Lesson 9 Framed tiles

# Episode 1

Reasoning Resources: Worksheet 1

Whole class preparation
Starting with pupils’ concrete experiences and engagement to focus attention on the mathematics.

Pupils are forced to make a distinction between the coverage for the area and the length around it for the perimeter.
Start by asking the class about what they know about ceramic designs and tiles. Lead to a mention of Islamic patterns in ceramic tiles as a feature of old buildings in the Middle East and Southern Europe.

Sometimes arrangements of small tiles are framed with ornamental ribbon or even with gold strips. Hassan is given the job as an expert tiler to complete a special assignment in the palace at Babylon. Certain rectangular areas have to be filled with identically sized square tiles and then framed with gold ribbon. Imagine you are his apprentice, and he asks you to guess which frame will have the longest border ribbon and which has the most tiles.

Give out Worksheet 1 to pairs. AskWhich rectangle do you think contains the most tiles? Which needs the longest border ribbon? Accept and record pupils’ initial suggestions. Ask the pupils to test these ideas or produce others without using any rulers. (They may use pencils.)
Pair and small group work
Without formal measurement pupils are forced to use non-standard units and make comparison of parts.

The pupils work in pairs to agree answers to the questions. Encourage pairs to rehearse explaining or showing their conclusions and reasons to the class. They can form groups of four to compare their answers before each group is asked to report their findings.
Whole class sharing and discussion
Noting and appreciating pupils’ strategies has both motivational and cognitive value. It highlights their resourcefulness, and that any agreed unit can be used to measure either the area or the length.

Pupils are pressed to further distinguish area and perimeter in measurements.
Groups are invited to report back. Ideas are recorded and disagreements noted. Pupils are asked to resolve disagreements if possible.

Different ways of estimating should be noted and compared. Some may use finger widths or some other object as a measuring tool. Others may imagine placing one shape on to the other to determine the greatest area. Some may make markings on the shape, splitting them to compare like with like.

The question: Would you expect the rectangle holding the greatest number of tiles to also have the longest border ribbon? can be used to focus on the difference between area and perimeter and to provide a move to more formal measurement.
Whole class reflection
Sources of error may be confusion between area and perimeter, and focusing on one feature, like length, ignoring another. Ask pupils; Why do you think some of the initial ideas were mistaken?