“Two minuses make a plus” - a lackadaisical phrase that takes responsibility for a popular misconception around arithmetic with negative numbers. Take for example the following:
-5 - 2
Ask a group of students to find the answers to this and a selection of answers can arise. 7 and 3 both crop up due to the aforementioned misconception (-3 is also a culprit but for different reasons) Why can this happen? Learners tend to fixate on this erroneous rubric rather than break down what they are trying to do. If they articulate the question to themselves as “minus five minus 2” then this can trigger the “two minuses make a plus” thought process which can become the sole focus of their solution, with everything else working around this – “I must end up with a positive answer, now how can I get to that?” This is where the tier three language becomes important for breaking down these misconceptions and embedding understanding. For example, if students were taught to articulate this questions as “negative five, subtract two” this differentiates the number (negative five) from the operation (subtract two). A number line can also be used to establish the nature of the question by identifying where negative five is and then subtracting two by moving to the left.
These misconceptions can arise due to students clumping rules for addition/subtraction of negative numbers with multiplication/division. This could be because they have just been taught them by rote which is why it is important to explore the underlying concepts behind arithmetic with negative numbers. If a student understands for example that “subtracting negative twelve is the same as adding twelve”, rather than “minus minus twelve is the same as plus twelve”, they are more likely to have success.
Subtracting a negative number can easily be conceptualised as adding debt, or the famous example of the effect on the temperature of a witches caldron when ice blocks are added to it. It can be more difficult however to fully understand the underlying concepts with multiplication of negative numbers. Students learn early on that multiplication is a method for repeated addition of positive numbers. This approach can help students understand what is happening when negative numbers are multiplied by positive ones, particularly if they understand how to add and subtract negative numbers. For example, means 4 lots of 3, or four 3s added together.
4 x 3 = 3 + 3 + 3 +3
I think most teachers would illustrate an example like this to demonstrate as well. Four lots of negative three should be four ( s added together.
4×(-3)=(-3)+(-3)+(-3)+(-3)=-3-3-3-3
A natural question to follow could be “well, what is then?” How do you explain negative four lots of 3? The easy get out to this remind students that multiplication is commutative, so
(-4)×3=3×(-4)
Which means we can think of it as three lots of , which they know how to comprehend. Unfortunately, this means that when the inevitable question of “so what is comes up, the reason why the answer is positive twelve often gets a less than satisfactory explanation. Mathematical symmetry is often cited using small tables or diagrams, and whilst this isn’t misleading in any way, it is inadequate to suggest that these rules linking addition and multiplication just break down when we start using numbers less than zero.
Let’s look in more detail at and interpreting this as four lots of three:
3+3+3+3
This implies that we start with a 3 and add three more 3s. However, it is more useful to think of repeated addition as starting with zero:
0+3+3+3+3
It is clear here that we are adding four 3s. Why is this helpful? There are many reasons, but for the purpose of this blog we will look at how this can help us conceptualise multiplying by negatives. Lets wind back to . Negative four lots of a number implies that we are repeatedly subtracting that number, from zero:
(-4)×3=0-3-3-3-3=-12
Having the zero as a starting point can assist with understanding that multiplying by a negative number is the same as repeated subtraction. This also gives us a wholly rewarding explanation as to why the product of two negative numbers is positive. Let’s take (-4)×(-3)
(-4)×(-3)=0-(-3)-(-3)-(-3)-(-3)
From previous knowledge that subtracting a negative number is identical to adding the positive of that number, students should deduce that the above can be simplified to
0+3+3+3+3=12
Garnering an understanding for the way negative numbers work within the rules of all the mathematics that students learn will lead them to have more fluency and confident with their maths. Like all topics, learning the skills purely by rote severely limits the ability to adequately apply these skills. It is vital to dive deep into the underlying fundamental concepts and master the topic to be able to apply the related skills to as many situations as possible. Negative numbers are no exception.