One problem with math education is that access is very uneven. In theory, the internet can level things out a bit. This post is the next in Occupy Math’s series of activities that parents and teachers can use for enrichment and enhancement. For free.
Here is an example of a “guess-the-next-number” puzzle — “what is the next number in the sequence 2,5,8,11,14,?” The answer is 17. The student should figure out that the terms of the sequence increase by three every time, and 14+3=17. This post is about constructing this sort of puzzle with some notes on how to make harder and easier puzzles. Puzzles like this are good arenas to practice math skills. They can be structured as contests which is motivational, and with the information in this post, they can be tuned to your student’s needs.
Generating formulas for sequence puzzles
This list below is a collection of different methods for generating sequences that you can use in guess-the-next-number puzzles. These can be used to populate worksheets, used for in-class contests; the harder ones can be used as long-term puzzles. Occupy Math has included some wild guesses at the appropriate grade level for these puzzles, but you should look at the puzzles as you generate them and provide your own reality check.
- Pick two numbers, start the sequence with the first and then keep adding the second to generate more terms. Make the sequence five terms long and ask students to guess the next term. The example of a sequence at the top of the post is like this. Level: second-fourth grade.
- This sequence also builds on technique #1. First generate a sequence of the sort described in #1. These numbers will be the differences between the terms of the new sequence. Pick a starting number for the new sequence and then generate next terms by adding successive numbers from your type #1 sequence. If the type#1 sequence is 1, 3, 5, 7, 9 (start:1, jump:2) and our starting number is 4, the new sequence is 4, 5, 8, 13, 20, 29. The numbers 1, 3, 5, 7, 9 are the jumps between the members of the new sequence. Level: fifth grade or higher.
- Pick two (small) numbers to act as multipliers and the first two numbers of the sequence. Start with the two numbers you picked. The next term of the sequence is your first multiplier times the number two terms back plus your second multiplier times the last term. If we pick all four numbers to be 1, we get the Fibonacci sequence. We start with 1 1 and then just add the last two terms so w get 1, 1, 2, 3, 5, 8,… If we leave the first two terms 1 1 and make the rule “add the term two back plus twice the last term” we get 1, 1, 3, 7, 17,… Making one of the multipliers negative (make it the smaller one) can ramp up the difficulty of these puzzles. Level: fourth-eighth grade.
- This is a small variation on #3. Pick one more small number and add that in as well when generating new terms. If we modify the Fibonacci sequence by also adding in a two each time, you get the sequence 1, 1, 4, 7, 13, 22,… This makes the sequence somewhat harder to figure out. Level: fifth-tenth grade.
- Pick a starting number, a (small) multiplier and an adder (possibly a negative number). To get the next term, multiply the current number by the multiplier and add the adder. If we start with 3, have a multiplier of 2, and an adder of -1 then the sequence is 3, 5, 9, 17, 33,… Level: third-eighth grade.
- Pick a starting number and a multiplier and an adder, both odd. Here is the rule for generating the next number. If the current number is even, divide it in half. If the current number is odd, multiply it by the multiplier and add the adder. A famous sequence, the hailstone sequence, is generated with this rule. If we start with 5 and use multiplier 3 and adder 1 we get 5, 16, 8, 4, 2, 1. Level: seventh grade or higher.
- Take method #3 and use three previous numbers instead of two. You may want to use more than five terms for this sort of sequence. This rule means you need to choose three starting numbers. If we start with 1,2,1 and add the last three numbers we get 1,2,1,4,7,12,23,42,… Level: fifth-ninth grade.
- Pick two rules for generating the next number in a sequence and use them alternately. If both rules are of the sort in #1, this is not too hard — for instance “start with 2 and alternately add 3 and 5” would give us 2,5,10,13,18,… If you do this with the harder rules, or use two very different rules, these sequences can be very hard. Add one to the higher grade level of the two methods used.
This is a short list of rules for generating problems. You can make up more fairly easily, like “add two sequences together”. There is another category of sequences that is important, but not really specified by a rule. Consider the sequence 2, 3, 5, 7, 11, 13, 17, 19, 23, … This sequence is the prime numbers. For sequences like this, that are famous, you just add those sequences into the mix when you are working guess-the-next-number problems. You might consider the digits of π as another example of this kind of sequence.
Something that is worth mentioning at this point is the grade level ratings given above, while mostly okay, do not apply to all individuals. There is a great deal of variation in which sorts of patterns are hard or easy for different students. One big factor is whether they have seen anything like this before. Another is if some part of the process of recognizing a pattern is something they have had problems with before and dislike. People also have different problem-solving skills, and those work well or poorly with different patterns. This is a reason to have students solve these guess-the-next-number problems in small groups.
Using these sequences
Occupy Math puts sequences like this on file cards with “easy”, “medium”, “hard” and “very hard” on the back. Students can pick how hard a problem they want or team representatives can pick problems for the whole class. This is done when the problems are used in team contests with small groups working them competitively. The cards can also be handed out to students who work a problem and come up. One way to structure this is to let a student be done when they get five problems right. The teacher can adjust problem difficulty (from their deck of pre-prepared cards) based on individual success. Watching students work these problems can also be diagnostic of which math skills they might be struggling with.
Guess-the-next-number problems are traditional. Many students will have encountered them before. In addition to encouraging practice with math skills, they give the students practice at pattern recognition, a generally useful skill. As with many of Occupy Math’s activity posts, these puzzles are intended to decrease the perception that mathematics is drudgery. Would you like Occupy Math to prepare some printable example sets of these problems? If so, e-mail him at dashlock@uoguelph.ca or speak up in the comments.
I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics
Occupy Math 