You will want to challenge your students with this problem for several reasons.
- First, it arises from something concrete.
- Second, although it is simple and easily grasped for very small numbers, students can't just count up any old way to get the answer when the number of participants is more than six. They have to use more mathematicseither counting systematically or generalizing that process and using multiplication.
- Third, even though students can use multiplication, the problem still isn't that simple. They have to account for not giving a valentine to themselves, either through "multiplying by one less" or correcting the "square" answer by subtracting the number of people. Therefore, students can't just plug in numbers and churn out an answer. They have to think.
- Fourth, there are many different and good ways to come up with a correct solution. While experienced students can explore why the different ways are equivalent, every student needs to know that there can be more than one way to solve this problem, and that choosing the best way depends on the individual and the particular numbers involved.
- Fifth, this activity is made easier by using common problem-solving strategies: using smaller numbers, looking for patterns, and even working backward (though this strategy is not included in the set of examples given).
- Finally, this problem arises frequently in other forms as students advance mathematically. It generates doubles of the triangular numbers (1, 3, 6, 10, 15,...), which themselves are a diagonal in Pascal's triangle. They often appear in problems defined by combinatorics, which means they surface in probability, statistics, and discrete mathematics. Knowing this problem thoroughly will save a student many hours in college. It is a "2 + 2 = 4" of advanced algebra.
You may also have seen this problem in another guise-that of the "handshakes problem." There are 30 people in a room; everyone shakes hands with everyone else. How many handshakes are made?
The relationship between the two problems is this: there are always twice as many valentines as handshakes. Think about it. If there were only two people, they would exchange one handshake but two valentines. The transaction between each pair is different in the two problems but the number of pairs is the same.
Very young students can handle this problem for small numbers by acting it out and using manipulatives. By fourth grade, students can begin to make some abstractions. By eighth grade, they can apply a formula after sufficient work with manipulatives, tables, and diagrams. And the problem, with arbitrarily large numbers, remains appropriate into adulthood.
The elaboration of NCTM Standard 2 (1998) at grades 6-8 contains the following text, for which work on this activity could be an exemplar.
All students should
analyze, create, and generalize numeric and visual pattern paying particular attention to patterns that have a recursive nature;
use patterns to solve mathematical and applied problems;
represent a variety of relations and functions with tables, graphs, verbal rules, and, when possible, symbolic rules.
Videotape 17 in the Teaching Math K-4 series shows Lilia Olivas's class of fourth graders in Tucson, Arizona, doing the valentine problem. You can find a description of the lesson and additional information on page H-19 of the Teaching Math guidebook.
If you are looking for resources on the Web, search for "handshake problem," since it is better known that way.
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