Conjecturing and generalising

Ben's class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

Investigate and explain the patterns that you see from recording just the units digits of numbers in the times tables.

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

What is the total number of squares that can be made on a 5 by 5 geoboard?

Can you use the diagram to prove the AM-GM inequality?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?