Transcript Introducing Trigonometry

Introducing Trigonometry
Introducing Trigonometry
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.
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?
Introducing Trigonometry
Height
(m)
120
100
80
60
40
20
0
0
2
4
6
8
10
12
14 16
18
Time in minutes
20 22
24
26 28
30
Introducing Trigonometry
Height
(m)
120
100
80
60
40
20
0
0
2
4
6
8
10
12
14 16
18
Time in minutes
20 22
24
26 28
30
Introducing Trigonometry
Height
(m)
120
100
80
60
40
20
0
0
2
4
6
8
10
12
14 16
18
Time in minutes
20 22
24
26 28
30
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
Introducing Trigonometry
Height
Can you now sketch a better representation?
(m)
120
100
80
60
40
20
0
0
2
4
6
8
10
12
14 16
18
Time in minutes
20 22
24
26 28
30
Introducing Trigonometry
The graph should look like this:
160
140
120
100
80
60
40
20
5
20
10
15
20
25
30
Introducing Trigonometry
Imagine a theme park…
…that also celebrates wildlife in
its ‘natural’ habitat.
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!)
AquaFerris
Entrance
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?
Introducing Trigonometry
Height
(m)
60
40
20
0
-20
-40
-60
0
2
4
6
8
10
12
14 16
18
Time in minutes
20 22
24
26 28
30
100
80
Introducing Trigonometry
The graph should look like this:
60
40
20
5
20
40
60
10
15
20
25
30
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?
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?
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?
Introducing Trigonometry
Description
1
2
3
4
5
6
Graph
Introducing Trigonometry
Description
Graph
1
C
2
3
4
5
6
Introducing Trigonometry
Description
Graph
1
C
2
F
3
4
5
6
Introducing Trigonometry
Description
Graph
1
C
2
F
3
D
4
5
6
Introducing Trigonometry
Description
Graph
1
C
2
F
3
D
4
What should be on
the5card?
6
Introducing Trigonometry
Description
Graph
1
C
2
F
3
D
4
5
6
The wheel rotates at
half its original speed
Introducing Trigonometry
Description
Graph
1
C
2
F
3
D
4
B
5
6
Introducing Trigonometry
Description
Graph
1
C
2
F
3
D
4
B
5
A
6
Introducing Trigonometry
Description
Graph
1
C
2
F
3
D
4
B
5
A
6
E
Introducing Trigonometry
Description
Graph
1
C
2
F
3
D
4
B
What should
graph B
5
look like?
6
A
E
Introducing Trigonometry
Height
(m)
60
40
20
0
-20
-40
-60
0
2
4
6
8
10
12
14 16
18
Time in minutes
20 22
24
26 28
30
Introducing Trigonometry
The graph should look like this:
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.
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°
Introducing Trigonometry
70
60
50
40
30
20
10
10
20
30
40
50
60
70
Where would these values be?
0° 30° 45° 60° 90° 180° 270° 360°
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.
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.
Introducing Trigonometry
Sheet
A
B
C
0°
30°
45°
60°
90°
Introducing Trigonometry
You should have noticed that it doesn’t matter
what size the wheel is, the sine values for these
angles are always as follows:
Angle
0°
30°
45°
60°
90°
Sine
0
0.5
0.71
0.87
1
Can you explain why the value remains the same
for a given angle?
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.
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)
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.
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?
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.
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.
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.
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.
Acknowledgements
Photo of Ferris Wheel in Paris taken by Ben and Kaz Askins Creative
Commons Licence Accessed 2/9/14