Suppose segment [math]AF[/math] were 5 units long and segment [math]FD[/math] were 2 units long. How long would segment [math]AD[/math] be?[br]
Suppose segment [math]BC[/math] were 10 units long and segment [math]BE[/math] were 6 units long. How long would segment [math]EC[/math] be?[br]
Suppose segment [math]AF[/math] were 12 units long and segment [math]FD[/math] were 5 units long. How long would segment [math]FE[/math] be?[br]
Suppose segment [math]AD[/math] were 9 units long and segment [math]AB[/math] were 5 units long. How long would segment [math]FD[/math] be?[br]
Draw another line segment that decomposes the [i]new[/i] rectangle into two regions: a square that is the largest possible and another new rectangle.[br][br]Keep going until rectangle [math]ABCD[/math] is entirely decomposed into squares.[br][br]List the side lengths of all the squares in your diagram.
The diagram shows that rectangle [math]VWYZ[/math] has been decomposed into three squares. What could the side lengths of this rectangle be?[br]
How many different side lengths can you find for rectangle [math]VWYZ[/math]?
What are some rules for possible side lengths of rectangle [math]VWYZ[/math]?
[size=150]In your rectangle, draw a line segment that decomposes the rectangle into a new rectangle and a square that is as large as possible. Continue until your original rectangle has been entirely decomposed into squares.[br][br][size=100]How many squares of each size are in your diagram?[/size][/size]
What is the side length of the smallest square?
[size=150]In your rectangle, draw a line segment that decomposes the rectangle into a new rectangle and a square that is as large as possible. Continue until your original rectangle has been entirely decomposed into squares.[/size][br][br]How many squares of each size are in your diagram?
What is the side length of the smallest square?
What do the fraction problems have to do with the previous rectangle decomposition problems?
What is the greatest common factor of 9 and 4? What is the greatest common factor of 27 and 12? What does this have to do with your diagrams of decomposed rectangles?
For example, here is a tiling of rectangle [math]KXWJ[/math] with 2 large squares, 3 medium squares, 1 small square, and 2 tiny squares.[br][br][img]data:image/png;base64,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[/img][br][br]Some of the squares used to tile this rectangle have the same size.[br][br]Might it be possible to tile a rectangle with squares where the squares are [i]all different sizes[/i]?[br][br]If you think it is possible, find such a rectangle and such a tiling. If you think it is not possible, explain why it is not possible. You may use the applet below to help you explain.