1. Building a Survival Shelter
A Project Based Learning Unit
For 8th Grade Mathematics
June, 2011

2. 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.

3. 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?

4. 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

5. 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

6. 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.

7. 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?

8. Sequence of Learning Experiences
“It’s all about the process.”

9. 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.

10. INVESTIGATION:
Group Activity
What is the role of right angles in construction ?

11. Pythagorean Theorem Puzzle
Discover the formula for Pythagorean Theorem using a geometric proof.

12. 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
hypotenuse c a
Longest side of the triangle
The legs form the right angle
Across from the right angle b
Leg

13. What is the Pythagorean Theorem?
Leg
hypotenuse c a b
The sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse (See Warm-up)
In any right triangle with legs a and b and hypotenuse c,
Leg
a2 + b2 = c2 .

14. Think-Pair-Share about each of the representations of Pythagorean Theorem below!
Leg
hypotenuse
a2 + b2 = c2

15. 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
a b2 = c2
(5)2 + (12)2 = (13)2
= 169
TRUE

16. Work with a partner. Determine which of the following drawings represent the Pythagorean Theorem.

17. 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.

18. Finding Missing Measurements Using Pythagorean Theorem
Do the Right Thing
Finding Missing Measurements
Using
Pythagorean Theorem

19. 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.

20. A spider is crawling on a 18” x 18” square window
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

21. The Pythagorean Theorem can be used to find unknown side lengths in right triangles.

22. Do The Right Thing a2 + b2 = c2 (12)2 + (16)2 = (c)2 144 + 256 = c2
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?
a b2 = c2
(12)2 + (16)2 = (c)2
= c2 c
400 = c2
To solve for c, do the opposite of squaring a number which is to find the square root.
400 = c2
= c 20
John has a 20-inch set.

23. Do The Right Thing a2 + b2 = c2 (8)2 + (b)2 = (10)2 64 + b2 = 100
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?
a2 + b2 = c2 b
(8)2 + (b)2 = (10)2
Form a ZERO PAIR to get b2 by itself!
64 + b2 = 100
b2 = 36
To solve for b, do the opposite of squaring a number which is to find the square root. = 36 b2
b = 6
The piece of lumber reaches 6 feet up the wall.

24. Finding Pythagorus Identify a Rectangular Shapes in the Room.
Practice finding missing measurements using Pythagorean Theorem.

25. 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.

26. Pythagorean Theorem Poster Individual Project
Model Geometric Proof
Examples of Solving Real-World Problems with Pythagorean Theorem
Pythagorean Spiral

27. Pythagorean Theorem Triples
Identify Pythagorean Theorem Triples.
Find missing measurements using triples

28. 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.

29. 5 12 13 3 4 5 9 12 15 10 24 26 Pythagorean Triples (3)2 +(4)2 = (5)2
What do you notice that’s similar between these sets of triples?
a2 + b2 = c2
(3)2 +(4)2 = (5)2
= 25
a2 + b2 = c2
(5)2 +(12)2 = (13)2
= 169
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.
a2 + b2 = c2
(9)2 +(12)2 = (15)2
= 225
a2 + b2 = c2
(10)2 +(24)2 = (26)2
= 676
Any Multiple of A Pythagorean Triple is also a Pythagorean Triple!

30. 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 So the Pythagorean triple is 10, 24, 26.
Proof:
= 262
= 676

31. 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.

32. What is a Blueprint? (Guest Speaker)
Develop a blueprint of your survival shelter.

33. 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