Lesson 14 Tents

# Episode 3

Reasoning Resources: Board compass or OHP compass
Whole class preparation
Mainly an extension episode.

Moving from the semicircle and radius to a circle.

Enclosing the circle with a square using the radius or diameter, and inscribing a hexagon using the radius.

Using the radius as a unit of measure for the perimeter of the polygon. Demonstrate, using a compass on the board or OHT (or a piece of string between a pen and a fixed point) how to draw a semicircle of given radius, then using the same radius to make a rectangle.The class should justify that the length of the curve is shorter than the sum of the two verticals and the horizontal top of the rectangle. They should also recognise that the sum of the two vertical ends and the horizontal top of the rectangle is four times the radius.

They could now move to a circle and a square, and talk about the perimeter of the square being four times the diameter. Then mark the curve, using the radius as a unit, so inscribing a half-hexagon with a circumscribed rectangle. Can they see, and justify to each other, that the semicircle curve is longer than three times the radius.

They can also see that the circle will fully inscribe a hexagon and a square circumscribes it. And that the perimeter of the hexagon is three times the diameter. Ask: Are there other regular polygons that help us calculate the length of the semicircle or circle from the length of the straight line side?
Pair and small group work

Sketching enclosing regular polygons and inscribed polygons allows thinking to move towards increasing the numbers of sides.
Encourage pupils to work with full circles and to double the number of sides of a square to make an 8-sided shape around the circle. Then double the number of sides of the hexagon to make a 12-sided shape inside the circle. Encourage sketching and rehearsing explanation rather than accurate measurements.
Whole class sharing and discussion
Collect ideas from groups, describing in their words that a regular polygon would envelop the circle touching at the middle of each side, and another polygon would be inside the circle with corners on the circle. Some pupils will attempt to describe increasing the number of sides, say by doubling, infinitely to reach the circle.

Hopefully pupils would discuss in their own ways how a circle can be understood as a limiting case of a regular polygon, something that is seen in practice in stitching patterns and LOGO designs.
Formalisation and reflection

Circle as a limiting case of enclosing or inscribed polygons.

Further work on numerical approximate value of Pi.

The puzzle of a clearly visible constant ratio that is not exact. Ensuring reflection is across several levels of thinking.
Summarise and discuss ideas from the lesson: There is a constant ratio between the semicircle curve length and its radius.The ratio is around 2 or 3.14. The semicircle length is the limit of an infinite-sided half-polygon. The perfect ratio, Pi, is true for a perfect circle which is never realised in practice.

Ask pupils: Why is this ratio useful? What is it about the semicircle arc length and its radius that causes a constant relationship? For many pupils, a question back to the beginning of the lesson would be useful for engagement and thinking: What helped you to decide which picture was a semicircle?
End of Lesson Reflection
Pupils reflect on the thinking and approaches that they have used during the lesson. They discuss how the development of mathematical relationships can be helpful.