Mathema'cal  Prac'ces  and  the  Development  of  Algebraic  Reasoning  and  Generaliza'on      

 YeukSze  Leong  

yeukszeleong@gmail.com  

Breakout Workshop

Goals    

•  Unpack  the  key  ideas  in  Math  Prac=ces  7  &  8    7:  Look  for  and  make  use  of  structure    8:  Look  for  and  express  regularity  in  repeated  reasoning  

 

•  Experience  the  prac=ces  by  doing  math.  

•  Share  instruc=onal  strategies  that  promote  the  development  of  these  math  prac=ces  

Big  Ideas  in  Math  Prac'ce  7  

(1) Consider  behavior  

(2) Surface  the  underlying  structure    (3)  Chunk        (4)  Connect  seemingly  disparate  objects  or  processes  (iden=fy  structural  similari=es)  3  

Big  Ideas  in  Math  Prac'ce  8  

(1) Look  for  repe==on  in  calcula=ons    

(2) Look  for  general  methods  or  shortcuts    

(3) Maintain  oversights  of  the  process  while  aPending  to  details.    

(4) Evaluate  reasonableness  of  intermediate  results  

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Problems/Tasks  

(1) Bare  Number  Tasks  

(2) Mathema=cal  Inves=ga=on  

(3) Word  Problems  

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Sum  of  Consecu've  Numbers  

 1  +  2  +  3  +  4  +  5  =      2  +  3  +  4  +  5  +  6  =    3  +  4  +  5  +  6  +  7  =    4  +  5  +  6  +  7  +  8  =    Write  at  least  one  observa=on  and  a  ques=on  you  have  as  you  work  on  these  problems.  

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What patterns do you see?

I notice …

I wonder …

     

Share  

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Do  you  have  a  shortcut?  

   

79  +  80  +  81  +  82  +  83    

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Building  on  …  

Would  your  shortcut  work  for  these  problems?    10  +  11  +  12    =  ?    3  +  4  +  5  +  6  +  7  +  8  +  9  =?    26  +  28  +  30  +  32  +  34  =  ?  

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What  about  when  adding  an  even  number  of  numbers?  

20  +  30  +  40  +  50  =  ?    1  +  2  +  3  +  4  +  …  +  98  +  99  +  100  =?  

How  about  1  +  2  +  3  +  4  +  …  +  n,    where  n  is  a  posi=ve  integer?  

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Hop,  Slide,  Traffic  Jam  Start  with  two  counters  on  one  side  and  three  counters  of  a  different  color  on  the  other  side  with  an  empty  space  in  between.            Your  task  is  to  move  these  counters  so  that  they  change  places  with  as  few  moves  as  possible.  You  can  either  SLIDE  a  counter  to  an  adjacent  empty  space  or  HOP  it  over  another  counter  to  get  to  an  empty  space.      

•  Independent  think/work  =me  

•  Small  group  work  

•  Whole  group  share/presenta=on  

•  Debrief  

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What is the minimum number of moves required to switch the blue and red counters? What if I have 3 blue and 4 red counters? What if if I have 10 blue and 11 red counters?

   

Share  

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Did  you  find  yourself  …  

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•  Considering  the  behavior  of  calcula=on  

•  Surfacing  the  underlying  structure  

•  Chunking  

•  Connec=ng  seemingly  disparate  objects  or  processes  (iden=fy  structural  similari=es)  

 

•  Looking  for  repe==on  in  calcula=ons  

•  Looking  for  general  methods  or  shortcuts    

•  Maintaining  oversights  of  the  process  while  aPending  to  details.    

•  Evalua=ng  reasonableness  of  intermediate  results  

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Word  Problems  

 Guess-­‐Check-­‐Generalize  

≠  Guess-­‐and-­‐Check  

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 Guess-­‐Check-­‐Generalize  

=  A  strategy  for  looking  for  and  

expressing  regularity  in  repeated  reasoning  

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Devon  exercised  the  same  amount  of  =me  each  day  for  5  days  last  week.      

•  His  exercise  included  walking  and  swimming  •  Each  day  he  exercised,  he  walked  for  10  minutes  •  He  exercised  for  a  total  of  225  minutes  last  week    

What  is  the  number  of  minutes  he  swam  each  of  the  5  days  last  week?  

Guess-­‐Check-­‐Generalize  

As  a  salesperson,  you  are  paid  $56  per  week  plus  $3  per  sale.    This  week  you  want  your  pay  to  be  at  least  $100.    Write  an  inequality  for  the  number  of  sales  you  need  to  make.  

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Guess-­‐Check-­‐Generalize  

Big  Ideas  of  MP7  &  MP8  

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•  Consider  behavior  of  calcula=on  

•  Surface  the  underlying  structure  

•  Chunk  

•  Connect  seemingly  disparate  objects  or  processes  (iden=fy  structural  similari=es)  

 

•  Look  for  repe==on  in  calcula=ons  

•  Look  for  general  methods  or  shortcuts    

•  Maintain  oversights  of  the  process  while  aPending  to  details.    

•  Evaluate  reasonableness  of  intermediate  results  

Instruc'onal  Strategies  

•  Use  low  threshold  high  ceiling  tasks  

•  Purposeful  grouping  of  tasks  to  highlight  mathema=cal  concepts  and  rela=onships  

•  Allow  =me  for  explora=on  

•  Don’t  rush  to  simplify  

•  Guess-­‐Check-­‐Generalize  

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Disclaimer The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.

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