Kristin A. Camenga 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

 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?

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?

Parallelogram 180 ○ rotation Rectangle 180 ○ rotation, 2 lines of symmetry (through midpoints of sides) Square 180 ○ rotation, 4 lines of symmetry Rhombus 180 ○ rotation, 2 lines of symmetry (diagonals)

If we rotate a parallelogram 180 ○ about therotate midpoint of a diagonal,  AC≅DB; AB≅DC  ∠B≅∠C, ∠A≅∠D  ∠BAD≅∠CDA, ∠CAD≅∠BDA  AB∥CD, CA∥DB

Why are the base angles equal?

 Traditional Method: Draw median and show triangles congruent.  Transformational method: Draw angle bisector and reflect triangle over it to see that angles coincide.

 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?

 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.

 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?

 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

Given: AB∥CD Prove: arc AC ≅ arc BD Idea: Reflect over the diameter perpendicular to CD.

 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!

 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 trick ustrick

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.

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’.

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.

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.

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.

 Transformations provide another way to help students make conceptual connections between ideas.  Examples: ◦ Quadrilateral classification ◦ Perpendicular bisector ◦ Definition using transformations

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?

 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.

 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.

Your turn!

 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.

 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.

 The ideas of symmetry and transformation have application in algebra as well.  This can help students connect algebra and geometry in a new way.

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! … … ………… …. … … … … … … …

 Translations and reflections of graphs  Odd & even functions  Circles: x 2 + y 2 = r 2  Unit circle trigonometry: sin(π/2-x) = cos(x)

 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 These slides can be found at