Lesson 14 Tents
|An activity approaches the ratio indirectly through exploring the relationship between the radius and half the circumference of a circle. This starts with strings and/or flexible strips rather than formal measurement. The issue of accuracy possible for arises naturally and allows consideration of its representation in calculators and computers, and irrational numbers.||Worksheet 1
Lengths of string or stiff but flexible materials for ‘measuring’ lengths. Rulers or measuring tapes
Worksheet 2 (Optional).Large paper circle (Optional). Compasses (Optional).Prepared semicircles (Optional)
Board compass. Calculators
|The tent in a semicircle|
|Many pupils’ difficulties with the circle result from premature formalisation. The symbol z of the circle ratio is sometimes ‘given’ to the pupils and used in an algorithmic manner when they haven't sufficient grounding in experience of the linear relationship it describes.
A real-life context of simple tents made of fabric stretched around a semicircular frame focuses pupils’ attention on the mathematical properties of a semicircle. The pupils identify what is and is not a semicircle, and hence define a semicircle and identify the radius and the half-circumference curve as the main variables.
|Radius to semicircle ratio|
|Working with different sizes of semicircle, pupils use string, tapes and flexible, springy strips of metal or plastic to compare the radius and the half-circumference directly without measurement. The value of this constant ratio as ‘a little larger than 3 is developed. Formal measurements are made and the calculated values of the ratio are considered. The issue of accuracy is then discussed but is not the major focus of the lesson. This can be left until TM19: Accuracy and errors, which introduces relative errors in measurement. It will be sufficient at this stage for pupils to realise intuitively that the bigger the semicircular arc, the less the relative error in the measurements, and the nearer the ratio of the measurements.|
|EPISODE 3 (EXTENSION)|
|A geometrical approach to the ratio|
|Pupils are encouraged to compare a semicircle, by drawing the circumscribed 2:1 rectangle and an inscribed half-hexagon, using the radius as a unit. This establishes that the ratio of the half-circumference to the radius lies between 3 and 4 and nearer 3, Sketching and thinking of other regular polygons allow the insight that the circle can be considered as a polygon with an infinite number of sides.|