Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

Can you cut up a square in the way shown and make the pieces into a triangle?

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

We can cut a small triangle off the corner of a square and then fit the two pieces together. Can you work out how these shapes are made from the two pieces?

Can you describe a piece of paper clearly enough for your partner to know which piece it is?

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

Make a cube out of straws and have a go at this practical challenge.

Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?

Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?

What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?

Can you make a 3x3 cube with these shapes made from small cubes?

Exploring and predicting folding, cutting and punching holes and making spirals.

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

What shape is made when you fold using this crease pattern? Can you make a ring design?

Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.

Can you visualise what shape this piece of paper will make when it is folded?

Can you split each of the shapes below in half so that the two parts are exactly the same?

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?

Make a flower design using the same shape made out of different sizes of paper.

Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

Can you fit the tangram pieces into the outlines of the people?

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

Can you fit the tangram pieces into the outline of Mah Ling?

How many different triangles can you make on a circular pegboard that has nine pegs?

Move just three of the circles so that the triangle faces in the opposite direction.

Can you fit the tangram pieces into the outline of this teacup?

Can you fit the tangram pieces into the outline of the butterfly?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

How will you go about finding all the jigsaw pieces that have one peg and one hole?

Reasoning about the number of matches needed to build squares that share their sides.

How many balls of modelling clay and how many straws does it take to make these skeleton shapes?

Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

Can you fit the tangram pieces into the outlines of the convex shapes?

Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will have holes drilled through them?

If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

This article looks at levels of geometric thinking and the types of activities required to develop this thinking.

Here are shadows of some 3D shapes. What shapes could have made them?

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?