The initial activity should be a free-draw for the students; that is,
they should create a random shape on a blank piece of paper, such as
To ensure that students do not draw a square, triangle, or some
other common shape, you may wish to model this by drawing an irregular
shape on the chalkboard or overhead projector.
Once all students have drawn a shape, ask them to estimate the
area of their shape using any method they choose. Some students may
overlay centimeter grid paper on top of the shape and then count
squares. Others may draw squares, rectangles, and triangles within
their shape and calculate the area of each polygon. Still others may
compare their shape with other objects for which they know the exact
dimensions and area, such as index cards or coins.
Allow students to compare their shape with a partner and
discuss how they estimated the area. After sharing with a partner, have
several students share their process with the entire class.
Next, distribute copies of the Polygons activity sheet, and have students cut out the shapes.
In their groups, ask students to create a design with their
shapes. The design should have no overlaps or gaps; in other words,
sides should touch to form an irregular figure. (Because students will
be handling these figures and moving them around, you might want to
copy them onto more durable paper.) Ask each group to discuss how they
would determine the area of the figure that they created.
Rather than arranging paper shapes, students can instead use the Patch Tool
to create a design. A printed copy can be used for measuring and
calculating the area. (Alternatively, students can determine the area
by considering the area of a triangle to be one unit, and all other
shapes can be described in a relative manner; for instance, a blue
parallelogram is equal to two triangles, so its area is two units.)
As a class, discuss the process of determining the area of
each shape and then adding the areas. Students should calculate the
area of each shape and record it directly on each shape, and then they
should determine the total area of their figure by adding the area of
all shapes that comprise their design.
Explain to students that composition is the act of putting
pieces together to form a whole, as they did with the shapes to create
a design. Conversely, decomposition is the act of breaking something
down into smaller parts, which is what they did to find the area of the
entire design. Tell them that they will use a process of decomposition
to determine the area of a larger space.
Identify a room in your school with an irregular shape (that
is, the room should not be completely rectangular; if possible, there
should be several angles that are not 90°.) Using that room as a basis,
pose the following problem, or a similar one, to students:
Our principal wants to know how much carpet (or how
many floor tiles) would be needed to cover the entire floor of this
room. He needs to know the exact area, because he doesn’t want to order
too much carpet and waste money, and he doesn’t want to order too
little and not be able to cover the entire floor. Your job is to
measure this room, determine the area of the floor, and write a letter
to the principal telling him how much carpet to buy and how you arrived
at your estimate.
If possible, take your students to the room around which the problem
is based and let them measure the dimensions. If that is not possible,
provide your students with a floor plan of the room. Alternatively,
instead of using a room within your school, you could choose a famous
room with irregular dimensions, such as the Octagon Room of the Royal
Observatory in Greenwich, England. A different example is shown below.
In pairs, students should calculate the area of the floor. Using
the ideas covered so far in this lesson, some pairs will naturally
decompose the room into simpler polygons and calculate the individual
areas. Other pairs will use other methods to estimate the area. After
all pairs have arrived at an answer, conclude the lesson by letting
students share their methods. Students who estimated will benefit from
hearing about the decomposition method. Students who decomposed the
room into simpler shapes will reinforce their estimation skills. Have
students compare the results with one another. How close were the
estimates to the actual area?
Questions for Students
1. What techniques did you use to estimate the area of the irregular figure that you drew? What other methods could you have used?
[There are many possible methods. One is to draw the figure on grid paper, or overlay the figure with transparent grid paper, and count the squares. Another is to approximate the shape with known figures such as triangles and squares, determine the area of each, and add them.]
2. What real-life applications are there to what was done in this lesson? What kinds of jobs or tasks would require someone to complete an activity like the one that you did?
[Any occupation that requires knowing the area of a surface would benefit from knowing how to decompose a shape into smaller figures to compute the area. Flooring specialists, carpenters, painters, .]
- When the students measured the dimensions of the room, how precise
were their measurements? Did they experience any difficulties while
measuring? What help did you need to offer students?
- Was your lesson appropriately adapted for the diverse learner?
- How did students demonstrate that they connected the area of
the individual shapes to the area of the larger shape? What words did
the students use to demonstrate that they understood the connection?
- How did your lesson address auditory, tactile and visual learning styles?