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Tessellations in Art and Geometry
Submitted by: John Hartsough
Geometry Teacher
Grade Level: 10th and 11th
Integration: Ten Lessons on Tessellations and Geometry
Gardner’s Intelligence(s): Spatial relationships, Bodily-Kinesthetic, and Linguistic
Bloom’s Taxonomy Category: Synthesis
Abstract:
This unit of study will enable the students to understand the relationship with constructions, parallel lines and the symmetrical uses of these shapes. The symmetry aspect will lead us right into the introduction of tessellations and will help the students analyze the difference between a tessellation and a symmetrical picture. At this point of the unit, the students will do a short project creating their own tessellation and reproducing it. They will be able to evaluate the use of the constructions to draw congruent shapes, angles and figures. The end result will be applying all of these skills and producing a regular sided birdhouse that is symmetrical and with all angles congruent and sides congruent.
Rational and Description of Unit of Study:
The purpose of this unit is to introduce the students to constructing their own shapes and parallel lines with a compass and a straightedge. They tend to use the easy way most of the time and use a protractor and a ruler, so this will show them how to do it with this hands-on method. The students will use this new knowledge to develop and create a project at the end of the unit. Throughout the unit they will use critical thinking skills and ask themselves higher-level questions.
Content:
The entire unit, falls into several of Gardner’s Intelligence list and will take approximately four weeks on block scheduling and 90 minute classes. There will be 10 lessons in all and there may have to be some flexibility on the time given for the projects during school time as well as time spent at home. Of course, the logical-math one on Gardner’s list is used but we will also touch on spatial, bodily kinesthetic, linguistic, and the naturalist area of thinking. Some of the activities are in the area of Art, creating a birdhouse. Writing will be used each class period by keeping a journal with new terminology and their uses and definitions. Reading the textbook, of course, but also reading on what birdhouse shape would be best for the bird you are making the house for and the size and dimensions of the house. Math, constructing and duplicating shapes and angles. These activities are only highlights of the unit of study of these different aspects of Geometry.
Goals/Content and Cognitive:
Students will learn to use a compass and a straightedge to construct congruent segments and congruent angles. Also they will bisect segments and angles only using these tools.
New Vocabulary:
Construction, straightedge, compass, perpendicular lines, perpendicular bisector, and angle bisector
Links to Curriculum Standards:
NCTM Standards: Geometry, Measurement, Problem Solving, Communication, Connections, and Representation
Guiding Question:
Can the students use a compass and straightedge effectively?
Can students construct the bisector of an angle to show its incidence and reflection? Can students construct a right triangle using the methods learned in this lesson?
Assessment:
Students will show their knowledge with a short lesson quiz (see bottom of this page) at the beginning of the next class period after they practiced doing their homework. The quiz will have six questions and give students the opportunity to show what they understand and have retained. Due to the block scheduling I try to open each period with a refresher of what the previous class was about and do this with a short quiz or with question/answer on the board or walking around watching them do one example on their own.
Learning Connection:
Construction methods are justified by postulates such as Euclid’s Fourth Postulate, that a circle can be drawn with any center and any positive radius, and by, for example, triangle congruency theorems. Compass-and-straightedge constructions provide a hands-on introduction to these postulates and theorems.
Learning Activities or Tasks:
• Construct a segment congruent to a given segment.
• Construct an angle congruent to a given angle.
• Construct the perpendicular bisector of a segment.
• Construct the bisector of an angle.
• Write the steps to each construction in your own words and be able to teach the steps to a fellow student.
Teaching Strategies:
• Walk around while the class as they are modeling the steps I have shown them and helping them with the constructions.
• Have students investigate what software can model compass and straightedge constructions. Some programs use a mouse and pointer to model the action of compass, straightedge and pencil.
• Model how to construct each thing with a huge chalk compass on the board and a yardstick.
• Include the students who cannot physically use a compass by putting them in pairs with other students to share the construction steps.
• Walk around the room reminding the students of how slow they need to be doing their constructions and remind them that a loose compass will not work for the precision they need.
Management:
Careful instructions must be given about the constructions and clear step-by-step guidelines need to be given. The students need to stay on task and model the behavior so they will remember how the constructions are done.
Materials and Resources:
• Journals or Notebooks
• Compasses
• Straight Edges
• White Drawing Paper
Tessellations Lesson 2
Gardner’s Intelligence(s): Spatial relationships, Bodily-kinesthetic, and Linguistic
Bloom’s Taxonomy Category: Analysis and Evaluation
Goals/Content and Cognitive:
Students will learn to identify angles formed by two lines and a transversal. They will also prove and use properties of parallel lines.
New Vocabulary:
Transversal, alternate interior angles, same-side interior angle, corresponding angles, two-column proof
Links to Curriculum Standards:
NCTM Standards: Algebra, Geometry, Measurement, Problem Solving, Reasoning and Proof, Communication, Connections, and Representation
Guiding Question:
Can the students visually tell whether or not two lines are parallel? Can they reason and deduce why or why not? Can the students tell what a transversal does to two parallel lines and explain why?
Assessment:
Students will show their understanding by doing examples from the textbook over that section. They will then at the beginning of the next class period work in groups of three to do a short worksheet for the first 20 minutes to check comprehension.
Learning Connection:
The Corresponding Angles Postulate is a variation of Euclid’s famous Parallel Postulate, which subsequent mathematicians vainly hoped could be proved as a theorem. In the nineteenth century, altering the postulate enabled the invention of hyperbolic and elliptic geometries. Any geometry that obeys the Parallel Postulate is now known as a Euclidean geometry.
Learning Activities or Tasks:
• Provide straws for students to use to model parallel lines and transversals, and provide protractors to test the postulate and theorems in this lesson
• Have partners discuss the vocabulary in this section and encourage them to use words like alternate, corresponding, interior, and exterior in non-mathematical contexts.
• Have students draw a diagram of parallel lines and a transversal, use numbers to label the eight angles formed, and color-code the angles to indicate which are congruent.
Teaching Strategies:
• Draw two parallel lines and cut it by a transversal
• Label the angles that are formed and discuss the similarities of this picture and possibly an airport runway
• Discuss with students the importance of Euclid’s Parallel Postulate to the development of hyperbolic and elliptic geometry
• State the steps to follow in a two-column proof. After the students have study the Plan for a proof, point out that working backward can help them plan a proof
Management:
The students really need to become comfortable with these new terms and need to spend time processing their meaning in order to be able to fully conceive what parallel lines and transversals can do. Make sure enough time is given to actually let it be absorbed and don’t rush the learning process.
Materials and Resources:
• Journals or Notebooks
• Straws
• Straight Edges
• White Drawing Paper
• Colored Ink Pens
Tessellations Lesson 3
Gardner’s Intelligence(s): Spatial relationships, and Linguistic
Bloom’s Taxonomy Category: Evaluation, Synthesis and Analysis
Goals/Content and Cognitive:
Students will learn to use a transversal in proving lines parallel. They will relate parallel and perpendicular lines to each other and understand the difference.
New Vocabulary:
Flow proof
Links to Curriculum Standards:
NCTM Standards: Algebra, Geometry, Problem Solving, Reasoning and Proof, Communication, Connections, and Representation
Guiding Question:
Why if we prove two lines are parallel will that help us with a problem? How can we prove that the opposite sides of a picture frame are parallel? What is the relationship between parallel and perpendicular lines? What is more beneficial a two-column proof or a flow-chart proof?
Assessment:
To check students’ understanding, the class will prove the Converse of the Corresponding Angles Postulate using the Converse of the Alternate Interior Angles Theorem as a postulate. The students will also name the three methods that this lesson gives to prove that two lines are parallel.
Learning Connection:
Euclid’s discussion of parallelism in The Elements introduced the topics in this text in a different order and proved the converse of the Corresponding Angles Postulate as a theorem. Thus the only postulate that required acceptance without proof for proving lines parallel was the Corresponding Angles Postulate.
Learning Activities or Tasks:
• Students will practice solving equations and writing the converse of conditional statements.
• Students will journal in their books about what a flow chart is and give an example of one.
• Students will apply what they have learned about parallel lines and come up with five tangible examples of when we know lines are parallel.
Teaching Strategies:
• To help students stop from incorrectly using conditional statements to justify an answer when its converse should be used. Lead a discussion as a class on which should be used and why.
• Whenever possible, mark the congruent angles on a diagram on the board to help students relate angles and parallel lines.
• Write the flow proof of the Alternate Interior Angles Theorem and write it as a paragraph proof.
Management:
Class will be broken into three 30 minute sections: the first to show parallel lines and proving them; the second in groups doing a practice worksheet; and the third writing in their journals and writing a flow chart.
Materials and Resources:
• Journals or Notebooks
• Worksheets
• Tangible examples around the room of parallel lines
Tessellations Lesson 4
Gardner’s Intelligence(s): Spatial relationships, Intrapersonal, Linguistic, and Musical
Bloom’s Taxonomy Category: Application, and Synthesis
Goals/Content and Cognitive:
Students will learn to classify triangles and find the measures of their angles. Students will relate exterior angles to the angles of a triangle.
New Vocabulary:
Acute triangle, right triangle, obtuse triangle, equiangular triangle, equilateral triangle, isosceles triangle, scalene triangle, exterior angle of a polygon, remote interior angles
Links to Curriculum Standards:
NCTM Standards: Algebra, Geometry, Measurement, Problem Solving, Reasoning and Proof, Communication, Connections, and Representation
Guiding Question:
Can the students tell the difference between all of the different types of triangles? Can the students relate why the angles of a polygon on the interior and exterior equal what they do with relationship to the triangles? Why is the Triangle Angle-Sum Theorem so useful in only Euclidean geometry?
Assessment:
Students will draw and label a triangle with an exterior angle at each vertex. They should measure and label each angle, classify the triangle, and then explain how the measurements illustrate the Triangle Angle-Sum Theorem and the Triangle Exterior Angle Theorem. They will also take the lesson quiz at the beginning of the next class period.
Learning Connection:
When alterations of Euclid’s Parallel Postulate lead to different geometries, the Triangle Angle-Sum Theorem appears strikingly different. In a hyperbolic geometry, the sum of a triangle’s angle measures is less than 180; in an elliptic geometry, the sum is greater than 180.
Learning Activities or Tasks:
• Students will draw and cut out a large triangle.
• Then number the angles and tear them off.
• Third, place the three angles adjacent to each other to form one angle that is 180-degree line.
• Students will compare their results with others and write down the observations.
• Students will make a conjecture about the sum of the measures of the angles of a triangle.
• Students will study the design in a Navajo-weaving blanket and realize that the textile art of some countries features patterns of triangles. Students could bring in photographs or other examples of this type of triangle patterns for extra credit.
• Students will get in groups of three or four and write a song or rap showing the different names of triangles and their definitions.
Teaching Strategies:
• Since proofs may be new to some of the students, a good introduction is to analyze a proof with missing parts and to recreate the reasoning behind each step.
• Students may confuse the Corresponding Angles Postulate and its converse. Ask them to identify the hypotheses and conclusions of each.
• Have students use geometry software to draw triangles and display the sum of their angles. Then have students manipulate the triangles and see that the sum remains 180.
• Show students shorthand ways to write names of proofs so that it is easier to remember the more and more they learn.
• Students may confuse angle measures with angle labels so point out that the measure of an angle never immediately follows an angle symbol.
• Explain what happens to the angle formed by the back of the chair and the armrest as you make a lounge chair recline more.
Management:
Repetitive teaching during this class will help them remember the differences of the triangles. Time given for each task needs to be enough but the class needs to move swiftly through the activities.
Materials and Resources:
• Journals or Notebooks
• White Drawing Paper
• Straight Edges
• Copies of a Navajo-weaving blanket
Tessellations Lesson 5
Gardner’s Intelligence(s): Spatial relationships, Linguistic, Bodily-kinesthetic, and Interpersonal
Bloom’s Taxonomy Category: Synthesis, and Evaluation
Goals/Content and Cognitive:
Students will classify polygons. Students will find the sums of the measures of the interior and exterior angles of polygons
New Vocabulary:
Polygon, convex polygon, concave polygon, equilateral polygon, equiangular polygon, and regular polygon
Links to Curriculum Standards:
NCTM Standards: Algebra, Geometry, Measurement, Problem Solving, Reasoning and Proof, Communication, Connections, and Representation
Guiding Question:
Can the students apply the triangles and angle-sum theorems to apply it to different polygons? Can the students understand how the number theory allows them to determine whether a regular polygon will tessellate a plane?
Assessment:
Students will draw a polygon with 5 sides and a polygon with 8 sides. For each polygon, students should show the exterior angles and find the sum of the interior angles and the sum of the exterior angles. Then they will write a paragraph explaining how the sums illustrate the Polygon Angle-Sum Theorem and the Exterior Angle-Sum Theorem.
Learning Connection:
Because each interior angle of a regular n-gon measures (180(n-2))/2, one can readily find the set of all regular n-gons that tessellate a plane. Combinations of regular polygons that tessellate a plane can likewise be found with a bit more work and application of some straightforward number theory.
Learning Activities or Tasks:
• Students will work in small groups making charts that display the different names of polygons. The charts will be displayed in the classroom with the name of each polygon, its number of sides, and an appropriate figure.
• Students will be able to name the polygon, identify its vertices, sides and angles.
• Students will make a list of buildings and sports stadiums that have polygons in their design.
Teaching Strategies:
• Remind students that there are also different ways to name sides and angles and give different examples.
• Connect the learning of convex and concave to Science and lens with rays of light and microscopes and telescopes and eyeglasses.
• Point out that the angles of a polygon can be called interior angles. Also, that the exterior angles of a regular polygon are congruent.
• Tell the students that it is usually easier to count vertices than sides of a polygon, because the number of vertices and sides are equal.
Management:
Give several examples of different types of polygons and keep students on task the entire period with the different activities.
Materials and Resources:
• Journals or Notebooks
• Posterboard
• Colored Markers
Tessellations Lesson 6
Gardner’s Intelligence(s): Spatial relationships, Linguistic and Bodily-kinesthetic
Bloom’s Taxonomy Category: Synthesis
Goals/Content and Cognitive:
Students will learn to identify transformations in tessellations, and figures that will tessellate. Students will identify symmetries in tessellations.
New Vocabulary:
Tessellation, tiling, translational symmetry, glide reflectional symmetry
Links to Curriculum Standards:
NCTM Standards: Algebra, Geometry, Measurement, Problem Solving, Communication, Connections, and Representation
Guiding Question:
Can the students apply the triangles and angle-sum theorems to apply it to different symmetrical shapes? Can the students develop their own tessellation? Can students recognize tessellations and understand when they are used?
Assessment:
Students will do a tessellation project.
Learning Connection:
The astronomer Johannes Kepler is believed to have been the first to investigate the possible ways of covering a plane with regular polygons. Analysis of tessellations using more complicated figures requires understanding of isometry and symmetry.
Learning Activities or Tasks:
• Students will tessellate which covers a figure without gaps or overlapping
• Students will pick a design
• Students will construct each fundamental region with the same size and shape
• Students must use different colors and may use old pictures, colored printer paper or construction paper but that is hard to cut
• Students will use an exacto knife not scissors
• Students will decide if they want construction marks to be seen or not. Either is fine but they must be consistent
• Project must be at least 40 pieces
• Shapes for the region cannot be a rectangle or a square
Teaching Strategies:
• Remind students that a drawing can only suggest that tessellations, like planes, extend without bound in all directions.
• Help students understand the patterns of a tessellation so that they can construct and design their own patterns.
• Use angle measures to explain how an arrangement of regular 12-gons and equilateral triangles tessellates a plane.
Management:
Careful instructions must be given about the designs and the students need to understand what is expected of them. Hopefully 90 minutes is enough to get this project started and the rest would be done outside of class.
Materials and Resources:
• Journals or Notebooks
• Colored printer paper
• X-acto Knives
• Construction Paper
• Straight Edges
• Old pictures of tessellations
Tessellations Lesson 7
Gardner’s Intelligence(s): Spatial relationships, and Linguistic
Bloom’s Taxonomy Category: Application
Goals/Content and Cognitive:
Students will graph lines, given their equations. The students will be able to write equations of lines. Students will review finding slope when given two points and graphing a line when given two points
New Vocabulary:
Slope-intercept form, standard form of a linear equation, point-slope form
Links to Curriculum Standards:
NCTM Standards: Algebra, Geometry, Problem Solving, Communication, Connections, and Representation
Guiding Question:
Can the students Assessment:
Students will do a tessellation project.
Learning Connection:
Rene Descartes published his philosophical treatise Discours de la methode in 1637. Its appendix Geometrie contained the first published record of methods of analytic geometry, permanently sealing the partnership between geometry and algebra. Within 30 years, the methods of analytic geometry would lead to the invention of calculus.
• Students will graph lines in slope-intercept form.
• Students will graph lines using intercepts.
• Students will transform a standard equation into the slope-intercept form and then graph it.
• Students will use point-slope form
• Students will write an equation of a line given two points.
• Students will be able to write in all three forms: point-slope, slope-intercept and standard form.
• Students will play a game to show which equation goes together using all three forms.
Teaching Strategies:
• Step by step instruction on finding slope and the intercept and the different forms.
• Have students use coordinate graphs to see that slopes of -4/3, 4/-3 and –(4/3) are the same.
• Discuss with students when point-slope form is easier to use than slope-intercept form.
Management:
Share plenty of examples and then have enough time to play a game with magnets on the board matching the three forms together for the same points and graph.
Materials and Resources:
• Journals or Notebooks
• Matching Index Cards
• Magnets
• Graph paper
• Straight Edges
Tessellations Lesson 8
Gardner’s Intelligence(s): Spatial relationships and Bodily-kinesthetic
Bloom’s Taxonomy Category: Application
Goals/Content and Cognitive:
Students will learn to relate slope and parallel lines. Students will be able to relate slope and perpendicular lines.
Links to Curriculum Standards:
NCTM Standards: Algebra, Geometry, Problem Solving, Communication, Connections, and Representation
Guiding Question:
Can the students apply their knowledge of slope and figure out the relationship with two parallel lines? Can the students apply that same knowledge to share the relationship with two perpendicular lines?
Assessment:
Students will do exercises from their text manipulating equations and stating whether or not the lines are parallel or perpendicular.
Learning Connection:
Slope is a fixed ratio that characterizes any non-vertical line. It is another example of a pure geometric concept, a line, described by algebraic methods. The product of the slopes of perpendicular lines being -1, although initially surprising, is merely the Pythagorean theorem in an analytic geometry setting.
Learning Activities or Tasks:
• Have students investigate how builders construct the walls of most buildings since they are perpendicular to the floors.
• Students will determine whether lines are parallel or perpendicular.
• Students will write equations of parallel lines and perpendicular lines.
Teaching Strategies:
• Explain algebraically why two lines cannot be both parallel and perpendicular.
• Ask students to describe unusual window designs they have seen elsewhere and discuss the parallel or perpendicular lines in them.
Management:
Keeping the students on task for the entire 90-minute period.
Materials and Resources:
• Graph paper
Tessellations Lesson 9
Gardner’s Intelligence(s): Spatial relationships and Bodily-kinesthetic
Bloom’s Taxonomy Category: Synthesis
Goals/Content and Cognitive:
Students will learn to construct parallel lines and perpendicular lines with a compass and a straightedge.
Links to Curriculum Standards:
NCTM Standards: Geometry, Problem Solving, Communication, Connections, and Representation
Guiding Question:
Can the students construct parallel lines given a line? Can the student construct a special quadrilateral? Can the student construct a perpendicular line from a point on a line? Can the student construct a perpendicular line from a point on a line?
Assessment:
Have student work in pairs to construct a rectangle. Their work should include a construction diagram and a written explanation of the steps they used.
Learning Connection:
The method in this lesson for constructing parallel lines is based on the Converse of the Corresponding Angles Postulate. An alternative method might base a construction on the Converse of the Alternate Interior Angles Theorem or on the theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. The method for constructing perpendicular lines is based on the method for constructing the perpendicular bisector of a segment.
Learning Activities or Tasks:
• Have Students work with partners to do the constructions taking turns explaining the steps in each one.
Teaching Strategies:
• Do the constructions on the board for this lesson.
• Make sure the students realize where to put their compass and what arcs should be made during the constructions.
• Explain how to construct a line parallel to a given line. Tell which theorem or postulate is used.
Management:
Careful instructions must be given about the constructions and give students plenty of time and opportunity to practice what they learn.
Materials and Resources:
• Compasses
• Straight Edges
• White Drawing Paper
Tessellations Lesson 10
Gardner’s Intelligence(s): Spatial relationships, Naturalist and Bodily-kinesthetic
Bloom’s Taxonomy Category: Synthesis
Goals/Content and Cognitive:
Students will combine all of their knowledge of angles, parallel and perpendicular lines, polygons and constructions to design and construct a birdhouse.
Links to Curriculum Standards:
NCTM Standards: Algebra, Geometry, Measurement, Problem Solving, Communication, Connections, and Representation
Guiding Question:
Can the students apply their knowledge to develop a useful and stable birdhouse?
Assessment:
Students will do a birdhouse out of wood and paint the exterior and make sure it is using the skills they have learned through all of the constructions.
Learning Connection:
What they can apply from math to science and nature.
Learning Activities or Tasks:
• Students will make a sketch of the birdhouse.
• Students will figure out the correct measurements.
• Students will cut the wood in shop class to the exact specifications.
• Students will put the birdhouse together using nails or wood glue.
• Students will paint their final project.
Teaching Strategies:
• Remind students to apply all of the facts they have learned to do this project.
Management:
Students will be doing different things at different times and they need to be focused and on task.
Materials and Resources:
• Wood
• Measuring equipment
• Wood Paint
• Nails
• Wood Glue
• Graph paper
Assessment Evaluation Sheet
Name ___________________________________________________
Constructions in Geometry
Tell 4 types of constructions:
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Tell about 3 of your best activities that you did in our study of constructions, tessellations and polygons:
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Name 5 different types of polygons:
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Describe the birdhouse and what birds will inhabit the one you constructed:
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TEACHER’S RESOURCES
1. Geometry - Prentice Hall Mathematics by Bass, Charles, Johnson and Kennedy
2. Informal Geometry Planning Guide – Prentice Hall Mathematics
3. Practice Workbook, Additional Practice for Every Lesson – Prentice Hall Mathematics
CONSTRUCTION
Websites
Geometric Construction ~ from Wolfram MathWorld
Ruler & Compass - This application allows you to simulate ruler and compass constructions on the net
Geometric Construction with the Compass Alone - Starting with two points, other points can be constructed with compass alone.
Critical Thinking Addressed by this Unit of Study in the Classroom:
In the unit: Constructions, Tessellations, Parallel and Perpendicular Lines in Geometry there was every critical thinking skill addressed. Learning the knowledge of each concept presented and beginning to understand it enough to apply it to projects, games and questions at the end of each lesson was used each period. Analysis, Synthesis and Evaluation were used quite a lot in this unit due to construction, design and evaluating. The students had to be able to analyze what shape, what line, what slope or what polygon they were given and they needed to know what to do with it. During this lesson not only were all of the critical thinking skills addressed but the multiple intelligences that Howard Gardner list were exhibited as well. Throughout each lesson both things are mentioned and posted specifically for that day. This unit is very helpful in using hands-on, visual and practical learners to a place where they can help each other and learn from each other. Constructions help students to increase their knowledge of how things have been created around them and it gives them a better understanding of what things go into simple constructions of buildings, birdhouses and just everything around them. The students will apply all of their knowledge as well as bring in skills to build things, art to decorate their birdhouse and the implementing of critical thinking skills enable the students to come away from this unit not only to be presented with a high level task but given the tools to do that task.
