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So I'm involved a fair amount with schools in the UK, and recently encountered a scheme for getting nine and ten and eleven year olds interested in mathematics (sadly what they do in class tends to be severely constrained by teacher knowledge and the demands of standardised testing).

I have various ideas to contribute - I'm looking for things which are not too abstract, and might lend themselves to activity and exploration. My tentatives include:

  1. Is there a number whose square is $2$ - because we can construct this geometrically, but our notation for numbers is limited (and introducing various schemes for closer approximation, and Pell's equation in simple form, and recurrence relations)

  2. Make a hexahexaflexagon for fun

  3. Use a slide rule as a mystery object (and introduce logarithms - how to multiply by adding)

  4. (Inspired by an IMO problem!) Explore large numbers and fast growth using a simple model - a series of trays with coins in - a move consists in either removing a coin from one tray and putting two in the next, or removing a coin from one tray and swapping the contents of the next two trays. [If you have $n$ coins in tray $4$ what is the most you can accumulate in tray $3$? - you'll be surprised, if you don't know]

Other ideas would be most welcome. What I am looking for is not stuff which will lead to straightforward answers, but stuff which catches the imagination with interesting and accessible questions (it is very easy to land youngsters in dead ends).

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    $\begingroup$ This may or may not be a reasonable idea but an activity could be to give them a list of the first 50 or 60 (more or less would probably be fine) positive integers and have them figure out which are prime. One way to do so is to cross off evens, then multiples of 3, then multiples of 5 and so on. It motivates a basic sieve for determining prime numbers. Another idea with number theory could be to motivate triangular numbers, squares, pentagonal numbers and more with pictures (using lots of dots!) and ask the kids to find a pattern with each. $\endgroup$
    – user328442
    Jan 13, 2018 at 17:41
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    $\begingroup$ @user328442 Put that as an answer so I can give it a proper vote ... $\endgroup$ Jan 13, 2018 at 17:56

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If you are happy to have the children work with paper and scissors a nice activity is on geometric dissections which involves the dissection of a number of geometric shapes before putting them back together into a shape that is different from what you started out with.

Perhaps the most famous example of this is Dudeney's dissection. Here one has four pieces that can be arranged into a square and an equilateral triangle. Many such dissections are possible. For example a hexagon can be dissected into a triangle, a square, or a pentagon. A list showing many of these dissections can be found here.

For younger children it probably works best if you give them one shape already, say a square in the case of Dudeney's dissection, with lines already marked on it and ask the pupils to cut the square along the already drawn lines, and from the resulting pieces, try to reassemble them back together so as to form an equilateral triangle. For older pupils maybe one can just give the pieces to them and ask them to try and form two different geometric shapes (perhaps what the actual shapes are can initially be given as they build confidence in performing such tasks).

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This may or may not be a reasonable idea but an activity could be to give them a list of the first 50 or 60 (more or less would probably be fine) positive integers and have them figure out which are prime. One way to do so is to cross off the evens, then multiples of 3, then multiples of 5 and so on. It motivates a basic sieve for determining primes.

Another idea could be to motivate triangular numbers, squares, pentagonal numbers and more with pictures (using lots of dots!) and ask the kids to find patterns with each.

Another one that I’ve heard is the “eggs in a basket” problem. Suppose that we have a basket full of eggs and if we grab eggs out in pairs then there will be one egg leftover. If we grab eggs in threes then there will be 2 leftover. If we grab eggs out in fours then there will be 3 leftover. If we grab them out in fives then there will be 4 leftover and so on. Continue this to something like 7 eggs with 6 leftover and see if anyone can find the smallest number of eggs in the basket that will work.

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•Dimensions: a flatland like approach could be very atractive for children.

•Fractals and space-filling curves, probably not giving so much importance to the theorical details but rather showing them some good animation (I think youtube have some) and explaining them the most impressive facts

•Also, I think infinity is always an atractive topic, maybe Hilbert's hotel could be a good starting point.

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