Lesson 9 Framed tiles

# Episode 2

Reasoning Resources: Worksheet 2, squared paper
Whole class preparation: story leads to testing numbers
Any size square paper could be used since the side of the given tile itself is used for measurements. Hassan now tells his apprentice that he has one batch of 12 tiles and another of 24 tiles and that he needs to know in each case the longest and shortest border ribbon lengths.

What answer do you think the apprentice worked out?
Pair and small group work on generating ideas
Approaching the idea that a square pattern has the shortest perimeter. Give pairs of pupils Worksheet 2 and some squared paper. Ask them to work on the 12 tile problem first and to share their ideas in a group of four. Teacher mediation is aimed at clarifying the reasoning, and pressing for checking alternatives.

Key question:How do you know that you have found the longest (or shortest) border ribbon length?
Whole class sharing and discussion
There us a move in though from physical tiles that would not remain square when cut to the idea of mapping unit squares onto a surface and adding all the part squares up to give an area in whole numbers plus a fraction if this is needed. Collect group ideas on the longest and shortest border ribbon lengths. Discuss how pupils could be certain that they had found the longest and shortest values. Allow ideas on the longest rectangle having the longest perimeter, and the ‘fatter’ having the shortest to emerge.

In discussing ‘Why that happens?’ prod for ideas such as ‘when the design is in one row the border touches each tile twice or three times".Discuss the answers to the final question, discussing the ideas of half tile and quarter tiles etc.

An extension could investigate the effect on tile numbers for a given area if the sides of the tiles are halved, or quartered or made one tenth size.