Measurement progression, 5th step
The purpose of the activity
In this activity, the learners develop an understanding of how to calculate the circumference of circles and cylinders by using the radius or diameter measure.
The teaching points
- The learners understand that the circumference of a circle can be calculated by multiplying the diameter by pi (π) or by multiplying the radius by two times pi (π).
- The learners understand the need for calculating the circumference indirectly from measurements of the radius and diameter rather than measuring directly.
- The learners know that ‘3.14’ (or 3) is an approximation for pi (π).
- The learners can measure length using millimetres.
- Discuss with the learners relevant or authentic situations where it is necessary to understand how to calculate the circumference.
- Rulers and measuring tapes (marked in millimetres).
- A selection of cylinders.
- Compasses and pencils.
- Adhesive tape.
The guided teaching and learning sequence
1. Begin the session by asking the learners to use a compass to draw the following circles:
- the smallest circle that can be drawn
- the largest circle
- four circles of different size that have the same centre point.
2. Discuss the features of a circle with the learners.
“Where is the diameter of a circle?” “Does it always go through the centre point?”
“Where is the radius?” “Does it always have one end on the centre point?”
“How could you find the diameter if the centre point wasn’t marked?” (Fold the circle in half.)
“How could you find the radius if the centre point wasn’t marked?” (Fold the circle in quarters.)
3. Ask the learners to work in pairs to collect together at least five circles or cylinders of different sizes (this could include the circles drawn previously). Ask them to measure the radius, the diameter and the circumference of each circle as accurately as they can and record their measurements in millimetres on a chart. Provide them with a collection of measuring tapes, rulers, string and tape.
4. If the learners have difficulty measuring the circumference, you may want to discuss their problems and their solutions. One way to measure the circumference is to wrap string (which is flexible) around the outer edge of the circle, using tape to hold the string in place and then removing the string to measure it against a ruler.
5. After the learners have recorded the measurements of several circles on the chart, challenge them to think about the relationship between the diameter and the circumference.
“Can you see a relationship between the lengths of the diameter and circumference?”
You may need to prompt further by encouraging the learners to use a calculator to divide each of the circumferences by the diameter and notice that the result is close to 3.
6. Ask the learners to divide each of their circumferences by the diameter and to record their outcomes on a ‘class’ number line marked in tenths from 2 to 4.
7. If the measurements have been reasonably accurate, the outcomes should be clustered around 3.1 and 3.2, and the learners will be able to ‘see’ the relationship.
8. Write the formula C = πd on the board. Ask:
“What do you think the symbols are?”
9. After the learners have given you their ideas, you may need to tell them that the symbol is read as ‘pi’ and that pi has the value of approximately 3.14.
10. Write the formula C = 2πr on the board.
“What might this be?” “Will you get the same answer as C = πd?” “Why?” Have the learners share their solutions.
Follow up by posing problems that require the learners to apply the formula. Conclude by posing a problem that requires the learners to apply the formula C = πd.
1. The largest Ferris wheel in the world is in Yokahama City, Japan. It is 100 metres tall. About how far do you ride when you go around this Ferris wheel once?
(C = 100 x 3.14, so one turn is approximately 314 metres.)
2. A large pizza has a circumference of 100 centimetres. What is the side length of the smallest box capable of holding the pizza?
(100 = 3.14 x d so d = 100/3.14 = 31.8 centimetres. The side length of the box needs to be at least 31.4 centimetres to hold the large pizza.)
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