Download Lesson Plan Title: Touching the sky Description – Students design

Lesson Plan
Title: Touching the sky
Description – Students design buildings with similar height but of varying depth of foundation and use their
mathematical skills to predict the depth the foundation of a building of a specified height should have.
Target grade: 9 and 10
Subjects covered: Math, Science
Goals –
(i) To be able to measure the angle “θ” by the help of sextant.
(ii) To be able to use trigonometric ratios.
(iii) To be able to measure the height of a building.
Prerequisites –
1.The students are familiar with right angled triangle
2. The students know about the trigonometric ratios of a triangle.
3. The students are familiar with values of tan θ for different angles.
Instructional Objective(s)
At the end of the lesson, students should be able to:
1: Measure the height of a building using mathematical formulae
2. Predict the depth foundation of a building of a specified height should have
3. Understand the criteria for stability of a building/ tower/bridge etc.
Material Needed
For each student: KWLH chart (provided )
For each group:
 Sextant
 Thermocol sheet
 Chart papers
 Glue
 Measuring tape
 Sketch pens
 Pencils
 Sharpeners
 Cutter
 Marker pen
 Wooden ply board
 Any other that the students need to create the structure
For the entire class:
Designing skyscrapers books, project book etc.
Instructional Resources
Introduction –
(i) Begin by asking few questions like:
 What is a skyscraper? (Expected ans- A tall building with continuous habitable floors that are
more than 40-50 in number)
 Which is the highest sky-scraper in the world right now? (Expected ans- Burj Khalifa in Dubai,
UAE)
 How do sky-scrapers stand so tall but still do not fall during winds and storms? What makes them
remain stand erect and strong? (Expected ans- by having a strong core and base, a deeper
foundation, stronger building material etc)
(ii) Giving pamphlets in which models and different skyscrapers are given so that they can think about it. Ask
them to go through this video showing the various skyscrapers.
https://www.youtube.com/watch?v=I6QOXOGN8I4
Engineering Designing
Introduction
Part 1i. Perform a simple activity with the students. Take a thin stick a few inches high and take a jar of soil. Embed
the stick lightly and superficially in the soil. Try to bend the stick, it’ll bend easily and might get uprooted
too. Now push the stick very deep and try pushing it again. Ask the students to observe carefully. The
stick doesn’t bend that easily. Explain that the deeper and stronger grip the foundation of a building has
on the frame of a building the higher it can stand.
ii. Divide the students into groups of 3-4.
iii. Instruct the students to assume the role of architects and design a skyscraper using the materials provided
(they’re free to use additional materials too). After designing, they must calculate the height of their
skyscraper and the depth of the foundation they used for the skyscraper to remain stable on gentle
pushing.
Part 2Recapitulate with the students the method of determining the height of a tall structure using mathematical formulae.
Take a simple example of a tree and tell them how its height can be measured. Explain thus:
(Source of information: http://www.tiem.utk.edu/~gross/bioed/bealsmodules/triangle.html )
Variables:
θ
an acute angle of a right triangle
x
side of a triangle adjacent to the
angle
y
side of a triangle opposite the
angle
z
the hypotenuse of a triangle
Method: Trigonometry is a branch of mathematics dealing with measurements of the angles and sides of triangles,
and functions based on these measurements. The three basic trigonometric functions that we are concerned with
here (sine, cosine, and tangent) are ratios of the lengths of two sides of a triangle. These ratios are the trigonometric
functions of an angle, theta, such that
where θ (theta) is the angle of interest, "opp" is the length of the side of the triangle opposite θ (y), "hyp" is the
length of the hypotenuse (z), and "adj" is the length of the side adjacent to the angle θ (x), as illustrated on the
triangle below.
If we know the lengths of two sides of a right triangle (recall from geometry that a right triangle has one angle that is
90 degrees [a right angle]), we can calculate the length of the third side using the Pythagorean theorem (opp 2 + adj2 =
hyp2, or y2 + x2 = z2). For the triangle below, the side opposite is three units in length, and the side adjacent to θ is
1.5 units in length.
The length of the hypotenuse is calculated 32 + 1.52 = hyp2 =11.25. Taking the square root of 11.25, we find that the
hypotenuse is approximately 3.35 units long. We now know the lengths of each of the sides of the triangle, and can
use these to find values for the trigonometric functions:
But what can we do with this information? Well, for one thing we can use it to find θ by taking the inverse of any of
the functions (for our purposes here can do this using a calculator). Doing this we see that θ is an angle of
approximately 63.4 degrees.
The drawing above shows a forester measuring a tree's height using trigonometry. Assuming that the tree is at a right
angle to the plane on which the forester is standing, the base of the tree, the top of the tree, and the forester form
the vertices (or corners) of a right triangle. The forester measures his or her distance from the base of the tree, and
then uses a clinometer (a small instrument that measures inclination, or angle of elevation) to look at the top of the
tree and determine θ .
Interpretation: In this situation, rather than knowing the lengths of all of the sides, we know θ and the length of the
adjacent side (x), and are interested in determining the length of the opposite side (y, the height of the tree). Which
of the three trigonometric functions deals with the adjacent and opposite sides?
We know θ , and we know "adj" (or x); multiplying both sides of the equation by "adj" or x and substituting our
known values, we get
The tree is approximately 44 feet tall. We can measure the height of a building in a similar manner.
Part – 3 – Ask the students to fill-in the KWL sheet and proceed with making the models of their skyscrapers.
Extensions:
Students can be asked to research on environment friendly skyscrapers. Which components are added/replaced to
make them eco-friendly and which components are retained?
Assessment and Rubrics
Pre activity –
(i) Discussion questions
 What is a skyscraper?
 Which is the highest sky-scraper in the world right now?
 How do sky-scrapers stand so tall but still do not fall during winds and storms? What makes them
remain stand erect and strong?
Activity Embedded –
1) You know from your geometry class that all the angles of a triangle must sum to 180 degrees. Therefore, the
third angle from the tree and forester triangle must be (180 degrees - 90 degrees - 31.8 degrees) = 58.2
degrees. Verify this using the three trigonometric functions (note that for this angle, y is the adjacent side
and x is the opposite side).
2) What would the angle of elevation (θ) be if the tree was 100 ft. tall?
3) Assessing students’ engineering design worksheets.
Post activity - RUBRICS:
Criteria
Design of structures
Excellent
Good
Average
The design matches The design somewhat The design doesn’t
the
criteria matches the criteria
match the criteria at
completely.
all.
Research work while The nature of design
reflects that extensive
designing
research
and
experimentation has
The nature of design
reflects that basic
research
and
experimentation has
The nature of design
reflects that almost no
research
and
experimentation has
gone into it.
gone into it.
gone into it.
Materials Used
Low
cost,
easily
available
and
appropriate materials
are
used
and
suggested for creating
the structure
Slightly expensive but
easily available and
appropriate materials
are
used
and
suggested for creating
the structure
Irrelevant
materials
are
used
and
suggested for creating
the structure
Collaborative
Approach
Active Involvement of Involvement of few Involvement of one or
all students
students
two students
Knowledge
skyscrapers
Mathematical
approach
about Extensive
Accurate
Average
Poor
Up to mark
Below mark
Suggested Further Reading for Students:
https://en.wikipedia.org/wiki/Skyscraper
www.dezeen.com/architecture/skyscrapers/
http://science.howstuffworks.com/engineering/structural/skyscraper.htm
http://www.explainthatstuff.com/howbuildingswork.html
Credits
(i)
(ii)
Mr. Bikas Chandra Jha, TGT (Maths)
JNV, Dumka. Jharkhand. Patna Region
Mr. Pawan Kumar, TGT (Maths)
JNV, Ranchi. Jharkhand. Patna Region