Prior to the lesson, ask students to bring in several flat, circular objects that they wish to measure with their classmates.
As a warm-up, give students an opportunity to estimate the area of
the circular objects that they have brought to class. Working in groups
and using the Area of Circles Activity Sheet, students should individually complete the first two columns:
- Description of the object
- Their estimate for the area of the object
(The other two columns will be completed later in the lesson.)
Area of Circles Activity Sheet
Students may use any method they like to estimate the area of their objects. Some possible methods include:
- Students can trace the shape of their object on a piece of
centimeter grid paper and count how many square centimeters make up the
total area of the circle.
- Students can divide the circle into wedges by drawing
various radii. They can approximate the area of each wedge using the
triangle formula. (This method is similar to a method used by
Archimedes, and it is the method that will be used later in this
lesson. For a connection to mathematical history, you may want to
include a brief overview of Archimedes and his method for calculating the area of a circle.)
- Students can inscribe the circle in a square, hexagon, or some
other polygon. Then, the same shape could be inscribed within the
circle. Students could determine the area of the inscribed and
circumscribed shapes to get lower and upper estimates, respectively.
(You may need to provide a sample drawing of this method, like the one
shown below.)
After students have estimated the area of several objects, allow
them to physically discover the area formula of a circle. Since this is
a whole-class activity, you may wish to enlarge the manipulatives and
display them on the chalkboard, or you can use them on the overhead
projector.
Distribute the Fraction Circles Activity Sheet.
Fraction Circles Activity Sheet
Have students cut the circle from the sheet and divide it into
four wedges. (This can be done if students cut only along the solid
black lines.) Then, have students arrange the shapes so that the points
of the wedges alternately point up and down, as shown below:
Ask, "When arranged in this way, do the pieces look like any
shape you know?" Students will likely suggest that the shape is
unfamiliar.
Then, have students divide each wedge into two thinner wedges
so that there are eight wedges total. (This can be done if students cut
only along the thicker dashed lines.) Again, have students arrange the
shapes alternately up and down. Again ask if this arrangement looks
like a shape they know. This time, students will be more likely to
suggest that the arrangement looks a little like a parallelogram.
Finally, have students divide each wedge into two thinner
wedges so that there are sixteen wedges total. (This can be done if
students cut along all of the dashed lines.) Allow students to arrange
the wedges so that they alternately point up and down, as shown below:
Ask, "When the circle is divided into wedges and arrange like
this, does it look like another shape you know? What do you think would
happen if we kept dividing the wedges and arranging them like this?"
Lead the discussion so students realize the shape currently resembles a
parallelogram, but as it is continually divided, it will more closely
resemble a rectangle .
You may wish to continue this activity by having students divide the wedges even further.
Ask students, "What are the dimensions of the rectangle that is formed?" From the previous lesson, students should realize that the length of the rectangle is equal to half the circumference of the circle, or πr. Additionally, it should be obvious that the height of this rectangle is equal to the radius of the circle, r. Consequently, the area of this rectangle is πr × r = πr2. Because this rectangle is equal in area to the original circle, this activity gives the area formula for a circle:
A = πr2The figure below shows how the dimensions lead to the area formula.
Allow students to return to the objects for which they estimated
the area at the beginning of class. They should measure the radius of
each object and record it in the third column on the Area of Circles
sheet. Then, students should use the formula just discovered, calculate
the actual area of each object, and record the area in the fourth
column.
Once all groups have completed the measurements and
calculations, a whole-class discussion and presentation should follow.
On the chalkboard, the presenter for each group should record the areas
for the objects. The students should compare the results of each group
and discuss the accuracy of the areas found.
The class should also compare their original estimates with
the actual measurements. On their recording sheets, have them highlight
the objects for which their estimates were very close to their actual.
Using a few sentences, have the students explain (on the recording
sheet) why some estimates were closer than others.
During the class discussion, the following are some key points to highlight:
- Emphasize that 3.14 is only one approximation for π. Refer to the previos
lesson, and discuss the various estimates that were found for π and
what caused these variations. Also explain that there are other
approximations, but typically 3.14 is used because it is accurate
enough for most situations and it is easy to remember. If students are
curious, other approximations for π are given on the Pi Approximation Activity Sheet.
- The total area is almost always an approximation. Because the
value of π can only be approximated, any time the area of a circle is
stated without the π symbol, it must be an approximation. For instance,
a circle with radius of 5 inches has an exact area of 25π in.2 and an approximate area of 78.54 in.2.
You might wish to hold a "mock debate" with one student taking each
position (yes, it’s always an exact value; no, it’s not an exact value)
giving examples and reasons to justify their position.
- Students should be able to calculate radius from diameter and diameter from radius. In particular, students should realize that d = 2r.
- Students should understand the area formula as described in
your curriculum. Slight variations are possible, so the version in your
textbook, standards, or other materials may be different from the
formula presented in this lesson.
Questions for Students
1. In your opinion, why did we use the properties of a parallelogram to discover the area formula for circles?
[Determining the area of a circle is difficult. By converting a circle to a parallelogram, we can use the formula for the area of a parallelogram to determine the area of the circle.]
2. When would it be necessary to know the exact area of a circle? When would an estimate be sufficient? Explain your thinking.
[Student responses may vary.]
3.Why did we approximate our answers for area? Can the area of a circle ever be exact?
[It is not possible to find an exact numeric value for π. Therefore, all calculations of area must be approximations (unless the answer is left in "exact form," which means using the symbol π to express the answer).]
Teacher Reflection
- When students were working in pairs to find the area of their
assigned circular objects, how precise were the students’ measurements
and area calculations? When the results were discussed as a class, did
those students who were not as precise while measuring demonstrate an
understanding of how to get more precise measurements? Or did all
students get basically the same results?
- Did students use both metric and customary units of measure?
With which were they more comfortable, and would future measurement
lessons make them comfortable with the other?
- Were concepts presented too abstractly? Too concretely? How would you change the presentation if this lesson were taught again?
- How do you know that students were actively engaged in the learning process?
- What content areas did you integrate within the lesson? Was this integration appropriate and successful?
- Did you find it necessary to make adjustments while teaching
the lesson? If so, what adjustments, and were these adjustments
effective?
