Intent

This first activity in the unit engages students in a series of questions for which there may be no familiar procedure or algorithm and for which there might be many solutions. It is students’ first opportunity to do mathematics together.

Mathematics

This activity introduces the mathematical idea of a sequence. Students are asked to find patterns that fit a given sequence and then to use these patterns to predict the next few terms of the sequence. The search for patterns is a recurring theme in this unit and throughout the IMP curriculum. These early activities also build a foundation for the concept of function, one of the truly big ideas of algebra.

Progression

After you have introduced IMP to students as a somewhat different kind of textbook, and have made students aware that their classroom working environment will have certain characteristics (see “Characteristics of the IMP Classroom” in the Overview to the Interactive Mathematics Program), this activity will be their first experience of IMP and the “IMP classroom.” Students will work on this activity in a small group of peers. They will be encouraged to be creative in describing their patterns and to share ideas with group members. The activity concludes with students discussing some of the patterns they identified.

Approximate Time

30 minutes

Classroom Organization

Small groups, followed by whole-class discussion

Doing the Activity

You might begin by asking one or more volunteers to read the instructions aloud. Emphasize that students are to do more than simply find the next few terms of each sequence; they are also to give a description of the pattern they see.

Offer some ideas about expectations for group collaboration and interaction—namely, that groups do both. Suggest a few methods to do so, such as occasionally asking what a neighbor found to compare to your result or what someone sees when you haven’t been able to notice a pattern. Emphasize that all students are expected to ask for help when stuck and to help others when asked.

Have groups begin writing down their ideas. Students might work individually on the first two questions and then discuss those in their groups before moving on to the next pair of questions. Stopping to share ideas lets the groups hear what everyone is thinking and see that there is more than one possible pattern or approach. Sharing also helps students learn to value each other’s thinking and to collaborate.

Circulate as students work and listen in on the discussions, limiting interventions in order to encourage students to rely on their own thinking and to work collaboratively within their groups. You might ask groups that need help some probing questions, such as those below.

In your own words, what is the pattern you found?

Does the pattern fit the terms of the sequence?

Does someone else have another way to describe this pattern?

Did you find other patterns that start the same way?

Can there be more than one correct pattern?

Remind students, if needed, to look for more than one possible pattern for each sequence or more than one way to describe a given pattern. If many groups seem stuck, you might interrupt and have a class discussion of the first question or two to clarify what is being asked.

When the majority of groups have finished Questions 1 through 6, you might bring the class together for discussion. As time allows, you can then have them turn to Questions 7 and 8.

Discussing and Debriefing the Activity

On the first day of the unit, it is important to establish a classroom climate where student thinking is valued and where it is safe to take the risk of sharing ideas.

Ask presenters to describe the patterns they found and the next terms these patterns led to. These descriptions can be very informal. Then ask for comments from the class. Have the class work together to make the pattern descriptions clear. (Over the course of this unit, you will ask students to make more precise statements, including algebraic descriptions of some patterns.)

It is important that students see different ways to describe a given pattern as well as different patterns that fit a given initial sequence. Ask if anyone found other ways to describe a given pattern. What other ways did you find to describe this pattern? For example, some students may describe the sequence in Question 4 (1, 2, 4, . . .) as “Double each term to get the next term,” while others may say “Add each term to itself to get the next term.” Although these two descriptions are different, they lead to the same continuation of the pattern, with the next three terms being 8, 16, and 32.

Did you find other patterns that start the same way? Some students may have discovered different patterns that fit the same opening terms of a given sequence. For instance, in Question 4 (1, 2, 4, . . .), the pattern could be “Add 1, then add 2, then add 3, and so on.” In this pattern, the next three terms would be 7, 11, and 16 (rather than the 8, 16, and 32 for a doubling pattern).

Question 5 also offers more than one option. For example, the sequence could repeat the opening terms, 1, 3, 5, 7, 5, 3, over and over again (so it goes 1, 3, 5, 7, 5, 3, 1, 3, 5, 7, 5, 3, 1, 3, . . .), or it could follow 1, 3, 5, 7, 5, 3 with 1, 3, 5, 7, 9, 7, 5, 3 and then 1, 3, 5, 7, 9, 11, 9, 7, 5, 3, and so on.

When students present more than one description of a pattern or more than one pattern, ask if both can be right. Bring out the idea that any description or pattern that fits the opening terms of a sequence is as correct as any other.

As time allows, ask students to present sequences that their groups created for Questions 7 and 8. You might display these on the board or have groups try to figure out each other’s patterns.

Key Questions

In your own words, what is the pattern you found?

Does the pattern fit the terms of the sequence?

Does someone else have another way to describe this pattern?

Did you find other patterns that start the same way?

Can there be more than one correct pattern?

Supplemental Activity

Keep It Going (reinforcement) asks students to use patterns to find the next few terms of four number sequences and to describe the patterns they found.