Student  Probe     Is  

Reasonable  Answers—Estimating  Fractions         At  a  Glance  

closer  to  0,   ,  or  1?  

  Answer:  1    

Lesson  Description   In  this  lesson  students  will  use  a  number  line  to   determine  the  placement  of  benchmark  fractions    and  then  use  these  benchmark  fractions  to   determine  reasonableness  of  answers  for  addition  and   subtraction  of  fractions.    

Rationale    

Just  as  using  estimation  to  check  reasonableness  of   answers  when  computing  with  whole  numbers  helps   students  become  more  mathematically  fluent,  thinking   about  the  reasonableness  of  answers  when  computing   with  fractions  increases  students’  conceptual   understanding  of  the  nature  of  fractions  and  their   operations.    Students  should  know  “about”  how  big  a   particular  fraction  is  and  be  able  to  easily  tell  which  of   two  fractions  is  larger,  but  that  is  only  the  beginning.     Students  need  to  intuitively  know,  for  example,  that   when  adding  

 a  reasonable  answer  is  closer  to  1  

than   .        

Preparation     Prepare  blank  number  lines  for  each  student.    Prepare  a   number  line  display  for  teacher  use  with  a  collection  of   10  fractions  written  on  the  display.     Example:  

   

 

What:  Using  a  number  line,  students  place   and  estimate  fractions  and  estimate   reasonableness  of  answers.   Common  Core  State  Standard:  CC.5.NF.2   Use  equivalent  fractions  as  a  strategy  to  add   and  subtract  fractions.  Solve  word  problems   involving  addition  and  subtraction  of   fractions  referring  to  the  same  whole,   including  cases  of  unlike  denominators,  e.g.,   by  using  visual  fraction  models  or  equations   to  represent  the  problem.  Use  benchmark   fractions  and  number  sense  of  fractions  to   estimate  mentally  and  assess  the   reasonableness  of  answers.  For  example,   recognize  an  incorrect  result  2/5  +  1/2  =  3/7   by  observing  that  3/7  <  1/2.   Matched  Arkansas  Standard:  AR.6.NO.3.4   (NO.3.6.4)  Estimation:  Estimate  reasonable   solutions  to  problem  situations  involving   fractions  and  decimals   Mathematical  Practices:     Use  appropriate  tools  strategically.   Attend  to  precision.   Construct  viable  arguments  and  critique  the   reasoning  of  others.   Make  sense  of  problems  and  persevere  in   solving  them.   Who:  Students  who  cannot  mentally  assess   reasonableness  of  answers  when  working   with  fractions.   Grade  Level:  5   Prerequisite  Vocabulary:  number  lines,   benchmark  fractions   Prerequisite  Skills:  locating  points  on   number  lines,  meaning  of  fractions,  finding   equivalent  fractions     Delivery  Format:  Individual  or  pairs   Lesson  Length:  15-­‐30  minutes   Materials,  Resources,  Technology:  Number   lines   Student  Worksheets:  None  

Lesson     The  teacher  says  or  does…    

1. We  need  to  mark  our   number  line.    On  the  left   end  we  need  to  place  a  zero   and  on  the  right  end  a  one.       (Model  for  students.)   2. Where  should  we  place     on  our  number  line?   3. Now  we  are  going  to  sort   the  fractions  I  have  written   on  the  board  into  three   groups:     a.  those  closer  to  0  

Expect  students  to  say  or   do…    

If  students  do  not,  then  the   teacher  says  or  does…      

In  the  middle.  

Model  for  students.    

 

 

b.  those  closer  to       c.  those  closer  to  1.     (Allow  time  for  students  to   place  each  of  the  10   fractions  into  one  of  the   three  groups.    Do  not  be   concerned  at  this  point   about  the  correctness  of   their  placements.)   4. Where  should    be  placed?   Closer  to   ,  because   Is  it  closer  to  0,   ,  or  1?   How  do  you  know?   5. Let’s  place  1/3  on  the   number  line  closer  to   .  

and  

.      

Write  1/3  on  their  number   line  close  to   .  

 

Guide  students  through   diagrams  or  pictures  to  see   this.  

Model  for  students.  

The  teacher  says  or  does…    

Expect  students  to  say  or   do…  

6. Where  should  we  place   ,  

Mixed  responses:  0  and   .  

closer  to  0,   or  1?  

   

How  do  you  know?  

and  

is  half  way  

between  0  and   .    Some  

students  may  say,  “

of  

7. Locate  

is    

Locate  

line  halfway  between  zero  

place.  

represents  zero  cents,     represents  50  cents,  and  1   represents  1  dollar,  where  is   ?”   Students  should  see  that    is   one  quarter  and  should  be   placed  at  the  midpoint  of  0  

 

 on  your  number  

If  students  do  not,  then  the   teacher  says  or  does…     Draw  this  or  make  the   connection  to  money.       Say,  “How  many  quarters  are   in  one  dollar?    If  zero  

and   .    in  the  correct  

Model.  

and   .     8. Where  should  we  place     on  the  number  line,  closer  

Halfway  between   and  

If  we  think  of  money,    is  3  

one.  

quarters  out  of  one  dollar  so  

to  0,    or  1?  

 would  be  the  midpoint  of    and  1.  

9. Locate  

 on  your  number  

line  halfway  between     and  1.   10. Continue  guiding  students   through  placement  of  the   remaining  fractions.  

Locate  

 correctly  on  the  

Model.      

number  line.  

 

 

The  teacher  says  or  does…    

11.  If  we  add  

,  do  we  

expect  an  answer  closer  to  

Expect  students  to  say  or   do…   1,  because  

 and  

that  is  almost  1.  

If  students  do  not,  then  the   teacher  says  or  does…     Model  the  addition  on  the   number  line  and  say,  “We  are   starting  with   ,  so  we  expect  

0,   or  1?  

our  answer  to  be  larger  than  

How  do  you  know?  

12.  If  we  subtract  

,  since  we  are  adding  more  

,  what  

is  our  answer  closer  to?  

0,  because   that  is  really  small.  

 and  

to  it.”   Model  the  subtraction  on  the   number  line  and  say,  ”Since   we’re  starting  at    and  taking   away,  we  expect  our  answer   to  be  less  than   .”  

13. Continue  guiding  students     through  similar  problems  as   needed.  

 

 

Teacher  Notes     1. Students  will  benefit  from  a  strong  number  line  model  for  fractions,  similar  to  the  number   line  model  used  for  whole  numbers  and  integers.     2. Once  students  are  comfortable  with  benchmark  fractions,  this  lesson  can  be  extended  to   other  fractions,  including  those  larger  than  one.         3. Once  this  activity  has  been  taught,  or  students  are  comfortable  with  benchmark  fractions,   sums  and  differences  of  fractions  can  be  used  as  an  oral  warm-­‐up  activity  or  transition  or   end-­‐of-­‐class  time  filler.    

Variations    

Use  linear  models  such  as  Cuisinaire  Rods  and  Fraction  Strips  in  addition  to  number  lines.        

Formative  Assessment         Is  

 closer  to  0,   ,  or  1?  

  Answer:  1  

References     Institute  of  Educational  Sciences  (IES).  (2010).  Developing  effective  fractions  instruction  for   Kindergarten  through  8th  grade.  Retrieved  September  2010,  from   http://ies.ed.gov/ncee/wwc/pdf/practice  gguides/fractions_pg_093010.pdf   Marjorie  Montague,  Ph.D.  (2004,  12  7).  Math  Problem  Solving  for  Middle  School  Students  With   Disabilities.  Retrieved  4  25,  2011,  from  The  Iris  Center:   http://iris.peabody.vanderbilt.edu/resource_infoBrief/k8accesscenter_org_training_resources_ documents_Math_Problem_Solving_pdf.html   Russell  Gersten,  P.  (n.d.).  RTI  and  Mathematics  IES  Practice  Guide  -­‐  Response  to  Intervention  in   Mathematics.  Retrieved  2  25,  2011,  from  rti4sucess:   http://www.rti4success.org/images/stories/webinar/rti_and_mathematics_webinar_presentati on.pdf Van  de  Walle,  J.  A.,  &  Lovin,  L.  H.  (2006).  Teaching  Student-­‐Centered  Mathematics  Grades  5-­‐8   Volume  3.  Boston,  MA:  Pearson  Education,  Inc.