Transcript this Up2d8 Maths resource

In January
2010 the
tallest
building in
the world
opened in
Dubai.
Wow, that’s huge!
Imagine standing at the
bottom of the tower and
looking up!
You’d have to look straight
upwards to see the top. I
wonder how far away I’d
have to stand to be able to
see the top without straining
my neck?
The Burj Khalifa (originally
called the Burj Dubai) is
more than 800m tall, about
twice the height of New
York’s Empire State Building.
It contains 498 floors of
offices and 1 044 residential
apartments. There are 57
lifts, the main service lift
rising 504 metres. The
outside of the building is
covered with 28 261 glass
panels.
The building can be seen
from a distance of 95km.
If I wanted to take
I fixed the angle on
The
last time
I was in
pictures
of some
my tripod and got a
London
I found
other very
tall a
great shot
spot where
buildings,
howI could
far
My tripod
got stuck at that
take
picture
of the
away awould
I have
angle! I really
top
of Canary
Wharfwant to use that
to
stand
from these
tripod to take a photo of the
with my tripod at the
Burj Khalifa. How far away
same angle?
would I have to stand?
Up2d8 maths
Burj Khalifa
Teacher Notes
Burj Khalifa
Introduction:
The tallest building in the world has just opened in Dubai. Work began on the 828m Burj Khalifa in September 2004 and
it was officially opened on 4 January (with a last minute name change from the Burj Dubai). The building breaks many
records:
•
tallest skyscraper to top of spire: 828 m (2 717 ft) (previously Taipei 101 – 509.2 m (1,671 ft))
•
tallest structure ever built: 828 m (2 717 ft) (previously Warsaw radio mast – 646.38 m (2,121 ft))
•
tallest extant structure: 828 m (2 717 ft) (previously KVLY-TV mast – 628.8 m (2 063 ft))
•
tallest freestanding structure: 828 m (2 717 ft) (previously CN Tower – 553.3 m (1 815 ft))
•
building with most floors: 160 (previously both 1 and 2 World Trade Center – 110)
•
world's highest elevator installation
•
world's fastest elevators at speed of 64 km/h (40 mph) (previously Taipei 101 – 16.83 m/s)
•
highest vertical concrete pumping (for a building): 606 m (1 988 ft) (previously Taipei 101 – 449.2 m (1 474 ft))
•
highest vertical concrete pumping (for any construction): 606 m (1 988 ft) (previously Riva del Garda Hydroelectric
Power Plant – 532 m (1 745 ft)
•
the first world's tallest structure in history to include residential space
•
highest outdoor observation deck in the world
•
world's highest mosque (located on the 158th floor)
•
elevator with the longest travel distance in the world
•
tallest service elevator in the world
•
world's highest installation of an aluminium and glass façade, at a height of 512 m (1 680 ft).
This resource allows students the opportunity to explore proportionality in similar triangles and, in doing so, could
provide an introduction to trigonometry.
Content objectives:
This context provides the opportunity for teachers and students to explore a number of objectives. Some that may be
addressed are:
•
know that if two 2-D shapes are similar, corresponding angles are equal and corresponding sides are in the same
ratio
•
understand and use trigonometric relationships in right-angled triangles, and use these to solve problems
•
use proportional reasoning to solve problems, choosing the correct numbers to take as 100%, or as a whole;
compare two ratios; interpret and use ratio in a range of contexts.
Process objectives:
These will depend on the amount of freedom you allow your class with the activity. It might be worth considering how
you’re going to deliver the activity and highlighting the processes that this will allow on the diagram below:
Activity: The activity explores similar triangles (although explicit mention of triangles is not made in the PowerPoint). The idea of a fixed angle
to observe a distant object is used as a context (a camera tripod at a fixed angle viewing the top of Canary Wharf) and students are given an
opportunity to explore how much further away they would need to be to use the same angle to photograph the top of the Burj Khalifa initially,
and then other tall buildings. The angle and distances (except for the height of the buildings) have deliberately been left blank to allow
you more flexibility in using the activity.
Differentiation: You may decide to change the level of challenge for your group.
To make the task easier you could consider:
•
fixing the angle of the camera and/or the distance from Canary Wharf for the initial photograph
•
suggesting that a scale drawing would be a useful strategy to start solving the problem
•
providing a diagram with the simplified situation modelled and asking students to place the other skyscrapers within the diagram. They
could then use their scale drawing to calculate the distances
To make the task more complex you could consider:
•
fixing the angle or distance at a ‘difficult’ number or allowing the students to set their own angle (maybe estimating the angle of the
camera from the quick flash of the photograph on the third slide)
•
encouraging the students to work in a general situation (for example, labelling the angle of the camera x)
•
asking the students to explore the situation if the tripod is fixed at a different angle. What’s the same and what’s different about the
solutions with these different angles?.
Outcomes: You may want to consider what the outcome of the task will be and share this with students according to their ability. This task
lends itself to a poster in which students are encouraged to explain their strategies
Working in groups: This activity lends itself to paired work and, by encouraging students to work collaboratively, it is likely that you will allow
them access to more of the key processes than if they were to work individually.
You will need to think about how your class will work on this task. Will they work in pairs, threes or larger groups? If pupils are not used to
working in groups in mathematics you may wish to spend some time talking about their rules and procedures to maximise the effectiveness and
engagement of pupils in group work (You may wish to look at the SNS Pedagogy and practice pack Unit 10: Guidance for groupwork). You may
wish to encourage the groups to delegate different areas of responsibility to specific group members..
Assessment: You may wish to consider how you will assess the task and how you will record your assessment. This could include developing
the assessment criteria with your class. You might choose to focus on the content objectives or on the process objectives. You might decide that
this activity lends itself to comment only marking or to student self-assessment. If you decide that the outcome is to be a presentation or a
poster, then you may find that this lends itself to peer assessment.
If you use the APP model of assessment then you might use this activity to help you in building a picture of your students’ understanding.
Assessment criteria to focus on might be:
•
understand and use trigonometrical relationships in right-angled triangles and use these to solve problems, including those involving
bearings (shape, space and measure level 8)
•
understand and use congruence and mathematical similarity (shape, space and measure level 8)
•
understand and use proportionality (numbers and the number system level 7)
•
use proportional reasoning to solve a problem, choosing the correct numbers to take as 100% or as a whole (calculating level 6).
Probing questions: You may wish to introduce some points into the discussion which might include:
•
what’s the same and what’s different when taking photographs of different towers?
•
is it always, sometimes or never true that, if you want to take a photograph of something double the height of the original, you have to
double the distance you are from it?
•
is it always, sometimes or never true that, if you want to take a photograph of something double the height of the original, you have to
double the angle of the tripod?
You will need:
The PowerPoint presentation. You might also like to use the worksheet which simplifies the situation. There are three slides:
The first two slides set the scene.
The third slide poses the first problem that, if my tripod got stuck at that angle, how far
away would I have to stand to take a photograph of the Burj Khalifa? and then goes on
to ask the question ‘How about if I wanted to take pictures of some other very tall
buildings? How far away would I have to stand from these with my tripod at the same
angle?’
The heights of the towers in the tasks are:
Canary Wharf 235m
Empire State Building 443m
Petronas Towers 452m
Sears Tower 527m
Taipei 101 509m
Burj Khalifa 828m