Intent This activity engages students in a problem that requires close reading. Students also examine a POW-style write-up for this problem, which will support their write-ups for POW 1: The Broken Eggs.

Mathematics Marcella’s Bagels gives students an opportunity to use a variety of problem-solving strategies. They might guess at the original number of bagels, examine what happens when they work through the steps in the problem, and then revise their initial guesses accordingly. The problem also lends itself to the powerful strategy of working backward. Thinking of the story as a movie, students can begin with the number of bagels Marcella has at the end and “run the movie backward,” undoing each action she took and arriving at the number of bagels she had at the start. At each step, Marcella gives away half of her bagels plus 2, so in reverse she would add 2 and then double the total.

Progression Students work backward or use other methods to analyze a complex word problem. They also examine a POW-style write-up for this problem.

Approximate Time 30 minutes

Classroom Organization Groups

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

Doing the Activity You might introduce Marcella’s Bagels by asking for volunteers to silently enact the events, with or without props, as you narrate the story. Afterward, ask students to restate the problem. As you direct students to work in their groups on the activity, encourage them to use the materials available to help them think through the problem. If students express that they are beyond using objects like beans or counters, assure them that doing mathematics involves using whatever it takes—pencil and paper, calculators and computers, models and manipulatives—to understand a situation or an idea.

Discussing and Debriefing the Activity Discuss the various methods students used to solve the problem. How did you find the answer? How do you know your answer is correct? The two most likely approaches will be (1) guessing the starting amount and then running the problem forward to see if the guess leads to the correct ending amount and (2) working backward. Some students might have tried to build an algebraic statement of the problem as it runs forward. Using x to represent the unknown starting amount, they may have written the following equation, which can be solved for x. Many students, however, will be struggling at this point in the course to make sense of order of operations and the algebraic thinking strategy of doing and undoing. x 2 − 2 2 − 2 2 − 2 = 2 size 12{ { { { { { {x} over {2} } -2} over {2} } -2} over {2} } -2=2} {} Next, students will focus on how to communicate how they solved a problem like this and how to show and explain their solutions. Tell students that this is what they are asked to do in their write-ups for POW 1:The Broken Eggs. Have students turn to the POW-Style Write-up of “Marcella’s Bagels” at the end of theunit, which uses Marcella’s Bagels to illustrate the POW write-up components. Ask them to read the example write-up on their own. When they have finished reading, ask, What did you notice that is helpful in this write-up? What is missing? What isn’t needed? Draw students’ attention to the ways the writer used the components in the POW write-up. Ask,How does the process used here differ from the solution?In particular, bring out the observation that the write-up describes how the writer thought about the problem; it doesn’t merely present the answer.

Key Questions How did you find the answer? How do you know your answer is correct? What is helpful in this write-up? What is missing? What isn’t needed? How does the process used here differ from the solution?

Supplemental Activity It’s All Gone (reinforcement) is a variation on Marcella’s Bagels, in which a man goes from store to store getting and spending money and winds up with no money in the end. Students are asked to determine how much money he had when he started.