Pythagorean Spiral
You will use compass constructions to create a poster of the Pythagorean spiral. The result needs to be
colored and may be creatively decorated. You will need to turn in your poster and a separate piece of
paper with all calculations.
Materials: + poster board or construction paper + ruler + compass + pencil, colored pencils or markers
Step 1: Place the poster board in landscape orientation. Locate the center of the page (explain how you
did this). This will be the starting point for your diagram. It will assure that your diagram stays on the
page.
Step 2: Using your ruler create a segment that is 1 cm long and perpendicular to the side of the poster.
Use your compass and ruler (without measuring) to construct a congruent segment that is perpendicular
to the original. Connect the endpoints of the two segments to create a right isosceles triangle.
Step 3: On a separate piece of paper, use the Pythagorean Theorem to calculate the length of the
hypotenuse. Show all work and write your answer in reduced radical form.
Step 4: Using the hypotenuse of the first triangle, create another right triangle on top of the previous
hypotenuse. The old hypotenuse will be the new base and construct a perpendicular segment to this,
with a length of 1cm. Then connect the two segments to form a new hypotenuse.
Step 5: On your separate piece of paper, show the calculations to find the length of the new
hypotenuse. Discuss what kind of numbers appear as the lengths of constructed hypotenuses.
Step 6: Continue to repeat this process of connecting and constructing new triangles with a side length
of 1 cm, using the previous hypotenuse as the other side. Continue to show your calculations on your
separate piece of paper. Construct triangles until you have formed a full spiral.
Step 7: Detail your Pythagorean Spiral with a design. Use color and a pattern to make a creative picture.
TO TURN IN:
Your poster with light pencil lines shown for constructions and color used to decorate the pattern, and
your work for each hypotenuse length on a separate sheet of paper should be turned in.
Transformations book
OBJECTIVE: Create a transfomations book that defines and illustrates reflections, translations, rotations,
and dilations.
MEANS OF COMMUNICATION:
Pop-up book
Comic book
Picture book
Textbook
Novel
Newspaper
Magazine
Video
Game
And the list goes on…
Your original work should include the following:
I. Cover/title page
II. Table of Contents
III. Preface/Forward to explain the details of the project
IV. A minimum of four chapters (one devoted to each motion). The chapters need to show the
development of the motion and convey an understanding of how it is used in Geometry. Each chapter
should exhibit four examples demonstrating how the motion is used in Geometry. A definition,
description and explanation of the motion should be included.
V. Epilogue about the author. You need to be specific about your feelings toward the project and what
you learned from your efforts.
House Making
Congratulations!! You and your business partner have won the contest to create your own dream home.
Now you must design it. In this project you will be creating a poster sized drawing of your dream home
(two dimensional) and a real house model (three dimensional). This dream home will include everything
and anything you desire. It will also include some of the geometrical shapes and solids we have been
studying (which is of course something you desire!!). You will be the architect and the construction
manager of this project.
PART I: FLOOR PLAN DRAWING
The first phase of your project will be to create a plan drawing of one floor of your home. It must
include distances and measurements and must also include at least one of each of these types of
shapes:




Square
Rectangle
Triangle
Circle (at least a portion of a circle, for example a semi circle)
Your drawing must have these elements:
 Each of your rooms must be clearly labeled (what type of room is it and what type of shape it is) and
neatly drawn.
 Rooms Required (At least):
 2 bedrooms
 2 bathrooms
 1 kitchen
 1 garage
 1 living room
 Your picture must be large and on poster size paper.
 You must use color and make this dream home appealing.
 You will use the scale for your drawing so that your drawing is realistic and life-like (1 inch = 5 foot).
 Your home should have realistic measurements, and should be something someone would actually
live in. You can refer to floor plans on the internet for assistance.
 Be CREATIVE! Have fun with the design and make it very imaginative.
PART 2: AREA CALCULATIONS
The second half of your project is to outfit your dream home. Your dream home must have some type
of flooring. You must choose at least 2 different types of flooring, to use in your home. In order to
determine how much flooring you will need you must calculate the area of each room and decide which
type of flooring you will use for each room.
You will need to produce clear calculations for the area of each room. Then using these areas and the
prices for flooring listed below, choose your type of flooring and calculate how much it will cost.
Carpet
Linoleum
Ceramic Tile
Hardwood flooring
$2 per square foot
$1 per square foot
$3 per square foot
$5 per square foot
PART 3: MODEL CREATION
The next portion of your project will take you into the third dimension. You will now create a 3-D model
of your dream home. You may use Styrofoam, cardboard, Popsicle sticks, or anything of your choosing
to create your model. Make it quality.
(Note: Your model must match your floor plan drawing!)
PART 4: TOTAL AREA AND COST
Your business team needs to figure out the total area of your dream house. You will also need to figure
out the total cost for the type of flooring.
OBJECTIVE: You and your business team will design your dream house with a floor plan drawing, area
calculations, model creation, and totals. You are to incorporate what we have learned about geometry
with houses.
GUIDING QUESTIONS: Here are some questions that will help you: creativity (Is this done with flare?),
visual (grabbing the attention of audience?), accuracy (going in the right direction?), evidence (obvious
steps?), correctly scales and models done well.
Tessellations
A tessellation is created when a shape is repeated over and over again covering a plane without
any gaps or overlaps. Another word for a tessellation is a tiling. A regular polygon has 3 or 4 or
5 or more sides and angles, all equal.
A regular tessellation means a tessellation made up of congruent regular polygons. Decide
which regular polygons tessellate the plane and explain why. You can also use a variety of
regular polygons to make semi-regular tessellations. A semiregular tessellation has two
properties which are:
1. It is formed by regular polygons.
2. The arrangement of polygons at every vertex point is identical.
Give couple of examples of semi-regular tessellations. Explain how you are certain that yours
will always work. To help you see why I ask, consider this tessellation:
1. There are other ways of constructing tilings: Begin with a square and design from one corner of
the square to an adjacent corner. (Do not draw diagonally). Do not stop halfway across!
2. Cut on the design line, being sure to have 2 pieces when done -the nibble and the rest of the sheet.
There should be no other pieces lying around.
3. Slide the nibble across the sheet to the opposite side and tape the straight edges together. Does
this new shape tile the plane? Why? What would happen if you attached the nibble to the
adjacent side? Could you tile? Why or why not? Explain?
Create your own Tessellation
The appearance of your tessellation should be neat! Your tessellation (pattern) should cover the ENTIRE
page (no gaps or unintentional white spaces). You are to turn in the template figure you used to create
your tessellation. Your project will also be based on creativity and difficulty. The more difficult and
complex the tessellation is, then the higher the grade. You may not simply take a polygon and slide,
rotate, or reflect it over and over again to create your tessellation. You must use a figure created using
one of the nibbling methods. Check out some of the famous Escher tessellations:
http://www.tessellations.org/tess-escher1.htm
Tangram projects
1.
Make your tangram puzzle. Use sturdy, construction, paper, and use the picture below as a
guide. Make your set of tangrams is from an 16 cm x 16 cm square. Find the length of each edge
of each tangram piece, the perimeter and area of each piece. Find the sum of the perimeters
and the sum of the areas of the seven pieces. For each piece determine what fraction of the
whole puzzle it is. Show carefully how you did it. No rulers should be used, only the knowledge
of how you constructed each piece:
Tangram pieces
Sides
Perimeter of
each one
Area of each one
Fraction
Small triangle (2)
Medium triangle
Parallelogram
Square
Large triangle (2)
SUM of all 7 pieces
Assuming that the square piece in the tangram set has an area of 1 square unit, how many square
units of area is in the entire puzzle? Make a drawing and show how you calculated your answer.
Once done, create 3 objects using your tangrams. One should have rotational symmetry, one
reflectional, and one should have no symmetries. Use your imagination only! You are welcome to
color and make them pretty 
Golden ratio – golden spiral
1. Construct a square with side a.
2. Draw a line segment from the midpoint of one side to an opposite corner (call its length c).
3. Use that line as the radius to draw an arc that defines the long dimension of the rectangle.
4. Use Pythagorean theorem to help you find the value of c and ultimately the ratio of longer side
to the shorter side of the rectangle. This ratio is called golden ratio, and is denoted  (phi).
Figures containing golden ratio are said to be particularly pleasing to the eye.
5. Construct a large golden rectangle on a large piece of construction paper. Inside it, construct a
square. The leftover space is a rectangle. What kind? How does it relate to the original
rectangle? How do you know? Within that rectangle construct a square. There is a new
rectangle inside it. Keep on going.
6. Once done, use your compass to draw quarter circles connecting opposite vertices of each
square and doing so construct a golden spiral. All the constructions should be visible, but you
are welcome to color the final piece of art.