tessellation notes

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TargetStrategies © 2008 Evans Newton Incorporated AR04MGE040403-1Last printed 10/31/08

TargetStrategies®

Aligned Mathematics Strategies

Arkansas Student Learning Expectations

Geometry

ASLE Expectation: AR04MGE040403

R.4.G.3Analyze characteristics and properties of two- and three-dimensional

geometric shapes and develop mathematical arguments about

geometric relationships: Identify and explain why figures tessellate

Focus Objective: The student will identify and explain why figures tessellate.

Level: Analysis

Strand: Relationships between two- and three-dimensions

Prerequisite Skills:

• describe relationships among types of two- and three-dimensional objects using their

defining properties (MG.11)

• describe sizes, positions, and orientations of shapes under transformations: translations,rotations, reflections, and dilations (MG.12)

• identify the line or rotational symmetry of objects using transformations (MG.15)

(Coding refers to an applicable TargetFundamentals™ lesson.)

Related Expectations:

AR04MGE010103 Develop the language of geometry including specialized vocabulary,

LG.1.G.3 reasoning, and application of theorems, properties, and postulates:

Describe relationships derived from geometric figures or figural patternsAR04MGE040402 Analyze characteristics and properties of two- and three-dimensional

R.4.G.2 geometric shapes and develop mathematical arguments about geometric

relationships: Solve problems using properties of polygons: Sum of the

measures of the interior angles of a polygon; interior and exterior anglemeasure of a regular polygon or irregular polygon; number of sides or

angles of a polygonAR04MGE050507 Specify locations, apply transformations and describe relationships usingCGT.5.G.7 coordinate geometry: Draw and interpret the results of transformations and

successive transformations on figures in the coordinate plane:

Translations; reflections; rotations (90°, 180°, clockwise andcounterclockwise about the origin); dilations (scale factor)




INSTRUCTIONAL PREPARATION

Materials:

• bag of geometric tiles of various shapes (one per pair of students)

• construction paper (2 sheets per student, optional)

• scissors (one pair per student, optional)

Duplicate the following (one per student unless otherwise indicated):

• Vocabulary reference sheet

• Tessellations reference sheet

• Creating a Tessellation worksheet

• Tessellate the Figure worksheet

• Tessellating Figures worksheet

Prepare a transparency of the following:

• Vocabulary reference sheet

• Tessellations reference sheet

• Creating a Tessellation worksheet

• Tessellate the Figure worksheet

Display the Focus Questions:

• What are the characteristics indicating a figure that tessellates?

• Why would a figure not tessellate?

INSTRUCTION

1. To begin the lesson, ask the students to name any geometric figure they can recall, and draw

the figure on the classroom board. After they name a few shapes, draw any common figuresthey forgot to mention on the classroom board, including various regular polygons. Then ask

the students the following question:

What properties lead you to identify one figure differently from another? (Answers

may vary but might include that the number of sides can be used to classify figures,e.g., a triangle has three sides, a rectangle/square has four sides.)

Discuss students’ responses. Explain that, while all the figures appear different, there are

properties about each that help identify them, such as the number of sides or interior angles.

Explain to the students that some geometric shapes can be placed and connected together ina very symmetrical pattern that covers a plane without any gaps or overlapping regions,

similar to the tile on a kitchen floor. Discuss with the class which shapes on the classroom

board might produce this type of pattern, and have volunteers come to the classroom board

to sketch the patterns to test their predictions. Explain to the class that with regular polygons, whether or not a figure tessellates, or produces this type of pattern, is actually

related to the interior angle measures of the figure. Tell the class that in today’s lesson they

will be investigating shapes that tessellate and how to determine if a figure will tessellate.




2. Distribute copies of the Vocabulary reference sheet and display the transparency. Review

the term polygon, which refers to any closed plane figure that includes three or more linesegments that meet at their endpoints. Tell the students that there are many other polygons

besides the shapes included on the reference sheet and those that were drawn on the

classroom board during the introduction. Refer the students to the definitions of hexagon,

octagon, and pentagon. Ask the following question:

What other figures can you think of besides these that represent polygons? (Answers

may vary but should include triangle, square, rectangle, trapezoid, parallelogram,

pentagon, and decagon.)

Discuss students’ answers. Instruct the students to list those figures not included on theVocabulary reference sheet on the reverse side of the reference sheet. Examples include the

following:

Square: Rectangle:

Trapezoid: Decagon: 10 sided figure

Parallelogram:

Review the term rotation on the reference sheet. Rotation is when a figure is turned in a

certain direction (clockwise or counterclockwise) around a given point. Explain to the class

that one type of tessellation involves rotations. Read over the definition of translation,

explaining that tessellations can also be produced by translating a figure vertically andhorizontally. Allow time for the students to review the remaining terms on the reference

sheet, and answer any questions they may have.

3. Distribute copies of the Tessellations reference sheet and display the transparency. Read the

introduction box aloud while the class follows along. Explain to the students that atessellation is best referred to as tiling, similar to the tile on a kitchen floor. Ask the students

the following question:

What is a common characteristic of tile? (It is commonly placed in a pattern where no

tiles overlap and there are no gaps between tiles.)

Remind the students that, just like tile, tessellations can be formed from a variety of

different shapes. Continue reading the information on the reference sheet, defining

translation tessellation and reviewing examples 1 and 2. Point out to the students thatexample 1 is a parallelogram, and figure 2 is an irregular dodecagon. Both shapes weretranslated, or “moved” horizontally and vertically. Explain to the students that example 1

did not contain any gaps or overlaps in the tiles after the figure was tessellated.

Refer to example 2 and identify both the horizontal and vertical translation of the figure.

Show the students the three gaps that were created between the tiles. Explain that this

example is not considered a tessellation because of the gaps that were created in the tiles




after the figure was translated. Answer any questions the students may have about

identifying translation tessellations. Ask the following question:

On the bottom of page 1, list any other regular geometric shapes that could form a

tessellation? (Answers will vary but may include square, rectangle, triangle,

hexagon.)

Discuss students’ responses, making sure to explain why one figure could tessellate and onecould not. Use all student examples given, and draw each figure in a tessellating pattern on

the classroom board if this shape was not used in step 1 of the Instruction component. Use

the drawing to verify or refute whether a shape can produce a translation tessellation.

After reviewing translation tessellation, ask the students to turn over their reference sheet

and draw a translation tessellation using the following two figures:

After allowing enough time for the students to finish, go over the answers below. Students’

answers may vary slightly, but should show that a rhombus can tessellate and a heart cannot.

4. Display the transparency of the second page of the Tessellations reference sheet. Read over the definition of rotation tessellation, and review examples 3 through 5. Refer to all three

examples and make sure the students can identify the original figure, identified by theshaded gray. Point out to the students that example 3 is a hexagon, figure 4 is not a common

shape, and figure 5 is a pentagon. Point out how all three shapes were rotated around the

original figure. Explain to the students that examples 3 and 4 did not contain any gaps or overlaps in the tiles after the figure was tessellated. Identify on the figure that every tile is

connected on all sides, which identifies this as a tessellation. Refer to example 5, identify

the rotation around the original figure, and make note of the gaps that were created. Pointout to the students that every side is not connected to the next tile, and for this reason

example 5 is not tessellation. Answer any questions students may have. Ask the following

question:

On the bottom of page 2, list any other regular geometric shapes that could form

rotation tessellation. (Answers will vary but may include square, rectangle, triangle,

hexagon.)

Discuss students’ responses, making sure to explain why one figure could tessellate and one

could not. Use all student examples given, and draw each figure in a rotating tessellating




pattern. Use the drawing to test whether the suggested figures tessellate when they are

rotated. Answer any questions students may have.

After reviewing rotation tessellation, ask the students to turn over their reference sheet and

draw a rotation tessellation using the following two figures:

After allowing enough time for the students to finish, go over the answers below. Students’answers may vary, but should show that the arrow cannot tessellate and a triangle can.

Continue reading the box on the bottom of the page. Explain to the students that in order for a regular polygon to tessellate, the measures of the interior angles must be a factor of

360 degrees. Remind the students that to calculate the value of one interior angle, you divide

the sum of the interior angles by the number of interior angles. The sum of the interior angles for any polygon is s = (n – 2)180 where n is the number of sides of the figure. Using

this equation we can find:

square/rectangle has interior angles of 90°, a factor of 360

equilateral triangle has interior angles of 60°, a factor of 360 a regular hexagon has interior angles of 120°, a factor of 360

a regular pentagon has interior angles of 108°, not a factor of 360

a regular octagon has interior angles of 135°, not a factor of 360 a regular decagon has interior angles of 144°, not a factor of 360

Have the students get with a partner, and distribute a bag of geometric tiles to each student

pair. Have the students use the tiles to verify that, for regular polygons, figures tessellateonly when the measures of the interior angles are factors of 360°. Then ask the following

question:

Is it possible for a regular nonagon (9 sided figure) to tessellate? (No it cannot

tessellate because the interior angles are 140 degrees, not a factor of 360.)

Answer any questions the students may have about determining whether a regular polygon

will tessellate. Make sure the students can calculate the interior angle measure of a regular polygon.




5. If there is sufficient time, distribute copies of the Creating a Tessellation worksheet. Have

the students do the activity using a pair of scissors and two sheets of construction paper.

6. Display the focus questions. Give the students time to think about them. Facilitate a

discussion that includes the following points:

Polygons come in many forms, identified by the number of sides on the figure.

Translation tessellations occur when a figure forms a pattern by translating either horizontally or vertically.

Rotation tessellations occur when a figure rotates about its vertices.

A figure can only tessellate when there are no gaps or overlaps between tiles. For a regular polygon figure to tessellate, the measure of its interior angles must be a

factor of 360 degrees.

7. Distribute copies of the Tessellate the Figure worksheet and display the transparency. Readthe directions aloud and answer any questions students have about the worksheet. Assign the

worksheet to be completed individually. When enough time has been allowed to completethe assignment, review the answers with the class, using the Teacher’s Answer Key.

8. To conclude the lesson, instruct the students to put away their worksheet and reference

sheets. Have them answer the focus questions in their math journal or on a separate sheet of paper. They should include examples of translational and rotational tessellations, as well as

how to calculate the interior angle measures of a regular polygon.

ASSESSMENT

Distribute copies of the Tessellating Figures worksheet to be completed individually.

ALTERNATIVE INSTRUCTION

Review with the students several different geometric shapes. Draw each of the following shapeson the classroom board: square, rectangle, parallelogram, isosceles triangle, trapezoid, pentagon,

hexagon, heptagon, octagon, and decagon. Make sure the students can identify the properties of

each shape. Briefly review with the students the definition of tessellation and the difference between translational and rotational tessellation.

Place the students into groups of three or four, and hand each group a bag of different tiles.Using the tiles, the students will find the matching tiles and try to create a pattern that tessellates.The students will do this with many different shapes in the bag and make notes of which shapes

can tessellate and which cannot. Ask the students to identify whether it is translational or

rotational tessellation as they work through the activity. Give the students enough time to getthrough the entire bag of tiles, as you answer any questions students may have. Once finished,

discuss as a class which shapes tessellate and which cannot. Make sure to explain why each

shape is considered tessellation or not.




After discussing the various shapes graphically, explain how to mathematically calculate whether

a regular polygon will tessellate using the interior angle measure. Have the students practicecalculating interior angles, and using that answer, determine if various different regular polygons

could tessellate.

ALTERNATIVE ASSESSMENT

Prepare additional assessment questions in the same format as the Tessellating Figures

worksheet.




Vocabulary

Geometry

The study of the properties andrelationships of points, lines, angles,

surfaces, and solids in space

Hexagon

A polygon with six sides

Octagon

A polygon with eight sides

Pentagon

A polygon with five sides

Plane figure

Any two-dimensional figure

Polygon

A closed plane figure formed from

three or more line segments that

meet only at their endpoints

Rotation

A transformation in which a figure is

turned a given angle and directionaround a point called the point of

rotation

Translation

A transformation in which a figureis vertically and/or horizontally shifted

A

A′

A A′




Tessellations

Tessellations are patterns of identical shapes that can completely cover a

plane with no gaps and no overlaps. These patterns can be used to create

artwork, and they are used in designs for tile, wallpaper, and clothing.

Types of Tessellations:

Translation tessellation: A pattern formed by horizontal and/or vertical transformations.

Example 1:

Example 2:

Original figure

The original figure

transformed both horizontally(left and right) and vertically

(up and down). There are no

gaps or overlaps betweentiles, which make this figure

tessellate.

Original figure

The original figuretransformed both horizontally

(left and right) and vertically

(up and down). There are gapsin between tiles, which does

not make this figure

tessellate.

Gaps




Since tessellations meet at a corner with no gaps, for a regular polygon to tessellate, its

interior angle measures must be a factor of 360 degrees.

Examples of regular polygons that tessellate: equilateral triangle, square,

parallelogram, rectangle, hexagon

Examples of regular polygons that do not tessellate: pentagon, octagon, decagon

Tessellations(continued)

Rotation tessellation: A tessellation that is created by rotating a figure about its vertices.

Example 3:

Example 4:

Example 5:

Original figure

Original figure

Original figure

Examples 3, 4, and 5 rotationallytransformed around the original figure.

There are no gaps or overlaps between tiles

in figures 3 and 4, making the figurestessellate. Example 5 has gaps between

tiles, which is not a tessellating figure.Gaps




Name ________________________________________________________________________

Creating a Tessellation

Directions: Follow the directions below to create a translation tessellation.

1) Trace a design on one of the sides of the rectangle.

2) Cut out the rectangle. Then cut along the dashed-line segments, and attach the cut-out pieceto the opposite side of the rectangle.

3) Tape the pieces together, and trace several of these shapes onto a blank sheet of paper.

4) Cut out the traced pieces, and put them together to form a translation tessellation.




Creating a Tessellation(continued)

Directions: Follow the directions below to create a rotation tessellation.

1) Create a design along the top of the square.

2) Rotate the design about point B so the endpoint at A is moved to point C .

3) Rotate the design about point C so the endpoint for B is at point D.

4) Rotate the design about point D so the endpoint for C is at point A.

5) Cut out and tape the design in its appropriate locations.

6) Trace several of the figures. Cut them out and place them to create a rotation tessellation.




Name ________________________________________________________________________

Tessellate the Figure

Directions: Read each problem carefully and solve. Explain your answer fully.

1. Which of the following polygons could tessellate?

A B C D

2. Johnny wants to tile his patio using only one polygonal shape. Which regular polygon shapecouldn’t he use to tile the patio?

A B C D

3. In order for a figure to tessellate, the sum of the interior angles must be a factor of what

number?

4. Sara is making a quilt using one of the regular polygon shapes, and she wants to make atessellating pattern. Her choices are a pentagon, parallelogram, circle, or octagon. Which

one of these shapes would work for Sara?




Name ________________________________________________________________________

Tessellating Figures

Directions: Read each question. Circle the letter that contains the correct answer to the questionor complete the problem in the space provided.

1. Which is not an example of tessellation?

A. B. C. D.

2. Jean’s shirt has a pattern that was created using a single repeated shape with no holes or

gaps between shapes. All the following could have been used to create Jean’s shirt pattern

except

A. B. C. D.

3. Melinda wants to tile her bathroom floor using only one polygonal shape. Which of the

following shapes could she use to tile the floor?

A. B. C. D.




4. Mark is putting new tile in the shower in his bathroom. He wants to use colored tiles that are

regular-shaped polygons. He will tessellate the polygons for the pattern on the tile as theyform the walls of the shower. Draw and explain two shapes that will not be able to tessellate

for his bathroom design, and two shapes that will be able to tessellate.

5. Ron is tiling his kitchen; however, he doesn’t know which shape of tile to use. Help him

identify which shape would tessellate to fit perfectly on his floor without having to cut thetile.

A. B. C. D.

6. Which of the following is a requirement for a regular polygon to tessellate?

A. The sum of its interior angles must be 360°. B. The sum of its exterior angles must be 360°.

C. Each interior angle measure must be a factor of 360°. D. Each interior angle measure must be 360°.




7. Which of the following is an example of tessellation?

A. B. C. D.

8. If the following shapes were drawn in a connecting pattern, which one would not tessellate?

A. B. C. D.

9. Danny is building a new home. He is laying shingles on the roof and wants to find a newshape to use. The new shape must tessellate in order to fit the entire area of the roof, and he

cannot cut the shingles. Identify a shape that would tessellate and fill the area of the roof below. Sketch in the figure to demonstrate how it could tessellate to cover the roof.

10. Which of the following is not an example of a tessellation?

A. B. C. D.




TEACHER’S ANSWER KEY

Tessellate the Figure1. B

2. B3. 360°

4. Parallelogram

Tessellating Figures

1. D

2. C

3. C4. 0-4 points: 4 points for correctly stating the figures that would not work include regular

pentagon, octagon, and decagon –explaining that any figure with interior angles that are not

factors of 360° would not tessellate – stating that figures that would work include triangle,square, parallelogram, rectangle, and hexagon – and explaining that any figure with angles

that are factors of 360° would tessellate; 3 points for three of the above components;2 points for two of the above components; 1 point for one of the above components; or

0 points for no correct components or for no response.

5. C6. C

7. A

8. D9. 0-2 points: 2 points for correctly answering hexagon and sketching hexagons in the diagram;

1 point for stating the correct answer with no drawing; or 0 points for an incorrect answer or

for no answer.10. D

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