You can create:
- A poster with your own collection of multiplication table patterns
- Your own unique plan for memorizing times tables
- Conjectures about algebraic rules related to times tables
If you are planning on memorizing the times table, this activity is the first step. Print out or create a standard times table. Then stare at it for a while. Do you see any groups of facts that are very easy for you? Color each group's cells with its own colored pencil, so you will still see the facts. Or you can draw little symbols in the cells - blue stars for times ten, red cats for doubles and so on. Symbols are better, because you can put several of them in each cell.
Each person will notice different things in times tables. I have done this activity with adult mathematicians and with four year olds, and everybody notices some patterns - different ones! Here are just a few examples:
- Times one and times ten are very easy.
- Times nine is a pattern - there is also a finger trick for it.
- Doubles (times two) are easy for most people and you can color them
- Times five have a pattern - some people think of them as half of times ten
- Off-diagonal numbers are one off complete squares (for example, 5*5=25, but 4*6=6*4=24)
- Actually, most multiplication facts come in pairs - like 3*7 and 7*3 - and you only need to memorize one fact out of each pair
The facts you color, those you think are easy for you, will be the facts you won't have to memorize. It is a bit scary to look at the monster table with its 100 facts (some memorize facts up to 12, though it's more rare now). As you notice patterns and color or mark facts, you will see the parts you still need to memorize shrinking. It is a great feeling of relief.
You can even color the facts you remember from silly jokes or rhymes, like "56=7*8" (counting five, six, seven, eight). A few of those are fine. I am scared of systems that use extrinsic mnemonics for many multiplication facts. There is a psychological danger there, called "runaway imagery" - the mnemonics can later block algebraic understanding.
Of course, the most interesting part, as far as math goes, is to figure out why each pattern works. For example, why are off-diagonal numbers one less than complete squares (numbers multiplied by themselves) on the diagonal? Figuring this out by making charts, experimenting with counters or doing algebra can lead to many investigations.
Because this activity helps you see many patterns in times tables, and patterns are algebra. Also, the activity makes you feel better about memorization, and organizes facts mathematically, by their properties and relationships. The activity is interesting for mathematicians who find more complicated patterns. And, last but not least, you can make a beautiful poster out of your decorated, colored and marked times table.
Higher and deeper
- Describing your patterns in general terms lead to algebra or number theory
- You can create times tables in different bases and see how patterns apply. Which facts are easy in base five?