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Pondering Patterns

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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.


RalphConnellyRalph 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. 

I haven't started to work on these yet, but I especially am interested in pattern 1. Do the dashes and the arrow indicate the number of numbers in between the first number and the last number shown? and couldn't anything work for the last three since nothing else is provided but the first number? The directions for this seem to refer to all in the list as a pattern; not separate patterns. I think I may be confused about this one.
Posted by: DonnaS_59217 at 5/3/2014 4:18 AM

Greetings, Donna, and thanks for posting! My apologies for my "graphically inept" attempt at drawing arrows, and the resulting confusion! "Pattern One" has 5 "going to" 4 ; 36 "going to" 8; 11 "going to" 1; 53 "going to" 7; and 942 "going to" 13. In extending the pattern, the question is basically what does 18 "go to"? What does 49 "go to"? and What does 371 "go to"? Hope this helps!
Posted by: RalphC_79522 at 5/3/2014 3:01 PM

I too thought that the number of dashes had something to do with figuring out Pattern 1, but then realized, as Ralph suggested, that looking at the numbers helped me find what came next. I enjoyed Pattern 2 and found it more challenging than Patterns 1 and 3.
Posted by: Sorsha-MariaM_71599 at 5/6/2014 6:00 PM

I love patterns and love to look for them everywhere. SO, I think the first number digits sum to one more the the second number. 5 is one more than 4. so 18 goes to 8 49 goes to 12 and 371 goes to 10.
Posted by: JoanneS_84718 at 5/6/2014 8:45 PM

Hope I didn't ruin the challenge for anyone by sharing my thought on the first pattern-- but i was so excited to get the other two patterns too. But I won't share them just now. However, I will challenge my students -with these and some other famous patterns- on a bulletin board.....! Thanks.
Posted by: JoanneS_84718 at 5/6/2014 8:54 PM

It is so great to see people commenting on the blog--thank you so much! And your comments have helped me to always be aware of different ways to "see" a problem! It never occurred to me that my "arrow dashes" could be interpreted as missing numbers in a pattern, but Donna and Sorsha-Maria helped me see how confusing that could be--again my apologies--the intent was NOT to throw you off the track! :) Joanne, it's great that you'll be able to use these patterns with your class, and don't worry about sharing your solution for the first pattern--glad you were excited at figuring them out! You mentioned you might share "some other famous patterns" on a bulleting board with your class--by all means share them here, too, if you'd like and are willing! Thanks, everyone!
Posted by: RalphC_79522 at 5/7/2014 5:18 PM

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