A: About two years ago I was asked about operations involving integers and following my reply I was sent a learning "tool" called a Negator, which was invented by Peter Shannon, principal teacher of mathematics from Boroughmuir High school in Edinburgh. I had a quick look at the tool and moved onto other readers' questions. Shame on me! I have recently been using the Negator to teach operations with integers very successfully.
The Negator has a moveable central disk, which allows for the dialling of numbers to carry out operations. The rules are simple, and there are only three.
First, at the beginning of a "sum" the arrow has to point at "0".
Second, if you are carrying out an addition, then you dial from the zero to the indicated number each time, even if you are adding a negative number.
Third, for subtraction you dial from the number to zero. So for 5 + 3 you would dial from "0" to "5", then put your pen in hole at "0" and dial to "3". The answer is given alongside the "8", indicating that 5 + 3 = 8. 5 - 3 would be dial from "0" to "5", then put your pen in "3" and dial to "0" giving 2.
What about 5 - - 3? Which can be thought of as 5 - (-3). In this instance dial from "0" to "5" as before. This time the operator is subtract, so move from the number to zero, and the number is - 3. So put your pen in - 3 and dial to "0" giving an answer of 8.
The same activity could be created using a number line with pupils moving along it following the same rules to create an understanding of operations on negative numbers, this just simplifies the process!
Having demonstrated how the Negator works. I would suggest that pupils carry out an investigation using the Negator. Make a selection of "sums" one per white label. Maybe five each of the types: * a + b; a
0 and b
0, eg 3 + 5, 6 + 7, 2 + 1, 3 + 4, 10 + 1
* a + b; a LESS THAN 0 and b
0, eg - 3 + 5, - 8 + 4, - 2 + 4, - 6 + 3, - 5 + 10
* a + b; a
0 and b LESS THAN 0, eg 1 - 4, 8 - 6, 9 - 10, 5 - 2, 7 - 15
* a + b; a LESS THAN 0 and b LESS THAN 0, eg - 4 - 5, - 6 - 8
* a - b, a
0 and b LESS THAN 0, eg 5- - 3, 7- - 8, 2- - 5 etc Have the 25 labels shuffled in an envelope and give out five pieces of A5 paper for each group to stick the sums on. Share out the labels between pairs of groups of pupils. Get them to sort them into matching sets, letting them know there are five groups. When these have been checked for correctness, ask them to stick them onto the five pieces of A5 paper, one group per sheet.
Next they use the Negator to find the answers to each "sum". Finally,they look at their answers and see if they can write a rule for times when they do not have the Negator to hand. This activity could also be done with a calculator, but kinaesthetic and visual learning taking place on turning the dial must not be underestimated. Multiplication can also be investigated, using the same three rules.
Consider 3 x - 2 as three lots of - 2; that is, - 2 + - 2 + - 2. So you dial from "0" to "-2" three times, giving - 6. Next, consider - 3 x 2 as minus three lots of 2, rather than - 3 twice. As we are asking for three subtractions of 2, we dial 2 to "0" three times, also giving - 6. In a similar way, consider - 3 x - 2 as minus three lots of - 2. We dial from - 2 to "0" three times, giving us +6, great!
* For details about the Negator, email: peter. email@example.com Hugh Beere has created a Negator for you to try which is available at www.mathagonyaunt.co.uk
A site that describes the use of red and blue chips for the understanding of addition and subtraction of integers
A java activity based on a similar idea using coloured chips for addition and subtraction for the interactive whiteboard can be found at
An interesting idea using the idea of digging holes and filling them in can be found at www.nzmaths.co.nz numeracyOther%20material LessonplansTeaching%20Integ
* Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.
Email your questions to Mathagony Aunt at firstname.lastname@example.org Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX