# 7 mosaic designs

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```					                                      MOSAIC DESIGNS

Enduring Understanding: Develop a better understanding volume and surface area of a rectangular
prism and finding volume or surface area within the context of a situation. Develop a better
understanding of how a change in one linear dimension affects the volume and surface area of a
figure. Develop a better understanding of how to convert within the US or metric system to maintain
an appropriate level of precision. Develop a better understanding of how to use formulas to
determine the measurements of prisms. Develop a better understanding of how to draw, describe and
label 1-dimensional, 2-dimensional, and 3-dimensional figures, including prisms. Develop a better
understanding of how to identify a pattern. Develop a better understanding of how to draw a
conclusion and support the conclusion using inductive or deductive reasoning. Develop a better
understanding of how to use everyday and mathematical language and/or notation to explain and/or
describe complex mathematical ideas and information in ways appropriate for audience and purpose.
Develop a better understanding of how to apply concepts from two or more of the mathematics
content strands in a given problem situation.

Essential Questions:

   How does the change in one linear dimension affect the volume and surface area of a figure?
   How does one calculate the volume and surface area of a rectangular prism?
   How can a conclusion be supported using mathematical information and calculations?
   How does one draw a net of a figure?
   How are units converted within the US or metric system?
   How is a proposal written?

Lesson Overview:

   Before allowing the students the opportunity to start the activity, access their prior knowledge
regarding surface area and volume of a solid and what affects those measurements of a solid?
   Students will need grid paper and square tiles, centimeter cubes, or other manipulatives to
complete these activities.
   A good warm-up for this activity is Geometry Gift.
   What happens to the volume and surface area of a rectangular prism when one linear
dimension of the rectangular prism is changed?
   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 do students make their thinking visible, clear, easy to follow?

Length of the Lesson: 250 minutes

EALRs/GLEs:

1.2.1-- Understand the relationship between change in one or two linear dimension(s) and
corresponding change in perimeter, area, surface area, and volume.
1.2.5-- Use formulas to determine measurements related to right prisms, cylinders, cones, or
pyramids.
1.3.1-- Understand the properties of and the relationships among 1-dimensional, 2-dimensional, and
3-dimensional shapes and figures.

Mosaic Designs                                                                                        1
1.5.1-- Apply knowledge of patterns or sequences to represent linear functions
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.
3.3.3-- Validate thinking about mathematical ideas.
4.2.1-- Organize, clarify, and refine mathematical information relevant to a given purpose.
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: ME01; ME02; ME03; GS01; AS01; SR02; SR04; SR05; CU02; MC01

Assessment: Use the multiple choice and short answer items from Measurement and Geometric
Sense 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.

Mosaic Designs                                                                                        2
MOSAIC DESIGNS
Activity 1
1. Ian is an interior decorator who is constructing rectangular mosaic designs of different areas using
colored ceramic square tiles. All the ceramic square tiles are congruent (same shape and size). The
length of one side of a tile is one unit. The tiles are glued together with the side of one square tile
connecting to another tile’s complete side. Tiles cannot just connect corner to corner.

Or

When the rectangular designs are completed, Ian wraps the perimeter of the mosaic with a thin plastic
piece. He wants to investigate the relationship between the number of tiles used in the rectangular
mosaic and the length of a plastic piece needed for his designs.
For one tile, the maximum length of a plastic piece needed to wrap the perimeter of the tile is
4 units. The minimum length of a plastic piece needed to wrap the perimeter of the tile is also
4 units because there is only one tile. Look at the two configurations for 8 tiles:

Perimeter = 18 units
Perimeter = 12 units
Both configurations use 8 tiles, but the two perimeters are different.
Find the maximum and minimum lengths of plastic pieces needed for two tiles through sixteen tiles.
You may use grid paper and manipulatives to help you. Remember the mosaics must be rectangles.

Record the information in the table on the next page. Look for patterns to see if you can make
predictions about the maximum and minimum lengths of plastic pieces needed.

Mosaic Designs                                                                                        3
Maximum length of      Minimum length of
Area
plastic piece needed   plastic piece needed
(Number of tiles)
(Perimeter)            (Perimeter)
1                 4 units                4 units
2                 6 units                6 units
3                 8 units                8 units
4
5
6
7
8                 18 units               12 units
9
10
11
12
13
14
15
16

Mosaic Designs                                                                     4
2. Pick three different areas for a mosaic and sketch a rectangular arrangement for the maximum
perimeter and minimum perimeter. Do not use any of the examples on the previous page. Be sure
to label the length, width, and area of the rectangles with a number and a unit of measure.

Mosaic 1
Maximum                                                     Minimum

Mosaic 2
Maximum                                                     Minimum

Mosaic 3
Maximum                                                     Minimum

3. Is there a mosaic for which the perimeter is an odd number? If so, draw the arrangement. If not,
explain.

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Mosaic Designs                                                                                        5
4. Predict the maximum and minimum perimeters for a mosaic with these number of tiles:

a.    35 tiles
Maximum Perimeter: ________________________________________________________
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Minimum Perimeter: ________________________________________________________
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b.    56 tiles
Maximum Perimeter: ________________________________________________________
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Minimum Perimeter: ________________________________________________________
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c.    64 tiles
Maximum Perimeter: ________________________________________________________
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Minimum Perimeter: ________________________________________________________
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5. Write a rule for finding the maximum perimeter for any rectangle with a given area.

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Mosaic Designs                                                                           6
Activity 2: Investigating Area
Ian has several different lengths of plastic pieces he can use for his mosaics. He wants to know the
maximum area and the minimum area of a rectangular mosaic that can have a specific length of
plastic around the perimeter.

1. Find the maximum and minimum areas for the perimeters (lengths of plastic pieces) given in the
table. You may use grid paper and manipulatives to help you. Remember the mosaics must be
rectangles.

Perimeter = 4 units
Perimeter = 8 units

Record the information in the table. Look for patterns to see if you can make predictions about the
maximum and minimum areas of the mosaic.

Maximum number   Minimum number
Perimeter
of tiles         of tiles
(Lengths of plastic pieces)
(Area)           (Area)
4 units                    1                 1
6 units
8 units                    4                 3
10 units
12 units
14 units
16 units
18 units
20 units
22 units
24 units
26 units

Mosaic Designs                                                                                         7
2. Predict the maximum and minimum areas for a mosaic with these lengths of plastic pieces:

d.    30 units
Maximum Area: ____________________________________________________________
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Minimum Area: ____________________________________________________________
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e.    40 units
Maximum Area: ____________________________________________________________
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Minimum Area: ____________________________________________________________
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f.    88 units
Maximum Area: ____________________________________________________________
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Minimum Area: ____________________________________________________________
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Mosaic Designs                                                                                8
3. Write a rule for finding the maximum area for any rectangle with a given perimeter.

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4. How could you support or prove your rule using data?

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Mosaic Designs                                                                           9
Activity 3: Application

1. Kelly is making a rectangular mosaic out of colored glass squares in art class. The area of each
glass square is 1 cm 2 . Kelly’s first design is a rectangle that has an area of 24 cm 2 .

What are the possible perimeters of her mosaic?

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2. The plastic piece that will go around the perimeter of the mosaic costs \$2.40 per meter.

How much would the minimum and maximum lengths of plastic pieces cost?

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Mosaic Designs                                                                                        10
3. Kelly decides to purchase 72 centimeters of plastic to go around the perimeter of a second design.

What are the minimum and maximum areas of the designs Kelly can create?

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Mosaic Designs                                                                                     11
Activity 4: Macaroni and Cheese Box
Miller’s Fine Foods wants to create a box for macaroni and cheese that will enclose 24 cubic inches.
They call on your industrial engineering firm to design a package. The project is assigned to you.

1. Draw 3 different rectangular prisms that each have a volume of 24 cubic inches. Be sure to
label the height, width, and length of each rectangular prism with a number and a unit of
measure.

Rectangular Prism 1

Rectangular Prism 2

Rectangular Prism 3

Mosaic Designs                                                                                     12
2. The boxes will be cut out from a flat piece of cardboard and shipped flat to the Miller’s Fine
Foods Company. The company wants to know what each of the boxes will look like as a flat piece of
cardboard. Draw a net for each of the rectangular prism you drew in question 1. Be sure to label all
dimensions of the net with a number and a unit of measurement. Also label where the flat cardboard
can be folded to create the rectangular prism.

Net of Rectangular Prism 1

Net of Rectangular Prism 2

Net of Rectangular Prism 3

Mosaic Designs                                                                                    13
3. Miller’s Fine Foods wants to use the least amount of cardboard for their macaroni and cheese box.
You are asked to find the surface area of each of the rectangular prisms you drew in question 1.

Surface Area of Rectangular Prism 1

Surface Area of Rectangular Prism 2

Surface Area of Rectangular Prism 3

4. Which one of your boxes would use the least material to make? __________________________

Mosaic Designs                                                                                    14
5. You found a low bid for cardboard that costs \$0.0035 per square inch. How much would your

macaroni box cost to produce? ________________________________________________________

6. Miller’s Fine Foods wants to ship their macaroni in containers that will hold 24 boxes of the size
you chose for #2.

a. What could be the dimensions of the shipping box be? ______________________________

b. How many cubic yards would that box hold? ______________________________________

7. Miller’s Fine Foods also markets a family size box of macaroni and cheese in a box that has to
hold 72 cubic inches. How would you have to change the dimensions of the box you originally
designed to hold that amount?

Mosaic Designs                                                                                          15
8. How much cardboard would your new, larger box take to make? __________________________

9. How much would it cost to make? ___________________________________________________

Mosaic Designs                                                                         16
10. Prepare a proposal showing the CEO of Miller’s Fine Foods your three examples and your final
products. Give the CEO a complete statement showing the cost of cardboard and the dimensions of
the box necessary for shipping. Write a paragraph explaining why you chose the boxes to present to
her.

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Mosaic Designs                                                                                   17
14. Tina has built a window flower box. The rectangular box measures 2 1 feet long,
2
1
2   foot deep,
and 1 foot wide.

Which is the volume of potting soil she will need to completely fill this flower box?

 A.        1 1 cubic feet
4

 B.        2 1 cubic feet
4

 C.       4     cubic feet
 D.        4 1 cubic feet
4

15. Simoné has two cylinders and wants to compare the volume of cylinder A to the volume of
cylinder B.

Which statement accurately compares the volume of cylinder A to the volume of cylinder B?

 A.       The volume of cylinder A is greater than the volume of cylinder B.
 B.       The volume of cylinder B is greater than the volume of cylinder A.
 C.       It is not possible to compare the volumes.
 D.       The volumes of cylinder A and cylinder B are the same.

Mosaic Designs                                                                                           18
16. An outside concrete playing surface is being added to a school. The contractor dug a rectangular
hole 40 feet long, 40 feet wide, and 6 inches deep.

Which is the volume of cement needed to fill the rectangular hole the contractor dug?

 A.       400 ft 3

 B.       800 ft 3

 C.       4800 ft 3

 D.       9600 ft 3

17. Sonya wants to buy ribbon to wrap around the rectangular-prism-shaped present shown.

She also needs 25 inches of ribbon to make a bow.

Which expression represents the minimum length of ribbon Sonya needs to buy?

 A.        l  w  h  25
 B.       2l  2w  2h  25
 C.       2l  2w  4h  25
 D.       4l  4w  4h  25

Mosaic Designs                                                                                    19
18. Damon wants to fertilize his lawn for the spring season. The dimensions of his lawn are shown.
Johnson’s Garden Shop sells fertilizer in 6-pound bags that cover an area of 500 square feet.

Which number of 6-pound bags of fertilizer will Damon need to completely fertilize his lawn?

 A.        2
 B.       3
 C.       4
 D.       5

19. The floor of Dwylene’s bedroom is shaped like a rectangle. She has a rectangular-shaped rug in
the center of the floor with the dimensions shown.

Which expression represents the area of the bedroom floor NOT covered by the rug?

 A.        22x square units
 B.        2x 2 square units
 C.        2 x 2 18 x  28 square units
 D.       18x  28 square units

Mosaic Designs                                                                                       20
20. Sarah told Eddie that the formula for finding the perimeter of a rectangle is P  2L  2W where
W is the width of the rectangle, L is the length of the rectangle, and P is the perimeter of the
rectangle.

Eddie told Sarah that there are equivalent ways to represent the formula for finding the perimeter of a
rectangle.

Which equation is equivalent to P  2L  2W ?

 A.       P  2 L W 

 B.       P  2  L  4W 

 C.       P  2  2L  2W 

 D.      P  L 2 W 

Mosaic Designs                                                                                        21

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