**Learning Outcomes**

Students will be able to

- find the rule that governs a recursive relationship
- solve a complex problem by creating and solving easier, related problems

**Common Core State Standards:** 7.EE.B.4, HSF-BF.A.1, HSF-BF.A.1a

**Vocabulary:** Rule, pattern, recursive

**Materials:** Diagram Designers Worksheet for Group A; Valentine Cards classroom resource for Group B

**Preparation:** Copy the worksheet for Group A and cut out the valentines from the classroom resource for Group B.

**Procedure**

**1. Introduction (5 minutes, whole group)**

Begin by presenting the following scenario to the class: It is Valentine’s Day, and everybody in class is exchanging cards. Each student gives one valentine to everyone else in the class. How many valentines are exchanged among 5 people? Among 10 people? Among all the people in this class? Can you find a *rule* that could tell you how many valentines would be exchanged for any number of people?

Explain that one way to solve complex problems like this is to solve an easier, related problem. Construct a quick table as shown and tell students that finding the missing data can help them find a *pattern*.

Tell students that they will now use different methods to explore the pattern that arises from exchanging valentines.

**2. Activity (15 minutes, small groups)**

Break students into small groups, as described below. Class size will determine the number of groups there are, although the number of students in each group should remain as listed.

*Group A: Diagram Designers (3 people in group)*

Hand out the worksheet. Students will use it to draw lines between different people and track the number of valentines given. Start with 2 people, then 3, 4, 5, etc. Each time one more person is added, how many more valentines are given? How does this help students think of a rule?

*Group B: Valentine Actors (5 people in group)*

Hand out the cut-out valentines. This group will act out the giving of valentines, first with 2 people, then 3, 4, 5, etc. (One person in the group should be responsible for making a table charting the number of people and the total number of valentines needed.) How many valentines does each student need if there are 5 people in a group? 6 people? How does this help students think of a rule?

*Group C: Computer Codebreakers (2 people in group)*

Have this group use the interactive by going to Task 1 and clicking “Add a Student” successively. As the group works through the problem, the students will learn one way of modeling the problem. They will also see how many valentines are needed for 5 people. Can they use this information to figure out how many valentines are needed for 10 people? How can they use the pattern to figure out a rule?

**3. Conclusion (10 minutes, whole group)**

Bring everyone back together and ask groups to report what they found. How many valentines are needed for 5 people? 10 people? Everyone in the class? Did anyone think of a rule that governs the pattern?

Return to the table from earlier. Fill in the number of cards needed for 2–5 students, and then add data for 6–10 students as well. Ask students to identify the pattern in the number of valentines. Tell them that this is called a *recursive* pattern; they can find each successive value, in this case the number of valentines, by applying a rule to the value before.

Ask students how solving a simpler problem—How many valentines are exchanged among 5 people?—helped them think about the more complex problem. Note: It’s not essential to share the rule for the number of valentines for *n* people, *n(n-1)*, if students are more interested in finding out how many valentines are needed for the class.