CREATING A LOGO
Enduring Understanding: Develop a better understanding of how to use the properties of special
triangles and regular polygons to answer a question. Develop a better understanding of how to use
direct measurements to calculate area measurements. Develop a better understanding of how to
develop a persuasive mathematical argument both orally and in written form.
Essential Questions:
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What is a regular hexagon?
What is an itemized bill?
What are the properties of an equilateral triangle?
What are the properties of a 30°- 60°- 90° triangle?
What are the properties of a 45°- 45°- 90° triangle?
How does one convert units within a system of measurement? How could you convert square
inches to square feet?
What are the properties of a regular polygon?
How can you use transformations (rotations, reflections, translations) to create congruent
figures?
How do I describe a transformation?
What is the difference between area and perimeter?
Which units are used for area and perimeter?
How is a persuasive argument developed?
How can I prove a certain solution or answer meets a set of conditions?
Lesson Overview:
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Before allowing the students the opportunity to start the activity: access their prior knowledge
regarding how to work with the properties of triangles and regular polygons. How many have
given persuasive oral presentations? How many have written a persuasive argument? Have
you ever tried to persuade your parents and/or friends to allow you to do something that they
didn’t want you to do? How did you go about trying to persuade them?
A good warm-up for this activity is Landscape Architect.
Some artistic license can be allowed to assist the students’ ability to be more creative.
Give the students a presentation rubric prior to their development of their presentations. The
rubric can include eye contact, clarity of speech, presentation to the entire audience, clarity of
information, use and quality of visual aides.
Introduce this activity on a Monday with the turn-in and presentations on Friday. Allow for
some time during class during the week for the students to work on this activity.
Provide the students with several copies of a template of the regular hexagon. One is
provided. The template given to the students should be on card stock so that the hexagon is
sturdy when the students trace the template onto a sheet of blank paper.
How is a problem situation decoded so that a person understands what is being asked?
What mathematical information should be used to support a particular conclusion?
How will the students make their thinking visible?
Use resources from your building.
Creating A Logo
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Materials: Graph Paper—preferably 1 cm grid paper
Crayons
Colored Pencils/Markers
Rulers
Regular Hexagon
Length of the Lesson: 100 minutes: Suggest assigning the Stained Glass Window as an overarching project of 3-5 days.
EALRs/GLEs
1.1.6-- Complete multi-step computations with combinations of rational numbers using order of
operations and addition, subtraction, multiplication, division, powers, and square roots.
1.2.3-- Apply unit conversions within measurement systems, U.S. or metric, to maintain an
appropriate level of precision.
1.3.1-- Understand the properties of and the relationships among 1-dimensional, 2-dimensional, and
3-dimensional shapes and figures.
1.3.4-- Apply multiple transformations – translations, reflections, and/or rotations to 2-dimensional
figures.
2.2.1-- Select and use relevant information to construct solutions.
2.2.2-- Apply mathematical concepts and procedures from number sense, measurement, geometric
sense, probability and statistics, and/or algebraic sense to construct solutions.
2.2.3-- Apply a variety of strategies and approaches to construct solutions.
2.2.4-- Determine whether a solution is viable, is mathematically correct, and answers the
question(s).
3.2.1-- Draw and support conclusions, using inductive or deductive reasoning.
3.3.1-- Justify results using inductive or deductive reasoning.
3.3.2-- Evaluate reasonableness of results.
3.3.3-- Validate thinking about mathematical ideas.
4.2.1-- Organize, clarify, and refine mathematical information relevant to a given purpose.
4.2.2-- Represent mathematical information in graphs or other appropriate forms.
4.2.3-- Use mathematical language to explain or describe mathematical ideas and information in ways
appropriate for audience and purpose.
5.1.1-- Apply concepts and procedures from two or more content strands, including number sense,
measurement, geometric sense, probability and statistics, and/or algebraic sense, in a given problem
or situation.
Item Specifications: NS02; NS04; ME02; ME03; GS01; GS02; SR02; SR04; SR05; CU02;
MC01
Assessment: Use the multiple choice and short answer items from Geometric Sense and
Measurement that are included in the CD. They can be used as formative and/or summative
assessments attached to this lesson or later when the students are being given an overall summative
assessment.
Creating A Logo
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Activity 1: Create a Logo
You have been contracted to produce a logo for a new company. The company is so new that the
founder has not decided on a name. The company founder wants to represent the company with a
logo that must have these characteristics:
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The outline of the logo will be a square;
There must be at least two non-overlapping squares, two non-overlapping rectangles, and
two non-overlapping triangles inside the logo;
The triangles must be congruent 45°- 45°- 90° triangles;
All squares must be congruent to all other squares.
1. How are you going to know that a triangle in your logo is a 45°- 45°- 90° triangle?
Describe at least two characteristics of a 45°- 45°- 90° triangle.
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2. On graph paper or 1 cm grid paper, draw a square with side lengths of 20 centimeters. Within the
square, draw a logo that meets these requirements:
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The outline of the logo will be a square;
A plastic frame will surround the square that is 20 cm on each side;
There must be at least two non-overlapping squares, two non-overlapping rectangles, and
two non-overlapping triangles inside the logo;
The triangles must be congruent 45°- 45°- 90° triangles;
All squares must be congruent to all other squares.
**Before proceeding too far along with your design, check with your teacher (initials) ________
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3. You may notice that, since the triangles are all congruent and the squares are all congruent, you
can use transformations (reflections, rotations, translations) to move one of the figures to the location
of the other(s). This is a common way that computer programmers create congruent figures in
designs.
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For your logo, describe a transformation or combination of transformations that you could
use to move one triangle to the location of another triangle. If you used a transformation,
make sure you label your transformed figure.
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4. For your logo, describe a different transformation or combination of transformations that you
could use to move one square to the location of another square.
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5. For your logo, is it possible to use a transformation or combination of transformations to move
one rectangle to the location of another rectangle? ______________________
--If it is possible, describe a transformation or combination of transformations you could use.
--If it is not possible, describe why you cannot use a transformation or combination of
transformations to move one rectangle to the location of another rectangle.
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6. You need to be able to describe your design to the founder of the company over the phone before
you work on putting it into an electronic format for printing.
Write a description of your image. You must use correct mathematical language including language
about the transformations (reflections, rotations, translations) for your logo.
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7. The founder is going to make a glass version of your logo to hang in the window of her office.
She is going to use different colors of glass for the squares, rectangles, and triangles in the logo. The
different colors of glass have different costs per square centimeter. She will put a plastic frame
around the perimeter of the logo to keep the glass from chipping or breaking. She will also put lead
stripping between the various shapes of the glass within the logo.
The founder wants to know, for your design, the total area and perimeter of the outside of the logo,
and the perimeter and areas of each of the squares, rectangles, and triangles within the logo.
a. Describe a strategy you could use to determine the total area of the logo.
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b. What is the area of one of the triangles inside your logo? ___________________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
c. What is the total area of all of the triangles inside your logo? ________________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
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d. What is the area of one of the squares inside your logo? _____________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
e. What is the total area of all of the squares inside your logo? _____________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
f. What is the area of one of the rectangles inside your logo? _________________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
g. What is the total area of all of the rectangles inside your logo? ______________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
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h. What is the total area of the colored glass inside your logo? _________________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
i. Describe a strategy you could use to determine the total perimeter of the outside of the logo
and the total perimeter of the figures inside of the logo.
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j. What is the perimeter of one of the triangles inside your logo? _______________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
k. What is the total perimeter of all of the triangles inside your logo? ___________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
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l. What is the perimeter of one of the squares inside your logo? _____________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
m. What is the total perimeter of all of the squares inside your logo? ___________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
n. What is the perimeter of one of the rectangles inside your logo? _____________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
o. What is the total perimeter of all of the rectangles inside your logo? __________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
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p. What is the total perimeter of the figures inside your logo? _________________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
q. What is the perimeter around the outside of your logo? ________________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
r. The area within the logo that is not a square, rectangle, or 45°- 45°- 90° triangle of colored
glass will be filled with clear glass.
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Describe a strategy you could use to determine the area of clear glass that would be
used for your logo.
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s. What is the area of clear glass that would be used for your logo? _____________________
Support your answer using words, numbers and/or diagrams. Be sure to include
appropriate units.
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8. Here are the costs of the thin plastic frame, the clear glass, the different colors of glass and the
lead stripping that will go between the various colors.
Material
Cost
Plastic frame
$0.24 per cm
Clear glass
$0.88 per square cm
Purple glass
$1.42 per square cm
Green glass
$1.14 per square cm
Yellow glass
$0.96 per square cm
Blue glass
$1.35 per square cm
Red glass
$0.85 per square cm
Labor Cost
$0.36 per square cm
Lead stripping for
between the glass
$0.87 per cm
All of the squares within the logo must to be filled with one color of glass. All of the triangles must
to be filled with a second color of glass that is different than the color chosen for the squares, and all
of the rectangles need to be filled with a third color of glass, different from the triangles and the
squares.
a. Determine which colors you are going to use for the squares, triangles, and rectangles. On
your graph paper, color the logo with the appropriate colors.
b. Determine the total cost for your logo by filling in the Itemized Bill on the next page.
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Itemized Bill
Name: _______________________________________
Date: ______________________
Dimensions of the window: __________________________________________________________
Cost for the lead: ________________cm @ $0.87 per cm
Material Cost: ______________
Cost for the plastic frame: _______________ cm @ $0.24 per cm Material Cost: ______________
Cost for glass (Round measurement calculations to the nearest hundredth):
 Color __________________
________________sq. cm @ ______________ per sq. cm
Material Cost: ______________
 Color __________________
________________sq. cm @ ______________ per sq. cm
Material Cost: ______________
 Color __________________
________________sq. cm @ ______________ per sq. cm
Material Cost: ______________
 Color __________________
________________sq. cm @ ______________ per sq. cm
Material Cost: ______________
Cost for Labor: ________________sq. cm @ $0.36/sq. cm
Labor Cost: ________________
TOTAL COST FOR THE LOGO: _____________________________________
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Activity 2: Stained Glass Window
You are the great stained glass architects of the Western World. You have been commissioned to
create a hexagonal stained glass window for the National Mathematics Institute. You will design the
window to meet the Institute’s guidelines:
1.) The window design must be a regular hexagon;
2.) You must submit a drawing of the window design with hexagon side lengths of 4 inches;
3.) The only shapes that may be used in the window design are 30-60-90 triangles and smaller
regular hexagons;
4.) At least 4 different colors of glass must be used;
5.) The drawing needs to be colored to match the color of the stained glass window;
6.) You must include a scale factor of 2:3 for your drawing and stained glass window;
7.) You must submit an itemized bill using these costs:
Material
lead strips (between pieces of glass and around entire window)
blue/brown glass
green/purple glass
red/yellow glass
all other colors of glass
Cost
$3.08 per foot
$7.67 per square foot
$8.23 per square foot
$10.47 per square foot
$9.88 per square foot
labor for entire window
$30.36 per square foot
1. Design a drawing of a hexagonal stained glass window. You may use grid paper to help you.
 Be sure your drawing follows the first three of the Institute’s guidelines.
2. Determine all relevant measurements of your drawing to an appropriate level of precision.
Use those measurements to determine the measurements of the actual window.
 Be sure to follow the sixth guideline.
3. Determine the colors you will be using for your window design and color your window design
accordingly.
 Be sure to follow the fourth and fifth guidelines.
4. Write a report to present to the National Mathematics Institute. Be sure to include:
 How your window meets the Institute’s guidelines for the window;
 The process you used to determine the total cost of the window, including the cost of the
lead strips, the different colors of glass, and the labor for the entire window;
 A persuasive argument on why the Institute should choose your window design.
5. You must present an itemized bill to the Institute for the cost of your window. Use this
template and the costs from the seventh guideline to create your itemized bill.
6. At the same time that you turn in your written report, itemized bill, and design for a stained
glass window, you will give an oral presentation to the class to persuade them why the Institute
should purchase your window.
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Supporting Calculations
Calculations dealing with the lead (Be sure to convert units):
Calculations dealing with the glass (Be sure to convert units):
Calculations dealing with the labor:
Calculations dealing with determining the actual size of the window:
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Itemized Bill
Name: _______________________________________
Date: ______________________
Dimensions of the window: __________________________________________________________
Cost for the lead: ________________feet @ $3.08 per foot
Material Cost: ______________
Cost for glass (Round measurement calculations to the nearest hundredth):
 Color __________________
________________sq. feet @ ______________ per sq. foot
Material Cost: ______________
 Color __________________
________________sq. feet @ ______________ per sq. foot
Material Cost: ______________
 Color __________________
________________sq. feet @ ______________ per sq. foot
Material Cost: ______________
 Color __________________
________________sq. feet @ ______________ per sq. foot
Material Cost: ______________
 Color __________________
________________sq. feet @ ______________ per sq. foot
Material Cost: ______________
Cost for Labor: ________________sq. feet @ $80.36/sq. ft
Labor Cost: ________________
TOTAL COST FOR THE WINDOW: _____________________________________
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Activity 3: Multiple Choice and Short Answer Items
1. Aunt Cecilia wants to paint the walls and ceiling of her master bedroom. The dimensions of each
of the four walls are the same. The length of a wall is 7 m and its height is 5 m. The total area of the
door and two windows in the room is 24 m². A can (one gallon) of paint can be used to paint an area
of 34 m². The store clerk told Aunt Cecilia that the paint she wants will cost her $23.72 per can,
including tax. Aunt Cecilia has $150.00 with her.
How much money will she have left after paying for the required number of cans of paint?
Support your answer using words, numbers and/or diagrams.
How much money will she have left after paying for the
required number of cans of paint? _________________________
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2. The Burke Community Center improvement Committee plans to build a rectangular swimming
pool that meets the requirements below.
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The perimeter of the pool should not exceed 120 meters.
The pool should have the largest surface area possible.
The fence around the pool will measure 30 meters by 50 meters.
The fence around the pool should be at least 4 meters from each edge of the pool.
Recommend to the Burke Community Center Improvement Committee a plan for the
swimming pool dimensions and area that best meets the requirements listed above.
Include a sketch showing the pool dimensions and fence dimensions and any
mathematical explanations to help convince the committee members to use your plan.
Support your answer using words, numbers and/or diagrams.
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3. The diagram shows the ground plan of a business mall. The excavation for its foundation is to be
18 feet in depth.
How many cubic yards of soil must be removed?
Support your answer using words, numbers and/or diagrams.
How many cubic yards of soil must be removed? __________________________________
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4. To estimate the area of an island that he found on a map, Nick traced two copies of the shape of
the island on a piece of paper. Then he drew one circle outside the shape and another circle inside the
shape as shown below.
What is the difference between the areas of the two circles that Nick drew? Use 3.14 for 
Support your answer using words, numbers and/or diagrams.
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5. Mr. Brady asked his algebra class to make a scale drawing of the classroom, which is shaped like
a rectangle with a width of 20 feet and a length of 24 feet. Travis made the width of the classroom in
his scale drawing 5 inches.
Which should be the length in Travis’s scale drawing?
 A. 6 inches
 B. 8 inches
 C. 9 inches
 D. 10 inches
6. The figure shows the dimensions of a scale drawing of a house and attached garage. The scale is
1:60 for the drawing.
15 cm
Garage
House
20 cm
15 cm
30 cm
Which is the total area of the outline of the actual house and garage?
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49,500 cm 2
 B.
123,900 cm 2
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135,000 cm 2
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2,970,000 cm 2
7. One leg of a right triangle is twice as long as the other leg. The length of the shorter leg is x.
Which expression represents the area of the triangle?
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x2
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2x 2
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2x
 D.
1
2
x
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8. For an original graphic design, Lee charges a fixed fee of $50 plus $25 per hour he works. His
competitor charges a fixed fee of $40 plus $30 per hour she works on a design. Lee’s competitor
advertises that her rates are cheaper.
Evaluate Lee’s competitor’s claim that her rates are cheaper.
Support your evaluation using words, numbers and/or diagrams.
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9. A regular hexagon has the same perimeter as a square with side lengths of twelve inches.
Which is the length of each side of the hexagon?
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2 inches
 B.
3 inches
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6 inches
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8 inches
10. The grass in Lindsay’s lawn in dying, and she has decided to replant the lawn. She made a scale
drawing to help her determine the area of her lawn. The shape of Lindsay’s vegetable garden and
flower bed are congruent trapezoids.
Which is the area of her lawn?
 A.
350 ft²
 B.
425 ft²
 C.
475 ft²
 D.
600 ft²
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