On the grid provided, we can draw lines with different gradients. How many different gradients can you find? Can you arrange them in order of steepness?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

Anna, Ben and Charlie have been estimating 30 seconds. Who is the best?

How many different differences can you make?

What are the possible areas of triangles drawn in a square?

What happens when you add a three digit number to its reverse?

We started drawing some quadrilaterals - can you complete them?

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

Which of these pocket money systems would you rather have?

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Can you find triangles on a 9-point circle? Can you work out their angles?

Discs are flipped in the air. You win if all the faces show the same colour. What is the probability of winning?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

Can all unit fractions be written as the sum of two unit fractions?

Which countries have the most naturally athletic populations?

How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?

Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea?

Play around with sets of five numbers and see what you can discover about different types of average...

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.

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

There are lots of ideas to explore in these sequences of ordered fractions.

A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?

Can you work out what step size to take to ensure you visit all the dots on the circle?

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

A game in which players take it in turns to turn up two cards. If they can draw a triangle which satisfies both properties they win the pair of cards. And a few challenging questions to follow...

Different combinations of the weights available allow you to make different totals. Which totals can you make?

What's the largest volume of box you can make from a square of paper?

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?