Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
DeBellis & Rosenstein: Discrete Mathematics for K-8 Teachers – Draft 11/4/04 – Classroom Guide
Coloring Pictures and Maps Classroom Guide: Chapter One Topic A
Coloring Pictures and Maps
Figure 1
Figure 2
Figure 3
K-8 Topic Overview
K - 2 3 – 5 6 - 8
• identifies
number of regions in a figure
• colors two
regions with different colors if “they’re next to each other”
• articulates the
number of colors needed to color a page in a coloring book
• decides if
“corners count as a border or not” and colors pictures accordingly
• colors medium-
difficulty pictures and maps
• identifies the
least number of colors needed to properly color pictures & maps
• defines common
border to exclude regions that meet only at a point
• identifies
“problem spots” in simple figures that seem to require 3 or 4 colors
• begins to reason
why a particular picture or map can’t be colored using fewer colors
• colors
continental US Map as introductory activity and colors complex maps
• identifies
“problem spots” in complex pictures or maps that seem to require 3 or 4 colors
• provides reasons
why these figures can’t be colored using fewer colors
• designs maps
that require four colors
DeBellis & Rosenstein: Discrete Mathematics for K-8 Teachers – Draft 11/4/04 – Classroom Guide
Introduction to Vertex-Edge Graphs Classroom Guide: Chapter One Topic B Vertex-Edge Graphs
����� ���������� ���������������������������! #"�$ %'&�(')�(�*�+,(.- )�/0*�"1)�"2#3�465'798'3�:1;�<�=?>@;�:�A�7B<'CD5E;�F�3G�HJI�K�LMI�HNH#ODP'QSRUT K�V6I�WXO�Y,G.TZTG�LMI�HJP�G�WD[�Q�\�H�R9P�G�T]T \_^`O�a#Ib QcV6O�d6LMQ b QMIM^
Figure 5
Figure 6
K-8 Topic Overview
K - 2 3 – 5 6 - 8 • counts the
number of vertices and edges in simple graphs
• identifies all her
neighbors (while standing on a vertex in a floor graph)
• constructs
simple vertex-edge graphs from simple pictures or maps
• colors two
vertices with different colors if “they’re joined by an edge”
• articulates the
fewest number of colors needed to color a graph
• draws vertex-
edge graphs to represent concrete situations
• investigates
simple properties of graphs like degree
• colors medium-
difficulty graphs using fewest number of colors and finds the chromatic number
• constructs &
colors graphs that represent conflicts
• begins to look at
the structure of “problem spots” to explain why fewer colors can not be used
• begins to reason
why a particular graph can’t be colored using fewer colors
• represents
concrete & abstract situations using graphs
• articulates
reasons why maps & graphs can’t be colored using fewer colors
• designs maps
that require four colors
• shows the
relationship between coloring graphs & coloring maps
• understands &
demonstrates how to resolve conflicts using graph coloring
•
DeBellis & Rosenstein: Discrete Mathematics for K-8 Teachers – Draft 11/4/04 – Classroom Guide
Coloring Mathematically Classroom Guide: Chapter One ����� ������2WKHU�. ���&RORULQJ�
$FWLYLWLHV��• eSf.g f�h,i�j k�l�m�n�h�oEpq#r�s�t�u t�uBv�w�x�yzu w�t�vD{ | }�~
�M���9�#���J�#���B�'�X�0�c��� ���'�#������N�����,�����'�c� ���E�
• � ���'�#�0�0���0�,���B�#�0�'�0��������@�����N�@� �0��� ¡�¢z£6¤�¥.�B¦�§#���§�¨_�'�©���X�@� �#�D�§�¤.� ¥M�.ªM¦�£1�6¦D«#¦.¬ ¦�£�0��¤��
• ¯® °±�²�³#´Xµ'´�¶0·'¸�¹»º�²�¼½�¾ ½�¿�À�Á�Â�Ã�Ä�Å Â!ƯÁ�Ç�ÅBÈÉ�Ê#ËÍÌ�Ì�Î Ï�Ð ÑÓÒ9Ô�Õ#Ö0×Ø�Ù6Ú'Û�ÜBÝ.Þ�ß,à�ß�Ü�áØ.â ÞØ�Ü9ã0ä0å�Ù6ÞXæ'Ú�ÙNã�çSÜèâ Ý�Ù6Þ�éJä6Úã'Ú.â Ú�Ù
• ê Ú.â Ú�Ùß�ÜBáØ.â Þ�Ø�ÜBã#ä#å'Ù0Þ�ßæ�Ù6Ú�áëã#Ú.â Ú�Ù�Ü é'ì!íMÚ�Ú�î6ßâ�ÜZî#ÞJà�ß#Ú�ã�ã#Þ�ÙJí�à�â]â
• ï�ð�ñ0òcó!ô�õ�õ�ö�÷ ø.ùMú ûMü�ý�þÿ�� ����� ��� �������������� �������
� ������������� ����� �"!#�"� ���%$ ��'&������ ���(� ��)���+* �%$ �,�� -".�/1032 4�576+/98:�4�;)<.�=->?6@2 AB=�4�CED�.F-".32 .�=
.���&ODVVURRPV ��GIHIJ H?K�LNM�OQP?L+R�S�T%K"U3VWL+MXR�HIJ HBK�L+M�OQY�H�H[Z�V\R�]3MWY[U^]W_']�S�`�U?_']3S�LNR�]aJ�]3R�S�L+bBL�S�c
dM�H3S
ef�g�hFi^j3k"l3i3h�m+n�lXo%p�l[q^rtsumwvyxzk"lBp^j�i3pX{ lXi�gB|�lBx}ho^j�oIv o?k,iX~�mNj�h�f3k"lu�av@m�|�lXh�stm g�[�3���������^���B�3�#���X�"�1�\�����%���3�\���?���#�B�"�X�3�%�I�#�"�X���3�^�%���3���[�[�F�?�����W�����^���3�F����a� �����\�?�[���3�y�a��� �X�������a�B�����^ '���¡���I� �?�'�W¢?�+�����%�"�W���������a�+�[�+��£7�u�y�%¤��B�¥��¢��3���3�¦E§�¨�©Qªu«yª%¬�ªB^®�¯I« ¯B°�±X²3³Q´�±B§+�µX¶^·¸§y¹�¹3ª?°"ªB�¨ ®�¯a« ¯�°'¹�¯B°,ª3¶%®�©¡±�º�¶%®�ª[»X¼#«�« ¯�¦½¨�©[ªB¾¨ ¿XÀ�¿IÁ ¿�Â,Ã�Ä�ÅIÆÇÂ#È[Æ+À�Ã�É3Â"ÅÊÆ+Ë\Ã�ÄzÆyÌ�ÍÏÎ�ÐXÎ3Ë3ÑQÃ�Ä[ÅBË\Î�ÌBÒ¥Ã�Ä[ÅWÓ%É�Å3Ì�Ã�Æ+¿3ËaÔ�Õ×ÖaÎ3Ë Ð?¿%ÉØÉ�Ì�ÅÙ3ÚBÛ1Ú�ÜFÝ�ÞIß Þ?Ü�à�á?â1Ü"Ú?ã3Ú3ä�å�ÚBæçß èêéyÙ^ë3ÚBÝ�Ú3à�à�äBÜìèzí3î�ë�å3éïß�åð[Ú�è7Ý�ä%ë^Ý�Þ?ÜÜ"ÚBÝ�å3ß èñéNæaÚBë�å3éyÙ�èò�ó�ôWõ�ô%ö÷ô3ø�ò ù3ú�û�ü[ô�ýFþ�õ^ÿ�þ�� þ�ý�ø\ù3ôBÿ�ô3ø�ø��Bý������ òFò�ó��ò��þ �+ù�ò�����ø������ ó����aþ��?þ3ú������������ !�#"$�%'&� ���( �)&�$ * $,+��-��%�.�-��/�0 1�231,+4&�$ * $,+5.�6�7
�8 �9���:"�$�%���/:&����;*=<�+>1�� ?,�9�'�=.�@A1�.B�)�C$'.B�D �+D�) E&�$ * $,+B�9�B/: �&��!�9FG�9�H"�2I�9���: :J,�-&���%G+H1K�L�M�KON-P!Q�R S QBT�UOV>TWS M!K�N9QBTWS X�Y!T�Z[U!\BM�]!T�UG^E_`L,T�U�T�]!LaN;S baV>T�P�Z#N;ScS,N-PAN9K!N9M ScS X�P�T�T�bdKDRe!f!g�hAi5jBk>l�f�gdm`nAh�fEoGf!hOj!p�lGq�hsr=j�t oGq�l�kCh�gvuwf�hGx kyk>o�z{l�f!o!k�n,hAi#j!p�l�q�hA|�}�f!q!hk�n�r9j'q�o�f�q�hp!kOr jEe�f�g hAi�jBk>o!oGg�~�k�nAh:q�n�r;t=gai>h�f'm#r;tctG�Ah�i>h�l�g�:��oi4k�n,h�f!o!k�r9oGf'o!��#���O��� � �,���-���G���,�!�3�G�B�,�!�C��G�'�!��G�,�=�d�B�G�A�-�:�A�����-�E�I�9���v�#�G��������!�E�-�!��� � �B��>�W� ���!�9�B�W� �:���G���>�����-�����,�,���c�B�:���A����D�!���-�!�!�������:�����B���H� � � �E�G�!�d���A�������D���A�>�!����C�:�4�,�>�:���G�y�A�=�G¡¢�#�G���G£¤ �O�B��� ����������A�-�:�!����D�>�,�������!�������>�O� �=�,��)�����,�-�����������)���A�������B�C��5�'����!��¥���G�)�!�¥�:�!� �-���)� �E�!����D�����B���)�4�G���,���>���,�!�B�G�!�d��� �����-���G�9�B�¥����!��¦�9�!�!�,�>�������B��� ���H�O�A�������G�v�����#� �c�!�!�'���,�������B�C���H�G£�§y���A�,�I�B���������!�D�O���=�,�����G�¥�:�C�O���A�:�G�G�!����-�C���B�G�!�A�9����-��� �,�������a�-�!���B�������C�A��� �¥�A�-���!����D�H�'�¨�©�c��>�Gª������>�:�>�`�E�,�#��� �>� ���B�A� �,�H�G£«A¬W ®G¯[°�±>¬�°E²!¬G¯³¬G´ °Gµ)¶ =¬�·'®!²�°�¸B¹�º,¬�»!¹�¼-½d¾ ¿ À:ÁBÂ�ÃBÄ�Å�Æ!Â)ÇÉÈ,ÊBË�Ì
DeBellis & Rosenstein: Discrete Mathematics for K-8 Teachers – Draft 11/4/04 – Classroom Guide
Coloring Mathematically Classroom Guide: Chapter One
2WKHU�� ���&RORULQJ�
$FWLYLWLHV��• ÍDÎGÏ ÎCÐ�ÐHÑCÒ�Ó=ÎCÔCÕ¨Î>ÖI×HØ!Ñ
ÍDÎCÔC×CÓ Ô�Ñ�ÔC×ÚÙGÏ�Û{Ù�Ü4ÎCÖI×HØ!ÑÝaÞ!ß à5á�â�ãWàÚäCàHáCå#æHåBß ÞDçè>é æCêìë éGí é ê�å
• ë>êHá>äCà5á ä¥î{ä�ï4ðñß àHò'ó ô õ÷öê5áCçGß é Þ>å¨ðøò,ß9ë>ò è>é êHë>áCå3äïDä�ê�à�Þ�áBê3à é æ5åDá4àDò�ê5á�á é êù>ú�ûCüìýDúGþ úCü�ÿ�����ü������ � ���������� �����ú�ü������EýDúGþ ú�ü
• � þ ������� � ����� ��ÿìù�ü5ú��!�"�#%$&"�#('�)�*�+-, . / 0�1�2�354�6�2�78:9;=<�>5?�@�<�A�BC�CED�<GF%H&?B�> H&@�<• I J�K�L�M&N�O�K�L�P�Q�R�I OS�P�K�T�UVIWL�PXZY�[�\�]_^%`a\�b%c�d&efd%ga\�]�hi j Y [�k�efd%\�lm[mg&c�[�k�] nonb�p�h�p�\ [�\�haq�[�k�e \r[s p:[�ph%[�q�]ut�h�c(\�h�v
����&ODVVURRPV � w�xzy-{�|�}�~z�&��{ ����� ��}�~�� �%�u��� ���-�f}��V~_|����f~ � ���&��� �-{ ��� ~ ��� | � ~ ��� � ��� ~���|�����_����&���5�����&�%���-���-�W�����¡ ¢��������£-¤���¥V������¦&�����§��£z�5�¨��¥-���ª©m£��%�����«¤&��©¬��¥-�����¦��&�&©&�5��£z�§���¥V��®¬�¯&£����° ¢�¦���¥²±³®��V©��5���_��©a±���©��§����¤������´©�� µ����&£��¶� µz��&�£�¦�� �&��£�£���¦-�ª� µ�·¨¸ ¥-�&µ�£�¥-��¦V�f�¹��� £���ºV�¨�-º��f�_�����»¼¯V� ���5�¨½:¥�µ���¥�©%��������� ��©�£���©�������������²�&�-©m��®���¯¹�5�½:¥«�W��¥°��¥V�V©%�¨��©%����¥�©%���¹©%��¾��f��&£�����¥°���¿½:¥¼�5��¥º-�-©����V©�£��¥V������¥-�-©��%½ �¿�����¹£��§®�¶� ��©�� µ�½ ¥µz�&��¦�©������ �-©�£¿��©%�������������Z�&��©m�®���¯¹�5�¨½�¥¼�§��¥���¥-�V©%�_��©%�_���¦�©a©%��¾«�W���£¨������¥����¨½ ¥¼�5��¥�º-��©%� �-©�£���¥-�z����¥V��©À�Á�Â%Ã�ðÄÆÅ�ÇzÈ5É_À�ÁVÃzÊ�Å�˨Ì�Í�Î Ì�Ï�À�Á²Ð³Ê�ÃVÂ�È5Ñ�Å-ÒÔÓ
Õ¼ÖØ׶ÙfÚØÛ%Ü�Ý°×5Ý¿Þ-Û�ßÚ�Ü�à�á â ã¹ä&å-æçVèféëêVì_í�ê�èfìzî%æ�ï&æ�è æ-ð�ñmí�ò&ä�íó�éZï&æ�è æ-ð�î�åVìô ìVð�î�õ5ï&ì�ä¿æ�ö¿ä�õ§ñaò�èfì_÷-ð�í�ò�å�ä¨ç�ä�õ5ó&÷�î�åVì_ö�ìø ì�ä�î�ó�ç&ñ¿êVìVð�æ�öï&æ�è æ-ð�äuíó�éZî%æìù¼ò�è íVõ5ó¨ø:åúzæ�ó�ì�ö�ìø ì-ð�ï�æ�è æ-ð¨õ äó�æ�î¬ä�ç�ö&öVõ5ï�õ5ì�ó�î�ûýü åVì&ú�ä�å�æ�ç-èfé¹í�è ä æ²èfì�í�ð%óå-æ�øþî�æzç�ä&ì ô ì-ð�î�ìùÿï�æ�è æ�ð�õ5ó&÷zî�æ�ð%ì�ä�æ�è ô ì�ï�æó�öVè õ5ï&î�äû � å-ì-ð%ì�í&äï�åØõ¶èfé¼ð%ì�ó¹õ5óì�í�ð�è õ§ì�ð�÷-ð�íé�ì�ä�ö&æï�ç�ä�æó¨ï�æ�è æ-ð&õ5ó�÷°ò-õ§ï�î�ç�ð%ì�äuíó�éÿñ�íò&ä
�í�î¬÷�ð�í�é�ì�ä�� � ã²î�åVì
������ � ����������� ������������������� �!"�#��$�%"�#��&��'�(� )�*�*�,+-�%� �.��*�*�,(/0%1� �"�2�3�#��$�45�* ����(� 76����*48�9�"(� :�3��;�(<��������,�9�"(�+=�#�3�>+?��+ �*��+A@"(�'B���<��1� �3'C )�*(�(��1(���DE ��(* �(,6��#���� =���)�*�����;��#���#(� <FG�2��;���� ;�#��$��*���1� � �H ��*�*�7'�(*�� ���"�#��$"IJ4K�L�M���(N�H% �������(����O ���(�;�(N� ��%����"(��>'P+-�*�9�"(3+-���*�#��1�3'�(*�� �����#��$�.��*�Q�2+?%���'O���*���P ���'C����R�#�S?T0UWV:X Y Z�[\#]�^*\ _a`cb�d=e�]�f,g�hie�]�f�e�[*f*d7jCk�[<gW^9l0k�k`*m7eJhilNb�nAeJhG\c^�d oqp�rsrut7vxwyWz*{*|�}"z"~����#�)��zN� � � |������{"� �*z��,|"� {����*~O�����-��|�{��z*{������ ���#���&z*�"~C{��9z��9}�� �����#�*�:�*�{*|����#���H�����-�H��}�{=���*|���{1�����9���J�#z*����z*���������~�zN� {*���#�����K{*�9}"z3�-{��*�#|��)�&�<�*�9}"z"~|�����&z����P{3~�z*{������A{1� ���<z3�?�}3{��3�2�*z����#� p�r�r�t�w� zN� ���{�~�z���z�8z�1{�?�1��z*�)���={*����}�z������2|�� � �Q��������z���?����~��*�
DeBellis & Rosenstein: Discrete Mathematics for K-8 Teachers – Draft 11/4/04 – Classroom Guide
Coloring Mathematically Classroom Guide: Chapter One
2WKHU�� ���&RORULQJ�
$FWLYLWLHV��• �¡*¢ ¡9£s¤C¥�¦P¦9§9¤&¨�£C¦
�¡�§&¤&¨�§&¦�§&¤ª©¢9«¬©9B¡&®-¤�¥�¦¯ §�¨�¤°¦�±�²N¤ª©&¤C¦&³-´C³�¨ §Oµ®�¡�´&£¶ �¡*¢ ¡9£ª³s©&§9±=¦&·�*¢ ©¨ §¸º¹�»?¼�½�¾�¿�À�¿�Á?¿�Á&Â�¾�Ã�¿�Á�Ä
• ÄÁ&Å�à Æ9ÇKÀ�È�À�ÉB½&ÇKÀBÉ*Ã�Á�Ê�ÁË&ÌKÍOÎ�Í�Ï�ÐsÑuÒ Ó�Ô�Õ×Ö Ø Ö�ÙÐ�Ï&ÚÒ�Ë�Û ÜÝѺÔ�ÒHÞ�Ô�Ì�Ë�ÐCÞ�Ï&ÜsÎÍOÎ�аӰÛ�Ï�ÐsÓ�ËPß�ÜOÏ?Ì�Ë�ß�Ðà�á*â á9ãªäCåäCæNç�èÝç9é9ê�àOáâ á9ã&ëìuí ä�à�îOä�äKè�ã�á9è9ï�ã°ð í ï�äÝá�ñò�á9î&ã5êï&ä í ó éKðCô�ç&ðã�ï&õ�î í ã�ïBðCô�ï�î�ä&ïPá�ñKñ�á�î&ãà�á*â á9ãªä9ëKö*ç&éKäOðCî�ê�ï9é&ðªäê�ã÷ç&æ:ç?ø�ç�è-á&éKç�ù�ç ó ï�âðCô9ç9ð¶ã�ï&õ�î í ã�ï&äÝñ í ú ïû�ü*ý ü9þªÿ��
• ������� ������ ���������� ��� ���������� �������� �"!#�����$�%�&(' '�) * $�+�,�- .�%�-�+�/�) 0
�
����&ODVVURRPV � 132 4�57698(2�:<;=8?>A@�B�5C6:ED F GIH�JAK�LNMPORQNSUT9QVMPS<W�KUL�X�O7SAY�H#W�T�X�OZTXOIT�[�[CM \<]#SAY�W�S�^_ K7M KAY(`�X#a<W�KEH#K7M ]#SETE] T9Y(`=S�W�\bKcU[�Y�K�QVMPS�deH fEg JNS(\UH#JAK9LAMPOhQNSET9QVMPS<W�K<S�^i[CM TC`�Xjlk�m n�oAp�q�rCs�tvuer�w(xEy{z|s~}�o<q��No���tNs�q o9���3q�rq�oxvuer�w9��p�o��9�9s�p�o<�(���p�n#�7z ��p�xA��#x�s�t#�<q �No<n(�tn(o�w�q�x?���?�Ap�r9w� �(�C� �A�(���#�b�V�P�(�������(�9�����v�������� ��N�¢¡�£¤�(�9�¥(¦9§#¨©�ª�«#¥?¬�¯®l°²±Z³´b¨�µA¦<¦9«�¶�ª�·?¸N¹�º¶7¦<»�¼(¨�µA¦(´U¥(µAª�½N¾P¶¿µ�º�À#¦bµ9º�¶¦�Á�¦N¹(©�¦9«�§�¦¥b©=«E¶C¦¥9©P¸�«N©�«�¸b¨ µN¦7©�¹Ãª�ÄÅ«UÆeº�Â(¥?¨ µ�º�¨?¹�¦�Ç9½�©�¹�¦¥bÈbª�¹e¬h§#ª7¾ ªA¹É¥A¼N©�«§#ª7¾ ªA¹(©�«#¸<¨ µN¦UÀ#¦A¹�¨©�§(¦¥?ª�·E§(ª�Æ�ÂC¾P¦�Á�¸A¹�º9µ�¥A¼�©=«v¶l©�¥(§(½�¥�¥�©=«�¸v¥#¨(¹�º�¨ ¦¸7©=¦¥Ê·(ª�¹·A©=«�¶Ë©�«#¸<¨�µA¦b§�µ7¹ ª�Æeº¨�©=§?«½�Æ?ÌN¦N¹?©�«?¥(½�§�µ<¸N¹�º�Â�µ9¥A¼(º�«¶Í©�«v½#¥9©�«(¸v¸A¹�º9Â�µ§#ª7¾ ªA¹(©�«#¸<¨�ªb¹�¦¥#ª7¾ À#¦<§(ª«�·A¾|©�§(¨�¥�±ÏÎ�¶C¦º7¾Ð¾ ´�¼�¨�µA¦¥(¦<¦�Á7¦N¹(©�¦9«§(¦¥Ê¥�µ�ª9½A¾P¶Ñ¾ º�´<¨�µA¦·(ª9½�«�¶Nº�¨©�ª�«¢·(ªA¹e·�½�¹�¨ µN¦N¹�º�«#º7¾ ´�¥�©P¥?ª�·¢À#¦A¹�¨ ¦�Á¿§#ª7¾ ªA¹�©=«#¸<º«(¶ZÀ#¦N¹�¨�¦�Á Ò ¦9¶A¸�¦¸C¹�º�Â�µ�¥b©�«Uµ�©P¸Aµ<¥(§�µAª�ª7¾=± Ó ©�¶7¶Ô¾P¦U¥(§�µAª�ª7¾�¥#¨�½(¶7¦9«�¨�¥?Ä�©Õ¾Ð¾(¸9ªE·½�¹�¨ µN¦A¹ ¨�µ9º�«¢¸A¹�º�¶7¦¥bÈ Ò ®Ö¥#¨�½(¶7¦9«�¨�¥b©�«¨�µA¦7©|¹eº«(ºC¾ ´9¥9©P¥Êª·UÆeº�Âv§#ª7¾ ª�¹(©�«(¸�¼�º�¥?«�ª�¨ ¦9¶Í©�«?¥�½�Ì�¥#¦�Ç�½�¦9«(¨Ê§�µ�º�Â�¨�¦N¹�¥�±Ó ª�¹�¦�ª�À#¦A¹�¼�¨ µN¦(´U¥�µAª�½N¾P¶RÌN¦b¦�Ái¦9§(¨�¦9¶�¨�ªE¥�µAª�Ä׸A¹�¦º�¨ ¦N¹ØÆeº�¨�µA¦�Æ�º�¨©�§�º7¾¥#ª9µl©P¥#¨©�§�º¨�©�ª«C¼A©=«¢¨�¦A¹�Æe¥?ª�·UÂ9¹�ª�¦A¹Ã½#¥(¦vª·Z¾ º«#¸A½�º#¸�¦Uº«¶Z«�ª�¨�º�¨©�ª�«N¼(º«¶Ç9½�º7¾|©�¨�´<ª·Eª9¹�º7¾�º�«�¶ÖÄÙ¹(©�¨�¨�¦9«v¦ Á�ÂC¾ º«(º�¨©�ª�«#¥�±UÚÛµN¦U·�ª�§½�¥E§#ª�«(¨©�«�½(¦�¥¢¨�ª¢¥�µl©�·�¨·9¹�ª9Æܧ#ª7¾ ª�¹(©�«#¸UºUÆeº9ÂEª9¹Ý¸N¹�º9Â�µb¨�ªEº�«#º7¾ ´AÞN©�«(¸<§#ª7¾ ªA¹�©=«#¸�¥Êª·UÆeº9Â�¥¢ª9¹e¸A¹�º9µ�¥º«¶�¦�Á�ÂC¾ ºN©=«A©�«(¸UÄßµ�´I§(ªC¾ ª�¹(©�«�¸�¥E½�¥(¦U¨ µN¦bÆE©�«A©�Æ�½�ÆÜ«�½ÆÊÌN¦N¹�ª�·E§#ª7¾ ªA¹É¥�± ³C¦i¾ ª�įº9¹�¦<¨�ÄŪU¦�ÁCº�Æ�ÂC¾P¦¥?ª�·¢º½(¨�µA¦9«�¨�©=§và Ò »Z¥#¨�½(¶7¦�«(¨�ÄŪA¹#áâ