A Valentines Day Problem Source The teachers lab Patterns in Mathematics

The Problem There are 25 students in your classroom. On Valentine's Day, every student gives a valentine to each of the other students. How many valentines are exchanged? How would you solve this problem?

George Pólya, a mathematician famous for problem solving, said that when you have a tough problem you should make the numbers simpler. Twenty-five may just be too many! So let's change it……………………………………………..

There are five friends. On Valentine's Day, every friend gives a valentine to each of the other friends. How many valentines are exchanged?

What was your approach to solving this problem? Some of us may have used the same method, and some of us may have used a different one. There is no one right way. Let's look at eight different solutions. They're all correct. See if one of them matches yours.

The answer is 20.

List Them All Suppose the names of the friends are A, B, C, D, and E. Let's list the valentines that each friend gives, starting with A's valentines. A–B, A–C, A–D, A–E B–A, B–C, B–D, B–E C–A, C–B, C–D, C–E D–A, D–B, D–C, D–E E–A, E–B, E–C, E–D That's 20 valentines.

Do Addition Suppose the names of the friends are A, B, C, D, and E. A gives 4 valentines B gives 4 valentines C gives 4 valentines D gives 4 valentines E gives 4 valentines Add the numbers. The total is 20 valentines

Draw Points and Arrows Suppose the names of the friends are A, B, C, D, and E. Draw an arrow between each pair. Then count the arrows. There are 10 and each one represents 2 valentines. So that's 10x2 a total of 20 valentines.

Make a Star This solution is a lot like "Draw Points and Arrows," except we arrange the points differently. Suppose the names of the friends are A, B, C, D, and E. Every arrow represents 2 valentines and there are 10 double-headed arrows. Again, that's 10 twice— a total of 20 valentines.

Make a Grid Suppose the names of the friends are A, B, C, D & E. We make a grid of all the friends. Each pink square is a valentine. The grey squares show that each person does not send a valentine to himself or herself. Either count the pink squares (20), or notice that there are 5 x 5 = 25 squares in the whole grid, minus 5 grey squares. So = 20 valentines.

Look for a Pattern Here we look at even smaller numbers. What if there were only one person? Then no valentines are given. Zero. What if there were only two friends instead of five? Then there would be 2 valentines exchanged. Three friends, there would be 6 exchanged. Four friends, there would be 12. What's the pattern? 0, 2, 6, 12,...? Between the first and second numbers is a difference of 2. Between the second and third, a difference of 4. Between the third and fourth, a difference of 6. And so on. If the pattern were to continue, the next number would be a difference of — 8 so 20 valentines would have been exchanged..

Use a Formula You know that if there are n people, each will send out (n - 1) valentines. So the total number of valentines V is V = n (n - 1) Since n = 5 in this situation, V = 5 x 4 = 20.

Just Do It / Act it out Get four friends. Now there are five of you. Give valentines to each other. Then collect all the valentines and count them. There are 20.

Let's go back and try to solve the original problem. Let's see where we are now. We got 20 valentines as the answer to the problem when we simplified it, and we looked at eight different solutions. There are 25 students in your classroom. On Valentine's Day, every student gives a valentine to each of the other students. How many valentines are exchanged? That 25 makes a difference! How would you solve the problem now? Would you use the same method or a different one?

so the answer is 25 x 24 = Valentines are given out in a class of 25 people!!