Circles Lesson 2
Discover the area formula of circles by separating into congruent shapes and using their understanding of other polygons.
As an Introductory activity, distribute the Fraction Circles Activity Sheet (download from Materials section) to student pairs. Have students highlight each part of a circle they know and recognize using a different color. Be sure students are identifying the radius and the diameter. In particular, students should realize that d = 2r. Students should be able to calculate radius from diameter and diameter from radius. Monitor student progress to check for any misconceptions.
Strategy for differentiation:
If necessary, give some students a word bank with the vocabulary: circumference, diameter, and radius and discuss parts of a circle with students.
Give students an opportunity to estimate the area of the circular objects that they have brought to class.
Working in small groups and using the Area of Circles Activity Sheet (download from Materials section), students should individually complete the first two columns:
Note: The other two columns will be completed later in the lesson.
Strategy for differentiation: Another method would be to have students estimate the area of circles using centimeter grid transparencies and cut out circles. Cut out circles of various sizes and give a set to each small group of students along with centimeter grid paper or centimeter grid paper transparency. Students would be able to trace the circles using pencils or dry erase markers and approximate the area of each circle by counting the number of squares.
Students may use any method they like to estimate the area of their objects.
Some possible methods include:
In pairs or small student groups, have students cut the circle from the sheet and divide it into four wedges. (This can be done if students cut only along the bold, solid, black lines.) Then, have students arrange the shapes so that the points of the wedges alternately point up and down, as shown below:
Different parts of the circle (radius and circumference) should be highlighted in a color from the Introductory Activity.
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 longer and thicker dashed lines.) Have students try and arrange the smaller wedges into a polygon they are familiar with. Students may take some time in determining the polygon. Allow them to think about and engage in productive struggle with this part of the activity.
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:
Facilitate 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.
Students should realize that the length of the rectangle is equal to half the circumference of the circle, or πr. Additionally, students should recognize that the height of this rectangle is equal to the radius of the circle, r.
Have students try and generate a formula for area of this new rectangle formed by the pieces of the circle. 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 = πr2
The figure below shows how the dimensions lead to the area formula.
Have a class discussion with students explaining that total area is almost always an approximation.
Using the highlighted circle from the Introductory Activity will help students to more easily identify the dimensions of the newly formed rectangle.
Watch for possible misconceptions:
Difficulty using the variables C, d, and r; and students not recognizing that the base of the parallelogram is only ½ of the circumference.
When returning to large group discussion, verify students understand and can apply the appropriate formula for area of a circle A = πr2
Ask students to return to the objects they estimated the area of at the beginning of class. Refer to the Apple Pi Activity Sheets from the Circumference of Circles Lesson, and have students calculate the radius of each circle using the diameter. Then, students should use the formula just discovered, calculate the actual area of each object, and record the area in the fourth column.
Have a class discussion about similarities and differences of the areas of the various circles.
The class should also compare their original estimates with the actual measurements. On their recording sheets, have them highlight their objects' estimates that were very close to their actual. At the bottom of the recording sheet, students should explain why they thought some estimates were closer than others.
Students can solve the following practice problems:
Do the following lesson: The Great Cookie Dilemma
In this lesson, students explore two different methods for dividing the area of a circle in half, one of which uses concentric circles. The first assumption that many students make is that half of the radius will yield a circle with half the area. This is not true, and it surprises students. In this lesson, students investigate the optimal radius length to divide the area of a circle evenly between an inner circle and an outer ring.
Possible journal entry or small group task:
Given the circumference of a circular object, how can you identify the area of this object? Justify your answers with mathematical thinking.
Given area of a circular object, how can you identify the circumference of this object? Justify your answers with mathematical thinking.
Leave your thoughts in the comments below.
How can we derive the formula for area of circles?
CCSS, Content Standards to specific grade/standard
CCSS, Standards for Mathematical Practices
PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS