Try This With Your Students!
Have you ever been Booed? Booing your neighbors is a way to enjoy the Halloween season by secretly giving treats and scary trinkets before Halloween day. As the tradition goes, once your household gets Booed, you have to Boo two other households on the following day. A household can only be Booed once, and each household can only Boo two other households.
a. If only one person in the United States starts the Booing process, and everyone who is Booed follows all the rules, on what day should the first person start the Booing process so that every household in the United States gets Booed before October 31st?
b. Make a prediction. If two people in the United States start the Booing process on the exact same day, how will this impact your answer from part (a)? Check your prediction by determining what happens when two people start the Booing process on the same day.
c. If the Booing process begins on October 22nd, how many people need to start the Booing process so that every household in the United States is Booed before Halloween day?
Your students will need to seek out the number of households in the United States to completely answer part (a). Do not tell them that they need this information. Instead, allow them the opportunity to develop the need for the information and seek out a way to retrieve it on their own. For your reference, as of 2018, there were 127,590,000 households in the United States.
Depending on the grade level of your students, expect them to begin part (a) in several different ways. Students in later grades who attempt an equation right away may struggle. If this happens, ask, What does this situation look like? This may lead students to draw a diagram or to make a table. Encourage students to look for patterns regardless of the approach they take. Students in earlier grades will most likely start by drawing a picture or acting out the situation with their classmates. Encourage this sort of thinking and acknowledge both approaches as great solving strategies. Whenever possible, encourage visual representations of the situation to help students make sense of the problem.
Your students may ask about households that do not celebrate Halloween. If they are concerned about including them, ask, How would you determine the number of households to exclude? Allow them a few minutes to brainstorm a way to determine a reasonable number. If they cannot come up with something reasonable, such as only using the number of households with children, then just have them use the number that is available.
For part (b), make sure students commit to a prediction before determining the exact answer. If two teams have created similar diagrams or tables, they can place their work side-by-side to model the new situation. For students in later grades who were able to write an equation, they may be able to see this problem as the addition of functions. For instance, if the function that models part (a) is f(x), then a strategy to approach part (b) is to solve f(x) + f(x) = 127,590,000 or 2f(x) = 127,590,000. A common prediction is to think the process takes half of the time if two people start on the same day. Students who make this prediction will later realize this is not correct.
For part (c), students with more advanced algebraic solving strategies at their disposal may not be challenged by this problem as much as students without these skills. Assuming the 22nd of October leaves nine days to reach every household, students can solve 127,590,000 = p(2(2)9 − 2) for p, where p represents the number of people starting the Booing process. Otherwise, students can look for the number of Booed households after nine days in one of the representations they already have. They can then divide 127,590,000 by that number.