missing
two fingers. Shall we
have a go at it? Hang on.
You can't take three from two,
Two is less than three,
So you look at the four in the eights place.
Now that's really four eights,
So you make it three eights,
Regroup, and you change an eight to eight ones,
And you add them to the two,
and you get onetwo base eight,
Which is ten base ten,
And you take away three, that's seven.
Now instead of four in the eights place
You've got three,
'Cause you added one,
That is to say, eight, to the two,
But you can't take seven from three,
So you look at the sixtyfours.
"Sixtyfour? How did sixtyfour get into it?"
I hear you cry.
Well, sixtyfour is eight squared, don't you see?
(Well, you ask a silly question, and you get a silly
answer.)
From the three you then use one
To make eight ones,
And you add those ones to the three,
And you get onethree base eight,
Or, in other words,
In base ten you have eleven,
And you take away seven,
And seven from eleven is four.
Now go back to the sixtyfours,
And you're left with two,
And you take away one from two,
And that leaves...?
Now, let's not always see the same hands.
One, that's right!
Whoever got one can stay after the show and clean
the erasers.
Hooray for new math,
Newhoohoomath,
It won't do you a bit of good to review math.
It's so simple,
So very simple,
That only a child can do it!
Come back tomorrow night. We're gonna do fractions.
Now I've often thought I'd like to write a mathematics
text book someday because I have a title that I
know will sell a million copies. I'm gonna call
it Tropic Of Calculus.
That Was the Year That Was by Tom Lehrer
1966, reissued as CD in 1990; Wea/Warner Brothers;
ASIN: B000002KO7
$10.99 [amazon.com]/ £9.98 [amazon.co.uk]
 Although for learners it not
necessary, and may be confusing, teachers of mathematics
may well use technical terms for things like ‘the
number doing the dividing’, or ‘the number
being multiplied’.
So, for multiplication sums, multiplicand x multiplier
= product; and
for division sums, dividend ÷ divisor = quotient.
 Benedict of Clairvaux (1090 – 20 August 1153) The rule of Saint Benedict
for monasteries, a translation by Dom Bernard
Basil Bolton OSB, monk of Ealing Abbey, 1968.
