Developing an Understanding
of Area with Index Cards
Jim Rahn
www.jamesrahn.com
[email protected]
If we draw a rectangle on an index card
What do we mean by the area enclosed in the
rectangle?
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If we draw another rectangle on the same index card
How can we find the area if we don’t want to just count
the squares contained in the rectangle?
What do we mean by base x height?
What do we mean by base (or length)?
What do we mean by height (or width)?
Why can we multiple the base times the height (or the
length times the width) ?
height
base
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Take the card you were given and make one cut in it that
is not parallel to any of the sides of the card but passes
through two opposite sides.
Let’s rearrange the two pieces.
What shapes can you make?
parallelogram?
trapezoid?
triangle?
kite?
another shape?
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Let’s think what this says about their
area?
Let’s think about the parallelogram first. How was it
made?
We started with a rectangle with a base
and height.
height
base
We made a cut in the rectangle to form two
pieces.
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If we slide the brown section and placed it next to
the yellow section parallelogram – What happens?
How is area of the parallelogram related to the
area of the rectangle?
What happened to the original base of the rectangle?
Is the base of the parallelogram the same as the
base of the rectangle?
height?
How long is
the slanted
side? Does it
tell us
anything?
base
How many rows of boxes are in the parallelogram?
What measurement really describes the number of
rows of boxes in the figure?
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Has the area changed by making
the cut and rearranging the two
pieces?
What method would provide us
with the number of boxes
covering the parallelogram?
Let’s look at another way to
put the two pieces together
Let’s form a trapezoid
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A trapezoid can be made if we
reflect and slide the yellow piece
to the left.
How is the area of the trapezoid related
to the area of the original rectangle?
Where are
the base
and height
of the
rectangle?
The area is the same since the whole card
was just rearranged to make this new
shape.
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The height is running vertically through the
trapezoid from one base to another.
base2
But the 2 bases of
the rectangle have
been rearranged to
form two new bases
of the trapezoid.
height
So the length
of the rectangle
is half the sum
of the bases of
the trapezoid.
base1
The total of the two new bases is equal to the total
of the two rectangle bases.
Another way to say it
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If you add the two bases together you have
twice the length of the rectangle.
base2
height
base1
How would you determine the length of the rectangle
from the trapezoid?
How would you find the area of the rectangle from
the trapezoid?
Here is even another way to
look at it
Think about the formula for the area of a
trapezoid
A=
1
( b1 + b2 ) h
2
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base2
What will adding (base1 and base2 ) find?
base1
base2
base1
Base1 + Base2 = sum of 2 rectangle bases
So (base1 + base2 ) x height would be the
same as (2 rectangle bases) x height or the area
of two rectangles.
But we know the trapezoid is made from one
rectangle, not two. So the area of the trapezoid
must be one-half of (base1 and base2 ) x height .
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Let’s look at the trapezoid another way
Take two new index cards and place them on top of each
other.
Cut out a trapezoid from the two cards while holding them
together.
You have two trapezoids. Can you put these two
trapezoids together to form a shape we already know
how to find the area?
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base 2
+
base 1
height
base 1
+
base 2
Where are the original bases of the trapezoid?
How long is the new parallelogram? How many
squares would be along the bottom row of the
parallelogram?
What tells you the how many rows of squares will
fit in the parallelogram?
How will you decide how many squares are in the
parallelogram? (base1 + base 2) x height
How does the area of the parallelogram compare
to the original trapezoid?
How will you find the area of one trapezoid?
one half of (base1 + base 2) x height
base 2
+
base 1
+
base 2
height
base 1
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Another way to look at the trapezoid
Using a large index card, draw a trapezoid by using two of the
parallel blue lines for the two bases of the trapezoid.
Place two index cards together and cut out two copies of your
trapezoid.
Put them together
Can you re-arrange the two shapes to create a “new” shape for
which you already know how to find the area?
How do you find the area of this new figure? How is it’s area
related to the original trapezoid.
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How long is the new base of the parallelogram? How is it
related to the original trapezoid?
How tall is the parallelogram? How is it related to original
trapezoid?
How would you find the area of a trapezoid?
How does developing
the area of a trapezoid
help you with problems
like. . .
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A trapezoid has a short base of 5 and a long base of 8 and a height
of 5, how many squares does it contain?
A trapezoid has a bases of 4 and 6 and an area of 96 squares.
How long is the height of the trapezoid?
A trapezoid has a base of 5, a height of 8, and an area of 40
squares. How long is the other base?
Suppose we cut the index card like this?
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If we cut the rectangle like this we get two right triangles.
So what is the area of one of the triangles?
What would be a reasonable way to find the area of
this one right triangle?
The base of the rectangle is also the base of the
triangle.
The height of the rectangle is also the height of the triangle.
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But we can’t multiply base times height because
they would give us the total number of squares in
the whole rectangle.
It is clear that the area of the triangle would be
one-half the product of the base times the height?
There is another way to look at these two shapes.
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Let’s think about the two right triangle pieces being
rearranged to form a larger triangle.
From our last investigation we noticed that the
area of a triangle is equal to one-half the
product of the base times the height.
How does that fit with this drawing?
The area of the large triangle equals the area of
the rectangle.
So the area of this large triangle should be the
base of the rectangle times the height of the
rectangle.
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Let’s look at the measurements carefully.
2 bases
base
base
height
The base of the large triangle is really two bases
of the rectangle.
The height of the large triangle is the actual
height of the rectangle.
So if think about our triangle formula which said
that the area of the triangle is one-half the product
of the base and height . . .
base
base
height
This means we would have
1/ 2 x (2 rectangle bases) x height or
rectangle base x height
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Does it make sense?
base
base
height
rectangle base x height = area of rectangle
= area of large triangle.
Is there another way to put these two right triangle
together?
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If we rotate and then reflect the blue region we can
create a kite.
How is the area of the kite related to the area of the
rectangle?
Where is the base of the rectangle? Where is the
height of the rectangle?
base
height
base
height
How is the area of the kite related to the area of the
rectangle?
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This is a special kite where the shorter and longer
sides meet at a right angle.
To find the area of this kite you could just multiply
the rectangle base times the rectangle height.
This formula will not work all the time but an
important observation would be
the two right triangles form both a kite and a
rectangle whose areas are the same.
So maybe we could think about ways other kites
could be made.
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Try a more general situation with two small index
cards.
We’ll draw any scalene obtuse triangle on one card so
that one of the sides lines up with one of the lines on
the card.
Then we’ll place the two cards on top of each other and
cut out a duplicate set of triangles.
How many ways can we re-arrange the two
triangles to make a various quadrilaterals?
What can you form?
three different types of parallelograms
a kite
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Let’s Think about the
Parallelograms and the Triangles
First
Bases and Heights
How many bases does a triangle have?
How many heights does a triangle have?
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Method 1
Method 2
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Method 3
A different parallelogram is form each time two
congruent sides are matched up.
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If two triangles make a parallelogram, how do you
find the area of one triangle?
Try putting the triangles together for form a
different parallelogram.
Can you find the area of the parallelogram?
Can you find the area of the triangle?
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What does the base of a triangle really indicate?
What does the height of a triangle really indicate?
Each way we put the two triangles together –
they make parallelogram.
What measurements find the area of a parallelogram?
How does the area of a parallelogram help you find
the area of a triangle?
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Let’s make a kite
Rotate the pieces to form a kite.
First we might notice that two triangles make a
kite.
We could find the area of each triangle and then add
them together
Or find the area of one and double the area.
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The two altitudes of the two triangles are each onehalf of the short diagonal of the kite.
The base of the triangle is the long diagonal in the
kite.
If we rearrange the shapes again we see that we
have a parallelogram and so the parallelogram has
the same area as the kite.
The long diagonal would be the one base of the
parallelogram and the altitude of the triangle (1/2 of
the the short diagonal) would be the height.
This leads to a formula that says the area of a kite is
½ the product of the two diagonals.
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What have you learned?
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„
„
„
Do you understand more about finding the
area of a kite?
Do you understand more about find the area
of a trapezoid?
Is a formula absolutely necessary or is it
sufficient to understand more about the
concept of how to find the area of the
shapes?
Of the shapes studied, which two are the
most important?
What have you learned?
„
„
„
„
Do you understand more on how to find the
area of a rectangle?
Do you understand more on why a particular
method can be used to find the area?
Do you understand the relationship between
the area of a parallelogram and a rectangle?
Do you understand the relationship between
the area of a parallelogram and a triangle?
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How will an activity like this
help the student?
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„
„
„
Students can use the models to help
them solve problems.
Students understand where the
formulas come from.
Students see the relationship
between formulas.
Students see that formulas are found
by just rearranging the shapes.
Students can use this skill of rearranging
shapes to find the area of odd shapes.
Studying area like this helps students
develop the conceptual idea of area
‰
‰
It is finding the total number of squares covering a
surface.
It develops problem solving skills rather than just
memorization of formulas.
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Problems like this take on new
meaning
If a parallelogram had 15 squares along the base
and 90 total squares, how tall would the
parallelogram be?
If the two congruent right triangles can be put
together to make a rectangle, what do you know
about the area of one of the right triangles?
If the area of the right triangle is 30 squares and
the base is 6, how do you know the height is not
5?
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„
If a trapezoid containing 108 squares has
one base that is 12 and a height of 4, how
long is the other base?
If the kite had an area of 36 square units
with a long diagonal equal to 9 units, how
long is the short diagonal?
If the kite had long diagonal that was 12
units long and a short diagonal that was 6
units long, what would the area of the kite
be?
If the kite had an area of 36 square units
with a long diagonal equal to 9 units, how
long is the short diagonal?
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Why not give your students a
chance to visualize area
and develop their own way of finding
areas of rectangles, trapezoids, kites,
triangles, and even odd shaped figures?
Developing an Understanding
of Area with Index Cards
Jim Rahn
[email protected]
www.jamesrahn.com
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