How can we help students make sense of addition and subtraction of
negative numbers?
Perhaps the following can help...
Let this be $4$
and this
and this
and this
Students could be asked to suggest some other possibilities.
Can they explain why they all represent $4$?
Let this be $^-2$
and this
and this
and this
Students could be asked to suggest some other possibilities.
Can they explain why they all represent $^-2$?
This is $3$
When we partition it we can make
different sums:
 |
$2 + 1 = 3$
$3 - 2 = 1$
$3 - 1 = 2$
|
Top + Bottom =
Whole
Whole - Top =
Bottom
Whole - Bottom =
Top
|
If we partition it
differently we can make different sums:
 |
$4 + (^-1) = 3$
$3 - 4 =$ $^-1$
$3 - (^-1) = 4$
|
Top + Bottom =
Whole
Whole - Top =
Bottom
Whole - Bottom = Top
|
And we can partition it
differently again:
 |
$5 + (^-2) = 3$
$3 - 5 =$ $^-2$
$3 - (^-2) = 5$
|
Top + Bottom =
Whole
Whole - Top =
Bottom
Whole - Bottom = Top
|
This is $^-4$
When we partition it we can make
different sums:
 |
$1 + (-5) =$ $^-4$
$^-4 - 1 =$ $^-5$
$^-4 - (^-5) = 1$
|
Top + Bottom =
Whole
Whole - Top =
Bottom
Whole - Bottom =
Top
|
If we partition it differently we
can make different sums:
 |
$^-1 + (^-3) =$ $^-4$
$^-4 - (^-1) =$ $^-3$
$^-4 - (^-3) =$ $^-1$
|
Top + Bottom =
Whole
Whole - Top =
Bottom
Whole - Bottom =
Top
|
Students could be asked to create three different sums based
on their partitions of other sets of pluses and minuses.
How can this model help us make
sense of adding negative numbers?
We can add a negative number to a positive number and end up
with a positive solution:
| $6 + (^-2)$ |
= |
? |
 |
= |
 |
| $6 + (^-2)$ |
= |
4 |
We can add a negative number to a positive number and end up with a
negative solution:
| $4 + (^-6)$ |
= |
? |
 |
= |
 |
| $4 + (^-6)$ |
= |
$^-2$ |
We can also add a negative number to a negative number:
| $^-2 +
(^-3)$ |
= |
? |
 |
= |
 |
| $^-2 +
(^-3)$ |
= |
$^-5$ |
Students could be asked to create a few more additions like
the ones above.
Gathering various results may then offer an opportunity to ask
students what they notice.
Can they suggest how to add negative numbers without going to
all the trouble of drawing pluses and minuses?
How can this model help us make
sense of subtracting negative numbers?
We can subtract a negative number
from a negative number and end up with a negative
solution:
Consider $^-8 - (^-3)$:
| This represents $^-8$ |
 |
|
And this represents $^-8$
subtract $^-3$
|
 |
| Leaving us with $^-5$ |
 |
So $^-8 - (^-3) =$ $^-5$
Can we subtract a negative number from a positive number?
Let's consider $5
- (^-2)$ = ?
How can we take $^-2$ from
$5$?
 |
represents $5$ |
| but so does this |
 |
Now we can take $^-2$ from $5$ and we will be left with:
So $5 - (^-2) = 7$
We can subtract a negative number from a negative number and end up
with a positive solution!
Let's consider $^-4 - (^-7)$ =
?
 |
represents $^-4$ |
| but so does this |
 |
Now we can take $^-7$ from $^-4$ and we will be left with:
So $^-4 - (^-7) = 3$
Students could be asked to create a few more subtractions like the
ones above.
Gathering various results may then offer an opportunity to ask
students what they notice.
Can they suggest how to subtract negative numbers without going to
all the trouble of drawing pluses and minuses?
I am aware that there are many other models that teachers use to
help their students make sense of addition and subtraction of
negative numbers - hot air balloons, with the options of adding and
subtracting heat

and sand
bags

, is just one possible
example.
I'm
keen to publish teachers'
accounts of other models that they have used successfully with
their students. Anyone willing to share these can email us at
nrich@damtp.cam.ac.uk .
Any comments we receive will be included in the
Notes .
I'm indebted to Don Steward for
first introducing me to this idea, and to Jitka Holcova for her
assistance with all the images.