Squares

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?

How would you move the bands on the pegboard to alter these shapes?

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

Ayah conjectures that the diagonals of a square meet at right angles. Do you agree? How could you find out?

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

This problem challenges you to work out what fraction of the whole area of these pictures is taken up by various shapes.

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

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

What can you see? What do you notice? What questions can you ask?