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Transforming Geometric Proof with Reflections, Rotations and Translations. Kristin A. Camenga Kristin.camenga@houghton.edu Houghton College April 15, 2011 All information from this talk will be posted at the website listed on the handout.
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Transforming Geometric Proof with Reflections, Rotations and Translations Kristin A. Camenga Kristin.camenga@houghton.edu Houghton College April 15, 2011 All information from this talk will be posted at the website listed on the handout. campus.houghton.edu/webs/employees/kcamenga/teachers.htm
Motivating Questions • What is transformational reasoning? • Why is transformational reasoning important? • How can transformational reasoning be used in the high school curriculum? • What does transformational reasoning contribute to student learning?
What is transformational reasoning?
Quadrilateral Transformations For each of the following quadrilaterals, describe the rotations and reflections that carry it onto itself: • Parallelogram • Rhombus • Rectangle • Square What connections do you notice?
Parallelograms by Symmetry If we rotate a parallelogram 180○ about the midpoint of a diagonal, • AC≅DB; AB≅DC • ∠B≅∠C, ∠A≅∠D • ∠BAD≅∠CDA, ∠CAD≅∠BDA • AB∥CD, CA∥DB
The Isosceles Triangle Theorem Why are the base angles equal?
The Isosceles Triangle Theorem • Traditional Method: Draw median and show triangles congruent. • Transformational method: Draw angle bisector and reflect triangle over it to see that angles coincide.
Key ideas • Uses transformations: reflections, rotations, translations, dilations. • Depends on properties of the transformation: • Congruence is justified by showing one object is the image of the other under an isometry (preserves distance and angles). • Similarity is justified by showing one object is the image of the other under a similarity (preserves angle and ratio of distances).
Why is transformational reasoning important?
Common Core Standards • Congruence, similarity and symmetry are all defined in terms of transformations. • Triangle congruence criteria (SSS, SAS, ASA) are to be explained using rigid motions. • Standards for mathematical practice: • Construct viable arguments and critique the reasoning of others. • Look for and make use of structure.
Mathematical Reasoning & Structure • Justifies results often stated without proof. • Encourages flexibility of thinking and use of multiple methods. • Foreshadows definition of a geometry via transformations. • Elucidates connections between geometry and algebra.
How can transformational reasoning be used in the high school curriculum?
Informal Reasoning & Memory • Uses the visual, intuitive sense of how a transformation maps one shape to another. • Builds on ideas of symmetry from elementary grades and could be used in middle school. • Helps with recall of theorems. • Builds geometric visualization • Examples: Isosceles Triangle Theorem, Parallelogram rotation
Example: Arcs Cut by Parallel Lines (Informal) Given: AB∥CD Prove: arc AC ≅ arc BD Idea: Reflect over the diameter perpendicular to CD.
Example: ASA(Informal) • Given: ∠A≅∠A’, AC≅A’C’, ∠C≅∠C’ • Prove: △ABC≅△A’B’C’ • Idea: • translate A to A’ • rotate △ABC until AC coincides with A’C’ • reflect over A’C’ if necessary. Then the whole triangle coincides!
Rigorous Reasoning Pre-requisites • Transformations and their properties: • Isometries – reflections, rotations, translations • Preserve lengths • Preserve angles • Dilations • Preserve angles • Preserve ratios of lengths • Image lines are parallel to original lines • Symmetries of basic shapes (lines, circles) • Basic properties and axioms of geometry • Experience that our vision can trickus
Example: Isosceles Triangle Theorem (Rigorous) Given: △ABC, where AB≅AC • Draw AD, the angle bisector of ∠BAC. Therefore, ∠BAD≅∠CAD. • Reflect over AD. • AD reflects to itself. • ∠BAD reflects to ∠CAD since the angles are congruent and share side AD. • AB reflects to AC since they are corresponding rays of angles which coincide after reflection. • B reflects to C since A reflects to itself and AB≅AC so the lengths along AB and AC coincide. • BD reflects to CD since B reflects to C and D reflects to itself and two points determine exactly one segment. • Since AB reflects to AC, B to C and BD to CD, ∠ABD reflects to ∠ACD. • Therefore ∠ABD≅∠ACD.
Example: ASA(Rigorous) Given: ∠A≅∠A’, AC≅A’C’, ∠C≅∠C’ • Translate △ABC so that A coincides with A’. • Rotate △ABC so that ray AC coincides with ray A’C’. Since AC≅A’C’, C coincides with C’. • If B and B’ are on different sides of line AC, reflect △ABC over line AC. • Since ∠A≅∠A’ and AC and A’C’ coincide and are on the same side of the angle, ∠A coincides with ∠A’. • Since the angles coincide, the other rays AB and A’B’ coincide. • Similarly, since ∠C≅∠C’ and AC and A’C’ coincide, ∠C coincides with ∠C’ and the other rays CB and C’B’ coincide. • Since ray AB coincides with ray A’B’ and ray CB with ray C’B’and two lines intersect in at most one point, B coincides with B’. • Since all sides and angles coincide, △ABC≅△A’B’C’.
Example: Midline of triangle (Rigorous) Given: △ABC, D and E are midpoints of AB and AC respectively • Apply a dilation of factor ½ to △ABC from point A. • Then B’ = D and C’=E by definition of dilation (B’ on AB, C’ on CE). • Since dilation preserves ratio of lengths, DE = ½BC. • BC∥DE because the image DE is parallel to the original BC under a dilation.
Example: Arcs Cut by Parallel Lines (Rigorous) Given: AB∥CD Prove: arc AC ≅ arc BD Draw diameter EF perpendicular to CD, intersecting CD at H and AB at G. • Since AB ∥CD, EH⊥ AB since it makes the same angle with both CD and AB. • Since diameters bisect chords, CH≅HD and AG≅GB. • Reflect over EF. • Since EF is a diameter, the circle reflects to itself. • Lines CD and AB reflect to themselves since they are perpendicular to EF. • Since CH≅HD and AG≅GB, A reflects to B and C reflects to D. • Since the circle reflects to itself and the endpoints of arc AC reflect to the endpoints of arc BD, arc AC reflects to arc BD. • Therefore arc AC ≅ arc BD.
Example: Parallelograms(Rigorous) Given: Parallelogram ABDC • Draw diagonal AD and let P be the midpoint of AD. • Rotate the figure 180⁰ about point P. • Line AD rotates to itself. • Since P is the midpoint of AD, A and D rotate to each other. • By definition of parallelogram, AB∥CD and AC∥BD, ∠BAD≅∠CDA and ∠CAD≅∠BDA. Therefore the two pairs of angles, ∠BAD and ∠CDA , and ∠CAD and ∠BDA, rotate to each other. • Since the angles ∠CAD and ∠BDA coincide, the rays AC and DB coincide. Similarly, rays AB and DC coincide because ∠BAD and ∠CDA coincide. • Since two lines intersect in only one point, C, the intersection of AC and DC, rotates to B, the intersection of DB and AB, and vice versa. • Therefore the image of parallelogram ABDC is parallelogram DCAB. • Based on what coincides, AC≅DB, AB≅DC, ∠B≅∠C, △ABD≅△DCA, and PC≅PB.
Connecting Ideas within Geometry • Transformations provide another way to help students make conceptual connections between ideas. • Examples: • Quadrilateral classification • Perpendicular bisector • Definition using transformations
Definition Transformations can be used to define objects and illustrate the structure of mathematical reasoning. • Example: a parallelogram can be defined as a quadrilateral with 180⁰ rotational symmetry. • The standard properties of parallelograms follow almost immediately from the definition; based on these we can prove opposite sides are parallel.
What does transformational reasoning contribute to student learning?
Student Benefits • Builds on students’ intuitive ideas so they can participate in proof from the beginning. • Encourages visual and spatial thinking, helping students consider the same ideas in multiple ways. • Serves as a guide for students to remember theorems and figure out problems. • Promotes understanding by offering an alternate explanation.
Student Benefits • Reinforces properties of transformations. • Applies axioms or theorems we don’t use frequently. • Motivates changing perspective between piece-by-piece and global approaches. • Generalizes more easily to other geometries, which are characterized by their symmetries.
Try using transformational reasoning! • Vertical angles are congruent. • If the base angles of a triangle are congruent, then the sides opposite those angles are congruent. • If a quadrilateral has diagonals that are perpendicular bisectors of each other, then it is a rhombus. • SAS: If two right triangles have two corresponding pairs of sides congruent and the included angles congruent, then the triangles are congruent.
Ways to include transformational reasoning in your classroom • Ask students to look for symmetry regularly! • When introducing transformations, apply them to common objects and ask what the symmetry implies about the object. • Use transformations to organize information and remember relationships. • Share another method of proof for a theorem already in your curriculum.
Transformational Reasoning in Algebra • The ideas of symmetry and transformation have application in algebra as well. • This can help students connect algebra and geometry in a new way.
Example: Commutativity of Multiplication (Informal) Show mxn=nxm, • Represent mxn as an array of dots with m rows and n columns. • Rotate the array by 90 degrees and you have n rows and m columns, or nxm dots. • Rotation preserves length & area, so these are the same number! … … … . . . … … … … … … … … . . . … … …
Symmetries of Equations • Translations and reflections of graphs • Odd & even functions • Circles: x2 + y2 = r2 • Unit circle trigonometry: sin(π/2-x) = cos(x)
Resources • Wallace, Edward C., and West, Stephen F., Roads to Geometry: section on transformational proof • Henderson, David W., and Taimina, Daina, Experiencing Geometry • The eyeballing game http://woodgears.ca/eyeball/ These slides can be found at http://campus.houghton.edu/webs/employees/kcamenga/teachers.htm