I hope you’ve been enjoying *TCM’s* “Math Tasks to Talk About.” From those who understand a lot
more about how these things work, I gather the blog is getting a good number of
visits, which is really nice to hear, but not too many readers are taking that
next step and commenting on the task or the discussion of it. Since I’m now
“clear proof” that you don’t have to know anything about blogging to
participate in a blog, I’m hoping that folks will realize that all you have to
do to comment on the blog is log in as an NCTM member. I look forward to
hearing from some of you!

For this next math task, I’m going to venture away
from the “classic problems” of the last two tasks, but stick with something
they had in common—namely, looking for patterns. This is such a powerful
problem-solving strategy that it warrants a lot of attention. The Handshake Problem, when
exploring the number of handshakes for different-size groups, generated a
pattern of 1, 3, 6, 10, 15, 21, 28, . . . . The Squares on a Checkerboard task generated
a pattern of 1, 4, 9, 16, 25, 36, . . . in its solution. These patterns are a
bit more complex than something like an arithmetic progression pattern such as
1, 4 ,7 , 10, 13, 16, 19, . . . where the next term in the pattern can be found
by adding a specific number (in this case 3) to the number before it, or a
geometric progression pattern like 2, 10, 50, 250, . . . where the next term in the pattern can be
found by multiplying the number before it by a specific number (in this case
5). However, one thing all the above patterns have in common is that there is a
way of determining what a particular term in the pattern (say the fifteenth term) will be.

Since this will be the last Math Tasks to Talk About
problem that I will pose, I’m going to make it a “biggie” by challenging you
and your students to ponder several
possible tasks with the same theme—patterning. The discussion above has already
provided one such task, namely just asking students to find out what the fifteenth ^{}term (or whatever number term you choose, depending on the grade level of
your students) in each of the above patterns would be.

Here are some other patterns to ponder, but I should
warn you that you’ll have to “think outside the box” when you consider these
patterns. Unlike the previous patterns, there’s not necessarily any way to
determine a particular term in the pattern. You just have to look at the
numbers already in the pattern, and use what you see to find the next number:

**Pattern 1**

Complete the following pattern:

5 ----->4

36--> 8

11---> 1

53---> 7

942---> 14

18---> ?

49---> ?

371---> ?

**Pattern 2**

Study
the numbers below, and continue the pattern by listing the next five numbers in
the sequence:

1,
1, 2, 2, 8, 10, 3, 27, 30, 4, 64, 68, 5, __, __, __, __, __

**Pattern 3**

Study
the numbers below, and continue the pattern by listing the next five numbers in
the sequence:

13, 4, 17, 8, 25, 7, 32, 5, 37, 10, 47, 11, 58, ___, ___, ___, ___,
___.

I hope you and your
students will have some fun with these patterns, and I look forward to your
thoughts and comments. I’ll be back in a couple of weeks with my reflections on
these pattern tasks.

Ralph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of *TCM*, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees.