Transforming Geometric Proof

with Reflections, Rotations and Translations

Kristin A. CamengaKristin.camenga@houghton.edu

Houghton CollegeApril 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?

Motivating Questions

What is transformational reasoning?

For each of the following quadrilaterals, describe the rotations and reflections that carry it onto itself:ParallelogramRhombusRectangleSquareWhat connections do you notice?

Quadrilateral Transformations

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)

Quadrilateral Transformations

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

Parallelograms by Symmetry

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.

The Isosceles Triangle Theorem

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).

Key ideas

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.

Common Core Standards

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.

Mathematical Reasoning & Structure

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

Informal Reasoning & Memory

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!

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 us

Rigorous Reasoning Pre-requisites

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: Isosceles Triangle Theorem (Rigorous)

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’.

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: Midline of triangle (Rigorous)

Given: AB∥CDProve: 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: Arcs Cut by Parallel Lines (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.

Example: Parallelograms(Rigorous)

Transformations provide another way to help students make conceptual connections between ideas.

Examples:◦ Quadrilateral classification◦ Perpendicular bisector ◦ Definition using transformations

Connecting Ideas within Geometry

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.

Definition

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.

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.

Student Benefits

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.

Try using transformational reasoning!

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.

Ways to include transformational reasoning in your classroom

The ideas of symmetry and transformation have application in algebra as well.

This can help students connect algebra and geometry in a new way.

Transformational Reasoning in Algebra

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!

Example: Commutativity of Multiplication (Informal)

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Translations and reflections of graphs Odd & even functions Circles: x2 + y2 = r2

Unit circle trigonometry: sin(π/2-x) = cos(x)

Symmetries of Equations

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

Resources