*Photo of Javier times 4, by Javier Ignacio Acuña Ditzel, via flickr.*

If you remember, we are in the middle of an experiment in mental math. We are using the world’s oldest interactive game — conversation — to explore multiplication patterns while memorizing as little as possible. Talk through these patterns with your student. Work many, many, many oral math problems together. Discuss the different ways you can find each answer, and notice how the number patterns connect to each other.

So far, we have mastered the times-1 and times-10 families and the Commutative Property (that you can multiply numbers in any order).

## The Doubles

What else is relatively simple? Does your student know the doubles? Doubles are often considered easy, because children do so much counting and addition with numbers less than 20. Even if your child finds the doubles tricky, a little focused practice should fix these facts in mind.

**Here is your first memory task:***Learn the doubles!*

Practice doubling big numbers, too. Use silly numbers to help:

38 2

= double 3 tens + double 8

= “sixty-sixteen”

= 76

And:

2 56

= double 5 tens + double 6

= “tenty-twelve”

= 112

Go back and forth, inventing double-puzzles for each other:

3½ 2

= double 3 + double ½

= 6 + 1

= 7

And:

2 47,000

= (double 4 tens + double 7) thousand

= “eighty-fourteen” thousand

= 94,000

## Review Game: Once Through the Deck

The best way to practice the math facts is through the give-and-take of conversation, orally quizzing each other and talking about how you might figure the answers out. But occasionally you may want a simple, solitaire method for review. Here’s how:

- Shuffle a deck of math cards and place it face down on the table in front of you.
- Flip the cards face up, one at a time.
- For each card, say (out loud) the product of that number times the number you want to practice.
- Don’t say the whole equation, just the answer.
- Go through the deck as fast as you can.
- But don’t try to go so fast that you have to guess! If you are not sure of the answer, stop and figure it out.

Brian at The Math Mojo Chronicles demonstrates the game in this video, which my daughter so thoroughly enjoyed that she immediately ran to find a deck of cards and practiced her times-4 facts. (It’s funny, sometimes, what will catch a child’s interest.)

You can use Once Through the Deck as a final check that your student knows the fact family well enough to mark it on your chart. Remember to mark both the row and the column!

## The Times-4 Family

Notice that the answers in the times-4 row are exactly double the answers in the times-2 row. Can your child see why that makes sense? If you have two of something, and you replicate two more of it, then you would have four of that thing, whether it is minions or cookies or numbers.

This means you do not need to memorize the times-4 facts. Just double the number to get the times-2 answer, and then double it again. For example, would mean seven doubled, which is 14, and then that answer doubled again:

7 4

= 7 2 2

= 14 2

= 28

Practice double-doubling a bunch of numbers. Can you use the double-double trick to figure out something like ?

53 4

= 53 2 2

= 106 2

= 212

This may take a little more time to practice — but that is okay! Quiz each other with unusual numbers, using double-doubling to get the answer. Test yourself with Once Through the Deck, and when you are ready, mark the chart.

## The Times-8 Family

In the same way that times-4 was the double of times-2, it makes sense that times-8 is the double of times-4. If you have four of something, and you replicate four more of it, then you will have eight of the thing in all. It doesn’t matter whether the thing is books or aliens or numbers. Even fractions: If you have four 1/16 size slices of pizza, and you take four more pieces, then you will have 8/16 of the pizza.

If you need to calculate something times eight, you can double the something, then double again — that makes four times — then double it once more for your final answer:

6 8

= 6 2 2 2

= 12 2 2

= 24 2

= 48

I find it helpful to count on three fingers, to make sure I don’t forget any of the doublings. Remember to experiment with big numbers, too. Can you double-double-double to figure out ?

132 8

= 132 2 2 2

= 264 2 2

= 528 2

= 1056

It may take a few days or even weeks of practice before you and your student feel comfortable with these. Take all the time you need, and when you both are able to mentally double-double-double almost any number the other can pose, mark off the times-8 column and row on the chart.

## The Times-5 Family

Your daughter can probably count by fives, but many children get confused when trying to skip-count large multiplication problems. A more reliable number pattern for times-5 calculations uses the doubles in reverse. Two fives make ten, so any even number of fives will make exactly half that number of tens:

6 fives = 3 tens

18 fives = 9 tens

24 fives = 12 tens

450 fives = 225 tens

All the odd numbers times five will come out “somethingty-five,” and you can predict what the “somethingty” will be by looking at the next lower even number.

7 fives = (6 + 1) fives = 3 tens + 5

25 fives = (24 + 1) fives = 12 tens + 5

109 fives = (108 + 1) fives = 54 tens + 5

Practice intensively on these until you can both get the answer right every time. Test yourself with a round of Once Through the Deck, and then mark them off.

Wow! Look back and see how much you have learned. You started with 144 facts, and you have narrowed it down to only 21 — and all you had to memorize so far was the doubles! Your child could probably memorize the last 21 facts without too much trouble, but let’s see if we can find a few more patterns to make them easier.

This is the fourth post in my Times Table Series, which will eventually be included in one of my math books. To be continued…[Go to part 4.]

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I absolutely love the information in your blog. I’m a math tutor that developed my own method of teaching multiplication tables drawing from various sources. I’m constantly adding to it, and have added your tips on extending the doubling principal for times 4 and times 8 (when I think about it, this is the technique Greg Tang shows in his book “The best of times” but I never employed it before. Just like students, teachers need something new to spark their interest).

I’m working with a student right now that seems to have become stuck on times 8. We’ve been working on it for over a month now (off and on) and he just doesn’t seem to be able to recall that when he can’t remember a times 8 fact, that he can double 3 times. I’m just wondering if you (or anyone else with similar experience) have advice on whether to continue working on it until he gets it, or if it’s better to skip it, move on, then come back to try it again later.

In addition to “learning facts” I do practical applications, like multiplication war and DAMULT dice. He always gets stuck on times 8 no matter the application where it comes up. He resorts to guessing, or thinks that he has to double 2 times or 4 times when prompted “remember how we multiply by 8?”. Any further advice or suggestions would be appreciated!

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I have had success with the “backing off” method when my kids hit a wall. Sometimes just forgetting about the topic for awhile lets it simmer in the student’s unconscious mind, and when we come back to it in a month or two, everything flows easily.

But I’m not sure that would work so well in this case. I’m assuming the student needs to use times-8 facts in his schoolwork and can’t ignore them. Is he mature enough to catch himself when he starts guessing, so that if you give him a fool-proof technique, he could slow down and force himself to use it?

If so, here’s the technique:

* Write down 8 copiesof whatever number you are multiplying times 8, arranged like the dots on a domino. You are going to add these numbers up in pairs to find out how much all 8 of them are worth.To calculate 13 times 8, I would write:

13 13

13 13

13 13

13 13

* Then circle the numbers in pairs, writing their sum right next to the circle. This is the first doubling. Now you only have 4 numbers:26

26

26

26

* Now circle, and write their sum next to the circle. This is the second doubling, and it leaves you with only 2 numbers to add:thosenumbers in pairs52

52

* Finally, add the last two numbers(the third doubling) to get your answer.It takes some scratch paper and the discipline to resist wild guessing, but it will always work!

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Russian Peasant Multiplication would also work. It’s quicker, but it’s more abstract, which means it may be harder for a child to remember. In each step, you double one number and cut the other in half:

13 times 8 =

26 times 4 =

52 times 2 =

104 times 1 = 104

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Thanks for the tips! The student is in grade 6, but I’m not covering school work with him, I went back to just going over basic facts. Like most students I tutor, adding/subtracting isn’t a problem, but (basic) multiplication is, and doesn’t become noticeable until they start on long division (which is when the parent gives me a call to get help).

He’s a high energy little guy, and definitely a visual/auditory learner. Trying to use beads and cups to show grouping or having him write number bonds went disastrously. I like your first tip, and I’ll give it a try. If he rebels, I’m going to skip times 8 for now. We still have to work on times 3, 6 and 7. When an 8 times fact comes up that he’s stuck on, I’ll do what I always do, which is patiently explain it to him again. It’ll sink in eventually.

Thanks for all the resources on your blog, I can get lost in it reading everything and following the links. I look forward to your book coming out!

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