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POW 11: Eight Bags of Gold

Module by: Interactive Mathematics Program. E-mail the authorEdited By: Interactive Mathematics Program, Key Curriculum Press

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Intent

This POW gives students another opportunity to solve and formally present solutions to an extended problem.

Mathematics

The context of this activity relates to some of the unit’s tasks involving counterfeit or unbalanced coins. The most important content addressed by the POW is the problem-solving process itself. Students will try a variety of strategies for finding the one light bag of gold among eight bags using just a balance scale. They will search for a method that requires fewer than three weighings, and they will try to justify the claim that their method requires the fewest weighings.

Progression

Students are introduced to the POW. They present solutions in a week or so, with an opportunity for revision prior to that.

Approximate Time

15 minutes for introduction

1 to 3 hours for activity (at home)

20 minutes for presentations

Classroom Organization

Individuals, then groups, followed by whole-class presentations

Doing the Activity

Ensure that everyone knows what a pan balance is and how it is used to compare weights.

Students are scheduled to share their POW write-ups and write reviews of one another’s work in the activity POW Revision. They will then revise their POWs (if they want to), and presentations will follow that.

Discussing and Debriefing the Activity

Have at least three students present their work on the POW to the class. Here is one strategy someone might present.

If the first weighing compares three of the bags to three of the others, one of two outcomes is possible:

  • The two sets of three bags balance each other. This means the light bag is one of the two not yet examined, and a single additional weighing, comparing these two, is all that is needed to find it.
  • One of the groups of three is lighter than the other. This means that the light bag is in this group. Only one additional weighing, comparing any two of these three bags, is needed. If they balance, the third one is the lighter one. If they don’t balance, the lighter one will be evident.

Key Questions

Are you sure your scheme will always work?

If you got stuck working on this problem, how did you get unstuck?

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