Geometry, The Common Core, And Proof

Geometry, theCommonCore, andProofJohn T.Baldwin,AndreasMuellerGeometry, the Common Core, and ProofOverviewAreaIntroducingArithmeticJohn T. Baldwin, Andreas MuellerInterlude onCirclesProving thefield axiomsNovember 28, 2012

OutlineGeometry, theCommonCore, andProofJohn T.Baldwin,AndreasMuellerOverview1 Overview2 AreaAreaIntroducingArithmeticInterlude onCircles3 Introducing Arithmetic4 Interlude on CirclesProving thefield axioms5 Proving the field axioms

AgendaGeometry, theCommonCore, andProofJohn cingArithmetici) functionsii) defining addition: segments, pointsiii) defining multiplication: segments, pointsInterlude onCirclesProving thefield axiomsG-SRT4 – Context. Proving theorems about similarityParallelograms - napoleon’s theoremArea: informally and formalllyAreas of parallelograms and trianglesa bit more on parallel linesmini-lecture Geometry vrs arithmeticIntroducing Arithmetic89lunch/Discussion: How are numbers and geometry beingtreated than in high school textsInterlude on circlesi) How many points determine a circleii) Diagrams and proofs10Proving that there is a field

LogisticsGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsHow many people have had a chance to spend any time withthe materials on line?Is there a way they could be more helpful?

LogisticsGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsHow many people have had a chance to spend any time withthe materials on line?Is there a way they could be more helpful?Does anyone want a vegetarian sandwich for lunch?

Common CoreGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsG-SRT: Prove theorems involving similarity4. Prove theorems about triangles. Theorems include: a lineparallel to one side of a triangle divides the other twoproportionally, and conversely; the Pythagorean Theoremproved using triangle similarity.This is a particularly hard standard to understand. What arewe supposed to assume when we give these proofs.Should the result be proved when the ratio of the side length isirrational?

Activity: Dividing a line into n-parts: DiagramGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axioms

Sketch of ProofGeometry, theCommonCore, andProofJohn T.Baldwin,AndreasMueller1CBDA is a parallelogram. (today soon)2Each Bn BDAn is a parallelogram. (next week)3Therefore all segments Cn Cn 1 have the same length.OverviewAreaIntroducingArithmeticInterlude onCirclesProving thefield axiomsStep 3 is the hard part. We will spend most of today on it.The key is:

Side-splitter TheoremGeometry, theCommonCore, andProofJohn T.Baldwin,AndreasMuellerTheoremEuclid VI.2 CCSS G-SRT.4 If a line is drawn parallel to thebase of triangle the corresponding sides of the two resultingtriangles are proportional and ude onCirclesProving thefield axiomsCME geometry calls this the ‘side-splitter theorem’ on pages313 and 315 of CME geometry.Two steps:1Understand the area of triangle geometrically.2Transfer this to the formula A bh2 .The proof uses elementary algebra and numbers; it isformulated in a very different way from Euclid. To prove theside-splitter theorem next week, we have to introduce numbers.

Activity: Midpoints of sides of QuadrilateralsGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsLet ABCD be an arbitrary quadrilateral? Let DEFG be themidpoints of the sides. What can you say about thequadrilateral DEFG?I called this Napoleon’s Theorem in class. That was a mistake;there is somewhat similar proposition called Napoleon’stheorem.

Activity: Conditions for a parallelogramGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsTheoremIf the opposite sides of a quadrilateral are congruent, thequadrilateral is a parallelogram.Prove this theorem. What is the key step that is also neededfor Napoleon’s theorem?

Activity: Triangles with the same area: InformalGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsSee Handout

Activity: Scissor CongruenceGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axioms174-175 from CME

scissors congruenceGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsWhat does scissor’s congruence mean?

scissors congruenceGeometry, theCommonCore, andProofJohn T.Baldwin,AndreasMuellerOverviewWhat does scissor’s congruence mean?If it is possible to cut one figure into a finite number of piecesand rearrange them to get the other, then they are scissorcongruent.AreaHilbert defines:IntroducingArithmeticTwo polygons are equidecomposable if they can be decomposedinto a finite number of triangles that are congruent in pairs.Interlude onCirclesProving thefield axiomsTwo simple polygons P and Q are equicomplementable if ispossible to add to them a finite number of pairs Pi , Qi ofequidecomposable polygons so that the disjoint union of Qwith the Qi is congruent to disjoint union of P with the Pi

Activity: Prove Euclid I.35Geometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsTheoremParallelograms on the same base and in the same parallels havethe same area.Restate in modern English; prove the theorem.in groups (10 minutes)

Why did it work?Geometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsWhat are the properties of Area that are needed for thisargument?Did you use equidecomposable or equicomplementablepolygons?

Axioms for AreaGeometry, theCommonCore, andProofJohn T.Baldwin,AndreasMuellerOverview1Congruent figures have the same area.2The area of two ‘disjoint’ polygons (i.e. meet only in apoint or along an edge) is the sum of the two areas of thepolygons.3Two figures that scissor-congruent have the same area.4The area of a rectangle whose base has length b andaltitude is h is bh.AreaIntroducingArithmeticInterlude onCirclesProving thefield axioms

Deepening understanding across grade levelsGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axioms6th grade standard (CCSS 6.G.1)Solve real-world and mathematical problems involving area,surface area, and volume. 1. Find the area of right triangles,other triangles, special quadrilaterals, and polygons bycomposing into rectangles or decomposing into triangles andother shapes; apply these techniques in the context of solvingreal-world and mathematical problems.Do your students understand this?

Common NotionsGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsThese common notions or axioms of Euclid apply equally wellto geometry or numbers or area.Common notion 1. Things which equal the same thing alsoequal one another.Common notion 2. If equals are added to equals, then thewholes are equal.Common notion 3. If equals are subtracted from equals, thenthe remainders are equal.Common notion 4. Things which coincide with one anotherequal one another.Common notion 5. The whole is greater than the part.return

DistanceGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsWhat is the distance between a point and a line?

DistanceGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsWhat is the distance between a point and a line?DefinitionThe distance between a point A and a line is length of thesegment AB where AB is perpendicular to at B.

DistanceGeometry, theCommonCore, andProofJohn T.Baldwin,AndreasMuellerOverviewAreaWhat is the distance between a point and a line?DefinitionThe distance between a point A and a line is length of thesegment AB where AB is perpendicular to at B.IntroducingArithmeticInterlude onCirclesProving thefield axiomsTheoremTwo lines m and are parallel iff they are always the samedistance apart.What does it mean? Is there a better formulation?Prove it.

TerminologyGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsModern (new math) text books make a big deal about thedifference between congruence and equality. Numbers arecentral - so equalities are only between numbers while linesegments or figures are congruent.Geometry before NumberEuclid did not have rational numbers as distinct objects - He’dsay line segments are congruent (or equal) where we’d say havethe same length.So equality can be replaced by congruence in understanding thecommon notions.

Ways to think of geometryGeometry, theCommonCore, andProofJohn T.Baldwin,AndreasMuellerOverviewTwo descriptions of the same dichotomy.Area1synthetic /analyticIntroducingArithmetic2Euclidian / coordinate (Cartesian)Interlude onCirclesProving thefield axioms

Ways to think of MathematicsGeometry, theCommonCore, andProofJohn T.Baldwin,AndreasMueller1Until the 19th century: Geometry and Arithmetic arebased on different intuitions and have separatefoundations.2Late 19th century: Arithmetic is the most basic; geometryand analysis can be built on arithmetic (proved forgeometry by e onCirclesProving thefield axioms

Ways to think of MathematicsGeometry, theCommonCore, andProofJohn T.Baldwin,AndreasMueller1Until the 19th century: Geometry and Arithmetic arebased on different intuitions and have separatefoundations.2Late 19th century: Arithmetic is the most basic; geometryand analysis can be built on arithmetic (proved forgeometry by e onCirclesProving thefield axiomsWe now study the other direction (also do to Hilbert)Analysis (algebra and continuity of the real numbers) can bebuilt from geometry.

Systems of Synthetic GeometryGeometry, theCommonCore, andProofJohn cingArithmetic3TarskiInterlude onCircles4BirkhoffOverviewProving thefield axioms

Functions and Equivalence RelationsGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsSee activity.

Overview on our axiom systemGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsWe combine Euclid and Hilbert.Geometry is fundamental; we define numbers as lengths.LengthNote that congruence forms an equivalence relation on linesegments. We fix a ray with one end point 0 on . For eachequivalence class of segments, we consider the unique segment0A on in that class as the representative of that class. Wewill often denote the class (i.e. the segment 0A by a. We say asegment (on any line) CD has length a if CD 0a.In a second stage we will use a to represent both the right handend and the length - the modern number line.

From geometry to numbersGeometry, theCommonCore, andProofJohn T.Baldwin,AndreasMuellerOverviewWe want to define the addition and multiplication of numbers.We make three separate steps.1identify the collection of all congruent line segments ashaving a common ‘length’. Choose a representativesegment OA for this class .2define the operation on such representatives.3Identify the length of the segment with the end point A.Now the multiplication is on points. And we define theaddition and multiplication a little differently.AreaIntroducingArithmeticInterlude onCirclesProving thefield axioms

Defining addition IGeometry, theCommonCore, andProofJohn thmeticInterlude onCirclesProving thefield axiomsAdding line segmentsThe sum of the line segments AB and AC is the segment ADobtained by extending AB to a straight line and then choose Don AB extended (on the other side of B from A) so thatBD AC .

Defining addition IIGeometry, theCommonCore, andProofJohn T.Baldwin,AndreasMuellerAdding pointsFix a line and a points 0 on . We define an operations on . Recall that we identify a with the (directed length of) thesegment 0a.For any points a, b on ,OverviewAreaIntroducingArithmeticInterlude onCirclesProving thefield axiomsa b cif c is constructed as follows.1Choose T not on and m parallel to through T .2Draw 0T and BT .3Draw a line parallel to 0T through a and let it intersect min F .4Draw a line parallel to bT through a and let it intersect in C .