Exploring and noticing

Try out this number trick. What happens with different starting numbers? What do you notice?

Use the interactivity to move Pat. Can you reproduce the graphs and tell their story?

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a rhombus.

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

Use 4 four times in a calculation. What answers can you get?

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?

Choose the size of your pegboard and the shapes you can make. Can you work out the strategies needed to block your opponent?

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