What next?
Here is the beginning of a lesson plan that was copied from the internet.
"Write the numbers 1 3 5 7 on the chalkboard. Ask students what number comes next. Usually a student will correctly guess 9. Ask for the next number in the sequence. Ask the student who answers how she or he knew that was correct. Students will offer explanations such as, "You're skipping a number every time." If they don't bring it up themselves, point out that these are the odd numbers. ..."
Is the student's answer, 9, correct? Is the explanation, "These are odd numbers", convincing? Could the student's answer be plain wrong?
Activity.
Students work in pairs or in small groups. The most ingenious patterns should be shown to the whole class. Students use TI-34 II calculators or similar ones.
One student enters his/her secret operation into OP1 (he/she should be sure that the equals sign is darkened, so the formula doesn't show on the display). He/she also puts 1 on the display by pressing [1][=]. Another student from the group starts pressing OP1 again and again. But each time, before pressing the key, he/she and the other students from the group try to guess what the next number is going to be.
The operation must yield only whole numbers. The student who is preparing the operation should try to "trick" the rest of the students, namely, to prepare a rule that apparently seems to be obvious, but which suddenly gives unexpected answers.
Examples.
We give below a few simple but "interesting" rules. All of them except Example 4 will start with the sequence 1 3 5 7.
1.
OP1 = remainder(Ans,7)+2 [=][=]
1 [=]
Now repeatedly press [OP1].
The sequence is: 1 3 5 7 2 4 6 8 3 5 7 2 4 6 8 3 ...
2.
OP1 = remainder(Ans,12)+2 [=][=]
1 [=]
Now repeatedly press [OP1].
The sequence is: 1 3 5 7 9 11 13 3 5 7 9 11 13 3 ...
These two examples show how easy it is to construct a sequence that produces odd numbers as long as the designer wants, and then to have the pattern switch. But this simple remainder method is not hard to "outguess", after you notice the length of the period.
3. We show below a more perplexing pattern.
OP1=remainder(A,Ans)+Ans+2 [=][=]
5*9*11*13*17*19→A [=]
1 [=]
Now repeatedly pressing [OP1] yields the following sequence:
1 3 5 7 11 13 15 17 19 21 32 43 59 114 173...
Nine is skipped, and after 21 the pattern starts looking "irregular".
4. Let's use another program that involves a random number generator.
OP1=2iPart(RANDI(1,4)/4)+Ans+2 [=][=]
1 [=]
Now repeatedly press [OP1].
Two examples:
1 3 5 7 9 13 15 19 ...
1 5 7 9 11 15 17 21 ...
This program counts by 2's, beginning with 1; but randomly (with probability 1/4) it skips some numbers.
5. Finally, a person seeing 1 3 5 7 may think "odd primes" and predict 11 13 17 19 23... .
General remarks.
When guessing a pattern from examples, there is no "right" or "wrong" way. Any hypothesis about future values may happen to be wrong.
If a pattern is really generated by some rule, it is theoretically possible to form a correct hypothesis which will give correct predictions. But it is not easy, and it involves a lot of luck.
But some sequences can be generated. not by rules, but by some random processes, or by a mixture of rules and random choices. When this is the case, any hypothesis that predicts future values will be proven wrong sooner or later.
These questions were thoroughly investigated both from a theoretical point of view (in mathematics) and from more practical angles (in computer science).
Letting students believe that a few examples uniquely determine a pattern is incorrect, and it may reinforce sloppy thinking habits.