Multiplication: An Adventure in Number Sense
Introduction: Too Many Facts?
Student: I have not learned the multiplication table yet. It looks like such a big and boring job! Can you help me?
Mentor: We will look at the table closely, and you will see that there are only about 13 facts that you may want to memorize. Anyway, all the table fits into one credit card size picture (click and look for yourself)! No big deal. What do you want with the table, though?
Student: What do you mean? I have to know it, don't I? I want to know it.
Mentor: Well, there are different meaning to the word "know." Let us figure out what kind of "know" you want with the table. Once you know the table, how are you going to use it?
How do you use or are going to use multiplication of whole numbers? How are other people using it? Find as many uses for it as you can, for example, "To have fun with numbers." Interview several people about their personal uses of multiplication of whole numbers. Ask people to be honest rather than educational. Sample questions to ask: "When was the last time you used multiplication? How did you use it?" 
Student: I guess I will be able to multiply numbers from one to ten. Oh, I will also use this knowledge to multiply larger numbers and fractions.
Mentor: It will be also useful in division.
Student: I am sure it is useful, but look how many facts are there to memorize! And how boring it looks!
Mentor: If you look a little deeper, you will see a lot of logic and hopefully a lot of interesting things about the table. Let us start looking together, before you rush ahead and spend a lot of time memorizing.
Click on any underlined fact to learn about a reason why you do not have to memorize it. For most facts, there are several such reasons, so you will be reading just one example reason. You may want to simply continue reading these pages in order to learn more about these reasons and some patterns behind the multiplication table.

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
1 

2 

3 
3*3=9 

4 
4*3=12 
4*4=16 

5 

6 
6*3=18 
6*6=36 

7 
7*3=21 
7*4=28 
7*6=42 
7*7=49 

8 
8*3=24 
8*4=32 
8*7=56 
8*8=64 

9 

10 
Mentor: How many facts are there, by the way?
Find several ways to compute how many facts are there in the table. Which ways do you like better? Why? 
Student: I need to multiply to find out... Or I can count. Well, it looks like there are a hundred facts. Too many and all boring, just as I said.
Mentor: We will deal with both problems in a moment. Meanwhile, do you know what is multiplication?
What do you think about multiplication? How would you explain the word to a three year old child? To an adult who has never learned it? To a person who knows the idea but does not speak your language? There are many different ways to explain it (even though some ways may be incorrect); find or construct a definition you like. 
Student: Uh... Well... Hard to say! You take two numbers...
Mentor: Or more.
Student: Right, and then you multiply them.
Mentor: How would you explain it to somebody who never heard the words?
Student: I will use an example. See, if you have three rows of chairs, with five chairs in each row, the thing you do to find the total number of chairs is called multiplication of numbers 3 and 5.
Mentor: You can use our multiplication applet to explain this sense of the word, and to play...
© Copyright 1998 by Maria Droujkova and Dmitri Droujkov. All rights reserved. No part of these materials should ever be used in any situation that involves compulsory teaching. See also copyright notes and student rights. 