*Photo of Lex times 11, by Dan DeChiaro, via flickr.*

We are finishing up an experiment in mental math, using the world’s oldest interactive game — conversation — to explore multiplication patterns while memorizing as little as possible.

Take your time to fix each of these patterns in mind. Ask questions of your student, and let her quiz you, too. Discuss a variety of ways to find each answer. Use the card game Once Through the Deck (explained in part 3)as a quick method to test your memory. When you feel comfortable with each number pattern, when you are able to apply it to most of the numbers you and your child can think of, then mark off that row and column on your times table chart.

So far, we have studied the times-1 and times-10 families and the Commutative Property (that you can multiply numbers in any order). Then we memorized the doubles and mastered the facts built on them. And then last time we worked on the square numbers and their next-door neighbors.

## The Distributive Property

This brings me to one final trick: the Distributive Property. The Distributive Property converts a tough multiplication problem into an easier addition problem. Like square numbers, the Distributive Property will follow your son or daughter throughout their school math career. Now is a good time to figure out how it works.

Assume we have two numbers to multiply, say, 62 25. We can break one of the numbers into two (or more, but let’s not make it difficult!) chunks that are easier to multiply. We could break them into any chunks we wanted, such as 25 = 17 + 8 or 62 = 39 + 23.

Since our goal is to make the calculation as simple as possible, however, let’s try chunking it like this:

25 = 20 + 5

Which turns our multiplication problem into:

62 25 = 62 (20 + 5)

The Distributive Property says that the first product, 62 times the whole 25, is the same as the sum of the two partial products, 62 times each part of 25. This is how the distributive property turns multiplication into addition:

62 25

= 62 (20 + 5)

= (62 20) + (62 5)

[Do you remember that 62 fives = 31 tens?]

= 1240 + 310

= 1550

The Distributive Property is the key to understanding the traditional pencil-and-paper rules for multiplying large numbers. But it is also a very important principle for working with mental calculation. You can break all of our remaining times-table numbers into easily-multiplied chunks.

## The Times-3 Family

For instance: 3 = 2 + 1. If you have two of something, and you get one more of that thing, then of course you will have three of that thing. Three puppies are the same as two puppies plus one more puppy. Three pizzas are the same as two pizzas plus one more pizza. And three of any number is the same as two of that number plus one more of it.

3 6

= two 6s + one more 6

= 12 + 6

= 18

And:

12 3

= two 12s + one more 12

= 24 + 12

= 36

Take turns with your child, giving each other numbers to multiply by three. Be sure to include some bigger numbers.

3 55

= two 55s + one more 55

= 110 + 55

= 165

Or:

41 3

= two 41s + one more 41

= 82 + 41

= twelvety-three

= 123

## The Times-11 Family

Similarly, the Distributive Property tells us that 11 of anything is the same as “ten and one more” of that thing. Eleven tigers are ten tigers and one more tiger. Eleven cars are ten cars and one more car. Eleven pencils are ten pencils and one more pencil. It is the same with numbers.

11 11

= ten 11s + one more 11

= 110 + 11

= 121

Or:

11 15

= ten 15s + one more 15

= 150 + 15

= 165

Take turns quizzing each other on these facts, and be sure to try it with bigger numbers, too.

11 36

= ten 36s + one more 36

= 360 + 36

= 396

## The Times-9 Family

In the same way that 11 is one more than ten, nine is one less than ten:

12 9

= ten 12s – one 12

= 120 – 12

= 108

Big numbers will stretch your child’s mental math skills, but isn’t that the point? Remember your mental math tricks and subtract in chunks:

9 27

= ten 27s – one 27

= 270 – 27

= 270 – 20 – 7

= 250 – 7

= 243

## Times-6 and Times-7 Families

We left these fact families for last, since many children (and adults!) find them difficult. But notice: Most of the numbers in these families are already colored in. That means the Distributive Property simply deepens our understanding by giving us another way to view the facts we already know.

You can break both six and seven up into two easily-multiplied chunks:

6 = 5 + 1

7 = 5 + 2

So anything times six is the same as “five and one more” of that thing, and anything times seven is the same as “five and two more” of that thing. Since the times-1, times-2, and times-5 facts are among the easiest to remember, we can use the Distributive Property to make our calculations simple:

12 6

= 12 (5 + 1)

= (12 5) + (12 1)

= 6 tens + 12

= 72

And:

12 7

=12 (5 + 2)

= (12 5) + (12 2)

= 6 tens + 24

= 84

Take at least a week on each of these: one week for times-6 and one week for times-7. Build up your mental math skills with big numbers, too. Add in chunks, finding the highest place values first.

34 6

= 34 (5 + 1)

= (34 5) + (34 1)

= 17 tens + 34

= 204

Or:

128 7

=128 (5 + 2)

= (128 5) + (128 2)

= 64 tens + 256

= 896

With that, we have made it through the whole times table chart, and we only had to memorize the doubles and squares. We did everything else with logic and number patterns!

This is the last post in my Times Table Series.

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This is great – really inspiring! Thanks so much.

I think a generally easier way to multiply anything by 25 is to remember that 25 = 1/4 * 100. Mentally finding 1/4 of a number by halving it twice is easy, (and easier if the number is a multiple of 2 or 4.) So half of 62 is 31, half of 31 is 15.5, 100 times that is 1550. (Of course, I’m implicitly using the distributive property to get those halves.)

You’re right, Alexandre, that’s a nice trick for multiplying by 25. I do use it sometimes, and if I’d thought of it while I was writing, I would have put a different number in my example.

You are fast becoming my mathematical hero. I’ve read the whole series and it is brilliant! Most of the tips you give I had never heard of, only ever learning by rote. Thank you so much for posting!

You’re welcome. I’m glad you found the series helpful. 🙂