Teaching objectives
In this lab the student never picks between options to continue a series: they build it with their own hands. They switch on lamps, paint cells, lay matchsticks, or remove cubes to construct the next step of a visual sequence, and the lab itself checks whether what was built is correct. When they get it right, the earlier steps are hidden and they are asked for one more step based only on the one they just made: getting it right once can be luck, getting it right twice in a row is not.
What you'll learn
The lab trains students to spot the rule governing a sequence and apply it forward, rather than recognise it among several options. It also works on counting the same figure in two different ways, which for many students is their first real experience of algebraic equality: two expressions that look different but count the same thing.
- Spotting the formation rule of a sequence from its first steps, and applying it to build the next one (rooms 1-8).
- Recognising and applying rotations, reflections, and colour inversions of figures on a grid (rooms 3-5 and 8).
- Counting the same structure in two different ways and checking that both give the same number (rooms 9-14).
- Recognising triangular numbers and square numbers as sums of successive rows or layers (rooms 15-16).
- Observing multiplicative growth and self-similarity in figures that repeat at different scales (rooms 17-19).
- Inventing your own rule by chaining transformations, and figuring out the lab's secret rule by watching how it transforms a figure step by step (room 20).
Key mathematical ideas
- Regularity: an interesting sequence is not a list of loose drawings but the result of applying the same rule again and again. Seeing it built by hand, step by step, is more convincing than stating it.
- Transformations of the plane — rotation, reflection, colour inversion — as operations that can be chained together and whose effect can be predicted before applying them.
- Counting the same figure two ways (for example, the border of a square as 4N−4 or as N²−(N−2)²) is the entry point to the idea that two different formulas can be the same quantity: the seed of algebraic equality.
- Triangular numbers (1, 3, 6, 10, 15…) and square numbers (1, 4, 9, 16, 25…) as accumulated sums of rows or of L-shaped layers.
- Multiplicative growth and self-similarity: figures such as the Sierpinski triangle or a branching tree, where every part reproduces the structure of the whole at another scale.
Level: Primary school (ages 8-12). In class it works well projected for the whole group in the construction rooms with double iteration: the effect of "hide the earlier steps and ask for one more" explains itself and invites asking out loud "what rule are you using?" before the student writes it in their own words in each room's free-text field.
Room-by-room contents
Room 1 · The sliding window
A 3×3 grid numbered 1 to 9. A window of three consecutive circles lights up and moves forward one square at each step: the student sees the first five steps and switches on the circles of the sixth themselves, then, with the earlier ones hidden, the circles of the seventh.
Student tasks
- Switch on the three consecutive circles for step 6.
- With the earlier steps hidden, switch on the circles for step 7.
Room 2 · The L
An L drawn on a 4×4 grid that rotates 90° clockwise at each step. The student drags across the squares to paint step 4 and checks that step 5 matches the first one again: the shape completes a cycle of four rotations.
Student tasks
- Paint the position of the L at step 4 by rotating it 90° from step 3.
- Paint step 5 and check that it matches step 1 again.
Room 3 · How does the tree go on?
Three steps of a fractal tree drawn - 1, 2 and 4 new twigs - and four continuations to choose from. The three wrong ones look a lot like the right one: in one each tip opens into three, in another two tips carry straight on and 6 twigs come out instead of 8, and in the third the new branches do not shrink; a wrong pick says exactly what is wrong with it. Then comes the table of powers of 2 up to step 6 and, as a reward, the tree drawn with nine steps. It replaces the mirror room, which was not getting across.
Student tasks
- Pick which of the four drawings continues the tree.
- Fill in the table of new twigs: 1, 2, 4, 8, 16, 32.
- See where a whole tree comes from: from one tiny rule.
Room 4 · Stars
A 7×7 grid in which every painted square infects its four neighbours, like frost growing on a window. The size of the patch follows the series 1, 5, 13, 25, and the student paints the next step and, with the previous one hidden, one more.
Student tasks
- Paint the next step by infecting each painted square's four neighbours.
- With the previous step hidden, paint one more step.
Room 5 · The grid of faces
A 3×3 grid combines three types of nose and three of mouth without repeating a combination in any row or column, like a Graeco-Latin square. One face is missing and the student builds it feature by feature, rather than picking it from a list; a variant leaves two squares empty.
Student tasks
- Work out which nose and mouth are missing from the empty square and build them.
- Solve the variant with two empty squares.
Room 6 · Two rules at once
An arrow rotates 45° at each step while its colour alternates between two, following two independent rules at the same time. The student sets the next step with the direction and colour it should have, then one more based only on that.
Student tasks
- Rotate the arrow 45° and alternate its colour to build the next step.
- With the previous step hidden, build one more step.
Room 7 · Two metamorphoses
The same L that rotates 90° per step, but this time the painted and empty squares are also swapped. The full cycle takes four steps: the student discovers this by building it, without being told in advance.
Student tasks
- Paint the next step by rotating the shape and swapping painted and empty.
- Keep building steps until you confirm the cycle repeats every four.
Room 8 · The clock
A dial of twelve lamps in which the lit ones move forward one hour at each step and a new lamp is added, alternating between the 12 and 6 positions. Four steps are shown and the student switches on the lamps of the fifth and, with the earlier ones hidden, the sixth; a reverse variant asks them to work out an earlier step from a later one.
Student tasks
- Switch on the lamps of step 5 by moving step 4 forward one hour and adding the new lamp.
- With the earlier steps hidden, switch on the lamps of step 6.
- Solve the reverse variant by working out an earlier step.
Room 9 · Matchsticks: a row of squares
A row of squares made of matchsticks: each new square shares one side with the previous one. The student predicts how many matchsticks the fourth square will need (13), lays them out, and counts them two ways: by vertical matchsticks (n+1) and horizontal ones (2n).
Student tasks
- Predict the number of matchsticks for step 4 before building it.
- Lay out the matchsticks and count them by verticals and by horizontals, checking that both counts agree.
Room 10 · Matchsticks: triangles
Triangles flipping upside down every other one, sharing a side with the previous one: the matchstick series is 3, 5, 7, 9. A variant subdivides every triangle into smaller ones, with the series 3, 9, 18.
Student tasks
- Predict and build the number of matchsticks in the fourth triangle (9).
- Solve the subdivided-triangle variant by counting two ways.
Room 11 · The big triangle
The big triangle of side N cut into equal little ones: 1, 4 and 9 little triangles with 3, 9 and 18 matchsticks. Sides 1 and 2 are given built, and side 3 is shown in grey, at real size and with the lines only, so that it is clear what "side 3" means before counting it. The student predicts, lays the matchsticks out and counts them two ways.
Student tasks
- Predict how many matchsticks the side-3 triangle needs.
- Lay out the 18 matchsticks and count them by slants and by rows.
- Fill in the table up to the side-4 triangle, which is not drawn.
Room 12 · The frame
Only the border of an N×N grid gets painted. The student predicts how many squares that will be, paints them, and counts them two ways: counting the perimeter directly (4N−4) and subtracting the inner hole from the total (N²−(N−2)²).
Student tasks
- Predict the number of border squares before painting.
- Paint the frame and check the count against both formulas.
Room 13 · The skeleton of the cube
An N×N×N cube with only the edge and corner cubies left. Step 1 (2×2×2, 8 pieces) and step 2 (3×3×3, 20) are given built, with the count shown large. The student predicts how many are left in the 4×4×4 and, if wrong, carves it out by removing the face centres and the inside; the layer slider is essential to reach what is hidden. Two counts using the numbers of the cube on screen: 8 corners + 12 edges of 2 pieces, or 64 minus the inner block minus the six plates.
Student tasks
- Predict how many pieces the 4×4×4 skeleton has.
- Take away the extra cubies and find the hidden ones with the slider.
- Fill in the table up to the 6×6×6 cube and notice each step adds 12.
Room 14 · Triangles of dots
Triangular numbers 1, 3, 6, 10. The whole triangular lattice is visible, far bigger than what has to be lit, with the first two layers on: the student decides where the new row goes, it is not pointed out. Two steps in a row are built, the second with the earlier ones hidden.
Student tasks
- Place the row that continues the triangle.
- Do it again with the earlier steps hidden.
- Fill in the table up to step 7, which is not drawn.
Room 15 · Squares and layers
Square numbers 1, 4, 9, 16 seen as bracket-shaped layers. The whole square lattice is visible with the first two layers on; each finished layer gets its own colour and a panel builds up 1+3+5+7. The student places two brackets in a row.
Student tasks
- Place the bracket that continues the square.
- Do it again with the earlier steps hidden.
- Fill in the table up to step 7, which is not drawn.
Room 16 · The honeycomb
Centred hexagons 1, 7, 19, 37. The whole comb is visible with one extra ring and the first two layers on; the student places the complete ring that comes next. At the end, two counts on the same figure: by rings and by the six wedges plus the centre.
Student tasks
- Place the ring that continues the honeycomb.
- Do it again with the earlier steps hidden.
- Count the figure by rings and by wedges, and fill in the table up to step 7.
Room 17 · Counting triangles
The Sierpinski triangle: 1, 3, 9, 27 filled triangles. The student predicts how many iteration 4 will have and, if wrong, builds it by subdividing one by one from iteration 3; if right, it is shown done. Then they fill in the table of filled triangles and, with iteration 4 on screen, count the black little triangles (the holes) and find they make a series too: 0, 1, 4, 13.
Student tasks
- Predict how many filled triangles iteration 4 will have.
- Subdivide triangle by triangle until it is complete.
- Count the holes in iteration 4 and fill in the table of holes.
Room 18 · The branching tree
Two branches come out of every tip, both from the same point: one carries straight on at 0.6 of the previous length and the other opens 20° at 0.5. The student grows the tree by tapping the tips and cannot run ahead: next-level tips do not respond until every tip of the current level has opened. After three levels come the table of accumulated branches (1, 3, 7, 15, 31, 63) and the tree drawn with seven levels, which by then looks like a fern.
Student tasks
- Choose how many twigs each tip opens into.
- Open every tip of the iteration.
- Fill in the table of new twigs and predict an iteration without building it.
Room 19 · Painted cubes
It opens by asking how many 1×1×1 cubies a 3×3×3 cube is cut into. On answering, the cube opens into layers: whoever gets it right moves on, whoever does not counts one layer and adds 9+9+9. Only then comes the paint question: how many cubies will have 3 painted faces. After that the 27 are sorted by number of painted faces, but **one pile at a time**: the 3-face ones are marked and checked, and only then does the next pile appear. The cube is really painted - the faces that were on the outside show the paint colour - and the three layers come apart, so every cubie can be tapped. It closes with the balance 8 + 12 + 6 + 1 = 27.
Student tasks
- Say how many small cubies the cube is cut into and check it layer by layer.
- Predict how many will have 3 painted faces.
- Sort the 27 cubies and fill in the table for the 4×4×4 cube.
Room 20 · Sticks and clay
The wire frame of a cubic grid: clay balls at the knots and sticks along the edges. Step 1 is given built (8 balls, 12 sticks) and step 2 is drawn in grey, at real size, so it is clear what is being asked. The balls at the knots are counted one by one (27) and then counted two ways: by layers and by columns. It comes at the end of the lab because a 3D frame is the hardest thing to read.
Student tasks
- Predict how many clay balls the step-2 frame has.
- Mark the knots and count them by layers and by columns.
- Fill in the table of balls up to step 3, which is not drawn.
Room 21 · Your own rule
Two modes. In the builder, the student draws a figure on a 3×3 grid and chains transformations - turn, mirror, invert, shift; the series is generated on a 7×7 grid with the figure tucked into the top-left corner, so that shifting reads as a movement and not as a jump. In guess mode, the lab hides a chain of two transformations, shows four steps and the student builds the fifth and the sixth; getting both right reveals the chain written out in words.
Student tasks
- Draw a figure and chain transformations to generate your own series.
- Work out the hidden chain by building the next step.
- Do it again with the earlier steps hidden and compare with what you had in mind.
Rooms to project
The most striking ones to show and discuss in class.