Number Patterns

Here are some charmers of mathematics that depend on the surprising nature of its number system. Not many words are needed to demonstrate the charm, for it is obvious at first sight. Just look, enjoy, and share these amazing properties with your students. Let them appreciate the patterns and, if possible, try to look for an “explanation” for this.
12345679 x 9 = 111,111,111
12345679 x 18 = 222,222,222
12345679 x 27 = 333,333,333
12345679 x 36 = 444,444,444
12345679 x 45 = 555,555,555
12345679 x 54 = 666,666,666
12345679 x 63 = 777,777,777
12345679 x 72 = 888,888,888
12345679 x 81 = 999,999,999
In the following pattern chart, notice that the first and last digits of the products are the digits of the multiples of 9.
987654321 x 9 = 08 888 888 889
987654321 x 18 = 17 777 777 778
987654321 x 27 = 26 666 666 667
987654321 x 36 = 35 555 555 556
987654321 x 45 = 44 444 444 445
987654321 x 54 = 53 333 333 334
987654321 x 63 = 62 222 222 223
987654321 x 72 = 71 111 111 112
987654321 x 81 = 80 000 000 001
It is normal for students to want to find extensions of this surprising pattern. They might experiment by adding digits to the first multiplicand or by multiplying by other multiples of 9. In any case, experimentation ought to be encouraged.