Guess That Sequence! Lesson Plan

Guess That Sequence! is a game that can develop skills with finding patterns and understanding sequences and functions. The game is played in the following way: one student (call her Student S) in a group of three or more begins the game by announcing that she has come up with a sequence. Student S then says the first three elements of the sequence and asks the other students if they have any guesses for what rule she is using. After all the other students have made their guesses or declined the opportunity, the Student S either says which students got the sequence she was thinking of, or if no one guessed correctly, lists the next three elements of the sequence.

Play continues until one of the students correctly guesses the sequence or the number of revealed elements of the sequence reaches a certain predefined upper limit. The game can be played with or without this upper limit but its presence is recommended as it can be used to encourage stronger students to come up with sequences that are within the realm of capability of their weaker peers.

The first student(s) to correctly guess the sequence receive two points. If no students correctly guess the sequence before the upper limit has been reached, nobody gets a point.

The student activity sheet we include is geared toward high school students after their first exposure to geometric series, but Guess That Sequence! is an adaptable game and can be used with students of any age. Elementary school children, for instance, could make patterns using markers and paper and ask each other to guess the pattern. The fundamental goal of Guess That Sequence! is to help students learn to identify patterns; but beyond that, it can be used to give students practice with concepts (for instance, recursively defined functions) that they encounter in class.

During the game, if you find that play is stagnating in some group, try giving that group a sequence that is unlike the ones they have been coming up with. Some example oddball sequences:

• 4, 3, 3, 5, 4... (the number of characters in the names of each positive integer (“zero” = 4, “one” = 3, etc.)
• n! for n = 0, 1, 2, … if the students are about to start learning about recursive functions.
• 5(1/2)^(n-1) if n even, 5(1/3)^(n-1) if n odd, if the students are studying geometric series

At the end of class, have a student from each group volunteer to explain to the class how he guessed a sequence that group found particularly challenging. Alternatively, have each student complete a paragraph or two about a sequence he or she found particularly interesting or challenging. This will give even those students who were not very successful with the game an opportunity to practice describing rules and patterns.

Guess That Sequence! could be used to help students satisfy the following NCTM Algebra standards:

Pre-K-2 Expectations:

• recognize, describe, and extend patterns such as sequences of sounds and shapes or simple numeric patterns and translate from one representation to another
• analyze how both repeating and growing patterns are generated

Grades 3-5 Expectations:

• describe, extend, and make generalizations about geometric and numeric patterns
• represent and analyze patterns and functions, using words, tables, and graphs

Grades 6-8 Expectations:

• represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules
• relate and compare different forms of representation for a relationship

Grades 9-12 Expectations:

• generalize patterns using explicitly defined and recursively defined functions

Guess That Sequence! Sample Solutions

1.           9, 15 ,21, 27…

This sequence is:         arithmetic geometric neither

a0 = 9

an+1 = 6 + an

• 3, 6, 24, 192…

This sequence is:         arithmetic geometric neither

a0 = 3

an+1 = 3(2)n

• 0, 1, 1, 2, 3, 5, 8…

This sequence is:         arithmetic geometric neither

a0 = 0

a1 = 1

an+1 = an + an-1

BONUS: What is the name of this famous sequence?

The Fibonacci sequence

Look Back: What are some of the differences between arithmetic and geometric sequences? What qualities of the first and second list of terms helped you determine whether the sequence was arithmetic or geometric?

Arithmetic sequences have a constant difference between successive terms; geometric sequences have a constant quotient between successive terms.