This$is$a$tessellation.$Materials: For the teacher: A picture of the tessellation. For each student: pattern blocks, paper, and pencil. Discussion on the rug: (Teacher shows the picture

POM   Do  the  Tessellation   P  1  ©  Silicon  Valley  Mathematics  Initiative  2012.    To  reproduce  this  document,  permission  must  be  granted  by  the  SVMI  [email protected]    

This  is  a  tessellation.    

What  shapes  do  you  see?    Describe  them.    How  are  the  shapes  alike?    How  are  the  shapes  different?    

POM   Do  the  Tessellation   P  2  ©  Silicon  Valley  Mathematics  Initiative  2012.    To  reproduce  this  document,  permission  must  be  granted  by  the  SVMI  [email protected]    

What  happens  at  the  corners  (vertices)  of  the  shapes?    Are  all  corners  the  same?    Explain.    How  many  yellow  shapes  can  you  see?    How  did  you  figure  it  out?    How  many  different  shapes  can  you  see  in  all?  

POM   Do  the  Tessellation   P  3  ©  Silicon  Valley  Mathematics  Initiative  2012.    To  reproduce  this  document,  permission  must  be  granted  by  the  SVMI  [email protected]    

Level  B    The  top  of  a  table  is  comprised  of  different  colored  polygons.                                    1. Create  a  similar  design  using  construction  paper.    

2. Describe  all  the  symmetries  in  the  design.    3. Re-­‐arrange  the  shape  into  different  rectangles.    Show  your  solutions.      

 4. How  many  rectangles  can  you  make  using  all  the  shapes?  

 5.  Can  you  make  a  different  square  (different  design)  using  all  the  same  shapes?    Explain.  

POM   Do  the  Tessellation   P  4  ©  Silicon  Valley  Mathematics  Initiative  2012.    To  reproduce  this  document,  permission  must  be  granted  by  the  SVMI  [email protected]    

Level  C    Brenda  and  Nick  are  remodeling  their  floor.    The  floor  they  want  is  made  of  triangular  tiles.    They  want  to  pick  out  different  shaped  triangles  to  determine  which  triangle  they  would  like  to  use.    They  selected  five  different  types;  an  equilateral  triangle,  an  isosceles  triangle,  a  right  triangle,  an  acute  scalene  triangle,  and  an  obtuse  scalene  triangle.    Nick  believes  not  all  of  these  triangles  will  tessellate  (cover  the  plane  with  no  holes  or  overlaps).                Which  triangles  will  tessellate  and  which  will  not?    Justify  your  findings  by  providing  a  complete  rationale  or  a  proof  to  verify  your  conjecture.            

POM   Do  the  Tessellation   P  5  ©  Silicon  Valley  Mathematics  Initiative  2012.    To  reproduce  this  document,  permission  must  be  granted  by  the  SVMI  [email protected]    

Level  D    Consider  the  polygon  and  its  position  on  the  coordinate  plane.    The  objective  is  to  use  the  minimum  number  of  reflections  to  return  to  the  original  location  and  orientation  of  the  polygon.    The  rules  require  that  the  reflections  used  cannot  undo  the  previous  moves.    Show  all  your  work  on  the  graph.    Be  sure  to  explain  how  you  know  your  series  of  reflections  return  you  back  to  the  original  location  and  position.                                    Is  there  a  way  to  shorten  the  number  of  reflections  you  used?    Conjecture  what  would  need  to  be  the  fewest  moves  to  cover  the  pre-­‐image,  under  the  condition  stated  in  the  problem.              

POM   Do  the  Tessellation   P  6  ©  Silicon  Valley  Mathematics  Initiative  2012.    To  reproduce  this  document,  permission  must  be  granted  by  the  SVMI  [email protected]    

Level  E    M.C.  Escher  used  glide  reflections  to  create  designs.  

     Determine  how  this  tessellation  was  created.    What  basic  polygon  did  he  use  as  the  foundation  of  the  design?    How  many  steps  did  he  use?    Create  your  own  glide  refection  tessellation.    

POM   Do  the  Tessellation   P  7  ©  Silicon  Valley  Mathematics  Initiative  2012.    To  reproduce  this  document,  permission  must  be  granted  by  the  SVMI  [email protected]    

   Primary Version Level A    

Materials: For the teacher: A picture of the tessellation. For each student: pattern blocks, paper, and pencil.

Discussion on the rug: (Teacher shows the picture of the tessellation) “Look at this picture very closely, what different shapes do you see?” (Students volunteer the shapes they see). “What do you notice about the different shapes?” (Students respond with ideas about color, size, and orientation). (Teacher asks) “How can I use some of these to make a square?” (Teacher calls on students to demonstrate and explain). (Teacher asks) “How many pieces did we use?” How do we know it is a square?” “Can we make a square a different way?” (Class explores these ideas).

In small groups: (Each student has the materials). The teacher says to the class, “You have a set of blocks on your table, please try to build the pattern you see in the picture.” (Students work in pairs to build the tessellation). The teacher says to the class, “Draw the picture you see. Do your best to make your drawing look the same as the one

POM   Do  the  Tessellation   P  8  ©  Silicon  Valley  Mathematics  Initiative  2012.    To  reproduce  this  document,  permission  must  be  granted  by  the  SVMI  [email protected]    

you built? How many shapes are in your drawing? What are the names of the shapes in your drawing?”

(At the end of the investigation, have students either discuss or dictate a response to these summary questions). What shapes are in your drawing? How many shapes did you use? Tell me how you figured out how to make the picture.

POM   Do  the  Tessellation   P  9  ©  Silicon  Valley  Mathematics  Initiative  2012.    To  reproduce  this  document,  permission  must  be  granted  by  the  SVMI  [email protected]    

Primary Version Level A Tessellations    

POM   Do  the  Tessellation   P  10  ©  Silicon  Valley  Mathematics  Initiative  2012.    To  reproduce  this  document,  permission  must  be  granted  by  the  SVMI  [email protected]