Fold a piece of paper in half, in half again, and again. Ask students to predict the number of regions created by ech folding (2, 4, 8, 16...).
Record the number of folds and regions and discuss how many records are necessary to be sure a pattern has been established. (After 2 and 4, the next number could be 6 or 8....)
Invite students to explain the relationship between the number pattern and the physical folds made in the paper.
Draw a rectangle divided by one horizontal line and record that this produces 2 regions.
Examine and record the number pattern produced by counting the number of regions, when further vertical lines are added (retaining only one horizontal line). What if 10, or 20, vertical lines are added?
Explore the sequence of regions produced for 2, 3, 4... horizontal lines and 1, 2, 3... vertical lines.
Develop the idea of using a table to record results and seek patterns.
Encourage students to justify their conclusions and results.

During or After Viewing

Examine in detail Lisa’s table of results for the number of folds and creases (you could verify her results by folding if you wish).

Can students fill in the results for 8, 9, 10... folds?

Do they agree with Lisa’s conclusion (4095 creases for 12 folds)?

Can they relate the results to the physical effect of each new fold?

Explore how finding differences in the table helps to establish a pattern.

Develop the idea that it is useful here to seek a numerical pattern, because of the physical limitations of the experiment, and that this is often the case in other contexts.

Look in detail at the Guard’s problem. The squares on a chessboard are to be filled with 1, 2, 4, 8, 16, 32... grains of rice.

How far can students continue this sequence? How would they describe this sequence?

Explore the sequences that would appear from estimation. Lisa estimated that a mugful was 16,384 grains, weighing about 250g. The end of the third row was judged to contain about 9 million grains, roughly 128kg. On which square would the weight of rice first be more than a tonne?

Consider the number of grains on the final squares of the board. How large is the number 2⁶³? (Some students may enjoy calculating it!) Consideration of the number of digits of successive powers of 2 that can be generated on a calculator can lead to an estimation of the number of digits of 2⁶³.

The Wizard estimated that the thirtieth square would contain about half a billion grains (actually 536,870,912 grains) — taking a billion to be one thousand million. Discuss other terms that are used for very large numbers, and differences between naming systems (see ‘Comments’ below).

Can students say the numbers that appear in the sequence of powers of 2?

Compare the sequence of powers of 2 with the results found for paper folding. Many students should be able to appreciate the general rule that the number of creases for n folds is given by 2ⁿ — 1.

Discuss how to proceed with the wizard’s suggestion to investigate the number of ‘up’ and ‘down’ folds.

Is the orientation of the paper important? How can they ensure they maintain it correctly?

Can all students recognise which folds are down and which up?

How can they record the patterns they find?

Can they predict results for 12 folds? For 20 folds? For n folds?

Here is a possible representation:
For 1 fold there is 1 crease pointing downwards.
v
For 2 folds there are 3 creases, 2 down and 1 up.
v v ^
For 3 folds there are 7 creases, 4 down and 3 up.
^ v v v ^ ^ v

And so on.

Some students may wish to explore the ‘inverted’ symmetry about the centre fold (a down on one side corresponds to an up on the other).

Comments

if m is the number of vertical lines in the rectangle and n is the number of horizontal lines, then the number of regions is (m + 1) x (n + 1).

The following may help students to appreciate the size of very large numbers:

An estimate of the number of words printed since the invention of the printing press might be in the region of 10¹⁷.
The number of grains of sand on all the beaches of the world is probably about 10²⁴.
Current estimates for the number of elementary particles in the Universe are between 10⁷² and 10⁸⁷.
The total number of humans who have ever on the earth is estimated at less than 10 billion.

The British billion (10¹²) is one million million. But the American billion (10⁹) is widely used and accepted. The Americans call our billion (10¹²) a trillion, whereas the British trillion is 10¹⁸ (one million cubed). So, when the Chancellor talks about so many billion pounds spent on the Health Service, is he using the British or American billion?

For n folds there are 2ⁿ — 1 down creases and 2ⁿ — 1 — 1 up creases; so the total number of folds is 2 x 2ⁿ — 1 — 1 or 2ⁿ — 1.

The Worksheets

Worksheet 1: Palindromic Pursuits

This activity aims to provide a structured introduction to palindromes, while allowing the more curious students to explore further.

Manual calculation allows students to monitor what is happening more readily.

Students should recognise that, for example, 47 and 74 give the same result in the same number of steps.

89 takes 24 steps.

For any two-digit number where the sum of the digits is less than or equal to 9, the palindrome appears in one step. If the sum of the digits is 11, the palindrome is 121.

196 has not (yet) been found to produce a palindrome.

Challenge answers:

6247² — 6244² = 37473
(1497 + 1382) x 24 = 69096
(12³ — 10²) x 402 = 654456

(7³ — 7) x 14² = 65856
(6778² — 6775²) x 2 = 81318
76668 x 11 =843348

45³ + 3724 = 94849
(697² — 414²) x 2 = 628826
2938² — 2784² = 881188

Worksheet 2: Cuts Me Up

This activity aims to provide a context, similar to the one in the programme, in which students can use tables of results to explore number patterns.

The relationships are much simpler than those in the programme, and should allow most students to make some progress with finding and stating general rules.

The use of letters in the tables invites students to make algebraic generalisations like:

P = n + 1
P = 2n + 1
P = l + 2

The activity can easily be extended. What if you tie the ends of the string before looping around the scissors? What if you change the meaning of the word ‘cut’ — for example you could define the next ‘cut’ as the act of cutting all the pieces of string produced so far.

Worksheet 3: Chessboard Challenge

There are 204 squares on the chessboard.

Students working systematically should spot that the number of squares on an n x n board is the sum of the first n square numbers.

The activity can be extended to considering the number of squares in a rectangular grid.

For a general rectangle, the number of squares is as follows:

1
2
3
4
5
6
1
1
2
3
4
5
6
2

5
8
11
14
17
3

14
20
26
32
4

30
40
50
5

55
70
6

91

The following expressions are found for the number of squares:
1 x n
n squares
2 x n
3n — 1 squares
3 x n
6n — 4 squares
4 x n
10n —10 squares

And so on.

An alternative chessboard investigation is to consider the number of different ways in which a square board can be divided along its grid lines into two equal parts of identical shape (not counting rotations and reflections). This can be approached in a similar way to the above problem, by building up the size of the square board. Only even-sided boards are possible, but odd-sided boards can be divided if the central square is removed.

A 2 x 2 board can be divided in 1 way.
A 4 x 4 board can be divided in 6 ways.

Further Ideas

Students can create their own number patterns using interlocking cubes to build sequences of shapes growing according to a rule.

In addition to investigating and predicting the number of cubes for each model in the sequence, they can also generate patterns by investigating:

the surface areas (taking one face of a cube as the unit of area)
the number of faces
the number of vertices
the total length of the edges (taking the edge of a single cube as a unit length)

This approach provides opportunities to build up tables of values, to investigate differences and to make predictions and generalisations in a practical context.