Exploring and noticing

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

How many ways can you find to put in operation signs (+, −, ×, ÷) to make 100?

The large rectangle is divided into quadrilaterals and triangles. Can you untangle what fractional part is represented by each of the ten numbered shapes?

How many different number families can you find?

Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...

Follow this recipe for sieving numbers and see what interesting patterns emerge.

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

Can you deduce which Olympic athletics events are represented by the graphs?

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?