Measurement progression, 5th step
The purpose of the activity
In this activity, the learners develop an understanding of how to calculate the area of a circle. This activity should follow the Circumferences activity above.
The teaching points
- The learners understand that the area of a circle can be calculated by multiplying the radius squared by pi (π).
- The learners are able to estimate the area of a circle.
- The learners can identify the diameter, radius and circumference of a circle.
- The learners know that the radius is half the diameter.
- The learners know that the circumference of a circle is approximately three times, and exactly pi (π) times, the diameter.
- The learners understand that r squared (r2) means r x r.
- A variety of sizes of paper circles – a pair of each size.
The guided teaching and learning sequence
1. Ask the learners to discuss what they already know about the area of the circle. Record their responses on the board, including the formula A = πr2 if it is given.
2. Give each pair of learners two paper circles of the same size and ask them to draw a diameter and a radius on one circle.
3. Discuss what they have done:
“Does a diameter always pass through the centre point?” (yes)
“Does the radius always end at the centre point?” (yes)
“What is the relationship between the radius and the diameter?” (The diameter is twice the radius.)
“Identify the circumference of the circle.”
“Is there a relationship between the diameter and the circumference?” (yes)
“What is the relationship approximately and what symbol is used to describe it?” (The circumference is approximately three times the diameter and the symbol used to describe it is π.)
4. Ask the learners to discuss how they could estimate the area of their circle.
You may need to remind them that the area is the number of square units needed to completely cover the circle.
Methods of estimation could include:
- estimating the number of square centimetres that would fit on the circle
- estimating what fraction of a square metre the circle is
- calculating the area of the square that the circle fits within, the side being the diameter of the circle, as an estimation of the area of the circle.
5. Ask the learners to estimate the area of their circle and record their estimation.
6. Explain to the learners that there is a formula for calculating the area of a circle and refer to it if it was given at the beginning of the session. Say that the learners are going to undertake an activity to justify the use of that formula to calculate the area of a circle. Ask them to fold the circle in half, in half again, and then in half again. When opened there will be 8 ‘sectors’. Ask the learners to cut out the sectors and reassemble them to form an approximate parallelogram. Draw the required shape on the board. Alternatively you may demonstrate the process.
7. Ask the learners to compare their first circle with the parallelogram and ask:
“Is the area of the (approximate) parallelogram the same as the area of the circle?” (yes)
“How would you find the area of the parallelogram?” (Multiply the base by the height at right angles between them.)
“What part of the circle is the height of your parallelogram?” (radius)
“What part of the circle are the two long sides of your parallelogram?” (circumference)
“What part of the circle is one of the long sides of the parallelogram?” (half the circumference)
“If the full circumference is 2πr, what is half the circumference?” (πr)
“If the area of the parallelogram is found by multiplying the base by height, write this using π and r.” (π x r x r)
“In what other ways can you write r x r?” (r2)
“Do you accept that you can use A= π x r2 to find the area of a circle?”
8. Write the formula for the area of a circle A = πr2 and ask the learners to use this formula to calculate the area of their circle.
Pose problems that require the learners to use the formula discussed above.
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