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

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 .

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

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 a2 + b2 = c2 (5)2 + (12)2 = (13)2 25 + 144 = 169 TRUE

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.

Finding Missing Measurements Using Pythagorean Theorem 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 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.

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? a2 + b2 = c2 (12)2 + (16)2 = (c)2 144 + 256 = 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.

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 - 64 -64 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.

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.

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? 3 4 5 a2 + b2 = c2 (3)2 +(4)2 = (5)2 9 + 16 = 25 5 12 13 a2 + b2 = c2 (5)2 +(12)2 = (13)2 25 + 144 = 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. 9 12 15 a2 + b2 = c2 (9)2 +(12)2 = (15)2 81 + 144 = 225 10 24 26 a2 + b2 = c2 (10)2 +(24)2 = (26)2 100 + 576 = 676 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