# Digits/places/powers of ten

#### catfish_friend

My 5.5 year old has been playing Tap Fish 2 and has coins (points) around 50,000. There was a background she wanted to buy for 10,000 coins and she knew that 10,000 was less than 50,000 so she was excited that she could afford it.

I took the opportunity to explain the places/powers of ten though she did not ask. I showed her the ones digit and asked her how many ones there were. I asked her what the next place value was and she answered. I asked her how many ones made a ten, to which she answered ten. Then when I showed her the hundreds place, I asked her how many tens made a hundred and she didn't know. I counted by tens to a hundred on my hands and asked her again and she answered ten. I told her the next place value was a thousand and the one after ten thousand. I asked her how many hundreds were in one thousand and she asked, "ten?" and she guessed the same for how many thousands are in ten thousand.

Am I being too teacher-y?

I stopped there and we went back to playing Tap Fish 2. I had been planning on getting tons of change and one hundred one dollar bills to play bank or shop. My 5.5 year old already plays Monopoly with occasional assistance but I know she doesn't know exactly how place values work, so I took a stab at working that out with her.

I'm still deschooling and trying to wrap my mind around what being a true partner in learning looks like. Did I lead too much? Is it better to wait for specific questions and hold off on trying to explain when there's no specific prompting?

Thanks as always for feedback! I always learn so much here (so punny, I know)!

Ceci

P.S. "Powers of Ten" is a DVD I own and have been meaning to show her...just have to finish unpacking from our summer move!

Sent from my iDon'tAlwaysHaveItOnPhone

I took the opportunity to explain the places/powers of ten though she did not ask. I showed her the ones digit and asked her how many ones there were. I asked her what the next place value was and she answered. I asked her how many ones made a ten, to which she answered ten. Then when I showed her the hundreds place, I asked her how many tens made a hundred and she didn't know. I counted by tens to a hundred on my hands and asked her again and she answered ten. I told her the next place value was a thousand and the one after ten thousand. I asked her how many hundreds were in one thousand and she asked, "ten?" and she guessed the same for how many thousands are in ten thousand.

Am I being too teacher-y?

I stopped there and we went back to playing Tap Fish 2. I had been planning on getting tons of change and one hundred one dollar bills to play bank or shop. My 5.5 year old already plays Monopoly with occasional assistance but I know she doesn't know exactly how place values work, so I took a stab at working that out with her.

I'm still deschooling and trying to wrap my mind around what being a true partner in learning looks like. Did I lead too much? Is it better to wait for specific questions and hold off on trying to explain when there's no specific prompting?

Thanks as always for feedback! I always learn so much here (so punny, I know)!

Ceci

P.S. "Powers of Ten" is a DVD I own and have been meaning to show her...just have to finish unpacking from our summer move!

Sent from my iDon'tAlwaysHaveItOnPhone

#### Joy

I just want to share my observation of how my son plays with number. I heard children could learn math naturally but didn't really know how that would happen.

1. He learned percentage by watching the downloading process. Also he learned number to 100. This was when he was about 4.

2. He played "teach me kindergarten". He was totally motivated by the coins he earned so he could buy fish food. In the beginning he had to use the counting hints. This was when he about 4 or 5.

3. When he played minecraft, he needed to allocate how many blocks from the pile of 64 blocks. Also when he designed his structure, he needed to use numbers and math. This was when he was 5 and 5.5.

4. In the last 3 months, he showed intensive interest in math. He has been doing calculation a lot, all by himself. He would ask me sometimes to confirm or sometimes to inform me his answer.

Things like

When we headed to a place, he asked me how long it would take. I said, 15 minutes. He would say, oh, that is 900 seconds.

Or he would calculate how many years ago it was 1960. (must be some tv he watched about 1960)

He was really into big numbers, like billion, trillion. One day he said, mama, I love you one billion trillion. I said, wow, there are lots of zeros. And he said, yeah, 21 zeros. (he knew billion has 9 zeros and trillion has 12 zeros)

I offered to show how to do addition the way I was taught. He was not interested. I offered to show him multiplication, he refused. So far that was the only two times I tried to offer to teach. Both times he refused. He has been learning in his own way.

The other day, he asked why one week is seven days. I said, I have to look it up. But I guess we can make 10 days per week. So how many weeks will we have then. He said, 3.

I am amazed how he figured things out without being taught.

Now he knows all the money, time, dates...

Sometimes he can explain how he gets his answer. Most of the time he would say, my brain does the work or I don't know how to explain.

Jihong/joy

Sent from my iPhone

1. He learned percentage by watching the downloading process. Also he learned number to 100. This was when he was about 4.

2. He played "teach me kindergarten". He was totally motivated by the coins he earned so he could buy fish food. In the beginning he had to use the counting hints. This was when he about 4 or 5.

3. When he played minecraft, he needed to allocate how many blocks from the pile of 64 blocks. Also when he designed his structure, he needed to use numbers and math. This was when he was 5 and 5.5.

4. In the last 3 months, he showed intensive interest in math. He has been doing calculation a lot, all by himself. He would ask me sometimes to confirm or sometimes to inform me his answer.

Things like

When we headed to a place, he asked me how long it would take. I said, 15 minutes. He would say, oh, that is 900 seconds.

Or he would calculate how many years ago it was 1960. (must be some tv he watched about 1960)

He was really into big numbers, like billion, trillion. One day he said, mama, I love you one billion trillion. I said, wow, there are lots of zeros. And he said, yeah, 21 zeros. (he knew billion has 9 zeros and trillion has 12 zeros)

I offered to show how to do addition the way I was taught. He was not interested. I offered to show him multiplication, he refused. So far that was the only two times I tried to offer to teach. Both times he refused. He has been learning in his own way.

The other day, he asked why one week is seven days. I said, I have to look it up. But I guess we can make 10 days per week. So how many weeks will we have then. He said, 3.

I am amazed how he figured things out without being taught.

Now he knows all the money, time, dates...

Sometimes he can explain how he gets his answer. Most of the time he would say, my brain does the work or I don't know how to explain.

Jihong/joy

Sent from my iPhone

On Feb 19, 2012, at 12:34 AM, catfish_friend <catfish_friend@...> wrote:

> My 5.5 year old has been playing Tap Fish 2 and has coins (points) around 50,000. There was a background she wanted to buy for 10,000 coins and she knew that 10,000 was less than 50,000 so she was excited that she could afford it.

>

> I took the opportunity to explain the places/powers of ten though she did not ask. I showed her the ones digit and asked her how many ones there were. I asked her what the next place value was and she answered. I asked her how many ones made a ten, to which she answered ten. Then when I showed her the hundreds place, I asked her how many tens made a hundred and she didn't know. I counted by tens to a hundred on my hands and asked her again and she answered ten. I told her the next place value was a thousand and the one after ten thousand. I asked her how many hundreds were in one thousand and she asked, "ten?" and she guessed the same for how many thousands are in ten thousand.

>

> Am I being too teacher-y?

>

> I stopped there and we went back to playing Tap Fish 2. I had been planning on getting tons of change and one hundred one dollar bills to play bank or shop. My 5.5 year old already plays Monopoly with occasional assistance but I know she doesn't know exactly how place values work, so I took a stab at working that out with her.

>

> I'm still deschooling and trying to wrap my mind around what being a true partner in learning looks like. Did I lead too much? Is it better to wait for specific questions and hold off on trying to explain when there's no specific prompting?

>

> Thanks as always for feedback! I always learn so much here (so punny, I know)!

>

> Ceci

>

> P.S. "Powers of Ten" is a DVD I own and have been meaning to show her...just have to finish unpacking from our summer move!

>

> Sent from my iDon'tAlwaysHaveItOnPhone

>

>

>

> TODAY(Beta) • Powered by Yahoo!

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> A frustrating end to a five-year winning streak might prove beneficial in the long run.

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#### Joyce Fetteroll

On Feb 19, 2012, at 1:34 AM, catfish_friend wrote:

You're still thinking in terms of all the information and skills she "needs to" learn and how (supposedly) unnatural it all is.

Think about how many words she knows and how complex the sentences are she constructs from them. How much of that did you pour into her head? How much of that did she discover on her own by trying out and playing with the ideas that were swirling around her when interest and need struck her?

If she's playing games that involve numbers, she'll see and use and play with the concepts of place value without ever needing to know what it's called or how it works. She'll get a *feel* for how numbers work by using them. Those feelings will grow into understanding even if she doesn't realize it, even if she can't explain it.

Just as she's learned how to change the tenses of verbs to match time and person without being aware of what tenses and verbs are. And she can do it with regular and irregular verbs without giving it a thought. She has grown a feel for it from being immersed in language, trying words out. She picks up feedback -- even when you don't consciously give it -- to see how well it worked and she makes adjustments, most of the time without conscious thought.

You can show her cool things numbers do when you think of it, if you think she might be interested. Judge how much to go on by her engagement. Is she interested or just indulging you until you shut up and she can get back to what she was doing? ;-) Ask if you're not sure! :-)

Joyce

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> Did I lead too much? Is it better to wait for specific questions andThink more in terms of sharing what you find interesting in the world and what she might find interesting rather than getting important information into her.

> hold off on trying to explain when there's no specific prompting?

You're still thinking in terms of all the information and skills she "needs to" learn and how (supposedly) unnatural it all is.

Think about how many words she knows and how complex the sentences are she constructs from them. How much of that did you pour into her head? How much of that did she discover on her own by trying out and playing with the ideas that were swirling around her when interest and need struck her?

If she's playing games that involve numbers, she'll see and use and play with the concepts of place value without ever needing to know what it's called or how it works. She'll get a *feel* for how numbers work by using them. Those feelings will grow into understanding even if she doesn't realize it, even if she can't explain it.

Just as she's learned how to change the tenses of verbs to match time and person without being aware of what tenses and verbs are. And she can do it with regular and irregular verbs without giving it a thought. She has grown a feel for it from being immersed in language, trying words out. She picks up feedback -- even when you don't consciously give it -- to see how well it worked and she makes adjustments, most of the time without conscious thought.

You can show her cool things numbers do when you think of it, if you think she might be interested. Judge how much to go on by her engagement. Is she interested or just indulging you until you shut up and she can get back to what she was doing? ;-) Ask if you're not sure! :-)

Joyce

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#### Joyce Fetteroll

On Feb 19, 2012, at 2:10 AM, Joy wrote:

I do want to reassure anyone whose child *isn't* doing number gymnastics at 5 and 6 that it's perfectly normal! Some people have a facility for numbers. Some for words. Some for understanding the inner workings of people. Some for understanding the interactions between people. Your kids may be whizzes at stories, what colors go together, turning 3D objects around in their heads, seeing patterns in words or pictures, noticing changes in nature, remembering tunes.

There are loads of intelligences. Most of us are a mix. If you haven't read Howard Gardner, nows the chance :-)

http://en.wikipedia.org/wiki/Theory_of_multiple_intelligences

Unschooling works because it feeds who a child is , what their interests are and provides a rich, supportive environment for them to explore and discover what else interests them.

Joyce

[Non-text portions of this message have been removed]

> Sometimes he can explain how he gets his answer. Most of the timeExactly! Ask a child how to express run, read, walk in the past, present and future for various people, they might stumbled ... and perhaps get bored ;-) But they do it without thinking all the time. True understanding is being able to use something properly. Being able to explain it is separate and not necessary for understanding.

> he would say, my brain does the work or I don't know how to explain.

I do want to reassure anyone whose child *isn't* doing number gymnastics at 5 and 6 that it's perfectly normal! Some people have a facility for numbers. Some for words. Some for understanding the inner workings of people. Some for understanding the interactions between people. Your kids may be whizzes at stories, what colors go together, turning 3D objects around in their heads, seeing patterns in words or pictures, noticing changes in nature, remembering tunes.

There are loads of intelligences. Most of us are a mix. If you haven't read Howard Gardner, nows the chance :-)

http://en.wikipedia.org/wiki/Theory_of_multiple_intelligences

Unschooling works because it feeds who a child is , what their interests are and provides a rich, supportive environment for them to explore and discover what else interests them.

Joyce

[Non-text portions of this message have been removed]

#### Sandra Dodd

-=-I took the opportunity to explain the places/powers of ten though she did not ask. I showed her the ones digit and asked her how many ones there were. I asked her what the next place value was and she answered. I asked her how many ones made a ten, to which she answered ten. Then when I showed her the hundreds place, I asked her how many tens made a hundred and she didn't know.-=-

Pretty teachery.

When she didn't know the answer, if she was interested and curious, telling her ten might be fun, but if she didn't know and seemed impatient with the interruption or if she was wincing away from the question, it might've been a good time to just say "They're all ten of the one before it," and smile and dance away happily.

-=-I asked her how many hundreds were in one thousand and she asked, "ten?" and she guessed the same for how many thousands are in ten thousand.-=-

The story in writing doesn't give the important information: Was she having fun?

Sandra

[Non-text portions of this message have been removed]

Pretty teachery.

When she didn't know the answer, if she was interested and curious, telling her ten might be fun, but if she didn't know and seemed impatient with the interruption or if she was wincing away from the question, it might've been a good time to just say "They're all ten of the one before it," and smile and dance away happily.

-=-I asked her how many hundreds were in one thousand and she asked, "ten?" and she guessed the same for how many thousands are in ten thousand.-=-

The story in writing doesn't give the important information: Was she having fun?

Sandra

[Non-text portions of this message have been removed]

#### Sandra Dodd

-=-I do want to reassure anyone whose child *isn't* doing number gymnastics at 5 and 6 that it's perfectly normal! Some people have a facility for numbers. Some for words. Some for understanding the inner workings of people. Some for understanding the interactions between people. Your kids may be whizzes at stories, what colors go together, turning 3D objects around in their heads, seeing patterns in words or pictures, noticing changes in nature, remembering tunes.-=-

Yes, and some take longer to get to those points than others. Being "bright" early in one pursuit or another doesn't guarantee "being ahead" for life. Someone might learn to read early and not become "a great reader" who carries a book around all the time. Someone might be fantastic at sports as a little boy and not ever pursue team sports as a teen or adult. Someone who grows to adulthood in a city apartment taking public transportation might become a hiking guide or mountain climber.

Don't trust trajectories. :-) See them as explorations or fascinations, and let them go.

Sandra

[Non-text portions of this message have been removed]

Yes, and some take longer to get to those points than others. Being "bright" early in one pursuit or another doesn't guarantee "being ahead" for life. Someone might learn to read early and not become "a great reader" who carries a book around all the time. Someone might be fantastic at sports as a little boy and not ever pursue team sports as a teen or adult. Someone who grows to adulthood in a city apartment taking public transportation might become a hiking guide or mountain climber.

Don't trust trajectories. :-) See them as explorations or fascinations, and let them go.

Sandra

[Non-text portions of this message have been removed]

#### catfish_friend

> --- The story in writing doesn't give the important information: Was she having fun? ---She was excited to answer "ten" to my last question of how many thousands were in ten thousand. I read her expression as a small light bulb going off, that she recognized there was a pattern. We were examining how many coins she had -- and we went down the numbers -- 50 thousand, 4 hundred and 92. After that, she seemed ready to play more and it was then that I wondered if I had been too teachery.

My dad is a retired statistician and when I was a kid, going to Kmart meant I could get a math workbook (which I truly enjoyed). I didn't have or even want many toys. I have a math/logic-oriented brain and I always found math to be relaxing. I had noticed that my 5.5 year old seems to enjoy counting play money or even just count but I knew that she didn't know place values yet. I can see now that I'm jumping the gun a bit. She knew that 10000 was less than 50000. She's absorbing and learning and I didn't teach her that. It was probably a matter of time before she would intuitively comprehend place values. Or, if she were confused, she's the type to ask for exactly what she needs helps with.

Thanks for all the responses. I'm going to keep deschooling and think more about strewing rather than teaching. Playing bank or shop with real money and showing her "Powers of Ten" by Charles and Ray Eames...

Ceci

#### Pam Sorooshian

If I understood what you did with her, that wasn't "place values." Place

value means understanding that a numeral's position in a number determines

its value. So in the number 444, the numeral 4 takes on different values

depending on where it is positioned - the one on the right is 4 units, the

one in the middle means 4 tens or a value of 40, and the one on the left

means 4 one hundreds or 400.

I want to warn you about thinking she really fully comprehends things (like

place value, or powers of 10, or multiplication, or anything else mathy)

because she can repeat them or give right answers. There is a danger of

introducing the "formal" versions of these things too soon. Much better to

let them experience powers of ten in real life a whole bunch of times over

a period of years before you give it any kind of formal existence in her

mind. Same with adding, multiplying, fractions, decimals, percentages,

division, exponents, and on and on.

My sister adopted a child who had been severely abused and neglected and

deprived of ordinary childhood experiences. She was 6 years old and, in

school, was learning to read. She was able to sound out words, but they

often had no meaning for her. She could read out loud: "I saw a zebra." But

she did not know what a zebra was. Similarly kids can perform a LOT of math

without deeply understanding what they are doing. How do you divide one

fraction by another? I bet you think you know. (Flip the second fraction

and multiply them.) But why does that work and why would you ever do it?

You probably never even wondered about that, I mean, why would you? You

thought that dividing fractions was something you "knew."

It sounds like your daughter is a lot like my youngest was - she did really

enjoy manipulating numbers - playing around with how numbers worked.

Instead of responding to that by giving her little mini-lessons and

quizzes, introduce games that might interest her. A game related to powers

of ten that my daughter enjoyed a lot (which might sound a bit boring to

read about, but she really really liked it) was this:

Put a dollar bill out on the table. Put a pile of pennies and dimes out.

You need one die. Take turns rolling the die.

...If she rolls a 5, then she can take 5 pennies from the pile.

...Then it is your turn. If you roll a 2, you take 2 pennies.

...Her turn - if she rolls a 6, she can take 6 pennies. BUT now she has

more than 10 and must exchange 10 pennies for a dime. So now she has 1 dime

and 1 penny in front of her.

...Your turn. Let's say you roll a 1. Then you take a penny. You don't have

10 yet, so no exchanging for you.

Keep going - if someone gets 10 dimes they get to trade that for the dollar

and win the game.

You can let the winner keep the dollar, but if you play it a lot, remember

she'll win about half the time. So we'd just put the same dollar back and

play for it over and over.

If she really likes it and has the patience, you could put a bunch of 1

dollar bills out and make it so that a person has to get 10 one-dollar

bills to exchange for the 10 dollar bill. That would be a long game,

though, so don't start with that.

While you have a bunch of coins sitting out on the table, you might play

some NIM games, too. These are strategy games - and can be very simple or

complex. Here is a simple example:

...put 20 coins out all in a row.

...alternate turns

....person 1 can pick up either 1 or 2 coins.

...person 2 can pick up either 1 or 2 coins.

Keep doing that.

Winner is the person who picks up the last coin.

(Or for a slightly better game, the goal can be to force the other player

to pick up the last coin.)

You can make these NIM games much more interesting - but that's the basic

game.

Here is another good one:

Lay out 12 coins in 3 rows. Row 1 has 3 coins, row 2 has 4 coins, row 3 has

5 coins.

Alternate turns - on a turn you can take any number of coins but only from

ONE row on a turn.

Player who takes the last coin wins. (Or switch that and make the goal be

to force the other player to take the last coin.)

-pam

On Sun, Feb 19, 2012 at 10:13 AM, catfish_friend

<catfish_friend@...>wrote:

value means understanding that a numeral's position in a number determines

its value. So in the number 444, the numeral 4 takes on different values

depending on where it is positioned - the one on the right is 4 units, the

one in the middle means 4 tens or a value of 40, and the one on the left

means 4 one hundreds or 400.

I want to warn you about thinking she really fully comprehends things (like

place value, or powers of 10, or multiplication, or anything else mathy)

because she can repeat them or give right answers. There is a danger of

introducing the "formal" versions of these things too soon. Much better to

let them experience powers of ten in real life a whole bunch of times over

a period of years before you give it any kind of formal existence in her

mind. Same with adding, multiplying, fractions, decimals, percentages,

division, exponents, and on and on.

My sister adopted a child who had been severely abused and neglected and

deprived of ordinary childhood experiences. She was 6 years old and, in

school, was learning to read. She was able to sound out words, but they

often had no meaning for her. She could read out loud: "I saw a zebra." But

she did not know what a zebra was. Similarly kids can perform a LOT of math

without deeply understanding what they are doing. How do you divide one

fraction by another? I bet you think you know. (Flip the second fraction

and multiply them.) But why does that work and why would you ever do it?

You probably never even wondered about that, I mean, why would you? You

thought that dividing fractions was something you "knew."

It sounds like your daughter is a lot like my youngest was - she did really

enjoy manipulating numbers - playing around with how numbers worked.

Instead of responding to that by giving her little mini-lessons and

quizzes, introduce games that might interest her. A game related to powers

of ten that my daughter enjoyed a lot (which might sound a bit boring to

read about, but she really really liked it) was this:

Put a dollar bill out on the table. Put a pile of pennies and dimes out.

You need one die. Take turns rolling the die.

...If she rolls a 5, then she can take 5 pennies from the pile.

...Then it is your turn. If you roll a 2, you take 2 pennies.

...Her turn - if she rolls a 6, she can take 6 pennies. BUT now she has

more than 10 and must exchange 10 pennies for a dime. So now she has 1 dime

and 1 penny in front of her.

...Your turn. Let's say you roll a 1. Then you take a penny. You don't have

10 yet, so no exchanging for you.

Keep going - if someone gets 10 dimes they get to trade that for the dollar

and win the game.

You can let the winner keep the dollar, but if you play it a lot, remember

she'll win about half the time. So we'd just put the same dollar back and

play for it over and over.

If she really likes it and has the patience, you could put a bunch of 1

dollar bills out and make it so that a person has to get 10 one-dollar

bills to exchange for the 10 dollar bill. That would be a long game,

though, so don't start with that.

While you have a bunch of coins sitting out on the table, you might play

some NIM games, too. These are strategy games - and can be very simple or

complex. Here is a simple example:

...put 20 coins out all in a row.

...alternate turns

....person 1 can pick up either 1 or 2 coins.

...person 2 can pick up either 1 or 2 coins.

Keep doing that.

Winner is the person who picks up the last coin.

(Or for a slightly better game, the goal can be to force the other player

to pick up the last coin.)

You can make these NIM games much more interesting - but that's the basic

game.

Here is another good one:

Lay out 12 coins in 3 rows. Row 1 has 3 coins, row 2 has 4 coins, row 3 has

5 coins.

Alternate turns - on a turn you can take any number of coins but only from

ONE row on a turn.

Player who takes the last coin wins. (Or switch that and make the goal be

to force the other player to take the last coin.)

-pam

On Sun, Feb 19, 2012 at 10:13 AM, catfish_friend

<catfish_friend@...>wrote:

> I had noticed that my 5.5 year old seems to enjoy counting play money or[Non-text portions of this message have been removed]

> even just count but I knew that she didn't know place values yet.

#### ceci

--- Instead of responding to that by giving her little mini-lessons andquizzes, introduce games that might interest her. ---

Pam -- The games you suggested are EXACTLY the kind of thing that my 5.5 year old would LOVE!!!

I'm going to play them with her as soon as I can! Thank you so much for the suggestions!

What's NIM by the way?

Ceci

[Non-text portions of this message have been removed]

Pam -- The games you suggested are EXACTLY the kind of thing that my 5.5 year old would LOVE!!!

I'm going to play them with her as soon as I can! Thank you so much for the suggestions!

What's NIM by the way?

Ceci

[Non-text portions of this message have been removed]

#### Robin Bentley

#### Pam Sorooshian

On Sun, Feb 19, 2012 at 2:31 PM, ceci <catfish_friend@...> wrote:

Another way to play without any objects, like when you're in the car or

waiting in line somewhere, is to take turns counting up. First person says

either "1" or "1, 2" and then the second person can count up either one or

two more, and so on. The goal is to force the other person to say "20."

So it might look like this:

A: 1, 2

B: 3

A: 4

B: 5, 6

A 7, 8

B 9, 10

A 11, 12

B 13, 14

A 15, 16

B 17

A: 18, 19

B 20>>>>so "A" wins.

In all NIM games there is a strategy that will allow the person who goes

first to win every time IF he knows the strategy.

You could allow taking of 1, 2 or 3 objects, too, to make any of the games

more interesting.

You can change the "win" so you are trying to be the one to say "20." AND

with this one, you could count down starting from 20 and try to make the

other person say "zero."

-pam

[Non-text portions of this message have been removed]

> What's NIM by the way?I think it is a German word - Nimm? Meaning "take."

Another way to play without any objects, like when you're in the car or

waiting in line somewhere, is to take turns counting up. First person says

either "1" or "1, 2" and then the second person can count up either one or

two more, and so on. The goal is to force the other person to say "20."

So it might look like this:

A: 1, 2

B: 3

A: 4

B: 5, 6

A 7, 8

B 9, 10

A 11, 12

B 13, 14

A 15, 16

B 17

A: 18, 19

B 20>>>>so "A" wins.

In all NIM games there is a strategy that will allow the person who goes

first to win every time IF he knows the strategy.

You could allow taking of 1, 2 or 3 objects, too, to make any of the games

more interesting.

You can change the "win" so you are trying to be the one to say "20." AND

with this one, you could count down starting from 20 and try to make the

other person say "zero."

-pam

[Non-text portions of this message have been removed]