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Introducing Trigonometry

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The London Eye (originally called the ‘Millenium Wheel’) is a large Ferris wheel on the South bank of the River Thames. It is a huge structure which is 120m in diameter. It rotates so that it completes one full turn in 30 minutes. This is (usually) slow enough to allow people on and off without having to stop the wheel.

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Introducing Trigonometry Imagine you are going on the London Eye. You get in at ground level. What would the graph of the height of the pod above ground during those 30 minutes look like? The following slides show 3 possibilities; what are the good and bad points about each one?

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Introducing Trigonometry 024681014 16 18 2022 12 26 2830 Time in minutes Height (m) 120 80 100 20 40 60 0 24

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Introducing Trigonometry 024681014 16 18 2022 12 26 2830 Time in minutes Height (m) 120 80 100 20 40 60 0 24

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Introducing Trigonometry 024681014 16 18 2022 12 26 2830 Time in minutes Height (m) 120 80 100 20 40 60 0 24

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Introducing Trigonometry The Geogebra file ‘Pod on Circle’ shows a point moving round a circle at a steady speed. Watch this, and see if this helps you to decide what the graph should look like. Think about when the height of the pod seems to be increasing quickly and when it increases more slowly. Sketch your best guess for the shape of the graph

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Introducing Trigonometry Can you now sketch a better representation? 024681014 16 18 2022 12 26 2830 Time in minutes Height (m) 120 80 100 20 40 60 0 24

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Introducing Trigonometry The graph should look like this:

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Introducing Trigonometry Imagine a theme park… …that also celebrates wildlife in its ‘natural’ habitat.

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Introducing Trigonometry Imagine ‘AquaFerris’… …a Ferris wheel which is half above ground and half ‘below the sea’ allowing people on it to see into an aquarium. (It probably doesn’t really exist, but it could do!)

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AquaFerris Entrance

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Introducing Trigonometry What difference is there between ‘AquaFerris’ and the London Eye? What difference would this make to the graph? Assuming that it is the same size as the London Eye and rotates at the same speed, can you draw the graph?

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Introducing Trigonometry 024681014 16 18 2022 12 26 2830 Time in minutes Height (m) 60 20 40 -40 -20 0 -60 24

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Introducing Trigonometry The graph should look like this:

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Introducing Trigonometry Some people think that a Ferris Wheel is a waste of time because you spend a lot of time close to the ground and not very much time high in the air, so you don’t get a good view for very long. Are they right? During the first 15 minutes (the time that pods on AquaFerris are above ground), how long is spent above 30m?

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Introducing Trigonometry If the wheel were bigger, but still took 30 minutes to turn, what would happen to the graph? AquaFerris has a radius of 60m and on the previous slide you worked out how much time people would be above 30m… half of its radius. If a wheel had a radius of 100m, how long would people spend above 50m?

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Introducing Trigonometry There are some graphs and some descriptions of similar to AquaFerris, can you match them up? Some descriptions and graphs are missing, can you complete them?

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Introducing Trigonometry DescriptionGraph 1 2 3 4 5 6

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Introducing Trigonometry DescriptionGraph 1C 2 3 4 5 6

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Introducing Trigonometry DescriptionGraph 1C 2F 3 4 5 6

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Introducing Trigonometry DescriptionGraph 1C 2F 3D 4 5 6

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Introducing Trigonometry DescriptionGraph 1C 2F 3D 4 5 6 What should be on the card?

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Introducing Trigonometry DescriptionGraph 1C 2F 3D 4 5 6 The wheel rotates at half its original speed

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Introducing Trigonometry DescriptionGraph 1C 2F 3D 4B 5 6

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Introducing Trigonometry DescriptionGraph 1C 2F 3D 4B 5A 6

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Introducing Trigonometry DescriptionGraph 1C 2F 3D 4B 5A 6E

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Introducing Trigonometry DescriptionGraph 1C 2F 3D 4B 5A 6E What should graph B look like?

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Introducing Trigonometry 024681014 16 18 2022 12 26 2830 Time in minutes Height (m) 60 20 40 -40 -20 0 -60 24

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Introducing Trigonometry The graph should look like this:

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Introducing Trigonometry One way of thinking about trigonometry is by looking at the positions of a point on a circle. In mathematics, we take note of the angle the point has turned through rather than the time that AquaFerris has been turning.

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Introducing Trigonometry AquaFerris took 30 minutes to make a full turn; what angles should go on the axis on the next slide? In particular, where would these be?: 0° 30° 45° 60° 90° 180° 270° 360°

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Introducing Trigonometry Where would these values be? 0° 30° 45° 60° 90° 180° 270° 360°

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Introducing Trigonometry Trigonometry enables us to work out distances (heights and lengths), and we obtain special values from the motion of the point around the circle. To understand where the values come from, we’re going to look at the height of the pod above the ground for the first 90° of its turn.

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Introducing Trigonometry On the worksheet, measure the heights carefully and work out the ‘sine’ values. There are 3 different sheets, each with a different sized wheel/circle.

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Introducing Trigonometry Sheet0°30°45°60°90° A B C

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Introducing Trigonometry Angle0°30°45°60°90° Sine00.50.710.871 You should have noticed that it doesn’t matter what size the wheel is, the sine values for these angles are always as follows: Can you explain why the value remains the same for a given angle?

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Teacher notes: Introducing Trigonometry When the changes to the National Curriculum for Key Stage 4 take place, all students will need to have at least a basic understanding of Trigonometry and will need to know some key values. This activity is intended to give students an appreciation of trigonometry related to motion in a circle and some of the key trigonometric values; it stops just short of working out missing values in triangles. The intended next step is that students would go on to use the sine values in their tables to work out missing values in triangles with 30°, 45° and 60° angles. The realistic (though not ‘real’) context makes the activity accessible to a wide range of students.

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Teacher notes: Introducing Trigonometry »Students should have the opportunity to discuss this with a partner or in a small group »Students should sketch or calculate (as appropriate) Resources: Introducing Trigonometry _ Sketch graphs (PDF) Introducing Trigonometry _ Card sort (PDF): 1 copy has 2 full sets of cards Introducing Trigonometry _ Worksheet (PDF) Pod on Circle (GeoGebra file)

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Teacher notes: Introducing Trigonometry Slide 2: the picture of the London Eye has a link to a You Tube video of the London Eye. There are several of these freely available on the web, but this one shows the wheel side on. Students may be able to spot the ‘red pod’ which is sometimes visible when the light catches it. Slide 4: you might ask students to sketch their thoughts before showing them the shapes, working in pairs may help. (Use the first slide on ‘Sketch Graphs’) work in pencil or a range of pen colours, it is intended that students will improve their sketches. Slides 5-7: these are possible answers. Discuss the merits of them e.g. symmetry, maximum and minimum heights, straight lines or curves etc.

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Teacher notes: Introducing Trigonometry Slide 8: Open GeoGebra file ‘Pod on Circle’ – a very simple animation. The animation can be started and stopped by clicking on the ‘play’ or ‘pause’ symbol at the bottom left of the screen. Students could time a revolution (approximately 30 seconds) and use this to help them create a better sketch graph. It is also possible to place points on the circle Ask students to improve their sketches on the first slide on ‘Sketching Graphs’. Slide 9: Can be used to enable students to share their own responses. Use the ‘ink annotation’ tool (which becomes visible when the pointer is allowed to hover over the bottom left of the PPT slide). Annotations can be saved or discarded. Slide 10: Who’s got something close to this?

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Teacher notes: Introducing Trigonometry Slides 11 - 13: One of the key issues in using a standard Ferris wheel to model trigonometric graphs is the fact that it ends up with a graph of a + bcos (t+d), which can provide an excellent challenge for students, but can be less useful in helping them to understand where the values arise from. Hence, the invention of a ‘Realistic’ context of ‘AquaFerris’ which may help with this issue. Slide 14: the key difference is that half of the Ferris wheel is below ground, so the entrance (and hence ‘starting point’) is halfway up the wheel, in line with the centre. This also makes the maximum height above ground the same as the radius of the wheel, which proves helpful later. Slide 15: Students should use the second slide on ‘Sketching Graphs’ to show what they think the graph will look like.

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Teacher notes: Introducing Trigonometry Slide 17: The height would change, but the basic shape of the graph is essentially the same. It will be stretched vertically. Slide 18: Use the ‘Card Match’ activity. Slides 19 -30: Answers. Students can draw on slide 29 before slide 30 is revealed. Slide 31: Students can use their graphs or Slide 16 to work this out. It should be 10 minutes (2/3 of the time). Question: what angle has the wheel turned through when it reaches half its height? Slides 32- 34: Thinking about angles rather than times. Students can use Ink annotation to show where the key points are: 0° 90° 180° 270° 360° at crossing points and turning points; 30° 45° 60° spaced appropriately between 0° and 90°. Note the 30° and ‘half full height’ connection again.

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Teacher notes: Introducing Trigonometry Slide 17: Students can use their graphs or Slide 16 to work this out. It should be 10 minutes (2/3 of the time). Question: what angle has the wheel turned through when it reaches half its height? Slide 18: The height would change, but the basic shape of the graph is essentially the same. It will be stretched vertically. Slide 19: Use the ‘Card Match’ activity. Slides 20 -31: Answers. Students can draw on slide 30 before slide 31 is revealed. Slides 32- 34: Thinking about angles rather than times. Students can use Ink annotation to show where the key points are: 0° 90° 180° 270° 360° at crossing points and turning points; 30° 45° 60° spaced appropriately between 0° and 90°. Note the 30° and ‘half full height’ connection again.

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Teacher notes: Introducing Trigonometry Slide 35 & 36: Use the PDF ‘Introducing Trigonometry _ Worksheet’ There are three different sized wheels/ circles: A, B & C. Either hand a selection out amongst the class or ask the class to work in small groups and give each group all 3 to look at. Ensure that students realise which angle is which on the worksheet (can point them out on Slide 33) Slides 37 & 38: Values obtained by the class should be similar to the answers on slide 38. It is vital that students appreciate that the angle dictates the value of the sine. Where possible encourage members of the class to explain why this should be so. Posing questions about enlargements and the effect on side lengths and angles may be necessary.

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Acknowledgements Photo of Ferris Wheel in ParisPhoto of Ferris Wheel in Paris taken by Ben and Kaz Askins Creative Commons Licence Accessed 2/9/14Creative Commons Licence

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