The focus for this unit is the measurement of circles and the relationships to be found within them. Students investigate a range of circles, each with a different radius, diameter and circumference and assemble this information in their search for relationships.
- Measure the circumference of circles
- State the relationship between the circumference and diameter of a circle
In this unit the studentsinvestigate the properties of a circle. The students will identify parts of a circle and define the terms, radius, circumference and diameter.
In their explorations the students will "discover" that the circumference is about 3 times the diameter. They can then be introduced to pi, either through a library exploration or with your guidance.
Interesting facts about pi
- The shorthand for pi is π
- One approximation for pi is 22/7.
- In the Bible (1 Kings 7: 23), the approximation for pi is 3.
- Another approximation is 3.14. Actually pi is a non-repeating, non-terminating decimal meaning it goes on forever. 3.14159265358979323 is just the beginning! People have calculated pi to thousands of decimal places.
- Pi is actually called an irrational number. This means that it cannot be written exactly as a fraction. (22/7 is just one of a number of good rational (fraction) approximations to pi.)
- Pi is also a transcendental number. This means that there is no polynomial equation that has pi as a solution.
- The first theoretical calculation of pi seems to have been carried out by Archimedes of Syracuse (287 - 212 BC). He obtained the approximation: 223/71 < π > 22/7
Range of circles and cans
- We begin the unit by drawing large circles outside on the ground. Give each pair of students a length of string and a piece of chalk (use different lengths of string of about 2 metres long). Challenge the students to construct a circle using the string and circle. (Tie the chalk to one end of the string. One student holds the non-chalk end fixed to the ground while the other walks around the fixed point drawing a circle.)
- When the circles are drawn ask the students to label the parts of the circle: circumference, diameter, radius.
- After they have labelled the circle tell the students that you want them to "measure" these parts. Ask for ideas of how they might do this, without using a ruler or tape measure. (Perhaps use foot lengths, hand spans.) Let them measure the radius and circumference of their circle using their chosen measure.
- Ask each of the students to write a statement about what they found. They can then check out their statement by walking around the circles drawn by other pairs.
- Share the students’ statements. At this point don’t expect the students to be any more exact than saying that the circumference is about 6 times larger than the radius.
Over the next three days the students draw and explore a number of circles or circular objects to gain a more exact understanding of the relationship between the radius, diameter and circumference. Later in the week you might like to get the students to find out what they can about pi as a research topic.
Begin by giving the students some circle challenges:
- Draw the smallest circle you can with your compass.
- Draw the largest circle.
- Draw 4 circles of different sizes that have the same centre point.
- Ask the students, in pairs, to gather together at least 5 circles or circular objects of different sizes. 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 like that below.
Circle Radius Diameter Circumference 1 2 3
- As the students struggle with the difficulty of measuring the circumference, you may want to discuss their problems and their ideas as a class. One effective way to measure the circumference is to provide string for the students to wrap around the circle. It may help to have cellotape to hold the string in place.
- After the students have had a chance to enter the measurements of several circles on the chart, challenge them to think about the relationship between the diameter and circumference. You could do this by saying that you know a "trick" and that if they tell you either the radius or the diameter you can tell them the circumference. Have the students challenge you with a couple of examples.
- Calculators are useful for students as they try to find out your "trick". Encourage them to be exact to use decimal numbers between 3 and 4 to get closer as they try to discover your trick.
- At the end of each day give students the chance to try out their guesses of your trick with others. Ask them to record their ideas about circles and the relationship between circumference and diameter so that they can be shared and recorded on a class chart.
- Reinforcement problems, such as those found in textbooks and/or worksheets could be assigned or written on the board.
At the end of this series of activities we look at the mathematical formula and see how it links to the ‘tricks" we have used to find the circumference using the measurement of the diameter.
- Begin by asking the students to summarise the trick they used for finding the relationship.
- Write the formula for pi on the board: C = (pi)d. What do you think that the symbols are?
- After the students have given you their ideas you will need to tell them that the symbol is read as pi and that pi has the value of approximately 3.14. If some of the students have found out facts on pi these could be shared.
- Discuss how this formula links to their ideas about the trick. Write the formula C = 2 (pi)r on the board. What might this be?
Would you get the same answer using the trick? Why?
- Conclude by posing a problem that requires the students to apply the formula.
The largest Ferris wheel in the world in Yokahama City, Japan, is 100 metres high. About how far do you ride when you go around once?
- Share solutions.