I'm supposed to give a 30 minutes math lecture tomorrow at my 3-grade daughter's class. Can you give me some ideas of mathemathical puzzles, riddles, facts etc. that would interest kids at this age?

I'll go first - Gauss' formula for the sum of an arithmetic sequence.

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This and this previous mse questions and their answers might be relevant. –  jkn Jun 23 '13 at 14:39
I'm not from the US - I was unsure what 3rd grade translated to age wise! –  jkn Jun 23 '13 at 14:53
You may be able to give some variant of the pirate game: en.wikipedia.org/wiki/Pirate_game –  GovEcon Jun 24 '13 at 5:31
How did it go? :-) –  Ant Jun 24 '13 at 9:28
Overall, it was a big success (it's my first time in front of a class), we talked about couple of subjects: * Naughty 37 (37*3k = kkk) * Concept of sequences and the sum of an arithmetic sequence * Geometry: say you have two square cakes, how do you halve them with a straight line (no Zig-Zag's) * Example of an unsolvable question: Collatz conjecture Guys - thank you very much for all your help!!! –  Amihai Zivan Jun 24 '13 at 15:01

$3\times8$ means $8+8+8$.

$8\times3$ means $3+3+3+3+3+3+3+3$.

Why should those both be the same number?

Why should $a\times b$ always be the same as $b\times a$?

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Do you have a good answer for this that is at a third grade level? Perhaps appealing to geometry? –  Alex Becker Jun 23 '13 at 19:58
Easy answer to that is that 8 3s is really 8 1s... three times. –  Bill James Jun 23 '13 at 20:50
@BillJames 8 3s is really 8 1s... three times, while 3 8s is really 3 1s... eight times. I fail to see how this simplifies matters. –  Alex Becker Jun 24 '13 at 2:21
@AlexBecker I was describing why it should work, or why it makes sense that it works, that adding a set of 3s together should come up with the same total as adding a set of 8s together... "8+8+8" has a "hidden" 3, as "3+3+3+3+3+3+3+3" has a "hidden" 8. –  Bill James Jun 24 '13 at 3:01
The answer that appeals to geometry is this: $\begin{array}{cccccccc} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \end{array}$ –  Michael Hardy Jun 24 '13 at 14:42

The Seven Bridges of Königsberg is a nice one. From memory, third-grade is around the time when kids like to try those problems of "can you draw a house without raising your pen and only drawing each line once", etc. which is essentially what this problem is. The solution, I believe, is simple enough for them to understand.

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1. Discuss the birthday paradox if there are at least 23 people in the room. In fact, ask the teacher in advance if he/she knows from student records if two students share the same birthday (an illustration with the students in the room won't go over well if you try it and nobody shares a birthday).

2. Look at the sum of odd numbers: 1, 1+3, 1+3+5, 1+3+5+7,... until someone notices a pattern. This is basically your idea of the sum of an arithmetic progression, but made concrete in an easily grasped way.

3. The 3x+1 problem. This will show them an example of an elementary unsolved problem that they can experiment with.

4. Examples of patterns that break down: if a sequence starts off as 1, 2, 4, 8, 16, what is the next term? I would hope some of the students will recognize these as powers of 2. The point is to show them several examples of counting problems that all start off this way but the next term is not 32. See examples 1, 5, and 6 in my answer at http://mathoverflow.net/questions/52101/longest-coinciding-pair-of-integer-sequences-known.

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I think these are all a little too advanced for 3rd graders. –  Alex Becker Jun 23 '13 at 19:59
If they don't know how to multiply, then yes the 3x+1 problem is too advanced. Do 3rd graders not know any multiplication? I appreciated that the first and last may be inappropriate. Why is adding successive odd numbers too advanced? The sums come out as 1, 4, 9, 16,... and if someone knows the perfect squares then this is a surprising connection between odd numbers and perfect squares. –  KCd Jun 24 '13 at 1:12
I have learned that I usually vastly overestimate how much kids know. For all of these I'd say the prerequisites are 1) basic arithmetic and 2) mathematical sophistication. 3rd graders have very little of the latter. For example, the notion of the sum of arbitrarily many numbers (I predict) will go over all of their heads. –  Alex Becker Jun 24 '13 at 2:19

Let them try to create maps that need as few colours as possible. The rule is that two counties are neighbours if their borders meet in more than a finite number of points and neighbours should not have the same colour. Hopefully, this might lead to an interesting discussion about the 4-colour theorem.

Here are example "maps".

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"As many" or "as few"? –  apnorton Jun 23 '13 at 16:26
@anorton Good point :D –  AD. Jun 23 '13 at 17:12

Teach 'em how to play sprouts.

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Show them a pizza. Ask how many pieces they can make by cutting it once.

The answer should be 2. Now ask them for 2 cutting. The answer should be 4. Then ask them to try for 3 cuttings. I hope some of them will the correct answer, which is 7. Then explain them what is happening in each cutting, how the number of pieces are increasing and why.

Then help them to formulate the problem for n cutting. The formula is

$C(n) = \frac{n(n+1)}{2} + 1$

If they understand everything so far, you can then extend the problem from pizza to cake :-)

Hope they will have fun! :-)

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Who says you cannot cut a pizza horizontally? ;-) Oh, and you didn't say the cut has to be straight, therefore I can make as many pieces as I want with just one cut! –  celtschk Jun 24 '13 at 16:02
@celtschk: :-) I did not even tell that the pieces can not be moved from their place ;-) –  Naffi Jun 24 '13 at 18:49
I guess I'm not smarter then a 3rd grader. How did you get 7 pieces from 3 (straight?) cuts? –  Adrian Jun 24 '13 at 20:21
@Adrian: Who said the cuts have to meet in the center? –  Tim Pietzcker Jun 24 '13 at 20:47
@Adrian: try to maximize number of intersection with each new cutting. –  Naffi Jun 24 '13 at 20:55

I am bit undecided about the following riddles, whether they are too difficult for 9-year-olds. Here comes anyway.

1. A boy was selling eggs (replace eggs with whatever works locally) to people in a building with four floors. The person living in the fourth floor bought half the eggs plus half an egg. The person living in the third floor bought half of the remaining eggs plus half an egg. The same with the persons living the second floor and in the ground floor. After his tour the boy had sold all his eggs. How many did he have in the beginning? (ok, the fraction may be a dealbreaker here - anyway, the idea is to work it out backwards)
2. The combined ages (in full years) of a girl and her mother are 43, those of the girl and her grandmother 69, and the ages of the mother and the grandmother add up to 94. How old are they? (Don't use a system of equations, just add them all up, divide by two, and... The method by trial and error is not to be sneered at either.)
3. Have them draw a 4x4 checkerboard, and cut out two opposite corners. Can they cover the remaining 14 squares with 7 domino pieces? Each domino piece is the size of exactly two squares. How to do it, or why cannot we do it all? (a classic - possibly tough to figure out, but easy to grasp).

Hmm. 11-13 years might be closer to the mark with the above. Rewinding my tape further...

1. They are probably working their way throught multiplication tables. Ask them to calculate 3x3 and 2x4, then 5x5 and 4x6, then 8x8 and 7x9? Is somebody seeing a pattern? Prove it using a square! If somebody has a sparkle in his/her eye, also try 7x7 and 9x5.
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+1 for #3. ${}{}$ –  apnorton Jun 23 '13 at 16:29
#1: why don't rephrase it as "half the eggs plus one egg", no fractions then but same idea. –  Vadim Jun 23 '13 at 17:47

A couple of my previous answers to similar questions:

Find a rubber chicken(!) and introduce the kids to $\pi$: http://math.stackexchange.com/a/395933/409 (It won't take a half-hour, but it's a good way to warm-up the crowd. :)

This exercise is described with a focus on conceptualizing logarithms, but the exponential aspect alone could be engaging: http://math.stackexchange.com/a/129246/409

Also, you could take a sheet of paper and cut a hole big enough to walk through. While this is usually described as a simple trick (or bar bet), I like to look at it as a demonstration that (very loosely speaking) even a finite area "contains" infinite length ... a concept that becomes clearer as one investigates space-filling curves.

You might also confront the kids with Zeno's tortoise paradox. Kids love animal stories! :) Instead of just telling the story, though, you could have a couple of kids re-enact the parts. Relatedly, you could have kids do the "walk half-way to the door, then half-of-halfway, then half-of-half-of-halfway" thing as a first brush with limits. (You'll never get beyond the door that way, but at some step you'll pass any particular point before it; therefore, the journey must "converge" at the door.)

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I think most posters overestimate what you can learn a bunch of nine year olds in 30 minutes. Remember you are not 1-on-1 with the brightest student, but you need to have the full class being able to follow you. They maybe have had division and such in class but I imagine it isn't their second nature yet, so you can't depend on it for such a short lecture.

Personally I would try to learn them to count in base-9 and conversions to and from base 10.

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If the book $Watership Down$ were more popular with kids these days (and if it were appropriate for 3rd graders), I'd suggest a lesson based on the book's revelation that rabbits can't count past four (presumably because that's the number of paws). Of course, with place values, rabbits can actually count as high as they like: One, two, three ... four = one full rabbit (or bunny) = "one bun"; then, five = one bun and one = "bunty-one", six = bunty-two, seven = bunty-three, and eight = two buns = twunty, ..., sixteen = "one bundred", and so forth. –  Blue Jun 24 '13 at 12:50
(Part 2) I've actually done the rabbit counting lesson with 9th graders; it went okay. We spent a couple of days on it, learning to add, subtract, and multiply. After learning some rabbit arithmetic (base four), we went on to consider octopi counting (base eight), and alien counting (base twelve; aliens have six fingers per hand), though not in a whole lot of detail. The kids got the impression that bigger bases were for smarter beings, so I asked what they thought the "computer base" must be, since computers are so smart. "Millions?" "Nope: Two." And we talked about binary for a while. –  Blue Jun 24 '13 at 13:04

Probability, throwing dice and gambling. How likley is to get #6 throwing dice and what implications this have.

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Maybe without the gambling for 3rd graders. :) We want math to be a "good influence." –  apnorton Jun 23 '13 at 16:27
Maybe is this moment to explain why mathematicians dont go to casino. –  ralu Jun 23 '13 at 17:17
Reminds me "The Wire " season 4 when Pryzbylewski drew his class attention by teaching them probability... –  Amihai Zivan Jun 23 '13 at 17:48

Imagine a vertical sea cliff. Floating in the sea some distance away from the cliff there is a boat, which is attached to a rope which goes diagonally up to the top edge of the cliff and then continues on the field at the top to a tractor. (Draw a picture.) The tractor then moves away from the cliff edge, pulling the rope, and so pulling the boat towards the cliff.

For the whole class to discuss:

• Which moves faster, the tractor or the boat? Or do they move at the same speed?

For the clever ones

• If the tractor moves at a constant speed, does the boat accelerate towards the cliff or decelerate? Or does the boat travel at a constant speed?
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+1 good example, but can only "prove" to third-graders by physical example. –  thethuthinnang Jun 24 '13 at 1:50

Eeny-meeny-miney-mo is essentially counting up to $16$. I'm not sure if they'd be comfortable enough with multiplication and remainders to totally grasp it, but I think kids would like to be able to "cheat the system" and intentionally predict who to land on (by, say, realizing that 16 is one more than a multiple of 5).

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I don't know if it's too advanced but...

I'm 21 years old and I just recently learned about 'using your hands to find the multiples of 9'.

For those who don't know what I'm talking about, hold your two hands out in front of you.

Say we want to find 9 x 8. Starting from your left hand pinky, count off 8 fingers and put your 8th finger down. You now have seven fingers on your left hand side and two on your right. Put them together. Voila! 72.

Try again with 9 x 4. You now have three on your left hand side and six on the right!

While writing this, I just realized that this is taking one minus X and combining it with 10 - X.

9 * X = (X - 1) and (10 - X). It's fascinating me!

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How about giving various proofs that $1+\cdots+n = n(n+1)/2$? There are at least a few different geometric, easily understandable ones (although I'm not sure about how easy, when it comes to 3rd-graders. You'll be the judge of that.)

I saw the one below recently, and it blew my mind. Using this, you will need to introduce the binomialcoefficient as well. But I guess they will be able to understand the idea of choosing two balls from $n$ balls, in different ways.

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It took me 20 minutes to figure out this beautiful picture, I think it's a bit advanced for them. –  Amihai Zivan Jun 24 '13 at 15:03
Yes, it is not exactly easy to see by youself - especially not when you're a 3rd-grader. But it is very easy to see that each of the yellow circles corresponds to precisely one way of choosing two purple circles! Telling them, without giving any insight in why it is so, that $\binom{n}{2}$ describes the number of ways to choose two purple circles, would (should (could)) complete their understanding of this proof. Showing them, then, another simple geometric proof of this identity (does it have a name?) would really show the beauty of math; that the same thing can be said in many different ways –  torbonde Jun 24 '13 at 19:12

Describe multiplication on a 12-hour clock, pointing out that 3*4=12 makes it unlike regular multiplication. Ask them which clocks don't have two numbers that can be multiplied to get the number of hours on the clock.

Taking (topological) dual polyhedra by letting new points be face centers and connecting centers of faces that share an edge.

What happens when you place a black square on the top-left corner of a grid, then make any square on the next line black (repeating this line by line) when there's a black square either above or above and to the left, but not both.

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When I was in second grade, we did an experiment where we tossed a pin n times and counted the number of times the pin fell across some line, given that it fell between two other lines. Years later I learned to compute this - but the idea was there. My vote is for doing something related to probability and statistics.

Another idea is to do something related to computer science. Find some simple algorithm and show how the time (number of steps) to complete changes as the the data set grows. One variation of this is to have the students themselves sort cards (make it a contest), time them for different size unorganized stacks and record their results. Then ask them about their process.

Whatever you do, don't go above their heads by doing something to mathy. These examples assume they are smart, but don't assume they know anything.

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1. Make a bagatelle and drop balls down it. Use this to demonstrate the central limit theorem, suitably simplified to "Large amounts of random anything look like a bell curve." Then, measure your students' heights and compare.
2. Try logic puzzles and games.
1. Nim
2. The Monty Hall Problem
3. Dice probabilities. Don't be afraid of teachers who don't want gambling; the students play Monopoly and Settlers, don't they?
3. Show R(3, 3) = 6. They go to parties too.
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Nothing like a little Ramsey theory in the third grade. –  Alexander Gruber Jun 27 '13 at 14:43
Yes, but that example is understandable by 3rd graders. Draw a pentagon to show that $R(3, 3) > 5$. Use red and blue-colored chalk to give examples for six people. You don't have to give the full theory, but that small example works. –  Eric Jablow Jun 27 '13 at 15:45

may be a nice multiplication speed improvement class will be helpful. tell the students to use multiplication in day to day life using the distributivity property of multiplication

• multiply 19x15 : it can be changed to 15x(20-1) in mind and then 300-15 =285 such tricks will help speed up their mathematics calculation speed and will be fun to teach with many examples!!

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I think Map Colouring and Probability are a great ideas, as mentioned above.

I would also suggest finding the highest prime, it requires multiplication/division, which is always good to practice.

Otherwise you could consider having them enact Bubble Sort[1], based on their height or birthday.

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Knot theory can be fun for third graders; the trefoil knit can be formed with one person and a stick, and some knits can be formed with 2 people, so you could show a picture of a knot, and have them try and make it.

Or better yet, a lot of people play a game where you have a lot of people randomly grab hands and try to make a circle by unwinding. You could play this several times, and have people suspect that you always get the unknot. Then you can have them form a trefoil, and when they can't undo it, discuss why they can't.

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