**Times square** red set, green set, blue set

The children build the familiar multiplication square but graphed vertically, progressing from the natural numbers to the integers at experiment 30 and from the integers to the real numbers at experiment 36. Of the 4 standard models for multiplication, only one, repeated addition, is used here. This is the one children turn to most readily. The point of the exercise is to realise algebraic symmetry geometrically. Teachers may like to devise a parallel treatment for the addition square.

**Zoom** green set, blue set

On the one hand, any child pushing a toy car, playing with a doll or recognising Mummy’s photo in an album accepts the same object on different scales. On the other, the consequences for measurement of change of scale take us up to Level 9 on the National Curriculum. But a rich dynamic experience of scaling – not enlarging but zooming – may lead us to appreciate why we needn’t leave so much space when they deliver that ton of sand or those thousand bricks but why we’ll need a lot more wool for our sister’s cardigan than for little Penny’s. The treatment is qualitative except where the quantities are experienced but not abstracted or where they lead to a surprise. Indeed it is the mixture of recognition and surprise which makes this such a good topic. The maths implicit here is that area goes up as the square of the linear scale factor: volume as the cube. If you have friends with the equipment – or the school has it – the children should look through a zoom lens while zooming it and enlarge one of their pictures on a zoom copier.

**Slices and solids** blue set

The growing child starts with 3-D objects and later abstracts 2-D shapes. by looking at 2-D sections through 3-D objects we can move back and forth between the two worlds and enrich our experience of both. We look at the complete slice, approaching the shape … from the inside: … from the outside: … and from both: Abbott’s Flatland is the syllabus for this section. (See the original or Martin Gardner’s Further Mathematical Diversions ch 12.) A water surface, a slice of light, a rubber band, a sheet of cardboard, the junction between 2 layers of coloured plasticine, may all be used to define the plane of section. Note that in almost all cases the ‘slices’ are related by two kinds of transformation: affine – represented by sections through the general prism or cylinder, or, wore generally, projective – represented by sections through the general pyramid or cone.

**Left and right** red set, green set, blue set

This section deals with mirror symmetry. In these explorations of space the young child can make discoveries and the older person examine observations long taken for granted. (If the visitor has mixed eye-hand dominance, it doesn’t matter. We’re not using lateral discrimination here but studying the phenomenon of handedness itself. For the purpose of these exercises it’s of no concern which hand you call which. For ‘…left …, then right …’ on the caption cards, read ‘… one …, then … the other …’.) At the end of the sequence we extend the idea of reflection from that in a plane to that in a line and, finally, a point. In The Ambidextrous Universe Martin Gardner covers all this material and goes on to examine nature’s preference for one handedness or the other at a fundamental level.

**All sorts** blue set

In exploring different ways of sorting and representing data we move between the ‘table’ scale and the ‘room’ scale: between placing a counter in a drawn circle and standing in a rope loop, between following a flowchart with a finger and negotiating an obstacle course of tables and chairs, and so on. The syllabus comprises trees and flowcharts, Carroll and Venn diagrams (1-, 2- and 3-D), bar and pie charts, scattergrams and barycentric graphs.

**Packing shapes** green set, blue set

This sequence starts with ways of tiling the plane then advances one dimension to ways of filling space. It moves from an examination of atomic packing to an investigation of the shape of soap bubbles in a foam. Though, as elsewhere, the sequence is progressive, certain experiments can be performed by both first-year undergraduates and – with different motives, preconceptions and expectations! – pre-school children.

**Transformations** red set, green set, blue set

Felix Klein’s transformations of space get more and more general as conditions are relaxed. Thus you start with the isometries. then take these as a special case of the similarities, and so on up through affinities, projectivities and topological distortions. This is how the sequence Transformations develops but only in the loosest possible way. In fact the title is little more than an excuse for drawing attention to everyday but surprising ways in which one mathematical object changes into another. In every case, however, one or more quantities are invariant. Teachers may like to name them as they go through the sequence. Be aware of transformations in other sequences: translations, rotations and reflections: Packing shapes; reflections: Left and right; dilatations: Zoom; affinities, perspectivities: Slices and solids.

**Angle** blue set

Angle is a dimensionless measure – it must always be defined as a ratio (a fraction of a turn, arc: radius, . . .) – so already abstract to that extent, and children find it hard to deal with this quantity they can’t locate: The kinetic, operational treatment (angle as ‘turn’) now familiar through LOGO is the approach least prone to misconstruction. But the static manifestation (angle as ‘shape’) can’t be ignored. The procedure here is to establish frames of reference with spirit level, plumbline, compass and use the vocabulary which goes with them – ‘vertical ‘/‘horizontal ‘ – ‘steep/ ‘shallow’; ‘north’/’east’ – ‘north-east’, etc. -, then free the angles from their reference directions – walk around with a ‘right angle checker’, record angles found at the vertices of loose objects with an ‘angle indicator’ and use the corresponding terms – ‘perpendicular’, ‘parallel’, ‘inclined’; ‘acute’, ‘obtuse’, ‘reflex’. At sixth-form level the approach to trigonometry is the same: a ‘trigogram’ displays the 3 ratios on Cartesian axes in the standard way, from which thereafter they may be divorced.

**L.C.M.s** red set, green set, blue set

The lowest common multiple of 3 and 4 is 12. If we look along the number line we thus find multiples of 3 and 4 coinciding at multiples of 12. The number line is a spatial model but the same arithmetic can be modeled in time. In fact embodiments of this simple idea are many and diverse. The familiar Cuisenaire rods are out in the sequence but also gear wheels, a glockenspiel and acetate masks each of which lets through multiples of a particular number from the ‘times’ table.

**Dissections** red set, green set, blue set

Here is another subsequence which has outgrown its parent, in this case Transformations. The core is a set of dissection puzzles. To solve them quickly one must: a) predict the effects of grouping simple angles into compound ones and adding lengths, b) remember the effects of so doing – successful or unsuccessful For stage (a) careful, directed observation is a prerequisite, and in both stages the capacity to ‘visualise’ – form and manipulate mental images — is exercised and developed. In all these transformations area is invariant but, though the lengths and angles of individual polygons remain unchanged – i.e. the transformations they suffer are isometric – the composite shape is not preserved. The puzzles stress this independence. The sequence is extended 1 dimension by a group of solid dissections – including the celebrated SOMA cube on both the ‘table’ and ‘floor’ scale. A series of puzzles where a given polygon must be produced from the intersection of 2 (or more!) others demands the same skills.

**2-D to 3-D** blue set

This is a little exhibition of ways to simulate 3 dimensions in 2: anaglyphs, stereopairs, linear perspective. Once you’ve got your eye in, objects in a conventional projection like the isometric are seen in 3-D); but note that in certain cases there is a many-one mapping of points on the object to points on the drawing, making for ambiguity.

**Pascal’s Triangle** blue set

This important array embodies number sequences ubiquitous in mathematics – the successive orders of triangle numbers, the binomial coefficients, the Fibonacci sequence – and their relations. The sequence Pascal’s Triangle opens a few doors to this Alhambra. Like Times Square most of this sequence can be adapted readily for use in the classroom. Though written quite independently, Tony Colledge’s (photocopiable) book Pascal’s Triangle (Tarquin) is virtually a teacher’s guide to this sequence.

**Loci and linkages** green set, blue set

Loci – Sliding ladders, rolling wheels, … What paths will selected points upon them follow? Does the geometry of the mechanism contain simple features to help you make your prediction? Linkages – Though less visible than they were a century ago, link motions are essential to devices we take for granted: tool and sewing boxes, folding steps. umbrellas, floor mops, screw jacks, cupboard hinges, door closers. In this sequence we study how the properties of the rhombus and parallelogram are exploited.

**Pythagoras’ Theorem** blue set

There are many ways to demonstrate why Pythagoras’ Theorem holds: here we take just 3. Before moving to the general case we look at the 3, 4, 5 and 1, root 2, root 3 triangles. We also apply the converse to check some right angles.

**Symmetry** red set, green set, blue set

We meet line and rotation symmetry first separately embodied then combined. Next we make designs with our own choice of symmetries. And finally we are introduced to the idea of a group of symmetry operations by fitting solid shapes in holes.

**Weigh-In** red set, green set, blue set

Formerly under the Challenges topic but now a sequence in its own right, this includes a number of exercises for both the 2-pan and mathematical balances.

**Challenges** red set, green set, blue set

In other parts of the Circus it’s clear what sort of maths is involved. But in situations like that set up for ‘Grandpa’s Armchair’ it’s difficult to get a mathematical handle on the problem. That particular example comes from John Mason’s excellent introduction to the psychology of problem-solving, Thinking Mathematically (Addison-Wesley).