Teachers’ Notes Key Stage 2
The Calculating Mr One has been developed to support the teaching of numeracy in schools at
both Key Stage One and Key Stage Two and to reinforce much of the number work
tested in the SATS for years 2 and 6.
The main focus of the play is on number patterns, mental
arithmetic and approaches to the four functions, including work with the number
line, number bonds, the number and multiplication grids, and also including
reading and interpreting data from a table.
Each mathematical idea is built upon throughout the play
involving the audience directly in both the calculations and the methodology
employed in problem solving and encouraging them to use a variety of approaches
to achieve a single answer. Throughout
the play the work is put into a number of everyday contexts through which the
problems are explored.
The following pages provide a summary of the work covered
and examples of how it is put into practice in the play. At the back of this booklet there are
pupils’ worksheets designed to tie in with the mathematical concepts covered in
the play and these can be photocopied for use in the classroom.
ORDERING
NUMBERS
The
position of each digit is important. We
show that a number, such as 503 is not the same if the order of the numbers is
reversed: 503 does not equal 305.
Numbers
are created from three separate digits and are arranged in ascending and
descending order according to their value.
The language of ordinality is used i.e., which comes first, second etc.
The position of a digit in a
number gives its value and the HTU grid is introduced. We show that each position in the grid is
worth ten times more than the position on its right.
The children are asked to make a number ten times larger, a
hundred times larger and then the operations are reversed. In the grid we see that as a number is
multiplied by ten all digits move one space to the left, and by one hundred
they move two spaces. We show that when
dividing by ten the digits move one space to the right. If there are any zeros then they must move
also as they are part of the number.

Th 
H 
T 
U. 
t 



5 
0 
3 
X
10 

5 
0 
3 

X 100 
5 
0 
3 
0 

The number line is introduced and
numbers are placed in their position on the line. Simple numbers are used to begin with, then larger numbers are
introduced and finally negative numbers are added to the number line.
We count forwards and backwards in
2s, 10s and 5s to see how negative numbers are different from positive numbers
and how they behave on the number line.
We use a thermometer to illustrate how to move from a negative number to
a positive one by counting on.
ESTIMATION
Estimation is approached as a useful tool in several
ways: an approximate amount can be used
when an exact answer is not needed i.e., how many were present at a football
match, it can be used to give a guide to the final answer of a calculation or
to check it after a calculation has been made and rounding up can be used in
mental arithmetic to help with a calculation where figures are rounded up or
down to the nearest unit, ten, hundred etc.
Various numbers from the number line are given to the
audience who then have to round up or down e.g., 18 is rounded up to 20. When a number is half way between it is
always rounded up e.g., 15 is rounded up to 20.
Rounding up is then used to find an approximate answer for a
problem: There are 9 Rowntrees Fruit Pastilles in a packet. If I have one a day for every day of January
how many are eaten in the month? We
round up to 10 and down to 30, therefore about 300 are eaten in a month.
Rounding up is used to make mental calculation easier. If you need to add 9 or 19 etc. round it up
to 10 or 20 to make the sum easier and
then take away the 1 .
With addition, subtraction, multiplication and division the
method employed to solve the problem is taken to be as important as the
answer. Number bonds are extremely
important to help with mental calculations and times tables are the building
blocks of multiplication and division.
Addition is approached initially from the counting on
method. We see that often it is easier
to put the biggest number first and count on.
Calendars are useful to illustrate counting on: how many days to my birthday?
Number bonds are introduced and explored from 1 to 10 and
then to 20 and on to 100. In addition
it is useful to find pairs of number that add up to say 10, 20 or 100 and then add on the rest:
6+8+3+4+7+2+9 = ?
(6+4) + (7+3) + (8+2) + 9 = 39
In this way we see the commutative law (although the
terminology is not used in the play)
Another method for addition is to use the 100 number grid
which quickly shows the number patterns made when a number is, say, 9, 19 or 29 more than another. The number grid is then used to work out the
best strategies for adding a pair of two digit numbers together:
23 + 56 = ?
Start at 56 and move down the grid 20 spaces to 76 and add on the 3
45 + 28 = ? Start at 45 and move on 30 to 75 then
remember to take away the extra 2
The partition method of addition is also explored firstly
with two digit numbers where we practice adding first the 10s and then the
units of two numbers: 46+19 = 40+10+6+9
= 65
And then we move on to adding HTU + HTU
With written addition we use the vertical method and
approach it initially using an addition grid: 562 + 427
Then approach a more difficult sum using the traditional
method of ‘carrying’ the extra tens and hundreds over to their appropriate
column:
Subtraction is treated as the inverse of addition and
looking we see that ‘counting on’ using a number line is a good method to find
the answer.
Partitioning can also be used to solve a problem: e.g.,
7857 = 78507
= 21
We look at the connection between addition and subtraction
and see how to check an answer by adding.
Finally we use the HTU grid to work out more difficult sums:
817525
The units column is easy but the tens column is more of a
problem. By taking one of the hundreds
and adding it to the tens the sum is now possible: 112 = 9. As a consequence
the hundreds go down one to compensate for the ‘borrowed’ tens and the sum is
completed.
Multiplication is introduced as repeated addition and the
number grid is used to find patterns for various tables. Work on the times tables is then extended to
a multiplication grid.
When approaching long multiplication there are two methods
used:
Firstly a sum can be partitioned into smaller calculations
and the answers added together: e.g.,
12 x 33 = (10 x 33) + (2
x 33)
=
330 + 66
= 396
Or the sum can be approached like this:
Division is understood as sharing: there is a certain
number of buns for tea, but there may be different numbers of people coming to
tea. How can the buns be shared between
the different numbers of guests? How many are left over?
It is also seen as repeated subtraction:
9
– 3 – 3  3 = 0 so 3 can be
subtracted from 9 three times 9
÷ 3 = 3
and we see that division is the inverse operation of
multiplication:
9 x 8 = 72 so 72
÷ 8 = 9 and 8 x 9 = 72
so 72 ÷ 9 = 8
The connection between multiplication and division is
explored and we see that knowing your times tables helps greatly in solving
division problems:
42 ÷ 6 = ? We know that
6 x 7 = 42 So 42 ÷ 6 = 7
When we have a number which does not divide exactly we use
our tables to work out the remainder:
44 ÷ 6 = 7 remainder 2
We give the division a context to illustrate the significance
of remainders: if a packet of fruit pastilles has 20 sweets and must be
shared between 3 people then each person will have 6 sweets (3 x 6 = 18) and 2
will be left over (the remainder).
We identify odd and even numbers. All even numbers end in 0, 2, 4, 6, or 8 and are divisible by
2. All even numbers end in 1, 3, 5, 7
or 9 and cannot be divided by 2.
Square numbers and their inverse, the square roots should be
known to help solve problems.
Numbers that divide exactly into other numbers are shown to
be factors of that number.
Prime numbers have only two factors: they are divisible only
by themselves and 1 and prime numbers up to 20 are seen to be 2,3,5, 7, 11, 13,
17 and 19.
Number sequences are treated as a code which needs cracking
to see which number comes next in the sequence.