This theme has similarities with several other themes where we create a fraction by subdividing a square or rectangle several times - for example, D03, D10 and D11. The theme is distinctive in that we subdivide numerous times and often by repeating an operation, such as halving, to produce a regular pattern, as in the first task below.
TASK 19A: Here we make use of the classic 'diamond' method of halving the area of a square - by rotating a given square through 45˚ and shrinking it until it fits tightly inside the original square. The orange region is formed by halving the blue outer square and then halving and halving again. So its area is one eighth of the outer blue square. The lemon region is formed by taking one third of the blue outer square, then taking a third of the resulting strip. So its area is one ninth of the blue outer square.
TASK 19B: For the green outer square on the left, we apply the halving operation 4 times. This means that successive squares have areas that are ½, ¼, ⅛, and ¹⁄₁₆ of the area of the green outer square. The fact that the halving operation occurs so many times might nudge some students away from determining this string of individual fractions to using a more formal approach of operating on the operations; so they might construct an expression like ½×½×½×½ or (½)⁴ and calculate its value. The area of the lemon region is ¼ of ¼ of its green outer square, and so has the same area as the orange region.
TASK 19C: The area of the orange region is ½×½×⅓×⅓ of its blue outer square. The area of the lemon region is ⅓×⅓×½×½ of its blue outer square. So they are both ¹⁄₃₆ of the blue outer square.
TASK 19D: Here we take a slight detour by considering the 'diamond' shape that we met in earlier tasks, and exploring what happens when the square that envelops the shape changes into a rectangle with a smaller and smaller height.
It turns out that the area of the orange region is ½, ⁵⁄₁₂, ⅓ and ¼ of the area of the square and the three successive rectangles respectively. Beyond the fact that the fractions get smaller, pupils will probably feel that there is no simple pattern here. However, there must be a pattern, surely?!
It transpires that the fraction is equal to h/12, where h is the height of the rectangle. Some pupils might spot this, if given enough time and encouragement! We can derive it algebraically, of course:
the area of the 'diamond' is ½h², and so the fraction is ½h²/6h = h/12.
TASK 19E: A further detour. Here we can think of the 'diamond' shape as being formed by overlapping flaps of an 'envelope' rather than being enveloped by a rectangle!
The fraction is given by the expression (36 – 12h + h²)/12h, where h is the height of the rectangle.
TASK 19F: We return to the earlier form of task.
The orange region covers ⅟₈₀ of its outer green square. Pupils might find the fraction in a step by step way (such as ⅟₂₀ → ⅟₄₀ → ⅟₈₀) or by deriving and evaluating expressions such as ⅟₂₀ × ¼ or ⅕ × ¼ × ½ × ½.
The lemon region covers ⅟₈₁ of its outer green square. Pupils might find the fraction in a step by step way (though this could get quite complicated) or by deriving and evaluating an expressions such as ⅟₉ × ⅟₉.
TASK 19G: This is more straightforward. We can express the fractions as, say, ⅟₂₅ × ½ and ⅟₇ × ⅟₇ respectively. So the lemon region is slightly larger.
TASK 19H: A nice simple one to finish with. We have ⅟₂₅ × ¼, say, and ⅟₁₀ × ⅟₁₀, say. So they are both ⅟₁₀₀.
