6.9.3 Lesson: Rectangle Madness
Rectangle ABCD is not a square. Rectangle ABEF is a square.
Suppose segment were 5 units long and segment were 2 units long. How long would segment be?
Suppose segment were 10 units long and segment were 6 units long. How long would segment be?
Suppose segment were 12 units long and segment were 5 units long. How long would segment be?
Suppose segment were 9 units long and segment were 5 units long. How long would segment be?
Rectangle JKXW has been decomposed into squares. Segment JK is 33 units long and segment JW is 75 units long. Find the areas of all of the squares in the diagram.
Rectangle ABCD is 16 units by 5 units. In the diagram, draw a line segment that decomposes ABCD into two regions: a square that is the largest possible and a new rectangle.
Draw another line segment that decomposes the new rectangle into two regions: a square that is the largest possible and another new rectangle. Keep going until rectangle is entirely decomposed into squares. List the side lengths of all the squares in your diagram.
The diagram shows that rectangle has been decomposed into three squares. What could the side lengths of this rectangle be?
How many different side lengths can you find for rectangle ?
What are some rules for possible side lengths of rectangle ?
Draw a rectangle that is 21 units by 6 units.
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. How many squares of each size are in your diagram?
What is the side length of the smallest square?
Draw a rectangle that is 28 units by 12 units.
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. 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?
Accurately draw a rectangle that is 9 units by 4 units.
Accurately draw a rectangle that is 27 units by 12 units.
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?
We have seen some examples of rectangle tilings. A tiling means a way to completely cover a shape with other shapes, without any gaps or overlaps.
For example, here is a tiling of rectangle with 2 large squares, 3 medium squares, 1 small square, and 2 tiny squares.
Some of the squares used to tile this rectangle have the same size.
Might it be possible to tile a rectangle with squares where the squares are all different sizes?
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.