Questions
1. What were the benefits of approaching the video as a trip? How would it
have been different if we had just simply told students about trigonometric
ratios without making it interesting? What effect does this have on the
student’s likelihood to pay attention and learn the material?
2. How would the world be different without our knowledge of angles and
trigonometry? What would buildings look like today if there were no rules on
how to measure angles? What influences do angles have on our every day
lives, outside of buildings? Explain.
3. Go back to the video and pause during the discussion of the tower of Pisa.
(Pause at the 44 second mark, before we go to the pyramids). How would you
have handled explaining the Pythagorean theorem to someone who didn’t
understand it and was afraid of the tower falling on them? Define the
Pythagorean theorem using your own words. Can you think of a way to figure
out how far the tower could have to fall other than using the Pythagorean
theorem? Explain.
4. Pause the video at 1:47, after we visit the Pyramids. We proved that the
triangles are similar by AAS (angle, angle, side), but what are the other ways
in which you can prove triangles are similar? What was the advantage of
building the Pyramids using similar triangles? How would they be viewed
differently if they had been made of entirely different types of triangles?
Explain. Draw three pyramids using different types of triangles.
5. Play the video from 1:48 to the end. What is SOHCAHTOA and how is it used
to solve right triangles? Do you think builder knew these ratios when
constructing the Parthenon, or is it something we have discovered over time?
Why or why not?
6. How would the Parthenon be different if it were square instead of
rectangular? How would you find the length of the diagonal if it was square?
What do you think was the reason behind building the Parthenon
rectangular? Did it have symbolic, cultural, or religious significance? Explain
your answer.
If the Parthenon was square it would have sides of equal lengths, but every
angle would still be a right angle. If it was square, you could use the
properties of a 45-45-90 triangle to solve the length of the diagonal, which
would be the hypotenuse. With these special right triangles, the length of the
hypotenuse is equal to the length of the sides times radical 2. The Parthenon
could have been rectangular because that it might have been the typical style
of buildings in Greece at the time it was built. The Parthenon could have had
cultural significance since it was built at the height of Athens power as an
empire.
7. If you were conducting this tour of ancient civilizations, what building would
you visit? Why? What purpose would this building serve in teaching students
about trigonometry and angles? How are angles important in building
ancient structures?
I would have taken a tour of the Eiffel tower because it is an iconic structure
with lots of angles. The building could be used to teach about many
principles of geometry because it is made of multiple types of angles and
shapes including squares and triangles. Angles are extremely important
when building these structures because they keep the building from
collapsing on itself and even keep them standing straight. If all the angles in
the Eiffel tower weren’t measured correctly, it may not even be standing
today.
8. Go to the smartboard file. Go to the fourth slide of the coliseum. How is this
different from other buildings we have seen? What is the purpose in making
the structure this way? How could you use the protractor to measure the
angle at which it was built? Can you? Why or Why not? What does this tell us
about how some ancient buildings were built? Explain.
The coliseum is different from the other buildings we’ve seen because
instead of being triangular or rectangular, it is circular. The building was
probably made this way simply because this style was prominent at the time
it was built, and the coliseum was used for the spectator sport of gladiator
fights, and typically stadiums for spectator sports are circular. It’s impossible
to use the protractor to measure the angle at which the coliseum was built
because it was not built at angle. The coliseums shape tells us that some
buildings were not built using as many angles as others do, and the buildings
purpose can influence how the building is constructed.
9. On the smart board file, you are given a protractor to measure angles on the
buildings. In ancient times, did builder have tools such as these? If not, what
do you think they used? How did they measure angles in a precise way? What
would be some problems of not having a protractor or measuring device
when building such magnificent structures? Explain.
Ancient civilizations did not have protractors like the ones we use today.
Instead of using tools to measure angles, I think they used math and
geometry principles like sohcahtoa and the Pythagorean theorem to
accurately measure angles in a precise way. Some problems that may occur
when not having a protractor or other measuring device though could be
doing the math wrong, and measuring angles incorrectly could result in a
variety of issues, such as the building being unstable.
10. What would be the consequences of measuring an angle wrong when
building these structures? What effect would that have on the reputation of
the building and the builders themselves? What do you think the reputation
of the builders of the tower of Pisa is now that it is beginning to lean?
Explain. What would have happened in ancient times if a builder’s building
crumbled to the ground?
Measuring an angle wrong could result in building instability or even the
collapse of the building. I think if a building had poor structure, it would
reflect poorly on the builders, and no one would want to hire them to build
anything ever again. I think the builders of the tower of Pisa have a bad
reputation since they were not intelligent enough to build the tower on
stable soil. In ancient times, I think if a builder’s structure crumbled to the
ground it would have resulted in the builder being fired or maybe even killed
depending on how cruel the person who hired them was.