Strategies for Teaching Multi-digit Multiplication

Multi-digit multiplication can be challenging at first. It is quite easy to teach that multiplication means repeated addition of the same number. However, kids typically never understand the traditional method of multi-digit multiplication because their teachers themselves don't understand it. In fact, ask adults why the way they were taught to multiply actually works, and the vast majority won't know.

I first taught my daughter how to multiply using the traditional method. Unfortunately, it soon became clear that she did not understand why the method works despite the fact that she could carry out multiplications with an arbitrary number of digits.

Let me preface the following discussion by saying that multiplication with an arbitrary number of digits requires knowing the following:

1. Regrouping into powers of 10: into ones, tens, hundreds, thousands, etc. For instance, 15 tens = 1 hundred plus 5 tens.
2. Multiplication by powers of 10: multiplying by 10 adds a zero to the right. Multiplication by 100 adds to zeroes to the right. And so on, and so forth...
3. The distributive law: should be understood at a basic level. It is far more important for kids to understand this than to blindly multiply an arbitrary number of digits.

A bit of advice on the distributive law for multiplication. If your kid does not really get why a (b + c) = ab + ac, try the following. Suppose you have two boxes full of candies. It is the same to:

1. put all the candy together and then double the number
2. separately double the number in each box and then put them together

If your kid does not get it at first, have him or her try it with real candy. M&M come in very handy here. Think about it. If you are going to double a pile of things, you might as well double two small sub-piles. It makes no difference how you do it. You always get the same answer. This is precisely what the distributive law means, and the way kids should understand it. There is no sense in forcing them to blindly memorize that a (b + c) = ab + ac.

I used several strategies to teach my daughter the reasons why multi-digit multiplication works.

1. Multiplication as Repeated Addition
I gave her examples of the distributive law --without telling her what I was doing -- using numbers, variables, and a combination of variables and numbers. For example:

c (a + b) = ca + cb
5 (4 + 6) = 5 * 4 + 5 * 6
5 (x + 2) = 5x + 5 * 2

I explained that 5 (x + y) = 5x + 5y because 5(x+y) means to add x+y to itself 5 times. Hence, you must have 5 x's and 5 y's. I actually wrote out

5(x+y) = (x+y) + (x+y) + (x+y) + (x+y) + (x+y) = 5x + 5y.

I then explained that 23 * 5 is the same thing as 23 * 5 = (20 + 3) * 5 = 20*5 + 3*5 because the definition of multiplication means I must have five 20's and five 3's. Hence, we can do the following:

23 * 5 = (20+ 3) * 5 = 20*5 + 3*5 = 100 + 15 = 115.

2. Alternate Method Using Distributive Property
We proceed similarly to the above method, but this may be easier to understand at first. Let's repeat the above example. 23 * 5 is the same as

(20 + 3) * 5 = 20 * 5 + 3 * 5 = 15 + 100 = 115

Let's expand this the product of two 2-digit numbers and introduce some useful record keeping method. Consider 23 * 25. This is the same as
The above method is much less confusing for young kids. The only drawback is that it uses many more rows that the traditional, grade school algorithm.

3. Alternate Method With Counting Followed by Regrouping

This is my preferred method. It is the closest to the method taught in school. Let's give an example. Suppose that you want to multiply 23 and 45. The first number is 2 tens and 3 ones. The second number is 4 tens and 5 ones. This means that the product will contain:

1. 3 ones, five times = 15 ones
   
2. 2 tens, five times = 10 tens
   
3. 3 ones, 4 ten times = 12 tens
   
4. 2 tens, 4 ten times = 8 hundreds

It should be clear that you kid needs to understand that ones x ones = ones, ones x tens = tens, tens x tens = hundreds, etc. Putting the above information into my notational framework, one gets:
The final step above is to regroup into standard numerical notation. For instance, the 15 ones is equal to 5 ones and 1 ten. Hence, the 22 tens plus the 1 ten from the 15 ones becomes 23 tens. This is equal to 3 tens and 2 hundreds. The, 8 hundreds plus the two hundreds from the tens add up to 0 hundreds and 1 thousand.

One advantage of this method is that it spells out every step done in the traditional method, while coming as close to it as possible without carrying and multiplying simultaneously. It is an easy jump to the traditional method once this method is understood.

I hope that the above strategies help your kid to learn multi-digit multiplication. At the end of the day, it is of the utmost importance to understand the process. The actual method is irrelevant.

Patience: This is the key to teaching highly gifted children. Do not assume that they need no repetition. They do. They just need fewer drills than normal kids. Do not assume they will get every concept right away. They will not. These kids learn amazingly fast, but they tend to forget things unless they are absolutely passionate about them. Hence, you may have to explain things a few different ways.

Consistency: This is extremely important. You need to set a routine. Kids like structure and routine. Do not do too much in any one day. If you spend 4 to 5 days a week doing 30 mins per day, your highly gifted child could finish elementary school math in less than two years. Don't push it. Take a week off here and there. Don't study during the weekend if they are very young. Do a little every day and don't let too many days go by without doing a little bit of mathematics. Consistency is the name of the game.

Praise Them When They Work Hard. Don't Praise Them for Being Smart: Kids need to be praised when they work hard or accomplish something that is difficult for them. Kids should not be praised for being smart. There is nothing they can do about the way they were born. You job is to teach them how to set goals and work to accomplish them. Being smart is not an accomplishment.