Intent This activity builds students’ skill in making sense of information presented in context, working with numeric data, and finding maximums and minimums.

Mathematics Students look for minimal and maximal solutions that fit given numeric constraints. Because students are working with families of different sizes, they will not be able to check their work simply by comparing numeric results. Instead, they must develop a feeling for the mathematical process of obtaining their results.

Progression This activity is meant for homework amid group work on Creating Families; it can also be used afterward. After working individually, students have the opportunity to verify their solutions with others.

Approximate Time 5 minutes for introduction 20 minutes for activity (at home or in class) 10 minutes for discussion

Classroom Organization Individuals, then groups, followed by whole-class discussion

Doing the Activity You might ask students to make an initial guess of how many people will be in all the families of the class wagon train. (This is Question 3.) Then explain that they will use the conditions—that is, the constraints—given in Creating Families to consider how big and how small each type of family can be. (Students do not need to have completed that activity.)

Discussing and Debriefing the Activity Give students time to compare answers in their groups. Next, ask students to report the minimum and maximum number of members in their family units. The summary chart posted at the conclusion of Creating Families can be compared with results from this activity, confirming that no family of a given kind was smaller than the minimum number possible or larger than the maximum number possible. Students can also compare their estimates on Question 3 of this activity with the actual results. Keep in mind that there may not be clear-cut “right answers” for parts of this activity, because some of the instructions in Creating Families are ambiguous. Asking, What assumptions did you make? can draw out different solutions. Students should recognize that once these ambiguities are resolved, Questions 1 and 2 do have exact answers. Ask for justification of these values. How can you be sure about the minimum and maximum values? The justification process makes students articulate what they figured on their own, which can be quite a challenge. For instance, students should see that the small family must have at least three members because it has “at least one child” and “more adults than children.” But they should also explain how they know that a family of exactly three people is possible, and illustrate by giving an example and explaining the relationships that the people share. Question 3 involves estimating as to what the families for the whole class will look like. If these estimates are at all reasonable, students’ work should be considered correct. For example, they might use the average of their minimum and maximum numbers from Question 2 as the average for each group, and multiply this by the number of groups in the class.

Key Questions What are the minimum and maximum sizes you found for each type of family? What assumptions did you make? How can you be sure about the minimum and maximum values?