Word Search 2x5

Surprising Number Patterns IV

Now the logical thing to inspect would be the pattern of these strange
products.
1 x 8 = 8
11 x 88 = 968
111 x 888 = 98568
1111 x 8888 = 9874568
11111 x 88888 = 987634568
111111 x 888888 = 98765234568
1111111 x 8888888 = 9876541234568
11111111 x 88888888 = 987654301234568
111111111 x 888888888 = 98765431901234568
1111111111 x 8888888888 = 987654321791234568

Here are some more charmers of mathematics that depend on the surprising
nature of its number system. Again, not many words are needed
to demonstrate the charm, for it is obvious at first sight. These depend
on the property described in Unit 1.4 and the unusual property of the
number 9.
999 999 x 1 = 0 999 999
999 999 x 2 = 1 999 998
999 999 x 3 = 2 999 997
999 999 x 4 = 3 999 996
999 999 x 5 = 4 999 995
999 999 x 6 = 5 999 994
999 999 x 7 = 6 999 993
999 999 x 8 = 7 999 992
999 999 x 9 = 8 999 991
999 999 x 10 = 9 999 990

Again, the number 9, which owes some of its unique properties to the fact
that it is 1 less than the base 10, presents some nice peculiarities.
9 x 9 = 81
99 x 99 = 9 801
999 x 999 = 998 001
9 999 x 9 999 = 99 980 001
99 999 x 99 999 = 9 999 800 001
999 999 x 999 999 = 999 998 000 001
9 999 999 x 9 999 999 = 99 999 980 000 001

Surprising Number Patterns III

Here are some more charmers of mathematics that depend on the surprising
nature of its number system. Again, not many words are needed to
demonstrate the charm, for it is obvious at first sight. Just look, enjoy,
and spread these amazing properties to your students. Let them appreciate
the patterns and, if possible, try to look for an “explanation” for this. You
might ask them why multiplying by 9 might give such unusual results.
Once they see that 9 is one less than the base 10, they might get other
ideas to develop multiplication patterns. A clue might be to have them
consider multiplying by 11 (one greater than the base) to search for a
pattern.
0 x 9 + 1 = 1
1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1 111
1 234 x 9 + 5 = 11 111
12 345 x 9 + 6 = 111 111
123 456 x 9 + 7 = 1 111 111
1 234 567 x 9 + 8 = 11 111 111
12 345 678 x 9 + 9 = 111 111 111
A similar process yields another interesting pattern. Might this give your
students more impetus to search further?

0 x 9 + 8 = 8
9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8 888
9 876 x 9 + 4 = 88 888
98 765 x 9 + 3 = 888 888
987 654 x 9 + 2 = 8 888 888
9 876 543 x 9 + 1 = 88 888 888
98 765 432 x 9 + 0 = 888 888 888

Word Search 2x2

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