In the previous sections, you studied several aspects of reasoning and proof, looked at ways to encourage students to develop their ability to make conjectures and to justify and prove their ideas, and considered aspects of reasoning and proof in an upper elementary classroom.

Now that you are familiar with the Standard and some of its aspects, we'd like you to apply it. An upper elementary teacher, Ms. Lilia Olivas, asked her fourth-grade students to figure out how many Valentines would be needed if all 24 of the students gave cards to one another. The students used a variety of problem-solving strategies, such as collecting data for a simpler problem through enactment, making a diagram or a table of total cards for various numbers of students, and making an organized list of exchanges.

Your task is to think about how reasoning and proof come into play? First, consider Ms. Olivas's problem as well as an extension of the problem involving a general case. Consider the reasoning or proof aspects of the problem and of the sample solutions in the following activity. Spend this time applying and reflecting on what you've learned so far about reasoning and proof.

  Consider a problem