Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!
What does the overlap of these two shapes look like? Try picturing it in your head and then use some cut-out shapes to test your prediction.
What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?
Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.
Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.
Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?
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?
Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?
Move four sticks so there are exactly four triangles.
What shape is made when you fold using this crease pattern? Can you make a ring design?
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?
Make a flower design using the same shape made out of different sizes of paper.
Can you visualise what shape this piece of paper will make when it is folded?
Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?
A group activity using visualisation of squares and triangles.
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
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?
Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.
What is the greatest number of squares you can make by overlapping three squares?
Can you split each of the shapes below in half so that the two parts are exactly the same?
Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?
This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!
What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?
Can you describe a piece of paper clearly enough for your partner to know which piece it is?
What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?
Can you cut up a square in the way shown and make the pieces into a triangle?
Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!
Move just three of the circles so that the triangle faces in the opposite direction.
Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?
How many different triangles can you make on a circular pegboard that has nine pegs?
A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?
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?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
Make a cube out of straws and have a go at this practical challenge.
You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.
Reasoning about the number of matches needed to build squares that share their sides.
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?
In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?
What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.
This second article in the series refers to research about levels of development of spatial thinking and the possible influence of instruction.
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 ...
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
Exploring and predicting folding, cutting and punching holes and making spirals.
Why do you think that the red player chose that particular dot in this game of Seeing Squares?
This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
How many pieces of string have been used in these patterns? Can you describe how you know?
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?