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December 1, 2009 7:14 AM   Subscribe

What are some cool math and number facts that would blow the mind of a seven year old?

Yesterday at the dinner table my son says "Hey dad, did you know 111,111,111 x 111,111,111 = 12,345,678,987,654,321?"*

He thought this was awesome. I've previously told him the trick about multiplying by 9 (2x9 = 18, 1+8=9, 3x9=27, 2+7=9 and so on up the times table) and a couple of other fun facts.

I'd like to present him with some other neat numerical properties and math facts. He'd need to be able to grasp the concept, so something like the non-palindromic properties of 196 might be a bit beyond him. He's in (the equivalent of) 2nd grade at a Montessori school and math is his favorite subject. He's currently doing 2-digit division and he's pretty much mastered multiplication. I'm not afraid to go a bit beyond what he's learning but I don't want to go too far beyond it.

What can you tell him about other interesting numbers and how to get to those numbers?

Calculator tricks might be fun as well. I'm sure if I teach him the S8008 trick he'll be the King of the Playground. Also tricks that require props such as playing cards, Legos, or anything else are welcome, as long as they demonstrate the type of thing I'm looking for.

I'll take book suggestions as well. Christmas is coming...


*interestingly, the first Google hit for 12345678987654321 is a MeFi thread.
posted by bondcliff to Education (68 answers total) 76 users marked this as a favorite
 
.9999999999999999....=1
posted by milestogo at 7:18 AM on December 1, 2009


The Golden Mean is a very neat concept to explore because it just pops up everywhere in ways he can recognize and appreciate.
posted by Askr at 7:22 AM on December 1, 2009 [2 favorites]


I don't know one specifically or what they're called, but how about one of those "Take any number, add x, multiply by y, divide by z, subtract e, and you have your original number!" tircks? Pretty dumb, but I bet it'd go over well with him!
posted by InsanePenguin at 7:24 AM on December 1, 2009


Also, Futility Closet posts "math notes" which sound like what you want. For example.
posted by milestogo at 7:26 AM on December 1, 2009 [1 favorite]


He sounds like he's a little past this, but boy, did I love Square One when I was younger. Can you find it on DVD somewhere? It had such a great combination of stuff -- geometry, probability, real-life uses, etc.
posted by Madamina at 7:29 AM on December 1, 2009 [2 favorites]


the trick about multiplying by 9 (2x9 = 18, 1+8=9, 3x9=27, 2+7=9 and so on

I suppose you also showed him the multiplying by 9 on your hands trick? ...I'm 40 and I still love this one.
posted by applemeat at 7:30 AM on December 1, 2009 [17 favorites]


Infinity minus 1 is still infinity. Infinity minus 2 is still infinity. Infinity minus 100 is still infinity...
posted by kestrel251 at 7:30 AM on December 1, 2009


Geometric constructions are infinitely awesome.
posted by odinsdream at 7:30 AM on December 1, 2009


I'd get him Martin Gardner's Aha! books: Aha! Gotcha and Aha! Insight.
Guaranteed to take someone who is interested in math and make it their lifelong passion.
posted by vacapinta at 7:35 AM on December 1, 2009 [1 favorite]


Consider the Penguin Dictionary of Curious and Interesting Numbers. From -1 to Graham's Number via Fibonacci numbers, Hardy's Taxi, Mersenne primes and so on - some of it may be beyond him at the moment (especially in the higher numbers) but lots of it won't be.

As far as mathematical tricks are concerned: how about the cyclic decimal of 1/7? A little more advanced is an algorithm I learnt when I was maybe a bit older, that enables you to take any date and work out what day of the week it is. I'm long out of practice at it, but the process is detailed here.
posted by Electric Dragon at 7:37 AM on December 1, 2009 [1 favorite]


Vedic maths
posted by qxntpqbbbqxl at 7:38 AM on December 1, 2009


This might be fun.
posted by shazzam at 7:40 AM on December 1, 2009 [1 favorite]


Also: Sam Loyd's Cyclopedia of Puzzles
posted by vacapinta at 7:40 AM on December 1, 2009


Tricks used by lightning calculators are sometimes surprisingly easy to learn. Calculating the thirteenth root is perhaps not so playground-friendly, but things like the squares-around-fifty trick might be good (or anything based around a binomial expansion).

For the squares around 50 - take the difference from 50 (e.g. 48 differs by 2) and do
2500 + 100x[difference] + [difference^2]
e.g.
2500 - 200 + 4 = 2304 = 48x48.

A smart kid can take that further and do things like 48x47 = 48x48 - 50 = 2254, and if they can see why it works they're even better off, and can use it in all sorts of other places.

Googling for the tricks lightning calculators use might turn up other fun stuff, like how to work out what day a particular date is (which is probably easily checked by a decent mobile phone these days).
posted by edd at 7:41 AM on December 1, 2009


Found that one on the net somewhere, it's pretty neat:

The 6174 loop

1. Select a four-digit number. (Do not use 1111, 2222, etc.)
2. Arrange the digits in increasing order.
3. Arrange the digits in decreasing order.
4. Subtract the smaller number from the larger number.
5. Repeat steps 2, 3, and 4 with the result, and so on. What happens?

Let's try 7173

Arrange the digits in increasing order. 1377
Arrange the digits in decreasing order. 7731
Subtract the smaller number from the larger number. 7731 - 1377 = 6354
Repeat the process with 6354
6543 - 3456 = 3087
8730 - 0378 = 8352
8532 - 2358 = 6174
7641 - 1467 = 6174
7641 - 1467 = 6174
7641 - 1467 = 6174 (we're in a loop!)

Amazingly, all four-digit numbers (not multiples of 1111) end up in the 6174-loop. No reason has been found for this phenomenon.
posted by PontifexPrimus at 7:42 AM on December 1, 2009 [29 favorites]


Doh I'm an idiot, 48x47 is 48x48-48...
posted by edd at 7:43 AM on December 1, 2009


9, 18, 27, 36, 45, 54, 63, 72, 81, 90.

Multiples of nine. Notice for that 18 ( 9x2 ), when flipped is 81 ( 9*9 ). 27 ( 9*3 ) when flipped is 72 ( 9*8 ), Same for 36 and 45.
posted by jasondigitized at 7:46 AM on December 1, 2009 [2 favorites]


The division-testing tricks are fun, ie, if the digits of a number add to a multiple of 3, the original number is also a multiple a 3.

34242 = 3 + 4 + 2 + 4 + 2 = 15

34242 / 3 = 11414

I'm pretty sure there are more. There is also some good stuff in the rec.puzzles FAQ.
posted by jquinby at 7:51 AM on December 1, 2009


Gah! Also forgot the online-encyclopedia of integer sequences. Some of these are quite cool.
posted by jquinby at 7:53 AM on December 1, 2009


1/81 = 0.0123456789...
posted by dhoe at 7:53 AM on December 1, 2009


I suppose you also showed him the multiplying by 9 on your hands trick

I'm well past 2nd grade, and this blew my mind. Goddamnit, I love math.
posted by InsanePenguin at 7:54 AM on December 1, 2009


Here's another thought. You know your son's level, and this may be something he could have read to him now, or you may want to wait a year or two, but Norton Juster's The Phantom Tollbooth is a great book for kids who like numbers. He also wrote The Dot and the Line, which is more about geometry.
posted by gudrun at 7:56 AM on December 1, 2009


This guy is always endlessly entertaining: http://vimeo.com/6643331 (part 1 of 4)

And there's the rule that any 2-digit number multiplied by 11 = [digit 1][digit 1+digit 2][digit 2]

11 x 11 = 1 [1+1] 1 = 121
12 x 11 = 1 [1+2] 2 = 132 (apologies if this is egregiously nonstandard notation. IANAM.)

(if the middle digit is greater than 9, carry, e.g., 77 x 11 = 847)
posted by socratic at 8:02 AM on December 1, 2009


Casting out 9's is pretty cool.
posted by SLC Mom at 8:25 AM on December 1, 2009


If you can get copies of some old episodes of Square One, I'm sure he'd love them!

Especially all the various equations where no matter the number you pick, the answer is always the same.
posted by zizzle at 8:27 AM on December 1, 2009


I had a lot of fun playing with magic squares and Pascal's triangle when I was your son's age.

Also, to follow up on Madamina's suggestion, searching for Square One playlists on YouTube turns up a lot of great math videos for kids.
posted by homuncula at 8:31 AM on December 1, 2009


Divisibility rules.
posted by fings at 8:31 AM on December 1, 2009 [1 favorite]


This thread was about an older age group but might be useful.
posted by Jaltcoh at 8:32 AM on December 1, 2009


I loved Square One so much when I was that age. Also Math for Smarty Pants (which is the only reason I remember anything about factorials).
posted by Metroid Baby at 8:35 AM on December 1, 2009


From that thread: the Collatz conjecture, a.k.a. "half or triple plus one" (HOTPO).
posted by Jaltcoh at 8:39 AM on December 1, 2009


I was about seven or eight when I learned about pi. I thought it was the coolest thing ever, though possibly because I got in trouble at school for knowing it too early.

Also that the "squared" and "cubed" operations apply directly to squares and cubes.
posted by Jon_Evil at 8:44 AM on December 1, 2009


Yeah, he's going to love watching Square One. That's where I first learned about the Fibonacci sequence.
posted by Faint of Butt at 8:45 AM on December 1, 2009


My kids' favorite books about math, for reading aloud to them, at ages eight or nine:

The Number Devil: A Mathematical Adventure
The Man Who Counted
Math Trek
Math Trek II
The Adventures of Penrose the mathematical cat


Loads of fun, for them and for me.
posted by Ery at 8:47 AM on December 1, 2009 [1 favorite]


In middle school, we used to play this game called 'Mental Math' where the teacher would just run off a string of numbers, varying the arithmetic functions (i.e. 1 + 6 - 2 x 3 + 2 +1 divided by 3 plus 2, etc., etc.). This game obviously requires the 'leader' of the game to be able to do the math in their head as well--however, if you run into trouble when you're giving the problem, you just multiply by zero and start it over. I always liked these and my dad used to rattle them off in the car when we'd take long trips.
posted by arm426 at 8:49 AM on December 1, 2009 [2 favorites]


I remember loving "Donald Duck in Math Land" as a child. You can even find it on youtube. It talks about the Golden Mean, the geometry behind billiard balls and so on.
posted by of strange foe at 8:57 AM on December 1, 2009 [1 favorite]


Math for Smarty Pants is awesome. I knew about factorials and all kinds of other things long before I formally learned them. Can't recommend it highly enough.
posted by Tomorrowful at 9:01 AM on December 1, 2009


if you run into trouble when you're giving the problem, you just multiply by zero and start it over.

It is the ability to do things like this that make it great to be a parent.

Also, for the toddlers who want 2 cookies, just break one in half. They understand 2 full hands, they don't understand half-a-cookie.

God, I miss those days.
posted by SLC Mom at 9:01 AM on December 1, 2009 [1 favorite]


My favorite multiply by 9 trick.
You have 0-9:

0
1
2
3
4
5
6
7
8
9

and you have 9-0:

9
8
7
6
5
4
3
2
1
0

Combine the first set with the second set and you have

09
18
27
36
45
54
63
72
81
90

IE, the multiplication table for 9
posted by nikkorizz at 9:06 AM on December 1, 2009 [1 favorite]


When I was a kid my dad told me about Hilbert's hotel, and that just about blew my mind (I'm a mathematician now). Also things like how there are just as many even numbers as there are numbers in general.
posted by pewpew at 9:08 AM on December 1, 2009


One I've just remembered is "Russian peasant multiplication", which can then be used to introduce binary ("this is how computers do maths!").
posted by Electric Dragon at 9:10 AM on December 1, 2009


Also, any number is divisible by 3 if you add up all the digits and the total is divisible by 3.
For example, the number 8,234,583.
8+2+3+4+5+8+3 = 33
8,234,583 is divisible by 3 because 33 is divisible by 3.
posted by nikkorizz at 9:12 AM on December 1, 2009 [2 favorites]


Seconding Martin Gardner. My mom had a number of compendia of his SciAm columns from the 50s...Even when I didn't understand the math, it was a good read.

Following from that....hexaflexagons.
posted by notsnot at 9:13 AM on December 1, 2009


Two things I loved:

A calculator trick. Take any 3-digit number. Enter it in the calculator, then enter it again (123123). Divide by seven, hit equals. Divide by eleven, hit equals. Divide by thirteen? Your original number (123).

And this might be a little above him right now, but once you teach him about pi (and the golden rectangle would be interesting, too): Buffon's Needle. I was obsessed with Buffon's Needle for a couple of years. You can easily do the experiment with a toothpick, some construction paper, and a calculator.

(And Square One! Three things! Ah ah ah.)
posted by fiercecupcake at 9:14 AM on December 1, 2009


Nine is just a cool number.

I'm fond of the trick that for any 1-10 multiplied by 9, if you add the digits of the answer together, they equal 9. Also the first number will always be one less than the original non-nine number. This really helped me out in 2nd grade (or whenever it was that we learned multiplication).

So 2 x 9 = 18 (1+8=9)
3x9 = 27 (2+7=9)
etc
posted by ugf at 9:16 AM on December 1, 2009


This isn't really that high on the Wow! meter, but it's pretty useful and provides a neat introduction to bases, if he's ready for that. I read about it in a persuasive article by Frederick Pohl about why we should switch to base 2.

Did you know you can use your fingers to count to over a thousand? Simply use your fingers as digits --down is 0 and 1 is up-- and count using base two. For example, raising your right thumb is 10000, or 16. When you line up both hands, you have ten places so you can count up to 1111111111, or 1,023.
posted by ropeladder at 9:17 AM on December 1, 2009


Not obviously mathematics but I think the result of cutting a mobius strip down the middle lengthwise is extremely mind-blowing (and it's easy for kids to do, just give 'em some old newspaper, tape and scissors).
posted by Rash at 9:23 AM on December 1, 2009


Wow, that explanation of Buffon's needle is a little... complex. What I used in 5th grade was: Given a needle of length x and lines drawn on a piece of construction paper x width apart, when the needle is dropped onto the paper, the probability of the needle touching a line is roughly pi.

Easy to test with a simple chart and a calculator.
posted by fiercecupcake at 9:26 AM on December 1, 2009


The Films of Charles and Ray Eames, Vol. 1, has a great short film entitled, "Powers of Ten". Perfect for a bright seven year-old because it combines stunning visuals with a clever presentation. It's widely available for sale or rental.
posted by dinger at 10:09 AM on December 1, 2009


Amazingly, all four-digit numbers (not multiples of 1111) end up in the 6174-loop. No reason has been found for this phenomenon.

This is indeed awesome, but also (partly) explainable. See, e.g., this explanation of what's going on. For three digits, doing the same sequence of steps ("Kaprekar's operation") will always end up giving you 495; for more than four digits, you won't end up with one unique number -- you either get a cycle between two or more numbers, or an answer that just bounces around all over the place.
posted by chalkbored at 10:10 AM on December 1, 2009


This is a great question.

How about this: the sum of the first N odd numbers is a square (N2).

1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
...

neat, and can be demonstrated by drawing successively bigger "dot squares" on a paper.
posted by lex mercatoria at 10:21 AM on December 1, 2009


And btw: reading about Kaprekar's operation led me to the Collatz conjecture (which Jaltcoh linked to above), which I'd never heard of until today but which is astoundingly and completely awesome. So, thanks.
posted by chalkbored at 10:21 AM on December 1, 2009


This book is also fun, and explains things like magic squares (I think -- I haven't got the book in front of me) and the progression of numbers (integer, rational, real, imaginary).
posted by lex mercatoria at 10:29 AM on December 1, 2009


Write down 12345679 ( note the lack of 8 )

Ask him his favorite single digit number. If he chooses 3, for example, give him a calculator and ask him to multiply 12345679 and 9 times his favorite number.. in this case 27.

12345679 x 27 = 333,333,333
posted by Juggernut at 10:46 AM on December 1, 2009


Explain different number bases. Suppose we had no pinky fingers -- we'd count 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 20....

Then talk about if we had another three fingers. We'd need another symbol to represent one more than 9, and two and three more. Make some up. Use funny shapes or colors.

0 1 2 3 4 5 6 7 8 9 И ф Г 10 11 12 13 14 15 16 17 18 19 1И 1ф 1Г 20 21 22... 98 99 И0 И1 ...

Talk about you can't know how many the symbols "10" means without knowing what base the number system uses.

"1234" in base x means 1*(base^3), plus 2*(base^2), plus 3*(base^1), plus 4*(base^0) .

----

Then, on a new day, talk about discovering the divisibility of a number by 9: Add the digits together. If the result is more than one digit. Do that again, until you get a single symbol. If the symbol is "9", then the original number is evenly divisible by 9.

----

Then, talk about why the 9 trick works. Nine is one less than the base system. It would work with base 3, base 8, or base ф too. Explore each to convince yourselves.
posted by cmiller at 11:12 AM on December 1, 2009


I did not like math as a kid, and still don't, but I did like The I Hate Mathematics Book, which appears to be from the same series as the aforementioned Math for Smarty Pants.
posted by doift at 11:29 AM on December 1, 2009


Not a trick, but if he likes math, play 24.

It's a card game. Each card has 4 numbers from 1-12. You're supposed to add, subtract, divide, and multiply them to reach 24. For instance, if you have 1 2 7 9 you can do 1 + (2*7) + 9.

You don't even need the cards. Just make up some numbers and challenge each other to reach 24. I like trying to find sets that can only reach 24 if you add in exponentiation.
posted by valadil at 11:45 AM on December 1, 2009


Late to the party but this is a very cool book.

And pretty cheap.
posted by wittgenstein at 12:39 PM on December 1, 2009


2nding the recommendation for The Number Devil.
posted by emeiji at 1:07 PM on December 1, 2009


Here's some math I was obsessed with at around that age & a little older, thanks to a very mathy dad and some good books. I really liked things I could doodle in notebooks or turn into artwork.

- The fibonacci sequence/spiral
- Number series in general, especially if they got really big really fast
- Proofs of the Pythagorean theorem
- Fractals
- Tessellating patterns
- Geometric axioms
- Relationships between 2d and 3d shapes (the idea conic sections was confusing to me but I still thought it was very cool)
- Codes!
- The "four fours" problem - how many integers can you make using four 4s (4+4+4+4=16, 4*4 + 4 + 4 = 24, etc.) This is especially great because every time you learn about a new operation you get more!

Books:

- Stories/problems from the Smullyan books
- Phantom Tollbooth
- Flatland
- I really wish I could remember the name of the "women in math" book someone gave me around 3rd grade - it was history-based, but had a project for each person (including that thing where you estimate curves using tangent lines on a rubber band pegboard.)

Also nthing Square One!
posted by heyforfour at 2:46 PM on December 1, 2009 [1 favorite]


As a young child, my parents rented a and played "Donald in Mathemagic Land" for me. It was the first Disney movie I ever saw and it was completely awesome. That was at least 16 years ago now, probably more like 18 years ago... I still think it's on my top 10 list of Things I've Enjoyed Watching.
posted by Cygnet at 3:47 PM on December 1, 2009 [1 favorite]


I think it's really cool that you are looking for things to further your son's excitement. Personally, my love of math started with contests, so that's where I looked.

I can almost swear there was a contest for fifth-graders, but the most basic one I can find is this one for twelve-year-olds.

Maybe wait a couple years, but you might cherry pick the more fascinating problems and maybe.... maybe... with you sitting right beside him answering questions and being extra patient while he attacks a problem, he can feed his enthusiasm. Looking at this contest, I like questions: 2, 3, 11, 13, 15 (fun?), 17, 25 (most challenging, but most interesting too.)

These contests are meant to be done with about 2 minutes a question for twelve-year-olds. But, there's nothing stopping you from giving your seven-year-old son a day for each problem while you help him when he needs help and you're patient while he works through the problems.
posted by esprit de l'escalier at 5:25 PM on December 1, 2009


Maybe wait a few years though... I can't remember what seven years old is like.
posted by esprit de l'escalier at 5:27 PM on December 1, 2009


just to add on to my previous answer: almost everything on my list above was in the form of interesting pictures to draw, posters on my walls, blocks (especially pattern blocks!), and riddles/puzzles with my dad. I don't think I was familiar at the time with any 'actual math' beyond the typical elementary school curriculum for my age.
posted by heyforfour at 5:53 PM on December 1, 2009


Seconding the Phantom Tollbooth and the Number Devil.
posted by LobsterMitten at 11:34 PM on December 1, 2009


Maybe this math trick is up above but I didn't see it: I'm not what you call super math literate, but this one stuck with me over the years and I'm convinced actually won me a freelance job once:

Numbers that end in 5 can be multiplied by themselves very easily.
  • 25 x 25 = ?
  • Take the first number (2) and use the next number in sequence (3)
  • Multiply those two numbers together ( 2 x 3 = 6)
  • and then tack "25" onto the end
  • So 25 x 25 = 625
I think I learned this in 4th grade or so and every once in a while it strangely comes in useful. Once I was interviewing for a freelance position and in some discussion about a database the client mused out loud something like " I wonder how much 65 x 65 is" and looked around for a calculator, in a split second I blurted out "4225". It was a large group and when the answer was confirmed I had one of those (rare) moments of feeling like a genius. I'm sure there were other reasons, but I won the gig.
posted by jeremias at 12:59 PM on December 2, 2009 [2 favorites]


I can almost swear there was a contest for fifth-graders, but the most basic one I can find is this one for twelve-year-olds.

Continental Mathematics League has programs for 2nd grade and up. There are a few practice problems online and you can order practice books. Not sure if you can enter the competitions as an individual, but there is probably a homeschool loophole somewhere.
posted by Flannery Culp at 9:04 AM on December 4, 2009


Casting out 9's (already mentioned here and obliquely referenced in the question) has a parallel for 11.
posted by chairface at 5:08 PM on December 7, 2009


Wow, the nostalgia from these answers make me wish I was a kid again!

I'd like to second most of the things people have said (including pointing out that Donald in Mathmagic Land is available on YouTube, as is lots of Square One; they were both fundamental in my mathematical childhood.)

The "I Hate Mathematics Book" was also a personal favourite (IIRC), don't let the title put you off!

But in terms of tricks, the only one I like that hasn't already been mentioned is the trick for squaring (two-digit) numbers, as described here. (So, for instance, (25 * 25) = (20 * 30) + (5 * 5) = 625.)

And he may be too young just now, but I'd also recommend anything by Martin Gardner (even though it's not really arithmetic).
posted by insipidia at 1:40 AM on April 16, 2010


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