Factorial is a word that mathematicians use to describe a special kind of numerical relationship. Factorials are very simple things. They are just products, indicated by the symbol of an exclamation mark. The factorial function (symbol: !) means to multiply a series of descending natural numbers. For instance, "five factorial" is written as "5!" (a shorthand method) and means 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in determining the numbers of combinations and permutations and in finding probability.

Now all of that may seem above your mathematical head, but let me introduce you to the book

*Anno's Mysterious Multiplying Jar* by Masaichir and Mitsumasa Anno. It is a story about one jar and what is inside it. Anno begins with the jar, which contains one island, that has two countries, each of which has three mountains. The story continues like this until 10 is reached. The colorful pictures are arranged within borders on the page as many times as the number of objects being discussed. For instance when four walled kingdoms are introduced, four kingdoms are on the page.

The explanation of 10! in the back of the book is also very helpful. Even if children do not understand the concept being taught, they will certainly appreciate the detailed colored drawings and imaginative story! The book is best for kids who have been introduced to at least basic multiplication facts, but younger kids will enjoy counting and looking at the pictures even if the rest of it is over their heads; so, this book helps with multiplying skills as well as the mathematical concept of factorials.

You might give the students a worksheet to keep track of how many islands, rooms, etc. there are. The final question is how many jars are there. Hopefully there will some students who catch on to the factorial concept, find the pattern and discover the answer!

Here is an example of how you might use factorials in solving a word problem. How many different arrangements can be made with the letters from the word MOVE? Because there are four different letters and four different spaces, this is how you would solve the problem.

**____ ____ ____ ____ **

**Four Possible Spaces**

All four letters could be placed in the first space. Once the first space is filled, only three letters remain to fit in the second space. Once the second space is filled with a letter, two letters remain to write in the third space. Finally, only one letter is left to take the fourth and final space. Hence, the answer is a factorial (4!) = 4 × 3 × 2 × 1 = 24 arrangements.

Try some problems in your classroom. Start with an imaginary character, Cal Q. Late, who is working at an Ice Cream Store called Flavors. A hungry customer orders a triple scoop ice cream cone with Berry, Vanilla, and Bubble Gum ice cream. How many different ways could Cal Q. Late stack the ice cream flavors on top of each other?

You could answer the question by listing all of the possible orders of the three ice cream flavors from top to bottom. (Students could have colored circles of construction paper to physically rearrange.)

- Bubble Gum - Berry - Vanilla
- Bubble Gum - Vanilla - Berry

- Berry - Vanilla - Bubble Gum
- Berry - Bubble Gum - Vanilla

- Vanilla - Berry - Bubble Gum
- Vanilla - Bubble Gum - Strawberry

Or, if we use factorials, we arrive at the answer much faster: 3! = 3 × 2 × 1 = 6

Learning about patterns and the use of factorials will stretch a students' mathematical mind. Why not try a few problems in your classroom? And by all means, check out Anno's Mysterious Multiplying Jar.