Lesson 2

Finding Area by Decomposing and Rearranging

Let’s create shapes and find their areas.

Problem 1

The diagonal of a rectangle is shown.

rectangle on a grid with a diagonal drawn from top left to bottom right corner. base = 3 units, height = 2 untis.

  1. Decompose the rectangle along the diagonal, and recompose the two pieces to make a different shape.

  2. How does the area of this new shape compare to the area of the original rectangle? Explain how you know.

Problem 2

Priya decomposed a square into 16 smaller, equal-size squares and then cut out 4 of the small squares and attached them around the outside of original square to make a new figure.

How does the area of her new figure compare with that of the original square?

4 by 4 square on left. Irregular shape composed of unit squares on right.
A:

The area of the new figure is greater.

B:

The two figures have the same area.

C:

The area of the original square is greater.

D:

We don’t know because neither the side length nor the area of the original square is known.

Problem 3

The area of the square is 1 square unit. Two small triangles can be put together to make a square or to make a medium triangle.

Three figures. A square, a small triangle, and a medium triangle.

Which figure also has an area of \(1\frac 12\) square units? Select all that apply.

Four figures labeled A, B, C, and D.
A:

Figure A

B:

Figure B

C:

Figure C

D:

Figure D

Problem 4

The area of a rectangular playground is 78 square meters. If the length of the playground is 13 meters, what is its width?

(From Unit 1, Lesson 1.)

Problem 5

A student said, “We can’t find the area of the shaded region because the shape has many different measurements, instead of just a length and a width that we could multiply.”

A multi-sided figure.

Explain why the student’s statement about area is incorrect.

(From Unit 1, Lesson 1.)