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Predicting Terms in a Sequence
by Gail Englert

A response to the question:

I need help on how to teach my 4th graders about sequences.

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I need help on how to teach my 4th graders this problem: 67, 68, 70, 73, 77, ____, _____ . Fill in the rest. We are using Excel mathematics and I have no textbook for them to use. Any suggestion would be helpful because they are lost and I have run out of ideas.

Raina

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Dear Raina,

The problem you are sharing with us is an example of a "growing" pattern (at least, that is what my fourth and fifth graders would have called it.) We look for patterns in the arithmetic problems we explore, and when we find examples of ways the answers form patterns, my students are able to begin to make generalizations about how our number system operates. That is why identifying patterns is such an important topic.

You might begin by taking a number like 123, and systematically, adding 1 (keeping track of the sums you get). So, you would start with 123 + 1, then take that sum and add 1, then take that sum and add 1, and so on.

Have your students arrange the "answers" in a way that helps them see any patterns that might result. I like to guide my students to use a chart. They might put the problem in the first column, and the resulting sum in the second column, like this

      123 + 1      124
      124 + 1      125
      125 + 1      126
      126 + 1      127
      127 + 1      128
      128 + 1      129
      129 + 1      130
      130 + 1      131
      131 + 1      132
      132 + 1      133    and so on...
      

(repeat this at least 12 times, and they will be able to see that the tens digit changes eventually, too, and they might begin to realize that what happens to one place value might affect the other place values as well).

Ask your students to tell you what they notice about the information on the chart, and give them time to tell you whatever they see, even the details they might consider very obvious. Guide them to tell you that 1 was added each time, and that the resulting answers show a pattern in the ones place at first, and then in the tens place, too. Let them predict what the next sum will be, and then test it out to check their predictions.

Ask them if that same pattern would occur if you began with a different starting number and let them test that out.

They may think this a silly exercise at first, so it is important that you help them see that practicing a skill is sometimes first done with numbers we are familiar with. They are practicing recording and analyzing data, using problems they know the answers to.

You can do the same activity adding 2 or 3 or any amount instead of 1. Let students keep data, and explore what is happening. Then change the activity. Instead of repeatedly adding the same number, add a series of numbers such as 1, then 2, then 3, and then 4.

      123 + 1      124
      124 + 2      126
      126 + 3      129
      129 + 4      133
      133 + 5      138
      138 + 6      144
      144 + 7      151
      151 + 8      159
      159 + 9      168
      168 + 10     178     and so on.

Now ask your students to tell you what they notice and guide them to note that this time the sums grew much more quickly than the first time. Let them conjecture about why that happened. Ask them if they can predict what the next tens digit will be and if they think the hundreds digit will ever change. Be sure to have them explain WHY they think so, when they give their predictions.

If you think it will not confuse them, you could introduce clock arithmetic, where you arrange the digits from 0 - 9 in a circle. If you begin with 3 (the ones place digit in 123) and add 1 by moving clockwise around the circle, you end up on 4. Adding 2 more will move you to 6, and adding 3 to that will move you to 9. From the 9, 4 more will hop you past the 0, 1, and 2 to land on the 3. 5 more than 3 will land you on the 8, and so on.)

If you would rather concentrate on whole number sums, have them focus on the tens and ones digit together so they have 124, 126, 129, 133 ...

You can repeat this activity using different starting numbers, and different sequences of numbers (add only even numbers, add only odd numbers, etc.)

The goal is to get students to see that our number system is predictable, that it is sensible. They have power when they realize they can predict what will happen. Identifying patterns and creating series of numbers that have patterns will help them "see" the patterns we want them to find. The exploration will give them the confidence they need to explore our number system. Hope this gives you an avenue to try with your students.

-Gail, for the T2T service

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