Transcript Building a Survival Shelter

Building a Survival Shelter
A Project Based Learning Unit
For 8th Grade Mathematics
June, 2011
REI is offering a Wilderness Survival class and wants to provide instruction
in building a survival shelter that is elevated but is built without access
to measurement tools such as protractors and rulers. Students will
create a Guide for Building a Survival Shelter that is based on
Pythagorean Theorem. Optionally, the shelter construction will be validated
with a three-dimensional scale model.
Guiding Questions
• What led to the
development of
Pythagorean Theorem and
how can it be used to solve
real-world problems today?
• How can the Pythagorean
Theorem be represented
through models and
pictures?
Standards-Based Project
8.7 Geometry and spatial reasoning. The student
uses geometry to model and describe the physical
world. The student is expected to:
8.7B use geometric concepts and properties to
solve problems in fields such as art and
architecture.
8.7C use pictures and models to demonstrate
the Pythagorean Theorem.
8.9 Measurement. The student uses indirect
measurement to solve problems. The student is
expected to:
8.9A use the Pythagorean Theorem to solve
real-life problems
Survival Simulation Game
You and your companions have just
survived a small plane crash . . . Your group
of survivors managed to salvage the
following twelve items. List in order of
importance for your survival.
•
•
•
•
•
•
A ball of steel wool
A small ax
A loaded .45 caliber pistol
Newspapers
Cigarette lighter (without fluid)
Extra shirt & pants for each
survivor
•
•
•
•
•
•
20 x 20 ft piece of canvas
A small ax
An air map made of plastic
1 quart 100-proof whiskey
A compass
Chocolate bar for each
survivor
Request for Submissions
Guide for Building a Survival Shelter
Today, REI wants to add a Wilderness Survival
class to its Outdoor School offerings.
As part of that class, they want to provide
instruction for building an elevated survival shelter
that is built without access to measurement tools.
You will create a Guide for Building a Survival
Shelter that is based on the Pythagorean
Theorem.
What do you know?
What do you need to know?
• Why and what is REI requesting?
• What mathematical concepts are required
in building the structure?
• Why might this be a challenging task?
• What are some of the requirements of the
survival guide?
Sequence of Learning Experiences
“It’s all about the process.”
What is your idea for a
Survival Shelter?
Using chart paper, each group will
sketch their idea for a survival
shelter.
This will be the starting point for
your project and will be refined
over the next two weeks as you
gain more information.
INVESTIGATION:
Group Activity
What is the role
of right angles in
construction ?
Pythagorean Theorem Puzzle
Discover the formula for Pythagorean
Theorem using a geometric proof.
What is the Pythagorean Theorem?
The Pythagorean Theorem is a relationship among the
lengths of the sides of a right triangle.
What do you notice about the hypotenuse and the legs of a right triangle?
Leg
c
a
The legs
form the
right angle
hypotenuse
Longest side of
the triangle
Across from
the right angle
b
Leg
What is the Pythagorean Theorem?
Leg
c
a
In any right triangle
with legs a and b and
hypotenuse c,
b
Leg
hypotenuse
The sum of the
squares of the lengths
of the legs is equal to
the square of the
length of the
hypotenuse.
(
Think-Pair-Share about each of the
representations of Pythagorean Theorem
below!
Leg
c
a
b
Leg
hypotenuse
Quinton cut two pieces of wood, one 5 feet long, and the
other 12 feet long. If the third piece he cuts is 13 feet long,
could the three pieces form a right triangle?
3 sides: 5 feet, 12 feet, 13 feet
13 feet
5 feet
Longest
side
12 feet
Work with a partner. Determine which of the following drawings
represent the Pythagorean Theorem.
Making It Right
Group Activity
• Using the sticks provided, form as many
triangles as you can.
• Measure the length of the sides of the
triangle and fill in the table. Remember,
“c” must always be the longest side.
• Using your protractor determine if the
triangle is a right triangle.
• Complete the table with the triangles you
formed.
Do the Right Thing
Finding Missing Measurements
Using
Pythagorean Theorem
Creepy Crawlies
Warm-up
A spider is crawling on a 18” x 18”
square window. The path of the
spider is shown below. Calculate
the distance traveled by the
spider.
A spider is crawling on a 18” x 18” square window. The
path of the spider is shown below. Calculate the distance
traveled by the spider.
We know that each leg is 18” and we
are looking for the length of the
diagonal or the hypotenuse.
c
18
18
The Pythagorean Theorem can
be used to find unknown side
lengths in right triangles.
c
a
b
Do The Right Thing
Television sizes are described by the diagonal measurement across the
screen. The rectangular screen of John’s television set measures 12
inches by 16 inches. What is the size of his television to the nearest
inch?
To solve for c, do
the opposite of
squaring a number
which is to find the
square root.
John has a 20-inch set.
Do The Right Thing
A 10-foot long piece of lumber is leaning against a wall. The bottom of
the piece of lumber is 8 feet from the base of a wall. How high up the
wall does the piece of lumber reach?
Form a ZERO PAIR to get
b2 by itself!
To solve for b, do the opposite of
squaring a number which is to find
the square root.
The piece of lumber reaches 6 feet up the wall.
Finding Pythagorus
Identify a Rectangular
Shapes in the Room.
Practice finding missing
measurements using
Pythagorean Theorem.
Finding Pythagorus
Directions
Work in Pairs. Materials: tape measure or meter stick, calculator.
• Part One – Calculate the
hypotenuse
– Find a rectangle in the
room
– Measure the length and
width (a and b) in inches
– Draw a sketch
– Calculate the diagonal ( c )
and show work
– Check your answer by
measuring the diagonal
• Part Two – Calculate the side
– Find another rectangle in the
room
– Measure the width and the
diagonal ( a and c)
– Draw a sketch
– Calculate the length of
rectangle (b) and show work
– Check you answer by
measuring the length of the
rectangle.
Pythagorean Theorem Poster
Individual Project
• Model Geometric Proof
• Examples of Solving
Real-World Problems
with Pythagorean
Theorem
• Pythagorean Spiral
Pythagorean Theorem Triples
Identify Pythagorean Theorem
Triples.
Find missing measurements
using triples
Pythagorean Triples
When the three side lengths of a
right triangle are all whole numbers,
such as 3, 4, 5 or 5, 12, 13, the set of
three side lengths is known as
Pythagorean Triples.
Pythagorean Triples
What do you notice that’s similar between these sets of triples?
If you
multiply 3,
4, and 5 by
3, you will
get 9, 12,
and 15.
If you
multiply 5,
12, and 13
by 2, you
will get 10,
24, and
26.
3
4
5
5
12
13
9
12
15
10 24
26
Any Multiple of A Pythagorean Triple is also a Pythagorean Triple!
Generating Pythagorean Triples
There are an infinite number of Pythagorean Triples. Greek
philosopher Plato discovered a way to generate some of them.
For any number, n, the legs of a right triangle are 2n and n2 - 1
and the hypotenuse is n2 + 1.
For example, for n = 5, the Pythagorean Triple is 10 or 2 x 5, 24
or 52-1 and 26 or 52+1. So the Pythagorean triple is 10, 24, 26.
Proof:
102 + 242 = 262
100 + 576 = 676
Rope-Stretchers
(Monday)
In ancient Egypt there were men called “rope-stretchers.”
They discovered that if a rope was tied in a circle with 12
evenly spaced knots that it could be used to form a right
triangle. This technique enabled them to ensure that the
foundations of their buildings were square (90 degree angles
at each corner).
Work with your team to come up with an explanation of their
method.
What is a Blueprint?
(Guest Speaker)
Develop a blueprint of your survival shelter.
Guide for Building a Survival Shelter
(Culminating Product)
• Description of wilderness
environment
• Blueprint of the design
• Instructions for Assembly
• Suggested Material
• Model of Pythagorean
• Explanation of geometric
concepts used in design