The Logic of Space - Is it possible?

Grade Level: 6
Classroom Time: Two 90-minute sessions

Materials Needed:

Protractor
Pencil
Handouts
Scissors
Metric and Customary rulers
Graph paper, 2 per student (http://www.math.kent.edu/~white/graphpaper/ - non bold 1”, two or handout)
Tape
Blank Paper
Cardstock/Posterboard
Markers and/or map pencils

Objectives:
TSW: Observe, discover and define optical illusions
TSW: Experience translations, transformations, rotations and reflections
TSW: Examine, define and discover tessellations or tiling
TSW: Review and define regular polygons
TSW: Create tessellations using regular polygons
TSW: Create tessellations using a general definition
TSW: Problem solve real life situations involving perimeter and area

Session One

Distribute Handout 1 without any explanation - Students answer the questions quickly while displaying each figure individually for a few seconds.
Figure 1 – http://www.artlex.com/ArtLex/o/opticalillusion.html - this one isn’t necessary….
Figure 2 – http://www.michaelbach.de/ot/lum_scGrid/index.html
Figure 3 - http://www.artlex.com/ArtLex/o/images/optclillusn_circeye_lg.jpg
Distribute Handout 2, rulers, and protractors after Handout 1 is completed. Compare the two handouts. Were they surprised by their answers? Have they observed other types of illusions before? Why do the pictures appear different than what they really are?
Display and examine the following optical illusions: http://www.artlex.com/ArtLex/o/opticalillusion.html, http://www.michaelbach.de/ot/
Many of the optical illusions presented are tessellations. Examine and define tessellations.
1. Examples of optical illusion tessellations: http://gwydir.demon.co.uk/jo/tess/optical.htm
2. Define tessellations: http://www.gvsu.edu/math/students/adk/Tesselations.html , http://mathforum.org/sum95/suzanne/whattess.html , http://www.coolmath.com/tesspag1.htm
3. Discover the master of tessellations, M C Escher –
  1. Images - http://images.google.com/images?sourceid=navclient&ie=UTF-8&rlz=1T4DMUS_enUS234US235&q=escher&um=1
  2. Official website which includes a biography - http://www.mcescher.com/ , http://library.thinkquest.org/16661/escher.html
Review Regular polygons – squares, triangles, hexagons, octagons, etc.
1. Print template(s) - http://library.thinkquest.org/16661/templates/index.html
2. Measure the sides and the interior angles of the figures putting the information in a chart stating the number of sides, the lengths of the sides and the measures of the interior angles.
3. Discuss commonalities and differences using the chart – all the figures have equal angles and equal sides – Regular figures
4. Review optical illusions – were Regular polygons used? - http://gwydir.demon.co.uk/jo/tess/optical.htm

Session TWO

Optical illusions, tessellations and math – Handout 3
1. Tessellation websites - http://www.mathacademy.com/pr/minitext/escher/, http://mathworld.wolfram.com/Tessellation.html , http://www.cap.nsw.edu.au/bb_site_intro/stage2_Modules/tesselations/tesselations.htm, http://www.teacherlink.org/content/math/activities/skp-tessellation/guide.html
2. The scheme Escher used to create his tessellations - http://www.artlex.com/ArtLex/t/images/tessel_reduc.schem.lg.jpg (print)
  1. What shapes are used? Are they Regular?
  2. Is there a pattern?
3. Websites with directions on creating a tessellation: http://mathforum.org/~sanders/geometry/GP08CreatingTessellations.html, http://www.teachervision.fen.com/mathematics/printable/3523.html,
4. Optional demonstration, creating tessellations on the computer: Directions and examples: http://www.teachers.ash.org.au/geparker/
  creating_a_basic_tessellation_using_microsoft_paintbrush.htm, http://www.shodor.org/interactivate/activities/tessellate/
Discovery: Handout 4
1. Do all tessellations require a Regular polygon?
2. Beyond the paper – logistics and mathematics needed to create large masterpieces
Review the scheme Escher used to create his tessellations, http://www.artlex.com/ArtLex/t/images/tessel_reduc.schem.lg.jpg, students create their own tessellation using Escher’s scheme.
Optional: Students create optical illusion tessellations either by mimicking observed optical illusion tessellations or try creating one of their own.
Review: http://gwydir.demon.co.uk/jo/tess/optical.htm

National Math Standards

Geometry Standard: understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects
Measurement Standard: understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume

TEKS:

6(8) Measurement. The student solves application problems involving estimation and measurement of length, area, time, temperature, volume, weight, and angles.

(B) select and use appropriate units, tools, or formulas to measure and to solve problems involving length (including perimeter), area, time, temperature, volume, and weight;
( C) measure angles;

(6.6) Geometry and spatial reasoning. The student uses geometric vocabulary to describe angles, polygons, and circles.

(A) use angle measurements to classify angles as acute, obtuse, or right;

(6.11) Underlying processes and mathematical tools. The student applies Grade 6 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school.

(A)  identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics;
(B)  use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;
(C)  select or develop an appropriate problem-solving strategy from a variety of different types, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem; and
(D)  select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems

HISD CLEAR Mathematics Objectives:

MATH.6.6A : Use angle measurements and estimates to define and classify angles as acute, obtuse, or right.
MATH.6.8C: Measure angles using a protractor or a pictorial representation of a protractor and estimate the measure of given angles.
MATH.6.11A: Identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics.
MATH.6.11D: Select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems.
MATH.6.12A: Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models.

National Visual Arts Standards

Making connections between visual arts and other disciplines. - Students describe ways in which the principles and subject matter of other disciplines taught in the school are interrelated with the visual arts
Choosing and evaluating a range of subject matter, symbols, and ideas - Students integrate visual, spatial, and temporal concepts with content to communicate intended meaning in their artworks
Understanding the visual arts in relation to history and cultures - Students know and compare the characteristics of artworks in various eras and cultures

TEKS – §117.32. Art, Grade 6.
(2)  Creative expression/performance. The student expresses ideas through original artworks, using a variety of media with appropriate skill. The student is expected to:
(A)  express a variety of ideas based on personal experience and direct observations;
(C)  demonstrate technical skills effectively, using a variety of art media and materials to produce designs, drawings, paintings, prints, sculptures, ceramics, fiberart, photographic imagery, and electronic media-generated art.
(4)  Response/evaluation. The student makes informed judgments about personal artworks and the artworks of others. The student is expected to:
(B)  analyze original artworks, portfolios, and exhibitions of peers to form conclusions about formal properties and historical and cultural contexts.

HISD CLEAR Objectives:

ART.6.1.02: Manipulate two different media to effectively express ideas in a variety of art forms.
ART.6.1.04: Explore different and unusual ways of approaching two particular subjects.
ART.6.1.13: Recognize two artists who use different art forms and media to create artworks.
ART.6.2.02: Critically evaluate his/her work and the work of others for two artworks.
ART.6.3.01: Develop an understanding of each of the types of visual expression in two, two- and three-dimensional visual artworks.
ART.6.3.15: Analyze two unfamiliar artworks.
ART.6.6.01: Develop perception skills necessary for generating and organizing two original art creations.

Visualize
Handout 1

Vizualize

In figure 1, which horizontal line is longer, A or B? ___________________

In figure 2, how many black dots can you count? ____________________

What do you notice in figure 3? __________________________________

Visual Math
Handout 2

Vizualize 2

Find the lengths of the horizontal lines in figure 1. Which one is the longest?

A. Customary _________ Metric __________

B. Customary _________ Metric __________

Find the measures of the angles on the right end of each straight angle in figure 1. Find the

A. _______________ B. _______________

Find the measures of the two small angles at the end of the first horizontal line. __________ ___________

Find the length and width of the large rectangle in figure 2 using both customary and metric measures. State the units used.

Customary: l = ____ w = ____ Metric: l = ____ w = ____

Find the area of the rectangle in figure 2. State the units.

Customary: A = ___________ Metric: A = ______________

Find the area of one of the small squares in figure 2. State the units.

Customary: A = ___________ Metric: A = ______________

How many small squares are there in figure 2? __________

Add the areas of all the small squares in figure 2 together.

Customary: ____________ Metric: _______________

Observation:

Find the diameter of the small and large circle in figure 3.

Small circle: Customary ___________ Metric ___________

Large circle: Customary ___________ Metric ___________

Find the circumference of both circles.

Small circle: Customary C = ___________ Metric C = ___________

Large circle: Customary C = ___________ Metric C = ___________

Write down your observations when viewing each figure.

Math Creations – Creating A Tile and Tesselation
Handout 3

In the upper left hand corner of the graph paper draw a four by four square.

Leave three square grids blank; draw another four by four square. Draw the diagonal of the square.

Cut each Regular polygon out.

Starting with the first square, select one corner (vertex) of the square. Draw a freeform line to one of the adjacent corners. Keep the “line” fairly simple. Put an “x” on both pieces.

Cut along the created line; make sure it is a smooth cut. There should only be two pieces, no scrapes (all pieces of the square have to be used).

Match the cut piece to form the original square. Slide the cut piece over the square, (Translation), straight angles (straight sides) connect. Make sure both “x’s” are on top. Attach the cut piece with tape.

Turn the modified square over. The “x’s” are on the bottom.

Optional step, repeat the above cut and translation with the remaining opposite sides.

Trace the modified shape (tile) in the middle of a new sheet of paper.

Without flipping or rotating or leaving gaps or making overlaps, slide the tile to the left matching the sides, trace. Continue until the end of the paper. Repeat from the beginning going to the right. Do not worry about partial pieces at the end of the page, tessellations can go on forever.

Start at the middle again. Move the tile up and trace, move to the left and then to the right tracing each piece, repeating until the entire page is filled. (The plane of the paper has been tessellated.)

Use map colors or markers to color in the tessellation.

Cut out the second square, cut along the diagonal. Using one of the triangles, create another tessellation.

Optional, refer to Escher’s scheme, the pattern he used to create tessellations. Create a tessellation following this scheme.

Beyond the Classroom Walls
Handout 4

Cut out a rectangle, three squares by four squares, from the graph paper. Is a rectangle a Regular polygon? Why or why not?

Find the perimeter and area of the rectangle. P = ______, A = ________
Create a tile. Trace the tile below.

Find the area of the tile. A = _______. Compare the area of the rectangle to the area of the tile.

Is it possible to tessellate the tile you created? Why or why not?

Using your tile, create a tessellation on a piece of cardstock.

Find the perimeter and area of the piece of cardstock. P = ____, A = _______

Your tessellation was selected to be framed and hung in the school library. If the frame is one inch wide on all sides, find the length and width of the framed picture.

l = ____, w = _____

A piece of glass is needed for the frame. The glass fits snuggly inside the frame with the frame extending one-half inch beyond the glass on all sides. Find the dimensions, perimeter and area of the glass. (Make a sketch.)

l = ____, w = _____, P = ______, A = _____

Your teacher decided to mount all the tessellations on the wall without a frame. How many tessellations will be displayed? ____________
Describe how you arrived at your solution.

Your teacher would like to display all the student’s tessellations on the classroom walls. They have to “touch” each other. How many walls and/or what part of the wall will be covered. Describe _________________________________________________________

Find the perimeter and area of the covered walls. P = ______, A = _______

Is the wall a tessellation? Why or why not? ____________________________________

Create a new tile using a blank sheet of paper that can be tessellated. Trace it on a piece of cardstock.

Half of the ceiling will be covered with your tessellation; do not worry about light fixtures. How many tiles will be needed in order to fill half of the ceiling? ____________
Describe how you arrived at your solution.

If each tessellation used to cover half of the ceiling took 4 minutes to create, how long would it take to make each tessellation needed to cover half of the ceiling?

Grade 6 Lesson 2

The mission of Project GRAD is to ensure a quality public education for all students in economically disadvantaged communities so that high school and college graduation rates increase.