## Kiss those Math Headaches GOODBYE!

### Multiplication Trick: x 15 — How to Multiply by 15 FAST!

Here’s the second in my set of multiplication tricks. (The first was a trick or multiplying by 5.)

TRICK #2:

WHAT THE TRICK LETS YOU DO: Multiply numbers by 15 — FAST!

HOW YOU DO IT: When multiplying a number by 15, simply multiply the number by 10, then add half.

EXAMPLE:15 x 6

6 x 10 = 60

Half of 60 is 30.

60 + 30 = 90

That’s the answer:15 x 6 = 90.

ANOTHER EXAMPLE:15 x 24

24 x 10 = 240

Half of 240 is 120.

240 + 120 = 360

That’s the answer:15 x 24 = 360.

EXAMPLE WITH AN ODD NUMBER:15 x 9

9 x 10 = 90

Half of 90 is 45.

90 + 45 = 135

That’s the answer:15 x 9 = 135.

EXAMPLE WITH A LARGER ODD NUMBER:23 x 15

23 x 10 = 230

Half of 230 is 115.

230 + 115= 345

That’s the answer:15 x 23 = 345.

15 x 4

15 x 5

15 x 8

15 x 12

15 x 17

15 x 20

15 x 28

15 x 4=60

15 x 5=75

15 x 8=120

15 x 12=180

15 x 17=255

15 x 20=300

15 x 28=420

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly  like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com  Just click the links in the sidebar for more information!

### Memorizing those Times Tables

Eegads! The times tables.

Is there any area of elementary math more fraught with stress and anxiety, save, perhaps, long division? Probably not. But for good reason.

Despite what a tiny minority of conceptual-learning purists might say, the times table facts ARE critical. Let’s face it: you really DON’T want your children to spend the rest of their lives reaching for the calculator to figure out 6 x 7; a certain amount of math simply needs to become automatic, to allow students to succeed at higher math skills and and to gain higher math concepts. Not only that, but knowing the times tables is widely recognized as a crucial milestone in children’s elementary math development.

In my work as a tutor, I’ve used many approaches to teach the times tables over the years, and each of them has one benefit or another. But I’ve settled on one technique as my “old-faithful” approach. This technique combines elements of both play and discipline, and it also melds both the “conceptual” approach and the “pure memorization” approach.

This technique relies on a three-step process, and it’s easy to learn and teach.

The first step is to simply isolate a particular times table fact set you’d like your child to work on, for example, the 4s. This act of isolating itself is critical. The child knows that she or he is required to memorize a limited set of facts for now (not the entire times tables), and that narrowing of the task decreases anxiety.

Once you’ve settled on the fact set, the second step begins, and it can be quite fun. In this second step there should be no mention even made of the times tables. All you’re doing in this step is laying the foundation for times tables facts. What you do here is work with your students/children to help them learn to first COUNT UP by the number you’re dealing with. So for example, if you’re teaching the 4s, you simply teach children how to COUNT UP by 4s. What that means is that you teach your children how to think their way through knowing and saying the following with speed and ease:  0 – 4 – 8 – 12 – 16 – 20 – 24 – 28 – 32 – 36 – 40 – 44 – 48.

I’ve found that most children take well to this learning process if you approach it in the spirit of a game. You might, for example, start by saying 0 and then throw your child a ball. She or he will then say 4 and throw the ball back to you. You then would say 8, and then throw the ball back to your child. Keep going till you hit the peak number, 40, 48, or wherever you decide to stop.

Another way to make this into a game for young children is to make it into a game like “patty-cake.” Make up a set of hand gestures to which you, very quietly, say:  1-2-3, and then clap hands and loudly say “4!” Then use the same hand gestures to quietly say:  5-6-7, and then clap again and loudly say: “8!” There are many ways to make this process of counting by 4s game-like. And if you’re short on ideas, ask your children/students what would make it fun for them.

In any case, once your children can accurately COUNT UP by 4s, work with them in the same fashion to COUNT DOWN by 4s. Same idea, but now you start by saying 48, or 40, and then help them count DOWN:  44 – 40 – 36 – 32 –  28 – 24 – 20 – 16 – 12 – 8 – 4 – 0. This takes a bit more time, but it can be done — and more easily than you might imagine.

Once your child can count both up and down, she or he has the mental “scaffolding” on which the times table facts are hung, as it were.

And so the third step involves combining this “scaffolding” with the actual times tables. Here’s how.

Have your children memorize what I call THE THREE KEY MULTIPLICATION FACTS:
x 1,  x 5, and x 10.

For example, when learning the 4s, these key facts would be:
4 x 1 = 4
4 x 5 = 20
4 x 10 = 40

Once children memorize those three key facts, help them see that to find 4 x 2 and 4 x 3, they just COUNT UP by 4 once or twice, beyond the key fact of 4 x 1 = 4. Similarly, to find 4 x 6 and 4 x 7 they just COUNT UP by 4 once or twice, beyond the key fact of 4 x 5 = 20. And to find 4 x 11 and 4 x 12, they just COUNT UP by 4 once or twice beyond the key fact of 4 x 10 = 40.

Work on this first, and have them master it before proceeding.

Once a child knows these facts, she or he has 9 of the 13 key facts (going from 4 x 0 through 4 x 12).

To learn the four other facts, help children see that to find 4 x 4 and 4 x 3, they just COUNT DOWN by 4 once or twice, below the key fact of 4 x 5 = 20. And to find 4 x 9 and 4 x 8, they just COUNT DOWN by 4 once or twice, below the key fact of 4 x 10.

By breaking the process of learning the times tables into these steps, you make the process less daunting for children. By teaching students how to COUNT UP or COUNT DOWN by the number you’re learning, you help children develop many rich aspects of number sense. And by connecting the process of COUNTING UP or DOWN to the times tables, you help children learn these critical facts both solidly and with understanding.

My advice:  try it. I guarantee that you’ll like it.

Happy Teaching,

—  Josh

### Multiplication Trick #1 — Fun with the 5s

SPICE IT UP!

That’s my advice to teachers and parents who see students getting bored or frustrated as they try to learn their times tables.

As you help students learn these critical facts, it helps, from time to time — to work on multiplication in a fun and relaxing way.

This is the first in a series of blogposts that make it more pleasurable to learn multiplication facts — by teaching multiplication tricks. Each post will contain a complete lesson plan:  instruction, practice problems, and all answers.

The first such trick is for multiplying by 5.

TRICK #1:

WHAT THE TRICK LETS YOU DO:
Multiply numbers by 5.

HOW YOU DO IT (EVEN NUMBERS):
When multiplying an even number by 5, just take half the value of the even number, then put 0 at the end.  Ta da … that’s your answer.

Example:  5 x 14

Half of 14 is 7.

Put down the 7, then put a 0 after it, and you get 70.

That’s the answer:  5 x 14 = 70.

Can you believe that it’s that easy? Watch how you can do the same feat with larger numbers…

Another example:  5 x 48

Half of 48 is 24.

Put down the 24, then put a 0 after it, and you get 240.

That’s the answer:  5 x 48 = 240.

PRACTICE Set A:  (Answers at bottom)

5 x 8

5 x 16

5 x 4

5 x 28

5 x 36

5 x 84

5 x 468

HOW YOU DO IT (ODD NUMBERS):
When multiplying an odd number by 5, first subtract 1 from the odd number, thus making it an even number. Then use the trick (above) for even numbers. And here’s the new thing to know — instead of putting a 0 after the result, put a 5.

Example:  5 x 13

13 – 1 = 12

Half of 12 is 6.

Put down the 6, then put a 5 after it, and you get 65, That’s the answer:

5 x 13 = 65.

Another example: 5 x 29

29 – 1 = 28

Half of 28 is 14.

Put down the 14, then put a 5 after it, and you get 145. That’s the answer:

5 x 29 = 145.

PRACTICE Set B:  (Answers at bottom)

5 x 7

5 x 13

5 x 9

5 x 15

5 x 23

5 x 47

5 x 685

5 x 8  = 40

5 x 16  = 80

5 x 4  =  20

5 x 28  =  140

5 x 36  =  180

5 x 84  =  420

5 x 468  =  2,340

5 x 7  =  35

5 x 13  =  65

5 x 9  =  45

5 x 15  =  75

5 x 23  =  115

5 x 47  =  235

5 x 685  =  3,425

### How to Best Learn the Multiplication Facts

QUESTION:

Is there a particular order that is most beneficial to learning times tables? For example, I have heard that learning 1, 5, 10, 11, 2, 4, 8, 12, 3, 6, 9 is a good order. What say you?

There are many “schools of thought” on what makes the best order for teaching the times tables.

I encourage everyone out there who has a favorite order to share it with us in the comments. And feel free to explain why this is your favorite order.

Here is my preferred order — and a few notes on why:

1, 2, 4, 10, 5, 11, 3, 6, 9, 8, 12, 7

IMPORTANT NOTE: I teach the facts for the multiplicands 1 – 6 first. Only after
students have those down do I advance to the multiplicands of 7 – 12.

i.e.: When teaching the x 4 facts the first round through, teach 1 x 4 through 6 x 4. On the second round through, teach the 7s through 12s, i.e.: 7 x 4 through 12 x 4.

Using this staggered approach helps students avoid getting befuddled by the size of numbers they encounter early on. This approach builds success and confidence early on.

A few notes on my order:

1st) I teach x 2 second. That’s because you can teach multiplying by 2 as simply “doubling” the number, a concept almost all kids understand. The “doubling” approach gives teachers a good way to talk if you’re going to use manipulatives to reinforce the concept.

In suggesting that you use manipulatives, I mean that you may want to use counters such as tiddly winks, paper clips, pennies, etc. To demonstrate a fact like 2 x 7, you would first lay out a row of 7 counters, then underneath them put out another row of 7 counters, and you would have the student count them all up.

2nd) I put x 4 third because you can teach multiplying by 4 as doubling a number TWICE. e.g.,: 4 x 3: 3 doubled is 6. And 6 doubled is 12, so 4 x 3 = 12.

Again, it would be a good idea to use manipulative counters to demonstrate this concept. Here you would lay out four rows of three counters, to show the fact that 4 x 3 = 12

3rd) I teach x 5 right after x 10 because multiplying by 5 gives you a number that’s exactly half of what you get when you multiply by 10.

When teaching the 5s tables, use the trick that multiplying by 5 is the same as multiplying by 10 and cutting in half. (And, for even multiples of 5, first cutting the number in half, then tacking on a 0 at the end.)

e.g.: for 4 x 5: Half of 4 is 2. Tack on a 0, and you get 20 4 x 5 = 20 [yes, it is that easy!]

4th) You can teach multiplying by 8 in several ways, depending on what works for your children. One way is to see that multiplying by 8 is doubling a number three times. e.g., for 6 x 8: 6 doubled is 12. 12 doubled is 24. 24 doubled is 48. So 6 x 8 = 48

Another strategy for 8: Multiply by 10, then take away 2 x the number. e.g., for 6 x 8: 10 x 6 = 60, and 2 x 6 = 12. 60 – 12 = 48, so 6 x 8 = 48

5th) Multiplying by 12 offers you a great chance to introduce the distributive property for multiplication. That’s just fancy language for saying that when you multiply, say, 6 x 12, you can look at it like this: Since 12 = 10 + 2, 6 x 12 can be viewed as: (6 x 10) + (6 x 2) = 60 + 12 = 72

Most children can get used to this quickly, if they are taught it EARLY ENOUGH.

### Exploring Addition Facts

QUESTION:

I have been asked to help my grandson memorize the addition tables. If I ask him what is 8 + 5, I watch him doing it in his head. He gets the right answer but takes a long time, which may hurt him later when he is introduced to multiplication. Do you have any suggestions?

Great question.

Without knowing your grandson, it’s impossible to know what he is doing when he takes some time to figure out addition facts.

But I’d suggest is that you just ask him — in a happy and curious way — what thought steps he is going through.

You can tell him that you’ve heard that students do mental math in different ways, and that you’d be interested to know how he is doing it. You might also want to reassure him that there is no “wrong” way to do math in your head.

[Be aware: some children have trouble verbalizing what they’re “doing” when they do math mentally. If your grandson has trouble telling you, you might prompt him by asking if he’s using either of the strategies I describe below.]

It may be that your grandson is trying to retrieve a memorized fact, but it’s more likely that he is using some kind of mental operation to arrive at the answer.

For example, it may be that he is “counting up” 5 from 8, to get to the answer. (If so, it would be commendable that he can count up 5 in his head — without using his fingers. Not all students can do this.) It’s also possible that he is taking 5 from the 8, and giving it to the first 5 to make 10, and then tacking on the extra 3, to get to 13 (an advanced strategy).

The main point, though, is that you can’t know till you ask him.

And the other point is that it’s often fascinating to open up a dialogue like this with kids, to find out how they do mental math.

Once you get the dialogue underway, I’d suggest that you just follow wherever it leads. For example, if your grandson is using the second strategy I mentioned (making 10 and adding on), ask him if he can extend the process a bit, and do problems like 18 + 5, 28 + 5,
etc.

If, on the other hand, he is “counting up” 5 from the 8, see if he can use the second method, too.

Essentially, what you have here is a great opportunity to find out how your grandson does addition, and to explore the operation with him. And have fun doing it.

Please feel free to write back if you do open up this kind of dialogue. I’d be curious to know what happens.

And to get to the heart of your question, I would say: Yes, you do want your grandson to develop speed or “fluency,” as teachers like to say. But when he is first learning the facts, it’s critical that he think about the operation, not just memorize facts.

To help him gain speed, I would suggest that you use flash cards or fact worksheets (just google math addition worksheets, and you’ll find loads of them).

And to help him develop a range of good strategies to help him learn the facts with understanding,
I suggest the Facts That Last series by Creative Publications.

https://www.creativepublications.com/productfamily.html?PHPSESSID=4c695f197ef3330314c21e50c8b2d70e&familyid=27