On%perpendicularity % % XXX% - PowerPoint PPT Presentation

Timo%Tossavainen,%%Eger,%July%2013% On%perpendicularity % % XXX% –%and%a%few%words%about%parallelism,%too% http://www.youtube.com/watch?v=vnnwfcDcNlY% Teaching)perpendicularity)and) Properties)of)perpendicularity) parallelism) ! Symmetry:%if%%%%%%%%%%%%,%then%also% a ⊥ b b ⊥ a ! Are%perpendicularity%and%parallelism%so%trivial%or%familiar%concepts% from%the%real%life%that%every%student%learns%the%essential%facts%about% Irreflexivity:%for%none%of%elements,% a ⊥ a ! them%on%her/his%own?% ! Is%is%possible%to%teach%proper%axiomatic%thinking%for%senior%high% ! Transitivity?% school%students?% ! Some%other%property?% ! The%conceptual%and%procedural%knowledge%of%mathematics:%Should% we%first%understand%in%order%to%be%able%to%do%mathematics%or%is%it% % % % % %?% a ⊥ b ⊥ c ⊥ d ⇒ a ⊥ d vice%versa?%

An)axiom)system)for)(planar)) Three)alternatives) perpendicularity)and)parallelism) ! A%binary%relation%is%perpendicularity%if%it%is% A1:%%%%%%%% ∀ a : ¬ a ⊥ a symmetric%and%irreflexive.% A2:%%%%%%% ∀ a , b : a ⊥ b ⇒ b ⊥ a Perpendicularity%is%not%a%binary%relation%but,% ! A3:% ∀ a , b , c , d : a ⊥ b ⊥ c ⊥ d ⇒ a ⊥ d for%example,%a%trinary%relation.% A4:% ∀ a : ∃ b : a ⊥ b ! There%is%not%a%universal%perpendicularity%but% several%different%perpendicularities%in%different% A5:% ∀ a , b : a || b ⇒ ∃ c : a ⊥ c ⊥ b contexts.% A6:% ∀ a , b , c : a ⊥ b ⊥ c ⇒ a || c Some)results) Another)model) { } Theorem)1.) Parallelism%||%is%an%equivalence%relation.%%%%%%%% Example)1.)) Let%%%%%%%%%%%%%%%%%%%%and%% X = 0,1 Theorem)2.) % ∀ a , b : a || b ⇒ ¬ a ⊥ b . ⊥ 0% 1% 0% no% yes% ∀ a , b , c : a || b ∧ b ⊥ c ⇒ a ⊥ c . Theorem)3. % %% 1% yes% no% These%results%and%many%other%verifiable% propositions%in%this%axiom%system%are%compatible% ||) 0% 1% with%the%model%of%Euclidean%geometry%in%plane.%% 0% yes% no% 1% no% yes%

Another)axiom)system)for)(algebraic)) More)models) perpendicularity) { } , x ⊥ y ⇔ xy < 0, x || y ⇔ xy > 0. Example))2.)))))))) X = R \ 0 A1:%%%%%%%% ∀ a ≠ 0 : ¬ a ⊥ a { } , x ⊥ y ⇔ xy = 1, x || y ⇔ x = y . Example)3.) A2:%%%%%%% X = R \ − 1,0,1 ∀ a , b : a ⊥ b ⇒ b ⊥ a Example)4. %% In%the%set%of%all%lines%in%the%Euclidean%plane,% A3:% ∀ a : ∃ b : a ⊥ b define%that%two%lines%are%perpendicular%if%the%smallest%angle% A4:% between%them%measures%45 o ,%and%parallel%if%they%are%parallel% ∀ a , b : a ⊥ b ⇒ a ⊥ − b in%the%ordinary%sense%or%the%angle%between%them%measures% A5:% ∀ a , b , c : a ⊥ b ∧ a ⊥ c ⇒ a ⊥ ( b + c ) 90 o .% % Some)facts)about)algebraic) % Pedagogical)conclusions) perpendicularity) ! It%is%compatible%with%every%vector%space,%the%axioms% ! Axiomatic%approach%helped%us%to%find%new%aspects% are%derived%from%the%property%that%the%inner%product% and%even%new%results%even%on%very%old%concepts.%% for%two%perpendicular%vectors%is%zero.%% ! We%extended%our%conceptual%understanding%about% ! In%algebraic%context,%interesting%questions%about% perpendicularity%and%parallelism%through%a%procedural% perpendicularity%are%different%from%those%in%geometric% approach.%% context.%% ! On%the%other%hand,%the%operationalization%of%these% ! An%example%of%interesting%perpendicularity:%in%the%set% concepts%required%that%we%already%have%an% of%integers ," a )and) b )are)perpendicular)if)and)only)if) internalized%view%of%the%domain%of%possible%axioms% they)are)relatively)prime .% and,%in%general,%suitable%criteria%for%choosing%proper% axioms%etc.%

References) 1.%L.%Haapasalo%&%D.%Kadijevich%(2000).%Two%types%of% mathematical%knowledge%and%their%relation.% Journal"für" Mathematik3Didaktik %21(2),%139–157.% 2.%P.%Haukkanen,%J.%K.%Merikoski%&%T.%Tossavainen%(2011).% Axiomatizing%perpendicularity%and%parallelism.% Journal"for" Geometry"and"Graphics"15(2),"129–139." 3 .%P.%Haukkanen,%M.%Mattila,%J.%K.%Merikoski%&%T.% Tossavainen%(2013).%Perpendicularity%in%an%Abelian%group.% International % Journal"of"Mathematics"and"Mathematical" Sciences,"Volume"2013,"Article"ID"983607." %