Multiplication methods – introducing formal long multiplication

Today I was with a group and we were trying to formalise long multiplication methods.  They were fairly secure on 2 digit by 1 digit using formal method (not grid/box method).  i.e. they were happy doing this.

So after a while, I tried to introduce them to 2 digit by 2 digit.  It was quite a leap, so I’ve been thinking about the microsteps in between.

Before moving on, however, I really like some of these ideas on Don Steward’s Median (still my favourite maths website!)  They make for good extension work as they go deeper and involve problem solving before moving on the “next” thing.

2-digit x 1-digit to 2-digit x 2-digit

It’s a big leap, so I’m thinking about the microsteps.

1. Multiples of 10.  We can start with a discussion about 12 x 20.  Some might well be able to do that mentally, and writing out their explanations should eventually lead to:
   We can discuss the practice of “putting a zero in the ones column” which then enables us to just focus on the number in the tens column and multiply that by the number above.  Here is a worksheet of these that I created using Math-Aids.
   multi_digit_power_ten
2. I’m assuming that the idea of partitioning a number into its tens and ones is pretty secure by now, so I might then look at this.
   
   Here is a worksheet that I have created to practise these. It’s a spreadsheet that you can change if you want to alter the questions.  If you’d prefer just simple pdfs to print, here are the questions and these are the solutions.
3. So, now we are ready to combine it all together and introduce the efficient method. Effectively doing the same calculations but with less repetition in the writing. Some suggested language to help solidify the steps:
   • always start with the ones – this is what we use the first row for
   • put in a zero when you are ready to multiply the tens – the answer goes on the second row
   • keep your columns in order, think about the place value of each number you write down
   • use the final row to do your column addition

If you are looking for some worksheets, here are some which I like because they are on squared paper.  The early ones have the hint of putting the zero in, the latter ones don’t. If you don’t like those, a quick Google images search on long multiplication worksheet will soon get you the one you want.

I’m sure there are lots of nuances I’ve skipped over here, but if this triggers any thoughts about how you would teach it, please leave a comment below.