Conjecturing and generalising

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

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

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?

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

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?

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?