Fact Families: Part 1

Everyone says that fluency is important in mathematics, but when you ask about fluency in mathematics, what is often reached for is one of two things. Firstly, people bring up times tables or number bonds. Sure, this is important, but ask people to talk about fluency beyond that and they struggle. Then what is often brought up is procedural fluency – that is to look at a problem and ‘know’ what to do.

But what fluencies could we work on with good students in Secondary level mathematics? This is a series of blogs about ‘Fact Families’ and relationships that underpin much of secondary maths, where a good understanding of these ‘Fact Families’ supports future learning.

A fact family is simply this – it is a ‘family’ of maths sentences, where each tells the same story in different ways.

The Additive Family

In general terms, the additive family is such that, from the image above:
a + b = c
c – a = b
c – b = a

When I was at mathsconf19 in Penistone I got in a rather heated debate about how we would want students to go about solving equations such as:
15 – 2x = 7

The argument from the table I was on was that to solve equations we needed one clear method, and this required the first step on for this equation to be the subtraction of 15 from both sides. This was around the time of the current Ofsted framework coming in and everyone was in a bit of a flap about departments having consistent methods/lessons and the sort. I found the argument astonishing! It seemed like a dangerous path to go down to risk a load of negative problems.

A fluency with the additive family would have far better outcomes on this question, but also fits the bill of a ‘universal’ method to these sorts of questions (should that be desirable (??)).

A fact family here says;
If 15 – 2x = 7 is true, then 15 – 7 = 2x is true, so therefore 8 = 2x

The fact family approach would also work for more standard equations, eg;
If 3x + 15 = 24 is true, then 3x = 24 – 15 is true, so therefore 3x = 9

Most students will arrive in Year 7 with the understanding of this relationship. But ignoring this relationship risks losing lovely connections in the KS3 curriculum. I believe that fact families enhance understanding of negative numbers.

Take: (-3) + 5 = 2
The family here also contains:
(-3) = 2 – 5 and 5 = 2 – (-3)

Take (-3) + (-5) = (-8)
The family here also contains:
(-3) = (-8)-(-5) and (-5)=(-8)-(-3)

Take 3 – 5 = -2
The family here also contains:
3 – (-2) = 5 and (-2) + 5 = 3 (and 5 + (-2) = 3)

No matter which approach you take to introducing negative numbers, referring back to the other facts you get from the “family” always gives you at least three for the price of one.

My theory is that most students are roughly familiar with this family, but that they would benefit from having it pointed out. My basis for this theory is from work around percentages, where reverse percentages mistakes tend to be formed upon the idea that it is an additive relationship.

Knowledge of that additive relationship sustains throughout the curriculum and well into KS5. If some of my A-level students had a better fluency of the additive relationship they may more readily spot these families:

Can you think of other places that the additive family come up in the curriculum? How could a fluency in fact families aid understanding?