Some Starting Points...

NUMBER CRUNCHING

Multiplication Square

Make a 10 by 10 multiplication square. Investigate number patterns on it.

Subtraction Patterns

Write down four numbers in the first row of a grid. In the second row write down the differences between each number and the next – imagine the last number in the row “wraps round” to be next to the first. Each subsequent row is obtained by working out the differences in the row above. The pattern stops when the row consists of four zeroes. Investigate for different sets of starting numbers.

Mandelbrot

Apply the rule “square it and add n” to the number 0 and then recursively on each answer you generate. Try  n = 2  to start with, and explore  -2 <= n <= 2.

A programmable calculator or spreadsheet is recommended, although you can use the “ANS” function on most calculators as well.

Egyptian Fractions

Ancient Egyptian fractions consisted only of unit fractions (numerator = 1) and the fraction 2/3. Investigate ways of writing various fractions as sums of “Egyptian Fractions”. How might Egyptian Fraction arithmetic be carried out?

A WHIFF OF ALGEBRA

Squaring

Investigate how the identity   n^2 = (n+a)(n-a)+a^2 can be used to simplify mental and pencil-paper methods of squaring numbers. What numbers are particularly suitable for this method?

Transforming Numbers

Consider the transformation  (a/b) -> (a+2b)/(a+b) where a and b are whole numbers.

What is the effect of repeated application of this transformation?

Investigate for different starting numbers.

Mapping

Investigate the effect of repeatedly applying the rule “Divide by 3 then add 5”.

How can you take this further?

Steps

Start with A = 1 and B = 3. Make new values of A and B with this stepping rule:

A(new) = 2B

B(new) = 1 + 2 x A(new)

Try changing the starting numbers, the stepping rule, or even the number of variables.

SHAPE AND SPACE CADETS

Rail Tracks

I have pieces of a rail track kit which are all quadrants (quarter of a circle). Obviously my toy train needs a closed track – I could use just four pieces, but are there other possibilities? How might this investigation be extended?

Constructions

You can make many constructions with straight edge, fixed compasses, moveable compasses, straight edge with two marked points and a ruler with two parallel straight edges. What constructions are possible if you limit your choice of equipment?

Shadows

What shadows can you generate from a square piece of cardboard?

Surprise, Surprise!

Draw any polygon. Construct a square on each edge. Try joining centres, corners, midpoints. What happens? Experiment with other regular polygons constructed on the edges.

IT’S PROBABLY DATA HANDLING

Discs

Make four discs, with a different number on each one.   If I take two at random what possible totals can I make? Now make a fresh set, with different numbers front and back. This time take two at random and toss them like coins. What totals are possible? Most likely? What if I toss any random combination of up to four discs at once? Can you make a set where every total is equally likely?

Betting Odds

Bookmakers use betting odds to express both probabilities and cash payouts for winners. Investigate how they work. Is it possible to place your bets so that whichever horse wins, your winnings will pay for all the lost bets?

Breaking Sticks

A stick of length l is broken into n pieces. What is the average length of each piece?
If n = 3, what is the probability that the three pieces will make a triangle? For n > 3, what is the probability of finding pieces that will make a triangle? A quadrilateral?

Words In Different Languages

Analyse samples of text from similar sources in different languages. Compare letter frequencies, word lengths, sentence lengths etc.

LOGICALLY SPEAKING

Mongé’s Shuffle

Consider a pack of four cards a, b, c, d in that order. Deal card a into your hand. Card b goes on top. Card c goes on the bottom. Card  d goes on top. The new order is d, b, a, c. If you use more cards, simply alternate “one above, one below” all the way. Can you shuffle the pack back to its original order? In what order must you place the cards to be sorted after one shuffle? Two or more shuffles? Devise your own shuffling sequence and investigate its effects.

Bell Ringing

Three bells can be rung in six different orders. To ring the full “change” of six “peals”, no bell can move more than one position on successive peals of the three bells. (1, 2, 3) can not be followed by (3, 2, 1) because Bell 1 has moved two positions. Investigate.

Grundy’s Game

Cubes are joined together to make a rod. On your go you must split one rod into two unequal rods. If you can’t, you lose. Investigate strategies.

Codings

Investigate the structure of codes used in daily life – postcodes, telephone exchange codes, road numbers, car registrations, bar codes, ISBN numbers etc.

A LITTLE LIGHT LITERACY

Morphemes

A={Any, No, Some, Every}  and  B={thing, where, one, body}. How many words can you make by joining a morpheme from Set A with one from Set B ? Can you find better choices for Sets A and B ?

Vish

Choose a word and look up its definition in a dictionary. Now pick a word from that definition and look  that word up. Repeat the process. How quickly can you get back to the original word? Can you get to any word? How long a string can you make without repetition?

All About Myself

“Short” is a short word but “long” is even shorter. Investigate self-describing words and sentences eg “This sentence contains five words” and those which describe the opposite of themselves.

Reading Age Of Newspapers

How can you determine the reading age of a newspaper? How do different newspapers compare? Can you use the same method to judge the reading age of a book?