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.

 

 

PLACE VALUE

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.

 

Counting on in twos from the negative number −6

 

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 .

 

 

 

 

 

 

 

 

 

 

 

 

THE FOUR OPERATIONS

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

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

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.,

78-57 = 78-50-7

  = 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: 817-525

 

 

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:  11-2 = 9.  As a consequence the hundreds go down one to compensate for the ‘borrowed’ tens and the sum is completed.

 

 

MULTIPLICATION

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

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).

 

 

NUMBER PATTERNS

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.