Adding and Subtracting Positive and Negative Numbers

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

$4 + (^-1) = 3$ $3 - 4 =$ $^-1$ $3 - (^-1) = 4$

Top + Bottom = Whole Whole - Top = Bottom Whole - Bottom = Top

$5 + (^-2) = 3$ $3 - 5 =$ $^-2$ $3 - (^-2) = 5$

Top + Bottom = Whole Whole - Top = Bottom Whole - Bottom = Top

This is $^-4$

$1 + (-5) =$ $^-4$ $^-4 - 1 =$ $^-5$ $^-4 - (^-5) = 1$

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

How can this model help us make sense of adding negative numbers?

$6 + (^-2)$	=	?
	=
$6 + (^-2)$	=	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:

$^-2 + (^-3)$	=	?
	=
$^-2 + (^-3)$	=	$^-5$

We can subtract a negative number from a negative number and end up with a negative solution:

This represents $^-8$

And this represents $^-8$ subtract $^-3$

Leaving us with $^-5$

Can we subtract a negative number from a positive number?

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.