Why the Speed-Method of Multiplying two Numbers from 50 to 59 Works

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Why the Speed-Method of Multiplying two Numbers from 50 to 59 Works

This is the supplemental lesson to the Math Mojo Monthly (Issue #2), "Multiplication of the Week" chapter. Please refer to that issue as you read along here.

Don't let the following paragraphs (9 steps) scare you away. It is simply an explanation of how you aready to multiplication. If it appears too confusing, I agree with you - the normal way we do things is too confusing. This explanation is simply to help you understand why the easier way works. That will be explained right after. It will be a lot easier, I promise!

Let's use the following example:

	5	6
x	5	7

1) Ones (or units) column, first partial product:

You would normally multiply 6 in the ones column of the multiplicand by the 7 in the ones column of the multiplier. That would give you 42.
You would write the 2 in the ones column of the first partial product, and carry the 4.
We usually write the carries above the column where they will eventually go, if there are more numbers to multiply in the partial product we are working on.

Thousands	Hundreds	Tens	Ones
		4		carries for partial products
		5	6	multiplicand
	x	5	7	multiplier
			2	first partial product
				second partial product
				final product

2) Tens column, first partial product:

Multiply the 5 in the tens column of the multiplicand by the 7 in the ones column of the multiplier. That gives you 35, but you still have to add the carry the 4 from the last step. That gives you 39.
Write the 9 (of the 39) in the tens column of the first partial product, and carry the 3 to the hundreds column of that partial product. (You don't have to write it above that column, because there are no more numbers to multiply in the partial product you are working on.)

(This is a pain, isn't it? I'll bet you have forgotten how silly and complicated some of the ways you were taught were, didn't you? And we're only about a third of the way through!)

Thousands	Hundreds	Tens	Ones
		4		carries for partial products
		5	6	multiplicand
	x	5	7	multiplier
	3	9	2	first partial product
			0	second partial product
				final product

It is important that you realize that you are not really multiplying 5 x 7 in this step. You are multiplying 50 x 7. You should understand that "the 5 in the tens column" really means 50. So you are getting 350 in this step, not 35. This is something they sometimes mention in school, but they don't make you understand it enough.

If you consider these first two steps, what you have actually done is do (6 x 7) + (50 x 7). It is important to understand this, because it is the essence of understanding why multiplication works the way it does, and how to make it simpler.

3) Ones column, second partial product:

This is the only real easy part of the way you learned.You just put a zero in the ones column of the second partial product.

Thousands	Hundreds	Tens	Ones
		4		carries for partial products
		5	6	multiplicand
	x	5	7	multiplier
	3	9	2	first partial product
			0	second partial product
				final product

But do you know why you were taught to do this? What does that 0 mean?

If you understood step 2, you will easily see that in the next step, you will be multiplying 50 x 6, not 5 x 6 like you were taught. If you didn't write the 0 first, and then did 5 x 6, you would get 30. That would be wrong. 50 x 6 is 300, not 30. They taught you to write the 0 as a placeholder, but I doubt that they made you understand why.

4) Tens column, second partial product:

This is where you multiply the 5 x 6 (which is really 50 x 6). You get 30, (which is really 300).
So you write the 0 of the 30 in the tens column of this partial product, and carry the 3 to the hundreds column of this partial product.
You have to write the 3 above the column where it will eventually go, because you are not done with this partial product yet. You still have to do the next step.

Thousands	Hundreds	Tens	Ones
	3	4		carries for partial products
		5	6	multiplicand
	x	5	7	multiplier
	3	9	2	first partial product
		0	0	second partial product
				final product

Make sure you understand that a 3 in the hundreds column, a 0 in the tens column, and a 0 in the ones column is the same thing as 300.

So far you have done (6 x 7) + (50 x 7) + (50 x 6).

5) Hundreds column, second partial product:

This is the final multiplication. (Unfortunately, with the "old school" way of doing this, you still won't be done, because you still have to add the partial products and their carries!)

Multiply the 5 (which is really a 50) in the multiplicand by the 5 (which also is really a 50) in the multiplier, and get 25 (which is really 2,500).
You don't have to carry the 2 above, you can write it to the left of the 5, because there are no multiplications beyond this one.

Thousands	Hundreds	Tens	Ones
	3	4		carries for partial products
		5	6	multiplicand
	x	5	7	multiplier
	3	9	2	first partial product
2	5	0	0	second partial product
				final product

By now you know that the 25 is really 2,500, which is 50 x 50.
So far you have done (6 x 7) + (50 x 7) + (50 x 6) + (50 x 50).
Please make sure you understand this, because it will be the key to the speed-method, which will make this stuff old-school for you.

6) Adding up the ones columns of the partial products:

If you still add the way you learned in school, which is right-to-left, (starting with the chump-change first), you would add the ones column of the partial product (0 + 2), and get 2 for the ones column of the final product.

Thousands	Hundreds	Tens	Ones
	3	4		carries for first partial product
		5	6	multiplicand
	x	5	7	multiplier
	3	9	2	first partial product
2	5	0	0	second partial product
			2	final product

7) Adding up the tens columns of the partial products:

Thousands	Hundreds	Tens	Ones
	3	4		carries for first partial product
		5	6	multiplicand
	x	5	7	multiplier
	3	9	2	first partial product
2	5	0	0	second partial product
		9	2	final product

Now add the tens columns of the partial products. You may be tempted to add the 4 in the green box, because it is a carry, but don't do it! You already added this carry in step 2! (It is confusing, isn't it?) 9 + 0 = 9, so write the 9 in the tens column of the final product.

8) Adding up the hundreds columns of the partial products:

Thousands	Hundreds	Tens	Ones
1	3	4		carries for first partial product
		5	6	multiplicand
	x	5	7	multiplier
	3	9	2	first partial product
2	5	0	0	second partial product
	1	9	2	final product

Add the hundreds columns of the partial products. This time you do add the carry, because it hadn't been added to anything before. 3 + 3 + 5 = 11, so you write the 1 in the hundreds column of the final product, and carry the other 1 to the thousands column.

Some people don't write that carry way up on top, sometimes they write it in the space directly above the 2. Either way is ok, but neither way is as good as the way you are going to learn after this!

9) Adding up the thousands columns of the partial products:

Thousands	Hundreds	Tens	Ones
1	3	4		carries for first partial product
		5	6	multiplicand
	x	5	7	multiplier
	3	9	2	first partial product
2	5	0	0	second partial product
3	1	9	2	final product

The final step! Add the thousands column and the carry. 1 + 2 =3, which gets written in the thousands column of the final product, making the entire final product 3,192.

Now let's leave that mind-numbing boredom, and get to some insight!

We have seen that the above example boils down to (6 x 7) + (50 x 7) + (50 x 6) + (50 x 50).

In any case of numbers from 50 through 59 being multiplied by each other, you will always have this pattern:

( ones column of multiplicand x ones column of multiplicand )
+ ( 50 x ones column of multiplier )
+ ( ones column of multiplicand x 50 )
+ ( 50 + 50 ).

If you made a general diagram of it, you could make it look like the one below,

	5	x
x	5	y

where x and y could be any digits.

If you did that, you could simplify the pattern by describing it like this:

(x x y) + (50 x y) + (x x 50) + (50 x 50).

(If, at this point, there is anything in the above which you don't understand, do not hesitate to go back to the last place where you DID understand everything, and start again from there. I mean it - that is the way to get this stuff. It is important that you do that, because the equation above is the most important thing to understand about speed-multiplication in general. )

We can actually do some pretty cool things with the above example.

Here is how to do it the Math Mojo way:

1) To get the answer to the ones column:

Multiply x x y, which in the case of 56 x 57 would be 6 x 7 = 42, exactly as above. So just do it, and write the 2 in the ones column and carry the 4 ten fours (but this time carry it in your head).

2) To get the answer to the tens column:

If you know anything about the distributive law, you will see that (50 x y) + (x x 50) is the same as (50 x y) + (50 x x) which, according to the distributive law, can be written as 50 (x + y).

In our example since x and y are 6 and 7, x + y is 6 + 7, which equals 13, so 50 (x + y) is the same as 50 x 13, which is 650. That is the same as 65 tens. Add the carried 4 tens to that, and you will get 69. So, write the 9 in the tens column and carry the 6 to the hundreds column . Make sure you only carry the 6 only in your head.

3) To get the answer to the hunderds and thousands columns:

This is the (50 x 50) part. This will obviously always give you 2,500. That is the same as 25 hundreds. Add the carried 6 hundreds to this, and you will get 31. Write the 31.

You are done.

Use the Math Mojo method to do lots of examples, in order to lock in what you have learned. Show the method to other people. Sharing, explaining and teaching are great ways to help you practice what you have learned.

You may be forced to "write your work" on tests in school. OK, do it their way for them, because they are grading you. But when you do it, think about the two farmers in the story in the Math Mojo Monthly, Issue #1.

If you or someone you know would like to learn an even easier alternative to the "times-tables," click here to check out "Numbers Juggling - Basic Multiplication of One-Digit-Numbers." It also contains an amazing speed-math method for instantly multiplying any number by 5, from left-to-right!

Mastering multiplication of one-digit numbers is the basis for all higher arithmetic. If you are teaching a child, you should definitely know about the method in "Numbers Juggling - Basic Multiplication of One-Digit-Numbers."

Are you a teacher? Do you have any students at all who have problems with basic multiplication? There probably isn't an elementary school teacher in a thousand who doesn't have a student like that. I have even given lectures in high-schools (good ones, even!) and have always met teachers who told me of at least one student who never got his or her "tables" down. You can empower and inspire some of the most innumerate students with the methods in this booklet.

Special-ed teachers have even used this book with great success. If you have students you haven't been able to reach, please try the methods in "Numbers Juggling - Basic Multiplication of One-Digit-Numbers." It is only available through this website.

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