List of Activities for This Unit

MOSAIC DESIGNS

TEACHER EDITION

List of Activities for this Unit:

ACTIVITY / STRAND / DESCRIPTION
1 – Finding Perimeter / ME / (#1) Find perimeter, given area
2 – Finding Area / ME / (#2-4) Max/Min Perimeter
3 – Finding Max and Min Area / ME / (#1-4) Find area, given perimeter
4 – Kelly’s Glass Squares / ME / (#1-3) Kelly’s Glass Squares
5 – Macaroni and Cheese Box / ME / (#1-2) Macaroni and Cheese Box
6 – Miller’s Fine Foods / ME / (#3-10) Miller’s Fine Foods
7 - Mosaic Designs Multiple Choice / ME / Mosaic Designs Multiple Choice
COE Connections / Boxes
Space Station Window
MATERIALS / Calculators
Grid Paper
Algebra Tiles
Warm-Ups
(in Segmented Extras Folder) / Geometry Gift

Vocabulary: Mathematics & ELL

area(s) / design(s) / max / rectangular prism
cardboard / equivalent / maximum / represent
CEO / even number / min / rule
configurations / fertilize / minimum / shipping box
container(s) / fertilizer / mosaic / square inch
contractor / flower box / odd number / tiles
cubic yards / manipulatives / perimeter / wraps
data / materials / prism

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 andsquare 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?

Performance Expectations:

4.3.CDetermine the perimeter and area of a rectangle using formulas, and explain why the formulas work.

4.3.EDemonstrate that rectangles with the same area can have different perimeters, and that rectangles with the same perimeter can have different areas.

4.3.FSolve single- and multi-step word problems involving perimeters and areas of rectangles and verify the solutions.

5.3.ISolve single- and multi-step word problems about the perimeters and areas of quadrilaterals and triangles and verify the solutions.

6.4.BDetermine the perimeter and area of a composite figure that can be divided into triangles, rectangles, and parts of circles.

6.6.ECommunicate the answer(s) to the question(s) in a problem using appropriate representations, including symbols and informal and formal mathematical language.

7.6.ECommunicate the answer(s) to the question(s) in a problem using appropriate representations, including symbols and informal and formal mathematical language.

8.5.ECommunicate the answer(s) to the question(s) in a problem using appropriate representations, including symbols and informal and formal mathematical language.

Performance Expectations and Aligned Problems

Chapter 19 “Mosaic Designs” Subsections: / 1-
Finding Perimeter / 2-
Finding Area / 3-
Finding Max and Min Area / 4-
Kelly’s Glass Squares / 5-
Macaroni & Cheese Box / 6-
Miller’s Fine Foods / 7-
MC Pbms.
Problems Supporting:
PE 4.3.C ≈ 4.3.E ≈ 4.3.F / 1 / 2 – 4 / 1 – 4 / 1 – 3 / 1, 2 / 3 – 10 / 14, 16
Problems Supporting:
PE 5.3.I / 1 / 2, 4 / 1 – 4 / 1, 2 / 3 – 10 / 15 - 17
Problems Supporting:
PE 6.4.B / 1, 2 / 3, 4 / 16
Problems Supporting:
PE 6.6.E ≈ 7.6.E ≈ 8.5.E / 3, 5 / 3, 4 / 1, 2 / 3 – 10 / 14, 16, 17

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.

FINDING PERIMETER

1. Ian is an interior decorator who is constructing rectangular mosaic designs of different areas using colored ceramic square tiles. All the ceramic square tilesare 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 anothertile’s complete side. Tiles cannot just connectcorner to corner.

When the rectangular designs are completed, Ian wraps the perimeter of the mosaic with a thin plasticpiece. He wants toinvestigate the relationship between the number of tiles used in the rectangular mosaic andthe length of a plastic piece neededfor 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 wrapthe 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 andminimum 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 predictionsabout the maximum and minimum lengths of plastic pieces needed.

Area
(Number of tiles) / Maximum length of plastic piece needed (Perimeter) / Minimum length of plastic piece needed (Perimeter)
1 / 4 units / 4 units
2 / 6 units / 6 units
3 / 8 units / 8 units
4 / 10 units / 8 units
5 / 12 units / 10 units
6 / 14 units / 10 units
7 / 16 units / 12 units
8 / 18 units / 12 units
9 / 20 units / 12 units
10 / 22 units / 14 units
11 / 24 units / 14 units
12 / 26 units / 14 units
13 / 28 units / 16 units
14 / 30 units / 16 units
15 / 32 units / 16 units
16 / 34 units / 16 units

FINDING AREA

2. Pick three different areas for a mosaic and sketch arectangular 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

MaximumMinimum

Mosaic 2

MaximumMinimum

Mosaic 3

MaximumMinimum

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

_No! In this problem, the rectangular tiles when touching on a side will always eliminate 2 unit lengths and from the perimeter total because of the two sides touching. Since all mosaics are made from chains of rectangles (4 sided), then all perimeter will be chains of rectangles minue touching rectangle panes and therefore 4(n) = 2(touching rectangles) = Perimeter.______

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

35 tiles
Maximum Perimeter:____P = (36)2 = 72______

Minimum Perimeter:____P = 24______

56 tiles
Maximum Perimeter:___P = 2(56 + 1) = 2(57) = 114______

Minimum Perimeter:___P = 30______

64 tiles
Maximum Perimeter:___P = 2(64 + 1) = 2(65) = 130______
Minimum Perimeter:___P 2(16) = 32______

5. Write a rule for finding the maximum perimeter for any rectangle with a given area.

P = 2(n + 1)

FINDING MAX AND MIN 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 predictionsabout the maximum and minimum areas of the mosaic.

Perimeter
(Lengths of plastic pieces) / Maximum number
of tiles
(Area) / Minimum number
of tiles
(Area)
4 units / 1 / 1
6 units / 2 / 2
8 units / 4 / 3
10 units / 6 / 4
12 units / 9 / 5
14 units / 12 / 6
16 units / 16 / 7
18 units / 20 / 8
20 units / 25 / 9
22 units / 30 / 10
24 units / 36 / 11
26 units / 42 / 12

2. Predict the maximum and minimum areas for a mosaic with theselengths of plastic pieces:

30 units
Maximum Area:_____56 rectangles______
Minimum Area:_____14 rectangles______

40 units
Maximum Area:______100 rectangles______
Minimum Area:______19 rectangles______

88 units
Maximum Area:____484 rectangles ______

Minimum Area:____43 rectangles ______

3. Write a rule for finding the maximum area for any rectangle with a given perimeter.

If the perimeter is divisible by 4 then the formula is

______Max Area = , but only good for P/4______

There is not an easy pattern for the other max areas. If you create a table there is a pattern with successive differences.

How could you support or prove your rule using data?

______Rule  if P/4 than Max Area = (P/4)²______

______

______This is what the table has as max area values as a function of P.______

______Every other entry is always divisible by 4 (P grows by 2 at each stage). Therefore, every other P is evenly divisible by 4.______

KELLY’S GLASS SQUARES

Kelly is making a rectangular mosaic out of colored glass squares in art class. The area of each glass square is 1. Kelly’s first design is a rectangle that has an area of 24.
What are thepossible perimeters of her mosaic?

24 cm²2  2  2  3 are the prime factors of 24.

2  12  28 cm Perimeter

4  6  20 cm Perimeter

8  3  22 cm Perimeter

1  24  50 cm Perimeter

______Possible rectangles all have 24 cm² Areas______

The plastic piecethat will go around the perimeter of the mosaic costs $2.40 per meter.
How much wouldthe minimum and maximum lengths of plastic pieces cost?

Min 20 cm($2.40/meter) = (0.2 m)($2.40) = $0.48

Max 50 cm($2.40/meter) = (0.5 m)($2.40) = $1.20

Kelly decides to purchase 72 centimeters of plastic to go around the perimeter of a second design.
What are theminimum and maximum areas of the designs Kelly can create?

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.

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

A few sample answers.

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

Answers will vary.

Net of Rectangular Prism 2

Answers will vary.

Net of Rectangular Prism 3

Answers will vary.

MILLER’S FINE FOODS

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

Which one of your boxes would use the least material to make? _____6”  2”  2______

5. You found a low bid for cardboard that costs $0.0035 per square inch. How much would your macaroni box cost to produce? _____$0.196______

Support your answer using words, numbers and/or diagrams.

Not in Student Version

6. In #5 you calculated the theoretical surface area needed to build the box. Draw a net for your box that could be cut out of cardboard that includes the reality of needing overlapping sides to properly close the box. Be sure to include fully overlapping tabs that could be glued to hold your box together. Calculate the new area and new price to make the box.

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? __24″ h × 4″ wide × 6″ deep

Support your answer using words, numbers and/or diagrams.

b. How many cubic yards would that box hold? ____0.012 yd³____

Support your answer using words, numbers and/or diagrams.

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?

8. How much cardboard would your new, larger box take to make? ____120 in²_____

Support your answer using words, numbers and/or diagrams.

9. How much would it cost to make? ____$0.42______

Support your answer using words, numbers and/or diagrams.

10. Prepare a proposal to show the CEO of Miller’s Fine Foods about your three original box designs, the box you decided to use, the design of the case for shipping macaroni to the stores, and the family size box of macaroni. Give the CEO a complete statement showing the cost of cardboard and the dimensions of the different boxes. Write a paragraph explaining why you chose the boxes to present to the CEO.

______Answers will vary.______

MOSAIC DESIGNS – MULTIPLE CHOICE

11. Tina has built a window flower box. The rectangular box measures feet long, foot deep, and 1 foot wide.

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

 A. cubic feet

 B. cubic feet

 C. 4 cubic feet

 D. cubic feet

12. 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.

Support your answer using words, numbers and/or diagrams.

Vol = Bh

Cylinder A: V = 3² 5 = 45

Cylinder B: V = 5² 3 = 75

13. 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

 B. 800

 C. 4800

 D. 9600

14. 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.

 B.

 C.

 D.

15. 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

16. 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. square units

 B. square units

 C. square units

 D. square units

17. Sarah told Eddie that the formula for finding the perimeter of a rectangle is 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

 A.

 B.

 C.

 D.

Teacher Ch 19 “Mosaic Designs”