In my last series of blog posts I outlined how Siegfried Engelmann teaches the relationship between addition and subtraction using the number family.

This blog post was inspired by an epiphany whilst teaching a very weak pupil basic addition. I had taught her the following in a sequence of lessons:

  1. Adding integers with no carrying
  2. Adding integers with carrying
  3. Adding decimals with same number of decimal places (carrying)
  4. Adding decimals but with varying number of decimal places (carrying)

In the last half of a lesson I was teaching pupils how to attempt the following set of problems.


I explained using the last problem that when I am adding 2 in the units column to the 9 I am not really getting 1, instead I am getting 11, and the tens is being carried to the column on the left. All pupils besides two understood this. I then demonstrated using a number family that:

  1.  2 is the small number
  2.  I am adding 9 which is the second small number
  3. To get a big number. This big number must be bigger than the two small numbers.

I also showed a non-example of the number family:


  1. 2 is the small number
  2. I am adding 9 which is the second  small number
  3. I will have a big number at the end of the arrow. It can’t be 1 because the big number cannot be smaller than the two small numbers.

At this point, she got it. This is my weakest pupil in year 7. I definitely underestimated how powerful the number family structure can be!