Mathematics
Advanced
50 mins
Teacher/Student led
+65 XP
What you need:
IWB/Projector/Large Screen

Same Perimeter, Different Area

Investigate whether two shapes with the same perimeter must have the same area. Discover that a rectangle's area changes as its shape changes, even when the perimeter stays fixed, and find that the squarest shape holds the most space.

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    1 - Getting Started ~4 mins

    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.

    Key point

    So here is our real question for today: if the fence around two gardens is the same length, must the two gardens hold the same amount of space inside? Have a think, then hands up. What is your first guess, yes or no?

    2 - Watch and Notice ~8 mins

    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.

    3 - Draw Two in Your Copy ~3 mins

    COPYBOOK MOMENT

    Illustration for Draw Two in Your CopyIn your maths copy, draw two different rectangles that each have a perimeter of 10 cm. Then write the area of each one underneath, and circle the rectangle that holds more space.

    4 - Try It Together ~8 mins

    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.

    Watch this one

    Keep an eye on that very last one, the 4 × 4 square. Before we build it, tell me with your thumbs, do you think it will hold more space or less than the others? Let's build them all and find out.

    Shape Measurer

    5 - Class Challenge ~8 mins

    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.
    The big question

    Does the rectangle that holds the most space always turn out to be the one closest to a square?

    Shape Measurer

    Pupil practice
    Module 5 · Area, Perimeter and Volume Measures
    Lesson 61 · Same Perimeter, Different Area
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