Workshop  2:    Solving  Equa5ons  

 NCTM  Interac5ve  Ins5tute,  2015  

Name  Title/Posi5on  Affilia5on  

Email  Address  

Warm  Up  

List  these  expressions  from  least  to  greatest:  2n          2n  +  1          2(n  +  1)    2n  –  1          2(n  –  1)    

Reflec5on  

What  would  students  need  to  understand  in  order  to  solve  the  warm  up?  

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Common  Core  Standards  

This  session  will  address  the  following:    

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7.EE.1   Apply  properGes  of  operaGons  as  strategies  to  add,  subtract,  factor,  and  expand  linear  expressions  with  raGonal  coefficients.  

7.EE.4   Use  variables  to  represent  quanGGes  in  a  real-­‐world  or  mathemaGcal  problem,  and  construct  simple  equaGons  and  inequaliGes  by  reasoning  about  the  quanGGes.  

Solving  Equa5ons  

Think  about  the  instrucGonal  sequence  you  use  in  teaching  how  to  solve  an  equaGon.      What  do  students  do  in  the  first  lessons?  

     What  are  criGcal  benchmarks  or  ideas  that    students  progress  through  in  the    instrucGonal  sequence?  

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Algebra  Magic  •  Think  of  a  number.  •  MulGply  the  number  by  3.  •  Add  8  more  than  the  original  number.  

•  Divide  by  4.  •  Subtract  the  original  number.  

Compare  your  answer  to  others  at  your  table.  Why  did  this  happen?  Find  2  different  ways  to  explain  it.  6  

Algebra  Magic  

What  could  be  done  to  the  steps  in  order  to  get  the  number  you  started  with?    

•  Think  of  a  number.  •  MulGply  the  number  by  3.  •  Add  8  more  than  the  original  number.  •  Divide  by  4.  •  Subtract  the  original  number.  

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Wri5ng  Expressions  •  Enter  the  first  three  digits  of  your  phone  number.  •  MulGply  by  80.  •  Add  1.  •  MulGply  by  250.  •  Add  the  last  four  digits  of  your  phone  number.  •  Repeat  the  above  step.  •  Subtract  250.  •  Divide  by  2.  

Describe  the  number  you  have.  How  did  the  problem  work?  

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Algebra  Magic  

Which  of  the  following  steps  can  you  reverse  without  changing  the  result?  Why?  

1)  Think  of  a  number.  2)  Subtract  7.  3)  Add  3  more  than  the  original  number.  4)  Add  4.  5)  MulGply  by  3.  6)  Divide  by  6.  9  

Algebra  Magic  

The  following  trick  is  missing  the  last  step.  •  Think  of  a  number.  •  Take  its  opposite.  •  MulGply  by  2.  •  Subtract  2.  •  Divide  by  2.  •  ??????????  

Decide  what  the  last  step  should  be  for  the  given  condiGon  so  final  result  is:  a)  One  more  than  

original  number.  b)  Opposite  of  original  

number.  c)  Always  0.  d)  Always  -­‐1.  

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Matching  Expressions,  Words,  Tables,  &  Areas  

Work  collaboraGvely  with  your  tablemates.  •  Match  cards  to  make  a  set  with  an  expression,  words,  table,  and  area  card.    

•  If  there  is  not  a  complete  set,  make  a  card  for  the  missing  type(s)  with  one  of  the  blank  cards.  

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Matching  Expressions,  Words,  Tables,  &  Areas  Large  group  discussion:  

•  Which,  if  any,  of  the  groups  of  expressions  are  equivalent  to  each  other?    How  do  you  know?  

•  What  will  students  learn  as  a  result  of  this  acGvity?  

•  What  challenges  might  student  encounter  with  this  acGvity?  

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Expressions  to  Equa5ons  

8  +  4  =                  +  7    

What  responses  do  students  give  for  box?    

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A  major  misunderstanding  

•  Many  students  do  not  understand  the  equals  sign.    

•  They  believe  it  signifies  that  the  answer  comes  next.    

2x  –  8  =  4x  +  6  

Equal  Sign–Two  Levels  of  Understanding  

Opera5onal:  Students  see  the  equal  sign  as  signaling  something  they  must  “do”  with  the  numbers  such  as  “give  me  the  answer.”      

Rela5onal:  Students  see  the  equal  sign  as  indicaGng  two  quanGGes  are  equivalent,  they  represent  the  same  amount.  More  advanced  relaGonal  thinking  will  lead  to  students  generalizing  rather  than  actually  compuGng  the  individual  amounts.  They  see  the  equal  sign  as  relaGng  to  “greater  than,”  “less  than,”  and  “not  equal  to.”  

Knuth, E. et. al (2008). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School, 13, 514–519.

Why  is  understanding  the  equal  sign  important?  

Knuth, E. et. al (2008). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School, 13, 514–519.

Transi5oning  to  Rela5onal  Thinking  

True  or  False:  471  –  382  =  474  –  385  674  –  389  =  664  –  379  583  –  529  =  83  –  29  37  x  54  =  38  x  53  5  x  84  =  10  x  42  64  ÷  14  =  32  ÷  28  42  ÷  16  =  84  ÷  32  

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•  No calculators – No computations •  Use relational thinking to justify answer.

Transi5oning  to  Rela5onal  Thinking  

What  is  the  value  of  variable?  73  +  56  =  71  +  d  67  –  49  =  c  –  46    234  +  578  =  234  +  576  +  d  94  +  87  –  38  =  94  +  85  –  39  +  f  92  –  57  =  94  –  56  +  g  68  +  58  =  57  +  69  –  b    56  –  23  =  59  –  25  –  s    

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•  No calculators – No computations •  Use relational thinking to justify answer.

Solving  Equa5ons  

An  equaGon  states  that  two  expressions  are  equivalent  for  certain  values  of  a  variable.    

EquaGons  become  useful  in  invesGgaGng  relaGonships  between  two  expressions.    

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Solving  Equa5ons  

•  Many  curriculum  materials  begin  with  equaGons  like  this:  

 14  –  w  =  9  

   

Foegen, A. & Dougherty, B. J. (2013). Algebra screening and progress monitoring study.

Solving  Equa5ons  

 14  –  w  =  9  

 48%  of  students  (1615)  got  it  correct.  

(2nd  grade  CCSSM  standard)  

Foegen, A. & Dougherty, B. J. (2013). Algebra screening and progress monitoring study.

Solving  Equa5ons  

Rather  than  starGng  with  ‘easy’  equaGons  and  applying  algebraic  manipulaGons,  let’s  consider  a  developmental  approach.    

5  +  x  =  12  5  –  5  +  x  =  12  –  5  

x  =  7  

Solving  Equa5ons  

 5  +  x  =  12  

 What  number  added  to  5  equals  12?  What  basic  fact  do  you  know  that  could  tell  you  the  missing  addend?  

Solving  Equa5ons  

When  you  see  an  equaGon  like  this,  what  are  3  other  related  equaGons  you  could  write?    

5  +  x  =  12        

Solving  Equa5ons  

When  you  see  an  equaGon  like  this,  what  are  3  other  related  equaGons  you  could  write?    

5  +  x  =  12        

5 + x = 12 x + 5 = 12 12 – 5 = x 12 – x = 5

Solving  Equa5on  

Diagrams  with  manipulaGves  are  another  way  that  can  support  students’  understanding  of  solving  equaGons.  

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Solving  Equa5ons  

Work  with  a  partner  at  your  table  to  complete  the  lab.    Be  prepared  to  share  your  ideas.    

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Solving  Equa5ons  

3x  +  2  =  4x  –  3        

Solving  Equa5ons  

Graph                                    3x  +  2  =  4x  –  3    Use  your  graphing  calculator  to  graph  the  two  expressions.    How  would  you  idenGfy  the  soluGon?        

Solving  Equa5ons  

Graphing                                    3x  +  2  =  4x  –  3        

Solving  Equa5ons  

1.  Logical  reasoning/inspecGon  2.  Fact  families/inverse  operaGons  3.  Physical  materials/diagrams  4.  Tables  5.  Graphing  

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Solving  Equa5ons  

How  would  you  solve  3x  +  2  =  4x  –  3  using  algebraic  steps?        

Solving  Equa5ons  

 3x  +  2  =  4x  –  3  3x  +  2  +  3  =  4x  –  3  +  3  

3x  +  5  =  4x    3x  –  3x  +  5  =  4x  –  3x  

5  =  x    

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Solving  Equa5ons  

3x  +  2  =  4x  –  3  3x  +  5  =  4x          A3  

5  =  x          S3x        

Solving  Equa5ons  

A:    Add  S:    Subtract  M:    MulGply  D:    Divide  CLT:  Combine  Like  Terms  DPMA:  DistribuGve  Property  of  MulGplicaGon  over  AddiGon  

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Why  is  it  important  to  understand  solving  equa5ons  

Dan  challenged  Amy  to  write  an  equaGon  that  has  a  soluGon  of  3.  Which  equaGon  could  Amy  have  wriken?  

 a.  4  –  x  =  10  –  3x    b.  3  +  x  =  –(x  +  3)    c.  –2x  =  6      d.  x  +  2  =  3  

Sample  of  student  work  

Dan  challenged  Amy  to  write  an  equaGon  that  has  a  soluGon  of  3.  Which  equaGon  could  Amy  have  wriken?  

 a.  4  –  x  =  10  –  3x    b.  3  +  x  =  –(x  +  3)    c.  –2x  =  6      d.  x  +  2  =  3  

Reflec5on  

•  What  new  idea(s)  do  you  want  to  implement  into  your  classroom?  

•  What  challenges  did  you  encounter  during  this  session?    

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Reflec5on  

(Principles to Actions: Ensuring Mathematical Success for All [NCTM 2014], p. 47)

Reflec5on  

(Principles to Actions: Ensuring Mathematical Success for All [NCTM 2014], p. 48)

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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