WEEK 7

 

6.2 Perimeter, area and volume

6..2.1 Understand the concept of perimeter (1)

Here we look at the perimeter of shapes, especially ones made of unit square tiles, and we consider what happens to the perimeter as the shape is changed in some way.

 

Monday: There is plenty of scope for exploration here. Students might discover that there are some moves that increase the perimeter, plenty that keep it the same, and one that reduces it. What characterises the shape in each of these conditions?

In the examples below, tile B has been moved to a new position. In the first case the perimeter has increased - by how much? In the second case the perimeter has stayed the same - why?

 

 

Tuesday: This builds on Monday's task, where students might have gained some sense of the fact that the less 'jagged' a shape is, the smaller its perimeter will be. Here they might also see that the perimeter tends to be smaller if the shape is 'roundish' rather than elongated.

If we had 36 tiles, then one can fairly easily see (or verify empirically) that the smallest perimeter occurs when they are arranged in a 6 by 6 square. For 41 tiles an arrangement like this gives the smallest perimeter (26 cm):

Or one could also argue like this: If we had 42 tiles, then one can fairly easily see (or verify empirically) that the smallest perimeter occurs when they are arranged in a 6 by 7 rectangle. If we now remove one of the tiles, to make 41 tiles, the perimeter will stay the same (26 cm) if we remove a corner tile (with others it will increase).

Wednesday: Here we examine Tuesday's solution more closely, by thinking about what happens when we add one extra or two extra tiles.

Thursday: This provides an opportunity to make more explicit the ideas about non-jaggedness and compactness that might have emerged from the earlier tasks. However, care needs to be taken - the numbers in the table make a pattern of sorts, but it is easy to opt too hastily for patterns that turn out not to hold. For example, students might decide that '28 cm' should occur the same number of times as '26 cm'. Students should be encouraged to visualise the shape for each row of the table.

Friday: We broaden things out by considering a 'continuous' shape rather than one made of discrete tiles. The key idea, already nascent in Monday's task, is that when we remove part of a shape, we can leave the perimeter unchanged if each part of the edge that has been removed is replaced by a single part of the same length and orientation.