Area Gymnastics: Rearranging Shapes to Find Area

Mathematics Grades 7th Grade 3:57 Video
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Lesson Description

Learn how to find the area of irregular shapes by rearranging their parts into familiar shapes like rectangles.

Video Resource

Finding area by rearranging parts | Geometry | 6th grade | Khan Academy

Khan Academy

Duration: 3:57
Watch on YouTube

Key Concepts

  • Area of rectangles
  • Area of composite figures
  • Geometric transformations (flipping/rearranging)
  • Decomposition of shapes

Learning Objectives

  • Students will be able to identify that rearranging the parts of a quadrilateral does not change its area.
  • Students will be able to decompose a complex quadrilateral into simpler shapes (rectangles and triangles).
  • Students will be able to calculate the area of a quadrilateral by rearranging its parts into a rectangle.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the formula for the area of a rectangle (Area = length x width). Ask students if they know how to find the area of shapes that aren't rectangles. Briefly introduce the concept of composite figures and that you can break these down into simpler shapes, or rearrange them. Then, introduce the Khan Academy video.
  • Video Viewing (7 mins)
    Play the Khan Academy video 'Finding area by rearranging parts | Geometry | 6th grade | Khan Academy'. Instruct students to pay close attention to how the green trapezoid is transformed into a rectangle.
  • Guided Practice (10 mins)
    After the video, work through a similar example on the board. Draw a trapezoid or another irregular quadrilateral. Guide students to identify a way to divide the shape and rearrange the parts to form a rectangle. Have students calculate the area of the resulting rectangle.
  • Independent Practice (10 mins)
    Provide students with worksheets containing various quadrilaterals. Instruct them to rearrange the parts of each quadrilateral to form a rectangle and then calculate the area. Ensure to include various difficulty levels. For a challenge, ask if it's possible to re-arrange to form *any* rectangle, or if the re-arranged rectangle is unique.
  • Wrap-up (3 mins)
    Review the key takeaways from the lesson. Emphasize that rearranging parts of a shape doesn't change its area, and this technique can be used to find the area of complex shapes.

Interactive Exercises

  • Shape Rearrangement Applet
    Use an online geometry applet (e.g., GeoGebra) to demonstrate the rearrangement of shapes in real-time. Students can manipulate the shapes themselves to visually confirm that the area remains constant.

Discussion Questions

  • Does rearranging a shape change its area? Why or why not?
  • Can you think of other shapes besides quadrilaterals that we could rearrange to find their area?
  • What are some real-world situations where you might need to find the area of an irregular shape?

Skills Developed

  • Spatial reasoning
  • Problem-solving
  • Geometric intuition
  • Area calculation

Multiple Choice Questions

Question 1:

Rearranging the parts of a shape to form a different shape does NOT change the:

Correct Answer: Area

Question 2:

If a quadrilateral can be rearranged into a rectangle with a length of 8 units and a width of 5 units, what is the area of the original quadrilateral?

Correct Answer: 40 square units

Question 3:

Which of the following shapes is easiest to calculate the area of?

Correct Answer: Rectangle

Question 4:

What does it mean to 'decompose' a shape?

Correct Answer: To break it down into smaller shapes

Question 5:

A trapezoid is rearranged into a rectangle. If the area of the rectangle is 36 square cm, what was the area of the original trapezoid?

Correct Answer: 36 square cm

Question 6:

Which geometric figure has four sides and only one pair of parallel sides?

Correct Answer: Trapezoid

Question 7:

What is the formula for the area of a rectangle?

Correct Answer: Length x Width

Question 8:

When rearranging a quadrilateral into a rectangle, what property remains constant?

Correct Answer: Area

Question 9:

Why might rearranging a quadrilateral into a rectangle be helpful?

Correct Answer: It's easier to calculate the area of a rectangle

Question 10:

Which of the following tools would be helpful in rearranging shapes and calculating their areas?

Correct Answer: Ruler

Fill in the Blank Questions

Question 1:

The area of a rectangle is found by multiplying its _____ and width.

Correct Answer: length

Question 2:

Breaking down a shape into smaller shapes is called _____.

Correct Answer: decomposition

Question 3:

A quadrilateral with only one pair of parallel sides is called a ______.

Correct Answer: trapezoid

Question 4:

The area of a shape is measured in _____ units.

Correct Answer: square

Question 5:

Rearranging a shape changes its _______, but not its area.

Correct Answer: perimeter

Question 6:

If a rectangle has a length of 10 cm and a width of 6 cm, its area is ______ square cm.

Correct Answer: 60

Question 7:

A polygon is a closed shape made up of straight _______.

Correct Answer: sides

Question 8:

The amount of space inside a two-dimensional shape is called its ______.

Correct Answer: area

Question 9:

The process of finding the area by rearranging parts relies on the principle that the area remains ________.

Correct Answer: constant

Question 10:

A tool used for measuring length and width is a ______.

Correct Answer: ruler

Educational Standards

7.GM.2:Use mathematical models and problems to calculate and justify the area of trapezoids and the area and perimeter of composite figures with rational measurements. 7.GM.2.2:Find the area and perimeter of composite figures.

Teaching Materials

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