Multiplication: An Adventure in Number Sense

Times Nine Patterns

Student: But what else is there to color? No "twin facts," and I do not see any more "free facts." Give me a hint if you know some tricks!

Mentor: I know a lot of tricks. Look at the ninth row. Do you see any pattern whatsoever?

Student: Well, we count by 9's to get the next fact...

Mentor: Write down all these "times nine" facts and look just at their first digits.

Student: Here they are:

9*2=18

9*3=27

9*4=36

9*5=45

9*6=54

9*7=63

9*8=72

9*9=81

How many different patterns can you find in these facts? There are at least three patterns there! Why each of the patterns happens? Try to find several different ways of explaining each pattern.

Student: I see! The first digits of the answers go 1, 2, 3 and so on. Oh, and the second digits go 8, 7, 6, and so on, all the way down to 1.

Mentor: Do you see any more patterns?

Student: The first digit is one less than the number by which I multiply nine. When I multiply 9*8, the first digit of the answer is 7.

Mentor: There is a very logical explanation to that pattern. In your example 9*8, there are nine 8s. It is how many less than ten 8s?

Student: It is eight less, of course. What does it have to do with the first digit?

Mentor: Think. What kind of the first digit would ten 8s give you?

Student: Ten 8s in eighty, so the first digit is 8. But we do not have ten 8s, we have nine of them only. I see! When we subtract the extra 8, we get seventy something, so the first digit is 7. Oh, and it works for any whole number from 1 to 9 you multiply by 9.

Mentor: Why does the number have to be from 1 to 9?

Student: Because if it is more than 9, you have more than two digits in the answer, so the whole pattern is different.

Mentor: Actually, the easiest way to multiply a number by nine (that is, to add the number to itself nine times) is to multiply the number by ten and then to subtract the one extra number that you have.

Student: Let me try five: ten 5s is fifty, and nine 5s is fifty minus the extra five, so it is forty five. The first digit is four, because it is fifty minus something.

Mentor: Something that is small enough, to be exact.

Student: Anyway, I can use this rule very fast, if I just practice. I do not have to memorize times nine part, then.

To multiply a number by nine, multiply it first by ten (which is easy!) and then subtract the number.

Mentor: If you practice just a little, thinking about patterns all the time, then you will be better off than people who just memorize the facts. You will be able to use the facts fast, and you will also understand them. There is also funny way to do times nine on your hands.