IM: 6.9.3: Rectangle Madness (Lesson)

Rectangle ABCD is not a square. Rectangle ABEF is a square.

Suppose segment AF were 5 units long and segment FD were 2 units long. How long would segment AD be?

Suppose segment BC were 10 units long and segment BE were 6 units long. How long would segment EC be?

Suppose segment AF were 12 units long and segment FD were 5 units long. How long would segment FE be?

Suppose segment AD were 9 units long and segment AB were 5 units long. How long would segment FD 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 ABCD is entirely decomposed into squares.
List the side lengths of all the squares in your diagram.

View rectangle VWYZ.

The diagram shows that rectangle VWYZ 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 VWYZ?

What are some rules for possible side lengths of rectangle VWYZ?

Follow the following steps in drawing a rectangle and answer the questions.

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 the diagram shows that 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?

Follow the following steps in drawing a rectangle and answer the questions.

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 the diagram shows that your original rectangle has been decomposed into squares.

How many squares of each size are in your diagram?

What is the side length of the smallest square?

Write each of these fractions as a mixed number with the smallest possible numerator and denominator:

\frac{16}{5}

\frac{21}{6}

\frac{28}{12}

What do the fraction problems have to do with the previous rectangle decomposition problems?

Follow the following steps in drawing a rectangle and answer the questions.

Accurately draw a rectangle that is 9 units by 4 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 there?

What are the side lengths of the last square you drew?

Write \frac{9}{4} as a mixed number.

Follow the following steps in drawing a rectangle and answer the questions.

Accurately draw a rectangle that is 27 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 there?

What are the side lengths of the last square you drew?

Write \frac{27}{12} as a mixed number.

Compare the diagram you drew for this problem and the one for the previous problem. How are they the same? How are they different?

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.

Follow the following steps in drawing a rectangle and answer the questions.

Accurately draw a 37-by-16 rectangle.

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 there?

What are the dimensions of the last square you drew?

What does this have to do with 2 + \frac{1}{3 + \frac{1}{5}}

Follow the following steps in drawing a rectangle and answer the questions.

Consider a 52-by-15 rectangle.

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.

Write a fraction equal to this expression: 3 + \frac{1}{2 + \frac{1}{7}}

Notice some connections between the rectangle and the fraction.

What is the greatest common factor of 52 and 15?

Follow the following steps in drawing a rectangle and answer the questions.

Consider a 98-by-21 rectangle.

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.

Write a fraction equal to this expression: 4 + \frac{1}{1 + \frac{7}{14}}

Notice some connections between the rectangle and the fraction.

What is the greatest common factor of 98 and 21?

Consider a 121-by-38 rectangle.

Suppose segment AF were 5 units long and segment FD were 2 units long. How long would segment AD be?

Suppose segment BC were 10 units long and segment BE were 6 units long. How long would segment EC be?

Suppose segment AF were 12 units long and segment FD were 5 units long. How long would segment FE be?

Suppose segment AD were 9 units long and segment AB were 5 units long. How long would segment FD 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.

The diagram shows that rectangle VWYZ 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 VWYZ?

What are some rules for possible side lengths of rectangle VWYZ?

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 the diagram shows that 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 the diagram shows that your original rectangle has been decomposed into squares.

How many squares of each size are in your diagram?

What is the side length of the smallest square?

\frac{16}{5}

\frac{21}{6}

\frac{28}{12}

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.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 there?

What are the side lengths of the last square you drew?

Write \frac{9}{4} as a mixed number.

Accurately draw a rectangle that is 27 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 there?

What are the side lengths of the last square you drew?

Write \frac{27}{12} as a mixed number.

Compare the diagram you drew for this problem and the one for the previous problem. How are they the same? How are they different?

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?

Accurately draw a 37-by-16 rectangle. 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 there?

What are the dimensions of the last square you drew?

What does this have to do with 2 + \frac{1}{3 + \frac{1}{5}}

Consider a 52-by-15 rectangle.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.

Write a fraction equal to this expression: 3 + \frac{1}{2 + \frac{1}{7}}

Notice some connections between the rectangle and the fraction.What is the greatest common factor of 52 and 15?

Consider a 98-by-21 rectangle.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.

Write a fraction equal to this expression: 4 + \frac{1}{1 + \frac{7}{14}}

Notice some connections between the rectangle and the fraction.What is the greatest common factor of 98 and 21?

Use the decomposition-into-squares process to write a continued fraction for \frac{121}{38}. Verify that it works.

What is the greatest common factor of 121 and 38?

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.

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 the diagram shows that your original rectangle has been entirely decomposed into squares.

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 the diagram shows that your original rectangle has been decomposed into squares.

Accurately draw a rectangle that is 9 units by 4 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.

Accurately draw a rectangle that is 27 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.

Accurately draw a 37-by-16 rectangle.

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.

Consider a 52-by-15 rectangle.

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.

Notice some connections between the rectangle and the fraction.

What is the greatest common factor of 52 and 15?

Consider a 98-by-21 rectangle.

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.

Notice some connections between the rectangle and the fraction.

What is the greatest common factor of 98 and 21?