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POW 16: Spiralaterals

Module by: Interactive Mathematics Program

Intent

The last POW of the unit is a geometric investigation in which students look for patterns in the figures formed by line segments that reflect various sets of number sequences.

Mathematics

A “spiralateral” is the spiral-like shape formed by connected line segments generated by a sequence of numbers. In this open-ended investigation, students decide how and what to explore about these shapes and how they are related to the sequences that generate them. For example, they may decide to investigate the question of when and why a spiralateral returns to its starting point.

Progression

Students look for patterns and organize their information to make generalizations.

Approximate Time

15 minutes for introduction

1 to 3 hours for activity (at home)

20 minutes a week or so later for presentations

Classroom Organization

Individuals

Materials

Grid paper

Doing the Activity

Take some class time to illustrate how a spiralateral is made. Seeing an example will be much clearer for some students than reading a written description. Let students begin to explore the problem in their groups.

You may want to suggest areas of further exploration for interested students. For example:

  • What happens if 0 is used in the sequence?
  • Can negative numbers be used? What about fractions?
  • How does the analysis change if the angle of turn is 60°? 120°?

Discussing and Debriefing the Activity

Ask three students to make POW presentations. As presenters share their observations about spiralaterals, focus the class’s attention on explanations of the discoveries.

For example, if students investigated the question of when a spiralateral returns to its starting point, they may have seen that sequences of length 2, 3, or 5 always return, but sequences of length 4 do not necessarily return. If so, you might ask, Why might sequences of length 4 be different from sequences of other lengths?Are other lengths for which spiralateral sequences do not always return to the start?

After the initial presentations, ask other students to share any discoveries they made or any variations on spiralaterals that they investigated.

Key Question

Why might sequences of length 4 be different from sequences of other lengths?

Are other lengths for which spiralateral sequences do not always return to the start?

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