Intent In this activity, students explore what it means to extend a problem, which can help one gain a better understanding of the structure of a problem.

Mathematics This activity extends Marcella’s Bagels by posing the question of how altering the final number of bagels would change the initial number of bagels. This is a “functions” question: the starting number is a function of the ending number. Students are asked to find that functional relationship by trying several ending values, organizing their findings in an In-Out table, and then searching for a rule.

Progression Students use backward reasoning or other methods to further investigate a complex word problem.

Approximate Time 15 minutes

Classroom Organization Groups, followed by whole-class discussion

Materials About 100 beans, counters, or similar items per group

Doing the Activity Introduce the question that frames this extension to the original Marcella’s Bagels problem: How does the solution to Marcella’s Bagels depend on the number of bagels Marcella has when she gets home? Monitor group interaction. Encourage students to share ideas and to make sure everyone has an opportunity to contribute his or her ideas.

Discussing and Debriefing the Activity Have a group or student volunteers share the ideas they pursued and what they learned about the problem. An In-Out table with all their data might look like this. Number of bagels whenMarcella gets home Number of bagelsMarcella started with 0 28 1 36 2 44 3 52 4 60 5 68

It is not crucial that students develop a rule to describe this relationship. If they haven’t found a rule, you might post the In-Out table and invite students to continue to think about a rule and bring their ideas to you when they have time. Conclude this activity with some discussion of the idea that each Problem of the Week (POW) requires students to write out, and sometimes explore, an extension to the original problem. Mention that mathematics is at least as much about creating interesting questions as it is about answering them.

Key Question How does the solution to Marcella’s Bagels depend on the number of bagels Marcella has when she gets home?