Interactive Circles for Geometry and Statistics

2. A Useful Circle Resource • Cut out 2 circles of the same size but in different colours • Cut along the radius of each • Slot the two circles together and lay them flat • The circles should be able to slide round to show larger and smaller sectors of each colour

3. Adjust your circles to show me… • A ‘blue’ sector with an acute angle • A ‘blue’ sector which is 90° • A ‘white’ sector which is a reflex angle • A ‘blue’ sector which is 45° • A ‘blue’ sector which is 330°

4. Adjust your circles so that the ratio of blue: white is… • 1:1 • 3:1 • 1:2 • 3:5

5. Adjust your circles to show me an estimate for what the pie chart would look like for… • The number of boys and girls in this class • The number of people born on weekdays and the number born at weekends • How much of the time you spend awake and asleep

6. The World on Wight Would the entire population of the world fit onto the Isle of Wight? 30 years ago, they could have, but the world population has grown since then. Is it still possible?

7. The World on Wight Would the entire population of the world fit onto the Isle of Wight? What information do you need? You probably need the following: Information about how big the Isle of Wight is An estimate of the current world population How many people can fit into a certain size space OR What size space is needed for a certain number of people

8. The World on Wight Would the entire population of the world fit onto the Isle of Wight? Questions you might consider to refine your answer: Is every person in the world the same size and shape? Are there places on the Isle of Wight where people would not be able to stand?

9. Pentagon Puzzler Follow the instructions to fold a pentagon Start with a piece of A4 paper; fold so that the two marked vertices touch. (approximate fold line shown)

10. Pentagon Puzzler Fold the resulting shape in half along its line of symmetry. Unfold again so that it looks the same as the diagram below.

11. Pentagon Puzzler Fold the edge marked in red to line up with the fold line and repeat with the similar edge on the opposite side of the shape.

12. Pentagon Puzzler The resulting shape is a pentagon… but is it regular?

13. Pentagon Puzzler • Unfold the pentagon so that you can see the fold lines. • A piece of A4 paper measures 210mm by 297mm. • Prove whether or not the pentagon is regular.

14. A question of ‘roundness’ • What is meant by the term ‘roundness’? • It can be thought of as a shape which has a constant diameter. • There are many uses for shapes which are round but not circular; they are used in some car engines. • This activity explores ‘roundness’.

15. A question of ‘roundness’ A two point test for roundness • To check that a cylinder is round, you can put straight edges either side of it and measure the perpendicular distance between them. Like this: • Moving the straight edges round the cylinder, the measurement remains the same. • It is called a two-point test as it only uses two points on the circle.

16. A question of ‘roundness’ • There are many examples of ‘round’ objects which are not circular but do have a constant diameter. • Do you know of any? • Have a look at a handful of change… • What do these photos tell us?

17. A question of ‘roundness’ • A selection of shapes are available to cut out. • Use a 2 point test of roundness to work out which ones are round.

18. Platonic Solids and Duals • Solid shapes are all around us. Some are ‘regular’ and some are not. What does it mean for a 3d shape to be ‘regular’? • There are 5 regular 3d shapes, can you think what they are? Image from greatlittleminds.com

19. Platonic Solids and Duals For this activity you will need access to each of the shapes: Tetrahedron Cube Octahedron Dodecahedron Icosahedron • 3d shapes have faces, edges and vertices. • How many faces, edges and vertices does each one have? • Create a table to summarise your values • What is the connection between the values?

20. Platonic Solids and Duals

21. Platonic Solids and Duals

22. Platonic Solids and Duals • You might notice that some of the numbers in the table are repeated. • These shapes are duals of each other. • Look at the pictures on the following slide and work out what this means. Photos courtesy of Jeff Trim, Little Heath School

27. Teacher notes: This month there are 5 activities you might like to use, appropriate for a wide range or classes and students. The following is a guide only. • A useful circle resource is most suitable for KS3 and some KS4 students. Thanks to Debbie Barker for the resources and ideas for this activity. • Pentagon Puzzler is most suitable for KS4 students • The World on Wight is a modelling problem suitable for KS 3 and KS4 students. Thanks to Kevin Lord for the resources for this activity. • A Question of Roundness is suitable for KS 3 and KS4 students and is based on an activity developed by Phil Chaffé (which in turn was sparked by an idea from Chris Sangwin.) • Platonic Solids and Duals is an activity by Jeff Trim, which is also suitable for KS3 and KS4 students, although the extension question in the teachers notes would prove challenging for A level students.

28. Teacher notes: A Useful Circle Resource This resource simply requires two identically sized circles in different colours. It can be used for a wide range of ‘show me’ activities in class; 3 suggestions are given. Using 3 or more different colours enables more complex pie charts or ratios to be explored, but this might detract from the elegance and simplicity of the resource.

29. Teacher notes: The World on Wight Hints andPossible approaches Use the print out of the map (on the next slide) to estimate area of Isle of Wight using simple shapes. • How could you do this more accurately? Use some metre sticks to make a square metre and carry out an experiment to estimate the number of people who could fit in a square metre. • What is the average number of people per square metre? • How accurate do you think your value is? Use the internet to research the population of the world and verify the area of the Isle of Wight. • World Population in 2011 = 6.93 billion people • World Population in 1980 = 4.44 billion people • Area of Isle of Wight = 384 km2

30. Teacher notes: The World on Wight Hints andPossible approaches Discuss the validity of the result, by discussing again the assumptions you made at the start. For example: • Not all people are the same size, most would be bigger than children • It is possible to stand on all parts of the island Can you find other islands that could be home to the whole world? Mathematics skills that might be used: • Estimation and approximation • Area of rectangles and triangles • Compound measures like density • Standard form • Simple averages and using averages • Converting units – m2 to km2

31. Teacher notes: Pentagon Puzzler • The resulting pentagon is not quite regular, but is close enough to be used by younger students to create patterns. • To prove whether or not it is regular, one might look at the lengths of the sides or the size of the angles. • In this case, we will determine one of the angles of the pentagon by considering the shapes created after the first fold (diagram below). • The angle marked , plus 90° from the original vertex of the rectangular paper create one of the vertices of the pentagon. If this is 108°, then the pentagon might be regular, if it isn’t then the pentagon is clearly not regular. • We will determine this by finding the angle which would have to be 18° (i.e. 108 ° – 90°) • Mathematical skills required are: • Solving equations & expanding quadratic expressions • Pythagoras’ theorem • Trigonometry

32. Teacher notes: Pentagon Puzzler • Using the dimensions of the paper and the shapes created after the first fold (diagram below) gives the following: 297-x x

33. Teacher notes: Pentagon Puzzler Using Pythagoras’ theorem with the dimensions shown gives: m2 = x2+ 2102 m x 210

34. Teacher notes: Pentagon Puzzler 1 Using Pythagoras’ theorem with the dimensions shown gives: m2 = x2+ 2102 From previously, m = 297 - x So m2 = x2 - 594x + 2972 Equating and x2+ 2102= x2 - 594x + 2972 594x = 2972 - 2102 594x = 88209 – 44100 x = 44109 ÷ 594 x = 74.25757… For the marked angle: Tanθ = x ÷ 210 Tanθ = 74.25757… ÷ 210 θ = 19.47° Therefore, the pentagon is not regular. 2 2 1 m x 210

35. Teacher notes: A Question of Roundness This activity can be used simply to get pupils to think more deeply about roundness or Geogebra or other Dynamic Geometry Software can be used to create round shapes. For instructions on creating the Geogebra file, please download Phil Chaffé’s files from the 2011 conference website. The following slides can be printed onto card, the shapes cut out and tested for roundness. All of them are, in fact, round.

39. Teacher notes: Platonic Solids and Duals • Euler’s relation is: Faces + Vertices = Edges + 2 A dual is created by connecting the centre of each face of the 3d shape to adjacent centres of faces. • A tetrahedron is a dual of itself • A cube and Octahedron are a pair of duals • A dodecahedron and icosahedron are a pair of duals Extension question: Prove that there are only 5 platonic solids.