MMMS Script A011 Negative Numbers
www.MMMSRD.com
Negative Numbers
Negative Numbers symbols are introduced (like much else in MMMS) with display cards with
−
1 written on each one. It becomes an object to be dealt with like any other object (like Cups:
and that is the point of previous use of the cups). Once the object and the vocabulary has
been introduced, the objects serve to make addition, subtraction, multiplication and division in
Algebra accessible to all Pupils. But the special quality of ‘Negative one’ has to be introduced
in a way that expresses that special quality. It is represented as a hole in the ground drawn
the same size as a ‘lump of earth’. It is absolutely essential that negative numbers are not
introduced with temperature, bank accounts, blocks of flats or the ‘number line’. Each of these
is an application of the mathematical idea of ‘negative-ness’. The idea of a ‘hole in the ground’
is abandoned very quickly in favour of the symbolic cards – but the ‘holes’ are a necessary
introduction.
The Story
Figure 1 This is a picture of my garden: flat and level
Figure 2 Sometimes I see this in my garden: a lump of earth: I write in my Maths Book:
1
Figure 3
Sometimes I see this in my front garden and this in my back garden. I write:
and
Figure 4
1+1=2
… and sometimes I see this and I write
and
Figure 5
2+1=3
Sometimes I see this (nothing) and I write
0
Figure 6
…. and sometimes in see this (a hole) and I think “I can see one lump of earth missing”
And I write 1[write 1] lump of earth missing [insert the negative sign]
© Richard Dunne
–
1
MMMS Script A011 Negative Numbers
www.MMMSRD.com
And I say “I can see negative one lumps of earth”. [You must not say ‘Negative one lumps of earth
missing’ because the ‘negative’ enshrines the ‘missing-ness’. The form of language is crucial].
And when I see this )negative two and negative one) I write
Figure 7.
-
2 + -1 = - 3
Now use cards to represent the lumps and holes and perform ‘Resources Table / Maths Table’
movements for the following
1
1
and
-
-
and
1
-
1
1
1
1
2+1=3
-
-
1
1
-
-
1
and
Make a point about the cards:
T: When you see …
© Richard Dunne
3--1=-2
-
1
2 + -1 = - 3
1
2+1=-1
MMMS Script A011 Negative Numbers
www.MMMSRD.com
−
1
… you say
−
−
1
1
−
1
−
1
−
−
1
1
−
1
−
1
… you know …
this
is
nega
tive
one
this
is
nega
tive
one
this
is
nega
tive
one
this
is
nega
tive
one
this
is
nega
tive
one
this
is
nega
tive
one
this
is
nega
tive
one
this
is
nega
tive
one
−
Look at each card Look at each 1. Let us count them. One, two, three, four, five, six, seven,
eight. How much is there here? [Indicating all the cards together]. We say Negative eight.
Yes. Eight of those things called Negative one is called Negative eight. When we say
Negative eight we are thinking of eight of these things called Negative eight. That is [Sketch a
picture of a card and place an 8 in front of it] …
8
−
1
But when you write it in your Maths Book you write:
−
8
Now the four operations can be performed using the actions as in the first five Big Ideas in
MMMS.
© Richard Dunne