Pattern Matching & Systematic Reduction Continued: How I Learned to Multiply

Last time I talked about memory and memorization I discussed how I learned the Braille alphabet. We reduced the number of symbols to memorize from 64 down to 5, plus a few rules to transform those symbols. I mentioned I wanted to show how I skipped learning the multiplication tables.

In elementary school, I hated having to memorize the multiplication table. Teachers always tried to tell me different ways to memorize it, but I could never shove that knowledge in my head.

In university, I took a computer hardware course. We learned how computer processors work on an electrical level. We built individual components and tested them. As part of this we learned how a computer does multiplication: it adds over and over again.

Suddenly I didn’t need to memorize the multiplication table. I knew I could add repeatedly no problem. All I needed to do was remember the starting number and add something to it.

Let’s go through an example together. Let’s do 12×8.

Brute Force Example

The easiest way I multiply this number is to do the following:

Start = 0. Loop = 1. Add 12. Result: 12. Is loop < 8? Yes: continue.

Start: 12. Loop = 2. Add 12. Result: 24. Is loop < 8? Yes: continue.

Start: 24. Loop = 3. Add 12. Result: 36. Is loop < 8? Yes: continue.

Start: 36. Loop = 4. Add 12. Result: 48. Is loop < 8? Yes: continue.

Start: 48. Loop = 5. Add 12. Result: 60. Is loop < 8? Yes: continue.

Start: 60. Loop = 6. Add 12. Result: 72. Is loop < 8? Yes: continue.

Start: 72. Loop = 7. Add 12. Result: 84. Is loop < 8? Yes: continue.

Start: 84. Loop = 8. Add 12. Result: 96. Is loop < 8? No, stop.

I love knowing this technique because it means as long as I can do addition over and over, I’m guaranteed to get the right answer.

There’s another way to do this problem though!

Decomposition into Easier Problems Example

This is where I apply another shortcut that I learned in math class: multiplication is commutative. What does that mean? It means the following are the same problem:

8×12 = 12×8

In other words, I could run 8×12 or 12×8 and get the same answer. It also means I can apply another mental shortcut I take when multiplying.

If 12×8 = 8×12, I can decompose this problem.

I know that 8×10 = 80. Since multiplication is repeated addition, I can decompose 8×12 into (8×10) + (8×2), or 80 + 16. I can compute these problems in my head much faster than trying to multiply 12 by 8 (or add 12 to itself 8 times).

Transformation into Easier Problems Example

Another trick I use is to find a problem that is close to the one I want to solve and see if there’s a way to work backward.

For example, let’s take 8×9. You could add 8 to itself 9 times, but there’s an easier way to solve this one. 8×9 is close to 8×10. In fact, it’s one step before 8×10. I can decompose this into (8×10) – (8×1). My problem is now an easy one to compute: 80 – 8: 72.

I often find it easier to do a multiply and subtract when I’m dealing with a multiplication problem that’s close to 10.

Another example: 6×8. This is close to 6×10. It’s also equivalent to (6×10) – (6×2). These two problems are way easier for me to compute.

• (6×10) – (6×2)
• 60 – (6×2)
• 60 – 12
• 60 – 10 – 2 [Because 12 is 10 and 2, and to remove 12 from a number, we can remove 10 and then remove 2]
• 50 – 2
• 48

Let me be clear: I hate memorization for the sake of memorization. It annoys me. If you don’t give a piece of information context it will not be remembered by your learners. Humans are great at remembering rules. We suck at memorization for the sake of.