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Take some numbered cards; between ten and twenty should be enough
(you could use a suit from a pack of playing cards).
Shuffle the cards.
Then organise your deck of cards into numerical order.
What method did you use to put them in numerical order?
Can you think of any other ways you could have sorted them?
Here are some different sorting algorithms you could try. You may find it easiest to lay the cards out in a line to keep track of their order and see what's happening at a glance.
Try each algorithm a few times, and keep a record of how many 'moves' or 'swaps' you do. You could work with a friend and 'race' against each other to see who sorts their pack the quickest.
If you are struggling to make sense of the written algorithms, here are some videos showing each algorithm being performed.
Here are some questions to consider:
A sequence of numbers x1, x2, x3, ... starts with x1 = 2, and, if you know any term xn, you can find the next term xn+1 using the formula: xn+1 = (xn + 3/xn)/2 . Calculate the first six terms of this sequence. What do you notice? Calculate a few more terms and find the squares of the terms. Can you prove that the special property you notice about this sequence will apply to all the later terms of the sequence? Write down a formula to give an approximation to the cube root of a number and test it for the cube root of 3 and the cube root of 8. How many terms of the sequence do you have to take before you get the cube root of 8 correct to as many decimal places as your calculator will give? What happens when you try this method for fourth roots or fifth roots etc.?
Keep constructing triangles in the incircle of the previous triangle. What happens?
Vedic Sutra is one of many ancient Indian sutras which involves a cross subtraction method. Can you give a good explanation of WHY it works?