Hidden Figures Revealed | Self-Guided Maths Workshops

Choose your adventure

Two ways to uncover hidden mathematics

You do not need to rush. Read one step, try the online activity, record your thinking in the booklet, and mark the step complete when you are ready.

Workshop 1 · 180 minutes

Maths Shapes Everything

Investigate 2D and 3D shapes, measure perimeter and area, then plan, build and improve your own structure.

Explore shapes by their properties.
Turn and zoom high-quality 3D solids.
Use geometry as an engineer.

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Workshop 2 · 180 minutes

Map Maker’s Quest with Magda

Turn maps into networks, discover one-stroke routes, protect connections and solve a map that truly needs four colours.

Plan several moves ahead.
Discover the mathematics of connections.
Explain why an answer must be true.

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How to work independently

Your four-part learning rhythm

1Read
Follow the short instructions at the start of each step.
2Explore
Move, test and change the online mathematics.
3Record
Draw or write only the evidence that matters in your booklet.
4Finish
Use the self-check, then mark the step complete.

Before you begin

Prepare a clear workspace

Print a booklet

Use the HTML version for a browser print preview or the ready-made PDF.

Bring simple tools

Pencil, ruler, four coloured pencils and plain paper are enough for both workshops.

For Workshop 1

Add up to six A4 sheets, no more than 60 cm of tape, and a small test object.

Pause when needed

Two planned breaks are part of each workshop. Your online progress saves automatically.

Starting again?

Only use this if you want to erase all saved online work for both workshops.

0 of 7 steps completeYour work saves on this device.

By the end

You will be able to…

identify 2D shapes by their properties;
identify 3D shapes by their properties;
explain regular and irregular shapes;
find perimeter and area in square units;
follow design criteria to build a structure.

Engineers do more than name shapes. They look for properties that help a design stand, fit and carry a load.

Your final mission

Later, you will build a structure at least 30 cm tall. It must fit on a 20 cm × 20 cm base and hold a small object for 10 seconds.

Booklet: Write your name on page 1. Keep page 2 open for the warm-up.

Warm-up riddle

I have four straight sides. My opposite sides are equal. Every corner is a right angle. What could I be?

Rectangle Square Both

Ready to investigate the properties behind shape names?

A property is a fact that stays true for a shape. Select each shape and look for facts that prove its name.

Square

4 straight sides · 4 vertices · all sides equal · all angles equal

Regular polygon

Property check

A hexagon has six sides. If all six sides and all six angles are equal, which word describes it?

RegularIrregular

Booklet pages 2–3: Solve the shape riddle. Draw the regular and irregular hexagons in the blank frames. For the regular one, make the six sides and angles as equal as you can.

Can you name a shape and give a property that proves it?

A solid can turn without changing its mathematical properties. Drag the model to rotate it. Pinch, use the mouse wheel or use the buttons to zoom.

100%

Look closely:

Cube6 flat faces · 12 edges · 8 vertices

Solid check

I have two matching triangular faces and three rectangular faces. What am I?

CubeTriangular prismSquare pyramid

Net investigation

Can six squares fold into a cube?

A net is a flat pattern that can fold into a solid. Use the button to compare the flat pattern and the cube.

Net check

Which numbered face is opposite face 2?

146

Booklet pages 4–6: Record the flat faces, curved surfaces, edges and vertices. Then investigate the cube and triangular-prism nets.

Can you identify a solid without relying only on its picture?

Perimeter

Walk around the outside edge and count the unit lengths.

Area

Count the square units covered inside the shape.

Live measurements

Area0square units

Perimeter0units

Select squares that touch along a side to make a connected shape.

Your turn: Load Shape A, then Shape B. They have the same perimeter but different areas. Change one square at a time and watch both measurements.

What did you notice?

Open the measurement self-check after trying

Shape A: perimeter 20 units; area 24 square units. Shape B: perimeter 20 units; area 20 square units.

Booklet pages 7–8: Record Shapes A and B. Page 8 is optional if you would like a deeper challenge.

Can you explain why two shapes may share a perimeter but have different areas?

Your structure must…

be at least 30 cm tall;
fit on a 20 cm × 20 cm base;
hold a small object for 10 seconds;
use no more than six A4 sheets and 60 cm of tape;
show at least one named 2D shape and one named 3D shape;
be tested and improved once.

Make three decisions

Best base shape Main 3D form What will you test first?

Booklet page 9: Draw a labelled side view and top view. Write how your plan meets every design criterion.

Check that your drawing shows sizes, shapes and where the test object will rest.

Build safely: Use a clear table. Keep the test object small and unbreakable. Ask an adult for help if you need scissors.

Work in this order

1Build the main shape.
2Check the height and base.
3Add the small object for 10 seconds.
4Notice one weakness.
5Change one part and test again.

My evidence checklist

My structure is at least 30 cm tall. Its base fits inside 20 cm × 20 cm. It held the object for 10 seconds. I stayed within the paper and tape limit. I can name a 2D shape and a 3D shape in it. I changed one part after testing.

0 of 6 criteria checked

I noticed…I changed…

The new test showed…

Booklet page 10: Record your two test results and the deliberate improvement you made.

A design is ready when you have evidence, not merely when it looks finished.

Choose one piece of mathematical evidence that you are proud of.

One thing my design proves is…

Open the full worked self-check after trying booklet pages 2–8

Shapes and solids

Page 2: Both rectangle and square fit. A square is a special rectangle.
Page 3: Both drawings need six sides. The regular hexagon has six equal sides and equal angles.
Page 4: Cube 6 flat / 0 curved / 12 edges / 8 vertices; cuboid 6 / 0 / 12 / 8; triangular prism 5 / 0 / 9 / 6; square pyramid 5 / 0 / 8 / 5; cylinder 2 / 1 / 2 / 0; cone 1 / 1 / 1 / 1; sphere 0 / 1 / 0 / 0.

Nets and measuring

Page 5: Face 2 is opposite face 4. A cube has 6 faces, 12 edges and 8 vertices.
Page 6: The two triangles are the matching faces. A triangular prism has 5 flat faces, 9 edges and 6 vertices.
Page 7: Shape A has perimeter 20 and area 24. Shape B has perimeter 20 and area 20.

Workshop finish

You have thought like a mathematical engineer.

You noticed properties, measured evidence, followed criteria and improved a real structure.

0 of 10 steps completeYour work saves on this device.

By the end

You will be able to…

plan several moves ahead in a network;
use points and edges to show connections;
predict when a one-stroke route can work;
protect an important connection;
explain why a path or number of colours is necessary.

Magda needs to study how five town places are connected. Buildings and bends make the map busy. What can she remove without changing the journey choices?

What must stay?

Building coloursEvery road bendWhich places connect

Booklet page 2: Read the five network words and simplify an everyday network of your own.

A network keeps the choices of connection and removes detail that does not matter.

A point is called a vertex. A connection is called an edge. Tap two labelled vertices to add or remove an edge. Drag a vertex to change the layout.

Connected

A network is connected when you can travel along edges from any vertex to every other vertex.

Try this:

Make a connected network with exactly four edges.
Move vertex A without changing any edge.
Find the degree of each vertex.

Choose two:

Choose two vertices to add an edge.

Booklet page 3: Copy your network with a different layout, but keep every connection. Record each degree.

Moving a point changes the picture, but it does not change which labelled points connect.

A one-stroke trail uses every edge exactly once. Choose a starting vertex, then tap connected edges. Used edges stay coloured so you can inspect your route.

Start at:

Choose a network.

Prediction first: Try Mystery A and Mystery B. Before each second attempt, count how many edges meet at every vertex.

Booklet page 4: Record the even vertices, odd vertices and starting places that worked.

Keep failed routes as evidence. A failed start can reveal where a successful start must be.

An odd vertex has an odd number of edges meeting there. Predict whether Mystery C and Mystery D can work before tracing.

Start at:

Choose a network.

0 odd vertices

A closed trail can start anywhere and finish where it began.

2 odd vertices

Start at one odd vertex and finish at the other.

4 or more odd vertices

A one-stroke trail cannot use every edge exactly once.

Booklet page 5: Mystery C has four odd vertices and is impossible. Mystery D has two odd vertices, so it can work only from B to C or C to B.

Can you use the number of odd vertices to predict before tracing?

In 1736, the city of Königsberg had seven bridges. Could a walker cross every bridge exactly once? Change the picture without losing a single connection.

Predict before revealing the degrees:

Booklet page 6: Decide before tracing, record the four degrees, and explain what stayed the same as the map changed.

The impossible answer is still a complete mathematical solution when the reason is clear.

A town needs more than short routes. It needs backup routes. Tap a road or place to remove it and inspect which places can still reach one another.

Remove:

Try removing road B–D. Predict what will happen first.

Booklet page 8: Circle a road and a place whose removal causes a split. Then add one small repair.

A strong network has another route ready when one connection disappears.

Edges pair up in the middle

At a vertex in the middle of a trail, one edge brings you in and another edge takes you out. Those two edges make a pair.

At an odd vertex, one edge is left without a partner. That leftover edge must be at the start or finish.

Reasoning check

Why can a one-stroke trail have two odd vertices but not four?

The dots are drawn in the wrong places.Only the start and finish can use an unpaired edge.Some edges are longer than others.

Say the rule in your own short way

Booklet page 7: Pair the edges around the degree-4 and degree-3 vertices. Use the leftover edge to explain the rule.

You have moved from testing examples to explaining why the pattern must hold.

Neighbouring regions must have different colours. Regions are neighbours only when they share part of a boundary. Touching at one point does not count.

Why this map is special

Colour the outside ring A–E first.
Every ring region touches its two neighbours.
Centre F touches all five ring regions.

Choose a colour, then choose a region.

Open a clue after you have tried three colours

Important discovery: The odd ring needs three colours. The centre touches every ring region, so it cannot reuse any of those three. This map really does need a fourth colour.

Point or boundary?

Regions R and S touch only at one corner. May they use the same colour?

YesNo

Booklet pages 9–10: Colour the same four-colour map, count its ten shared ring boundaries, then investigate why a triangle needs three colours.

Can you explain why three colours fail, rather than only showing that four colours work?

A fair mystery has clear rules, enough information and an answer that the creator has already tested.

Booklet page 11: Draw the puzzle itself. Solve it, then copy your clear rule, clue and reason from the preview.

Ask another person to test the mystery if someone is available. Revise any rule that was unclear.

One choice that changed the rest of a network was…

Open the worked self-check after trying booklet pages 4–10

One-stroke networks

Mystery A: 0 odd vertices; a closed trail can start anywhere.
Mystery B: C and D are odd; start at one and finish at the other.
Mystery C: 4 odd vertices; no one-stroke trail.
Mystery D: B and C are odd; start at one and finish at the other. Example: B–A–D–C–E–B–C.
Seven Bridges: degrees 3, 3, 5 and 3; no one-stroke trail.

Smallest colour counts

Page 9: 4. The five-region ring needs 3 colours and centre F touches every ring region, so it needs a fourth.
Page 10 triangle: 3, because each region touches the other two.
P and Q: adjacent, so they need different colours.
R and S: not adjacent when they touch only at one point.

Workshop finish

You have thought like a network mathematician.

You planned ahead, found patterns in connections and explained why answers must work.