Foundation B: Spoken Multiplications

Some multiplications are much easier than others. You might know 4 × 7 and 15 × 3 from memory. You might be able to find 34 × 8 using a suitable method. But 34.56 × 8.765 clearly requires a more complex method and a lot more thinking!

Thankfully, in practical situations involving multiplication of numbers with lots of digits, you would either be satisfied with an estimate, or accept the use of a calculator for a precise answer.

So what difficulty of multiplications should we aim for? In my experience, it is important to solve multiplications of 2 digits × 1 digit—such as 34 × 8.

As I remarked in the section on spoken additions, we often need to solve this size of multiplications when the numbers are not shown to us visually. For example:

  • When someone tells us the numbers
  • When the numbers are a result of another calculation or using memorized values
  • When using the estimation technique (in another section) to convert a more complex multiplication into one of 2 digits × 1 digit

So our next foundation of mental math is to multiply a 2-digit number by a 1-digit number without necessarily seeing either number. As before, I’m calling these “spoken multiplications” to align with the main way that we will practise them.

 

Harder Multiplications

It’s an interesting challenge to solve spoken mutliplications of 2 digits × 2 digits—such as 34 × 78—but these are much more difficult. A typical person who can solve 2 digits × 1 digit in 3 seconds will take 20–30 seconds for these! I do have a full set of techniques for these that are not part of this section.

When the numbers are written down, there is another method called “criss-cross” or “cross-multiplication” that is used to solve multiplications of arbitrary size, such as 34561234 × 78563412. This is important in mental calculation competitions, but is not a foundation of mental math for our purposes!

 

Study Guide