How Size Corresponds with Area
Grade level:
7th grade (but could be modified
for lower or upper grades)
Mathematic Topic:
Geometry, specifically area
Materials Needed:
teacher:
overhead projector, transparency of triangular graph paper, scanner (optional),
computer overhead projector (optional), 3-D equilateral triangle manipulatives,
hand-out of area chart (see last page for copy of hand-out) (9)
student:
four pieces of triangular graphing paper, pencil
Prerequisite Knowledge and
Skills:
knowledge of area concepts (area
is length x width/base x height) (2)
Overview of Lesson:
Students will construct their own
shape using the triangular manipulatives. They will then double, triple,
and quadruple the size of their shape. Each time, they will be asked to
examine the area of their larger shape compared to the area of the original
shape. Students will begin to see a pattern emerging: i.e.--students will
see that when any shape is doubled, its area is always quadrupled; when
a shape's size is tripled, the area is 9 times as large as the area of the
original shape; when a shape's size is quadrupled, the area is 16 times
as large as the area of the original shape, and so on. (3, 4, 5)
Activities:
(6, 11, 12, 15)
1.) Have students pick up equilateral
triangle manipulatives as they enter the classroom. Encourage students to
individual "play" with the manipulatives, asking them what they
observe about them (size, shape, dimension, color, what they are made of,
etc.). (10)
2.) Ask students to construct a
shape, any shape (regular of irregular), using their equilateral triangles.
Recommend that the students use no more than 12 triangles for this first
shape. (1)
3.) Ask students to count the number
of triangles used in their shape. Have them record this number on their
area chart under "number of triangles in original shape". Tell
students that the number of triangles in their shape represents the area
of their shape. Have students graph their shape onto the triangular graphing
paper. If possible, scan some student's shapes and ask the class to determine
the shape's area.
4.) Now, ask students to double
the size of their shape. Before they begin, ask them what they think doubling
the size involves; i.e.: can you double the shape by doubling just one side?
Or is it necessary to double both sides? Watch carefully to make sure that
the student's are doubling their shapes. Did they double the perimeter of
both length and width? It may be necessary to show them an example of a
shape, then the shape incorrectly doubled and correctly doubled. Ask them
how they know that one has been doubled correctly. When this idea was tested
on a student, the student came up with the above shapes, incorrectly doubling
the original shape. These shapes can be used as an example (7, 14):
Ask the students which shape they
feel is correctly doubled. Why? (for teachers: example A is the original
shape, example B is incorrectly doubled, example C is correctly doubled).
5.) When the students have doubled
their shape properly, have students graph their shape and find the area.
Chart the area under "number of triangles in the doubled shape".
Ask what they notice about the area of the new shape compared to the area
of the original shape. If possible, scan a student's original shape (noting
the area) and then scan the student's doubled shape. Examine the difference
between the two areas. How much larger is the area of the doubled shape?
6.) Now, have student's triple the
size of their original shape. Once again, discuss how they will know when
the shape is tripled. Graph the shape and chart the area. Compare the area
of the tripled shape to the original shape. Students should begin to see
a pattern emerging.
7.) Have the students quadruple
their shapes (it may be necessary to group the students in pairs to allow
for enough triangle manipulatives to quadruple the shapes). Graph the shape
onto the triangle graph paper and chart the area under "number of triangles
in the quadrupled shape". Once again, note the difference between the
area of the quadrupled shape compared to the area of the original shape.
8.) Ask students if they see a pattern.
Why do they think this pattern is occurring? Is the pattern the same for
everyone, regardless of the shape they used? If students are slow to respond,
ask how they doubled their original shape. Did they double both sides? What
is two times two? Point out that the area of the doubled shape is
four times as large as the original shape. Do they see a similar
pattern for the tripled and quadrupled shapes? (8)
Wrap-Up:
(13)
Have students answer the last two
questions on the "Area Chart" hand-out. Ask students if they have
ever made an image larger on a copy machine. If so, do they think that the
machine actuallydoubled their image? As homework, have students make
a shape on the triangular paper. Ask them to copy this image on a copy machine,
giving the machine a command to make the image two times larger than the
orginal. What can they see? Did the machine double the image? Is the area
what they expected it to be (4 x's as large as the original)?
AREA CHART
1.) Area
of original shape: ____ triangles
2.)
Area of doubled shape: ____ triangles
3.) Area
of tripled shape: ____ triangles
4.)
Area of quadrupled shape: ____ triangles
What do you think the area
will be if you make your shape five times as large as your original shape?
How about ten times as large?
Triangular Graphing Paper
