Snowflake Geometry

Snowflake GeometryBig IdeasUnit of InstructionSnowflake GeometryGeometry Concept Geometric relationships found in a regular hexagonRationaleThis project brings out artistic inventiveness while providing a great review of geometricconcepts, such as lines, angles, bisectors, symmetry, and polygons, to mention a few.A real advantage of this project is that it takes almost no class time. Best of all, it iseasy for all students to succeed.NCTM 9-12 Standards Analyze characteristics and properties of two- and three-dimensional geometricshapes and develop mathematical arguments about geometric relationships.Apply appropriate techniques, tools, and formulas to determine measurements.Use visualization, spatial reasoning, and geometric modeling to solve problems.Make and investigate mathematical conjectures.Build new mathematical knowledge through problem solving.Recognize and use connections among mathematical ideas.Understand how mathematical ideas interconnect and build on one another toproduce a coherent whole.Use the language of mathematics to express mathematical ideas precisely.Use representations to model and interpret physical, social, and mathematicalphenomena.Idaho Content Standards G.4.1.3 Establish the validity of geometric conjectures.G.4.3 Apply transformations and use symmetry to analyze mathematicalsituations.G.4.3.1 Understand and represent translations, reflections, dilations, androtations of objects in the plane.G.4.4.1 Draw and construct representations of two-dimensional geometricobjects using a variety of tools.9-12.VA.3.2.4 Select and utilize visual, spatial, and temporal concepts toenhance meaning in artwork.

BackgroundVocabularyAcuteObtuseBisectorsOrganic shapeClassification of ometric shapeSymmetryHexagonVerticalKiteMath Instruction (pre- or post-project)This is a review of past concepts and vocabulary relating to the relationships betweenangles and lines.Art InstructionStudents will need to have an understanding of positive and negative space andgeometric versus organic shapes.Section 10 Page 141

Driving QuestionProject ObjectiveStudents will identify, by folding paper, geometric terms and relationships.Students will recognize that geometric relationships are not limited to the shape of theoriginal object.Questions to be AnsweredWhat is this particular geometric shape (after each fold)?What geometric characteristics define the shape?Do geometric characteristics always hold true?MaterialsMaterials Required 8 ½” white paper - each student may use up to six pages per day Colored paper- optional for second day of snowflake production Scissors School laminator String/thread/fishing line to hang snowflakes RulerReference MaterialsSnowflakes for All Seasons Cindy Higham (ISBN 1-58685-528-X)Snowflake Bentley Jacqueline Briggs Martin and Mary Azarian (ISBN owflakes.htmSection 10 Page 142

Lesson OutlineDescription of ActivityThe students will create a regular hexagon and then identify geometric relationshipswithin the folded hexagon. Then they will use the regular hexagon to create originalsnowflakes and finally thematic snowflakes. In conclusion, they will write a one pageessay that reflects what they learned in the experience.Day OneSteps 1-9 How to fold a hexagon Students will fold a sheet of paper to create a regular hexagon. Teacher will askquestions about each form that is created (see pages 9-18).Day Two In class, students will work on their projects in small groups, identifying the 50plus relationships in the hexagonal form (see pages 20-22).Day Three Teacher will demonstrate positive and negative space and geometric/organicshapes (see pages 6-8). Students will experiment with cutting, forming original snowflakes, perfecting theirstyle, creating a positive and negative image snowflake. Snowflakes are laminated and hung in the classroom.Note: we highly recommend visiting snowflakebentley site to demonstrate positive andnegative crystals.Optional Day FourStudents may choose a thematic snowflake design from the book Snowflakes for AllSeasons.Colored paper may be provided for this project.Section 10 Page 143

AssessmentRubricHexagon identification:1 point given for each relationship identified.Snowflake production: 10 points if they turn in a completed snowflake.Reflection paper:20 pointsIdeas for Further Independent Student ProjectStudents may create three-dimensional paper sculptures.Section 10 Page 144

Positive and Negative SpacePositive shapes occupy positive space. The area around positive shapes, the background, isnegative space. Artists think of the entire composition, the space between the shape and thespace around the shape in creating pieces of art. In the composition of the chairs, the spacearound the chairs, the negative space, is as clear as the positive space.In the ceramic vase below, the negative shapes on either side of the top half of the vase nearlymirror the lower positive shape of the vase.Section 10 Page 145

Geometric and Organic ShapesGeometric Shapes / RectilinearShapesNotice the shapes below and to theleft of the red line. Angular shapestherefore we call them rectilinearshapes.Most rectilinear shapes can bealso called geometric shapeshowever a couple of them are not.These are circles and ovals.Organic Shapes / Curvilinear ShapesNotice the shapes below and to theright of the red line. They are allcurving and flowing therefore we callthem curvilinear shapes.Curvilinear shapes could also becalled organic shapes. We can includecircles and ovals.Organic and curvilinear shapes areusually natural shapes. Think of leaves,Geometric and rectilinear shapes animal shapes, and plant shapes. Aren'tthey all organic?usually are man-made. A fewexceptions to the rule are crystalsand honey combs.So how would you answer this question?Circles and ovals are considered to be which? There can be more than one answer. Geometric ShapesRectilinear ShapesOrganic ShapesCurvilinear ShapesSection 10 Page 146

"Abstraction of a Cow, Four Stages" by Theo van Doesburg, Museum of Modern Art, NY Notice this 3-dimensional organic form of the cow in the first image. The cow becomes an abstract cow because it becomes geometric but isstill 3-D looking. Then it becomes flat 2-dimensional shapes that one can still see as anabstract cow. And the fourth stage is simply geometric shapes inspired by a cow. Section 10 Page 147

How to Fold a HexagonInstructions & QuestionsStep 1Start with paper crosswise (face down).ABDCQuestions (and possible answers)Q: What shape is this?A: RectangleQ: What makes it a rectangle?A: It is a parallelogram with 4 right anglesQ: What is a parallelogram?A: A quadrilateral with opposite sides parallelQ: What is a quadrilateral?A: A four sided polygonQ: What is a polygon?A: Many sided figure in a plane having 3 or more sidesQ: What makes a figure closed?A: A closed figure in a planeQ: What is a plane?A: Example: picture a flat piece of paper that goes indefinitely in all directionsQ: Why isn’t it a square?A: A square must have four equal sidesSection 10 Page 148

Q: Is a square a rectangle?A: AlwaysQ: Is a rectangle a square?A: SometimesQ: What is a right angle?A: An angle whose measure is exactly 90 degreesQ: Back to the rectangle, name the rectangleA: Rectangle ABCDQ: Can it be named another way?A: Rectangle BCDEQ: Can it be named by just three letters? Why not?A: Open ended responseQ: Is the rectangle horizontal or vertical?Q: What are the measurements of the rectangle?A: 8.5 X 11 inchesQ: What is its length?A: 11 inchesQ: What is its width?A: 8.5 inchesQ: What is its height?A: 8.5 inchesQ: Wait! I thought you said it’s width was 8.5 inches, now you say the height is 8.5inches, which is it width or height?A: It all depends on how you look at itQ: What is the difference between width and height? Is that true for all polygons?A: Width: breadth, distance across from side to side Height: measurement from top tobottomQ: How would you calculate the area of this rectangle?A: A bh or A lwQ: What is the area of this rectangle?A: A bh (11) (8.5) 93.5 square inchesSection 10 Page 149

Snowflake Geometry Instructions & QuestionsStep 2Fold in half, left over right.FAEDQuestionsQ: Name the new rectangle.A: Rectangle ADEFQ: What has happened to the area now?A: It is half the area from beforeQ: What happened to the rest of the area? Did it disappear? Where is it?A: On the other halfQ: Why don’t we count it?A: Area is calculated in a planeQ: Open it up. What does the fold make?A: A line segmentQ: So if we open it up what do we have?A: Two half planesQ: Name the line segment.A: FEQ: What is a line?A: A point moving through spaceSection 10 Page 150

Snowflake Geometry Instructions & QuestionsStep 3Refold your paper. Fold in half again, folding the top down.GHFAQuestionsQ: Name the new rectangle.A: Rectangle AFGHQ: What have we done to the area now?A: Halved itQ: What relation does the area of this rectangle have to the area of the originalrectangle?A: It is one fourth the areaQ: How do we know?A: ½ X ½ 1/4Q: How many folded edges are there? What makes a folded edge?A: Three folded edges. A folded edge is made by overlapping layers of paperQ: How many folded edges do we see?A: Two folded edgesQ: Name the folded edges that we seeA: FG and GHSection 10 Page 151

Snowflake Geometry Instructions & QuestionsStep 4Fold in half once more, folding the top down again.GHQuestionsQ: What have we done to the area now?A: Halved itQ: What relation does the area of this rectangle have to the area of the originalrectangle?A: It is one eighth the areaQ: Can you see a pattern? What is it?A: Every time one matches the edges it halves the areaQ: How far could we continue the pattern?A: To infinity theoreticallyQ: Could we actually do it with a piece of paper?A: NoSection 10 Page 152

Snowflake Geometry Instructions & QuestionsStep 5Unfold once.GHFAQuestionsQ: What does the middle folded segment create?A: A midsegmentQ: What is a median/midsegment?A: A segment made by connecting the midpoints of two sidesQ: Are the two rectangles similar?A: YesQ: What makes geometric shapes similar?A: Having the same shape but not necessarily the same sizeQ: Are the two rectangles congruent?A: YesQ: How do we know?Q: Can you prove it geometrically?A: Answers may varySection 10 Page 153

Snowflake Geometry Instructions & QuestionsStep 6Place lower left corner F on center fold and crease through upper left corner G.GHQuestionsQ: What shapes do we see now?A: Two triangles, two trapezoidsFAQ: Name a triangle.A: GFJJQ: What is a triangle?A: A polygon with 3 angles and 3 sidesQ: What is the height of the triangle?A: Get a ruler out and measure it if you do not know (5 ½ inches)Q: Are the two triangles similar?A: YesQ: What requirements have to be met for triangles to be similar?A: AAA similarityQ: Name the trapezoids.A: Can’t without designating another pointQ: What is a trapezoid?A: A quadrilateral with exactly two parallel sidesQ: How does a trapezoid differ from a rectangle?A: A trapezoid has one pair of parallel sides; a rectangle is a parallelogram with 4 rightanglesQ: What is the height of the trapezoids?A: 4.25 inchesQ: Why isn’t the height 5.5 inches?A: Because height must be perpendicular to the baseSection 10 Page 154

Snowflake Geometry Instructions & QuestionsStep 7Turn over, flipping the bottom to the top.JGAHQuestionsQ: What shapes do you see now?A: Three trapezoidsQ: What can you tell me about the middle folded line?A: It is a midsegment, corresponding angles, its length is half the length of the sum ofthe two bases (AJ GH)Section 10 Page 155

Snowflake Geometry Instructions & QuestionsStep 8BisectJGH by matching GJ to GH. Crease on GF.FGAHJQuestionsQ: What did we just do to angle JGH?A: We bisected itQ: What new angles did we form?A: FGD, a 30 degree angleQ: What is bisecting an angle?A: Dividing the measure of the angle in halfQ: What other shapes do you see?A: Two triangles, a quadrilateralQ: Is the quadrilateral a special type of quadrilateral?A: No, it is not a trapezoidSection 10 Page 156