Negative Divided by a Negative Equals Funny

"A negative times a negative is a positive". It's a hard one to explain. We all learnt it at school and practised it to the point of fluency, but it's not until we're asked why it works that we stop and think about it.

Numbers lines and visualisations are very helpful when teaching the addition and subtraction of negative numbers. But with multiplication and division it's not so clear.

Let's look at a few approaches and resources.

1 . Pattern Spotting
Draw a standard multiplication table and extend it backwards to include negative numbers. It's a straightforward pattern that all students should be able to spot and continue. Get students to do this using Colin Foster's activity on page 5 of his Negative Numbers chapter.

2. Multiplication Grids
Take two 2-digit numbers and multiply them together using grid multiplication. For simplicity, let's take 12 x 11:

Here we have written 12 as 10 + 2 and 11 as 10 + 1. But it would work just as well if we expressed those numbers differently. Instead, let's write 12 as 15 - 3 and 11 as 15 - 4. We should get the same answer:

This only works if -3 x -4 = 12.

Note that this explanation requires students to first understand that positive x negative = negative. This is relatively straightforward to explain in terms of repeated addition.

3. Proof
Here's a proof that is clear and accessible to us experienced mathematicians. I'm not sure how accessible it is to Year 7 students, but it's worth a go.

a and b are positive

a + (-a) = 0

[a +(-a)]•b = 0•b

a•b + (-a)•b = 0

a•b is positive. Therefore (-a)•b is negative

b + (-b) = 0

(-a)•[b + (-b)] = (-a)•0

(-a)•b + (-a)•(-b) = 0

Since (-a)•b is negative, we conclude that (-a)•(-b) is positive.

Perhaps start with a numerical example instead of a formal proof.

3 + (-3) = 0

Multiply everything by -4

3(-4) + (-3)(-4) = 0(-4)

-12 + (-3)(-4)  = 0

(-3)(-4) must equal 12 to make this statement true.

Further Reading
It's a good idea to read about a topic before you teach it, even relatively simple topics that you've taught many times before. Here are some helpful links:

• The FAQ pages on Math Forum always provide interesting answers:Why is a negative times a negative a positive?
• I enjoy James Tanton's curriculum essays: Why is negative times negative positive?
• I was inspired to write this post after watching this excellent video 'A negative times a negative is a...' by Mathologer (my new favourite YouTube channel!).
• The History of Negative Numbers from NRICH is worth reading.

I like this clip from Stand and Deliver:

The wording here is important. 'A negative times a negative equals a positive' is clearly preferable to 'two minuses make a plus'. The latter is confusing and may lead to misconceptions. An example of a common mistake is shown below (taken from mathmistakes.org via Nix the Tricks).

Tasks and Resources
Here are a few resource recommendations for this topic:

• I really like Don Steward's 'Directed Number Gaps' activity - it works well with any year group
• MathsPad has a great range of negative number puzzles including arithmagons (some of these resources are only available to subscribers)
• CIMT's negative numbers chapter has activities for practising multiplying negatives.

Colin Foster suggests that you ask students to make up ten multiplications and ten divisions each giving an answer of –8 (eg –2 × –2 × –2 or –1 × 8 etc).

The squaring and cubing (etc) of negatives is worth discussing - students should spot that an even power gives a positive value (eg what is the value of (-1)¹⁰⁰?).

It may be worth exploring calculator behaviour too (ie some calculators require brackets when squaring a negative). It's important that students know how to use their calculator properly. There's a great resource from MathsPad for this - Using a Calculator: Odd One Out.

This topic is revisited in later years when students are practising substitution. For example, if a = 3, b = -2 and c = -5, find the values of: abc; bc²; (bc)²; a²b³ and so on. This Substitution Puzzle from mathsteaching.wordpress.com gets quite challenging.

Do let me know if you use an interesting method or resource for teaching the multiplication of negative numbers.

"Minus times minus results in a plus,
The reason for this, we needn't discuss"
- Ogden Nash

olivierscimerial.blogspot.com

Source: https://www.resourceaholic.com/2016/08/multiplying-negatives.html

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