Here are two different rectangles. Walk your finger all the way around the edge of each one and you would travel exactly the same distance, twelve centimetres, on both. They have the same perimeter.

Three shapes coming up, each with the same 12 cm fence around the outside. Before each one, show me your thumb: up if you think it holds more space than the last, sideways if less. Then we count the squares inside.

Now we keep the fence a bit longer, exactly 16 cm, and we hunt for rectangles. We build these together on the board, in order: first the long thin 1 × 7, then the 2 × 6, then the 3 × 5, and finally the 4 × 4 square. Before each one, show me your thumbs guess for its space, then we build it and read the space inside off the tool.

This is our investigation. We fix the fence at 12 cm and find every whole-number rectangle that fits, then compare the space each one holds. A whole-number rectangle is one whose sides are a whole number of squares, like 2 by 4 (we don't use half-squares).

We work these in order on the board, one job at a time:

1. List all the whole-number rectangles with a perimeter of 12 cm.
2. Write the area of each one beside it.
3. Say which rectangle has the largest area and which has the smallest.