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How to See Why the Divisibility Trick for 3 Works


One of my subscribers asked why the trick for divisibility for 3 actually works. [If you missed the post on that trick, go here:]

But the gist of the trick is this:  3 divides evenly into a number if 3 divides evenly into the sum of the digits of the number.

I’ll prove this trick for a three-digit number, but you’ll see why the proof applies to numbers with as many digits as you’d like.

Let’s call our three-digit number cde, where c is the digit in the 100s place, d is the digit in the 10s place, and e is the digit in the 1s place.

We can state the value of our cde number like this:

cde =  (100 x c)  +  (10 x d)  +  e

This shows that the c-part of the number is made up of 100 groups of c. If you think about it, there’s no reason we can’t re-write the value of this digit as 99 groups of c plus 1c, or just:  (99c + c).

In the same way, the d-part of cde is made up of 10 groups of d, which we can re-write as (9d + d). And of course the e-part of the number is just e.

So now we have this:

cde =  (99c +  c)  +  (9d + d)  +  e

Using the rules of jolly old algebra, we can shuffle the terms around a bit to get this:

cde =   99c + 9d + (c + d + e)

Then, adding a set of parentheses for clarity, we get this:

cde =   (99c + 9d) + (c + d + e)

So far, so good. But what’s the point? Well, we’re just getting to that.

Let’s think a bit more about the 99c term. Factoring out a 3, we see that 99c =  3 x 33c. Since 3 is a factor of this expression, 99c must be divisible by 3. Aha, progress, right?

Factoring the d-term, we see that 9d = 3 x 3d. Again, since 3 is a factor of this expression, 9d is  also divisible by 3.

So we now know that both 99c and 9d are divisible by 3.

We can never forget the Divisibility Principle of Sums (DPS), which says:

If a number, x, divides evenly into both a and b, then it divides evenly into their sum, (a + b).

What does that mean here? It means that since 3 divides evenly into both 99c and 9d, it must divide evenly into their sum:   (99c + 9d).

And remember that our number, cde, equals nothing more than:  (99c + 9d) + (c + d + e)

So, we’ve just found out that 3 divides evenly into the quantity in the first parentheses:  99c + 9d

So to find out if 3 divides evenly into the whole number cde, all that’s left is to find out whether or not 3 divides into what remains, the quantity in the second parentheses:  c + d + e

But guess what? c + d + e is the sum of the digits for our number, cde. So this idea right here is the trick for divisibility for 3:  To find out if 3 divides evenly into a number, just add up the number’s digits and see if 3 goes into that sum.  If it does, then 3 does go in; if not, 3 does not go in.

So this proves the divisibility trick for a three-digit number like cde.

To see why the same trick works for numbers with four or more digits, keep in mind that the larger digits can similarly all be broken up as we broke up the digits of c and d. For example, if we have a four-digit number, bcde, then the value of the leading digit b can be viewed, first, as (1,000 x b). And then this can be split apart once more, into 999b + b. That way, this four-digit number fits into the pattern of the trick. And this same kind of split up can be done for any digit whatsoever.

So if you follow the logic of this proof, you now see that the divisibility trick rests on solid logical/mathematical ground.

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Divisibility, Divisibility by 3, Divisibility Proofs, Divisibility Tricks, Division, Fractions, Making Math Fun, Math Theories, Mental Math, Number Sense, Number Theory, Proofs, Times Tables, Uncategorized

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Times Tables, Learning the Threes


What’s more important for early math than knowing the times tables?

Not much, right?

Since the times table facts are so fundamental, and because many students struggle with them, I’d like to share a strategy I came up with today for
learning the 3s. This technique works particularly well with students
who struggle with memorizing apparently random facts. (We know these
facts are not random, but if learned with nothing more than flash cards,
they can appear random.)

The strategy involves three stages, each stage bringing the child closer
to being able to QUICKLY access the desired multiplication facts. Here are the stages, in order they should be taught.

STAGE ONE:  “Patty-Cake Threes”

What I do here amounts to a “patty-cake” approach to learning the threes, which works like this.

The student and I sit facing each other with our hands up. We hit our right hands together and say “one,” then hit our left hands together and say “two.” Then we
hit BOTH HANDS TOGETHER and say, “THREE.” When saying the “one” and “two,” we utter the numbers quietly. But when we say “THREE” and all successive multiples of three, we say these numbers loudly, almost (but
not quite) shouting.

After three, we continue:  “four, five, SIX … seven, eight, NINE, ten,
eleven, TWELVE … ” and so on. So this gives children a fun way to
hear — and get a feel in their body for — the multiples of three, in the proper
order.

Patty Cake

Image by davie_the_amazing via Flickr

STAGE TWO:  “Finger-Drumming”.

After the child has the rhythm of the number three, from the “patty-cake” approach, we do “finger-drumming.” To “finger-drum” the multiples of 3, the child makes a fist with one hand, and shakes it, saying with each shake, “one, two, THREE!” And when saying “THREE,” the child extends one finger from the fist. The child continues: “four, five, SIX,” and at “SIX,” he extends another finger, so he has two fingers out.

Then you ask the child, for example, “What is three times two?” Answer: the number he just said, “six.”

In this way, the child can “finger-drum” out all of the multiples of three. To
reinforce the times tables as you go, ask questions like:  “What is 3 x 4? What is
3 x 5? etc.” Each time you ask, the child must “finger-drum” till s/he gets the
correct answer. This flows very nicely from the “patty-cake” approach as it
builds on the rhythmic feel for counting in threes.

STAGE THREE:   “Finger-Skip-Counting”   The third stage follows “finger-
drumming.” To begin finger-skip-counting, the child must have done enough “finger-drumming” so s/he is quite familiar with the multiples in the correct order.

To “finger-skip-count,” 3 x 4, for example, the child holds out a fist and
runs through the multiples of 3, like this:  “Three (extending one finger), Six (extending two fingers), Nine (extending three), Twelve (extending four fingers).”  You ask, “So what is 3 x 4?” And the child answers:  “3 x 4 equals 12.”

I found it helpful to first just challenge the child with the multiples from 3 x 1 through 3 x 5. Once s/he develops competence there, proceed to “finger-skip-counting” the multiples from 3 x 6 through 3 x 10. Finally do 3 x 11 and 3 x 12.

Put all together, these three stages offer a fun and rhythmic way for children
to learn their multiples of three. I’m curious to find out if I can use a similar
approach for the 4s, and I’ll find out soon.

I can’t be sure, but it seems like children could probably learn their 4s
by jumping rope, or doing other activities with a rhythmic nature.

If any of you have used an approach like this one for learning the times
tables, feel free to share it.

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Games, Learning the 3s, Math Instruction Techniques, Multiplication, Multiplication Tricks, Rhythmic approaches to math, Times Tables, Uncategorized, Using GAMES when Teaching Math

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Multiplication Trick #3 — How to Multiply by 25 FAST!


Here’s the third in my series of multiplication tricks. The first was a trick for multiplying by 5. The second a trick for multiplying by 15, and now this one, a trick for multiplying by 25. Anyone see a pattern?

TRICK #3:

WHAT THE TRICK LETS YOU DO: Quickly multiply numbers by 25.

HOW YOU DO IT:  The key to multiplying by 25 is to think about quarters, as in “nickels, dimes, and quarters.”

Since four quarters make a dollar, and a dollar is worth 100 cents, the concept of quarters helps children see that 4 x 25 = 100.

Since four quarters make one dollar, children can see that twice that many quarters, 8, must make two dollars (200 cents). And from that fact children can see that 8 x 25 = 200.

Following this pattern, children can see that twelve quarters make three dollars (300 cents). So 12 x 25 = 300. And so on.

Fine. But how does all of this lead to a multiplication trick?

The trick is this. To multiply a number by 25, divide the number by 4 and then tack two 0s at the end, which is the same as multiplying by 100.

A few more examples:

16 x 25. Divide 16 by 4 to get 4, so the answer is 400. [In money terms, 16 quarters make $4 = 400 cents.]

24 x 25. Divide 24 by 4 to get 6, so the answer is 600. [In money terms, 24 quarters make $6 = 600 cents.]

48 x 25. Divide 48 by 4  to get 12, so the answer is 1200. [In money terms, 48 quarters make $12 = 1200 cents.]

Try these for practice:

20 x 25

32 x 25

36 x 25

16 x 25

24 x 25

44 x 25

52 x 25

76 x 25

Answers:

 20 x 25  =  500

32 x 25  =  800

36 x 25  =  900

16 x 25  =  400

24 x 25  =  600

44 x 25  =  1100

52 x 25  =  1300

76 x 25  =  1900

But wait, you protest … what about all of the numbers that are not divisible by 4? Good question! But it turns out that there’s a workaround. You still divide by 4, but now you pay attention to the remainder.

If the remainder is 1, that’s like having 1 extra quarter, an additional 25 cents, so you add 25 to the answer.

Example:  17 x 25. Since 17 ÷ 4 = 4 remainder 1, the answer is 400 + 25 = 425.

If the remainder is 2, that’s like having 2 extra quarters, an additional 50 cents, so you add 50 to the answer.

Example: 26 x 25. Since 26 ÷ 4 = 6 remainder 2, the answer is 600 + 50 = 650.

If the remainder is 3, that’s like having 3 extra quarters, an additional 75 cents, so you add 75 to the answer.

Example:  51 x 25. Since 51 ÷ 4 = 12 remainder 3, the answer is 1200 + 75 = 1275.

Now try these for practice:

9 x 25

11 x 25

14 x 25

19 x 25

22 x 25

25 x 25

34 x 25

49 x 25

Answers:

9 x 25  =  225

11 x 25  =  275

14 x 25  =  350

19 x 25  =  475

22 x 25  =  550

25 x 25  =  625

34 x 25  =  850

49 x 25  =  1225

Happy teaching!

—  Josh

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! 


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Divisibility, Learning Math Facts, Making Math Fun, Math Facts, Math Tricks, Mental Math, Times Tables, Uncategorized

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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.

PRACTICE Set:(Answers below)

15 x 4

15 x 5

15 x 8

15 x 12

15 x 17

15 x 20

15 x 28

ANSWERS Set:

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! 
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Learning Math Facts, Making Math Fun, Math Facts, Math in General, Math Tricks, Multiplication, Multiplication Tricks, Times Tables

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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


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Elementary Math, Learning Math Facts, Making Math Fun, Math Facts, Math Instruction Techniques, Mental Math, Multiplication, Times Tables, Using GAMES when Teaching Math

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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

 

ANSWERS Set A: 

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

 

 

ANSWERS Set B: 

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

 

 


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