Day  2  Mental  Images  and  Interpretations  of  Rational  Numbers  

I  can  solve  problems  applying  the  equal  sharing  (quotient)  interpretation  of  rational  numbers.  

Jared  Problem  31  students  are  going  on  field  trips  to  four  different  locations.  20  sandwiches  were  distributed  among  the  four  groups  as  described  below.  Each  group  shared  the  subs  equally  among  the  students  in  that  group.  Rank  the  groups  on  the  amount  of  sub  sandwich  each  student  would  get.  Draw  pictures  to  justify  your  thinking.      Minnesota Science Museum 12  students  9  subs  

Walker Art Gallery 8  students  6  subs  

Bell Museum 3  students  2  subs  

Frick Museum 8  students  3  subs  

   

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  2  

Why  are  we  here?              

 

Video  Snapshot    (Clip  342)  1  min  26  seconds  

 Jace  Grade  4  

Task  #1  Jake’s  Party:  5  kids  share  a  chocolate  bar.  Jace’s  Party:  8  kids  share  a  chocolate  bar.  Who  is  going  to  get  more  chocolate  bar,  you  or  Jake?  Explain  why.  

Task  #2  

Circle  the  larger  number:    !!      !

!  

Explain  your  answer.    

Anticipate  How  will  Jace  answer  each  task?    

Monitor  Response  What  does  Jace  know  about  equal  sharing?          What  does  Jace  know  about  fraction  symbols?          What  do  your  kids  think  about  fraction  symbols?              

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  3  

34  

           

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  4  

I.   Interpretations  of  a  Rational  Number  

I  can  identify  five  interpretations  of  a  rational  number.  

I  can  identify  the  interpretation  that  is  being  used  in  my  classroom.      A. Measure (length, area, volume) B. Quotient (fair sharing)

C. Ratio (part to part, part to whole) D. Operator (transforming)

E. Part-Whole Relationship

   

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  5  

II.   Naming  Fractions  with  Fraction  Circles,  Words,  Symbols,  Paper  Strips,  and  Number  Lines.  

I  can  name  fractions  using  fraction  circles  and  words.    

   

All  student  pages  can  be  found  at  http://www.cehd.umn.edu/ci/rationalnumberproject/rnp1-­‐09.html  

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  6  

Big  Ideas:  

 

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  7  

 

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  8  

Big  Ideas:  

     

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  9  

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  10  

I  can  name  fractions  using  paper  strips  and  symbols.      

symbol written name fraction circle paper strip

       

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  11  

I  can  make  connections  among  different  representations  of  fractions.  

Lesh  Model              

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  12  

I  can  name  fractions  on  a  number  line.    3.1.3.1    Read  and  write  fractions  with  words  and  symbols.  Recognize  that  fractions  can  be  used  to  represent  parts  of  a  whole,  parts  of  a  set,  points  on  a  number  line,  or  distances  on  a  number  line.      4.1.2.2    Locate  fractions  on  a  number  line.  Use  models  to  order  and  compare  whole  numbers  and  fractions,  including  mixed  numbers  and  improper  fractions.    5.1.2.3    Order  fractions  and  decimals,  including  mixed  numbers  and  improper  fractions,  and  locate  on  a  number  line.        A.   Norman  runs  along  Portland  Avenue.  He  stops  after  running  2 !

!  miles  from  home.  

Locate  where  he  is  on  the  number  line  below.          

           B.   Amy  needs  a  piece  of  wood  that  is  !

!  of  an  inch  thick.  Show  the  thickness  of  the  piece  of  

wood  on  the  number  line  below.          

           C.   Desmond  runs  around  the  half-­‐mile  track  at  the  fitness  center  five  times.  He  says  he  

ran  !!  of  a  mile.  Show  the  distance  he  ran  on  the  number  line.  

       

       

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  13  

Difficulties  with  the  Number  Line    Student  A  –  Show  where  three-­‐fourths  is  on  the  number  line.    

     Student  B  –  Show  where  three-­‐fourths  is  on  the  number  line.  

       Student  C  –  Write  the  names  for  the  tick  marks  on  the  number  line.        

     Student  D  –  Find  three-­‐fourths  on  this  number  line.    

   

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  14  

III.   Reconstructing  the  Unit  

I  can  identify  and/or  reconstruct  the  unit  in  various  situations.    A.   Label  !

!  of  the  rectangular  strip  below.  

       B.   The  rectangular  strip  below  is  !

!  of  a  rectangle.  Draw  the  original  rectangle.  

       C.   The  rectangle  below  is  !

!  of  some  rectangle.  Draw  the  original  rectangle.  

       D.   How  long  is  strip  B  compared  to  the  length  of  strip  A.    

   

A

B

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  15  

E.   Label  !!  on  the  number  line  below.  

 

     F.   Identify  where  the  following  fractions  are  on  the  number  line:    

1,    1 !!,    !

!  

   

     G.   Paul  walks  20  miles  from  his  school  and  is  !

!  the  way  to  Joe’s  house.  How  far  is  Joe’s  

house  from  school?  Joe’s  house  and  the  school  are  on  the  same  street.    

     

0

23

0

Paul

20 miles

School

0

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  16  

IV.   Ordering  Numbers  using  Manipulatives  

     

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  17  

   

   

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  18  

   

 

Video  Snapshot    (Clip  302)  4  min  31  seconds  

 Ally  Grade  5  

Circle  the  bigger  number  in  each  pair.    

!! !

! !

! !

!

1 !! !

!" !

!

!! !

!

   

Anticipate  How  will  Ally  circle  each  pair?    

Monitor  Response  Which  pairs  did  Ally  get  correct?        What  benchmark(s)  does  Ally  seem  to  rely  on?        What  does  Ally  know  about  fraction  symbols?        What  does  a  “bigger  number”  mean  for  Ally?              

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  19  

 Video  Snapshot    (Clip  338)  1  min  22  seconds  

Felisha  Grade  2  

Circle  the  larger  or  put  an  equal  sign  if  they  are  the  same.  

!! !

!

!! 1

!! !

!

 

Anticipate  How  will  Felisha  circle  each  pair?    

Monitor  Response  What  strategy  does  Felisha  use  to  determine  the  larger  fraction?    What  does  Felisha  think  of  when  she  sees  numbers  like  !

!?  

     

 Video  Snapshot    (Clip  343)  0  min  51  seconds  

Jacky  Grade  5  

Circle  the  larger  or  put  an  equal  sign  if  they  are  the  same.  

!! 1

!! 1

 

Anticipate  How  will  Jacky  circle  each  pair?    

Monitor  Response  What  does  Jacky  think  of  when  she  sees  numbers  like  !

!?  

 What  does  Jacky  think  of  when  she  sees  improper  fractions  like  !

!?  

         

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  20  

V.   Informal  Ordering  Strategies  

I  can  order  rational  numbers  using  manipulatives  and  mental  images.    Determine  the  larger  number  by  picturing  them  in  your  mind.  Circle  the  larger  number  in  each  pair.    

I. II. !!"

!!"

!! !

!

III. IV. !! !

!

!!"

!!"

           

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  21  

VI.   Formal  Ordering  Strategies    A. Common Denominator B. Cross Multiplying

C. Percents D. Difference Between Numerator and Denominator

           

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  22  

VII.   Deepen  your  Mental  Images    A.   What  happens  to  the  value  of  a  fraction  as  the  numerator  increases?            B.   What  happens  to  the  value  of  a  fraction  when  the  denominator  increases?            C.   What  happens  to  the  value  of  a  fraction  as  the  numerator  decreases?            D.   What  happens  to  the  value  of  a  fraction  when  the  denominator  decreases?            E.   What  happens  to  the  value  of  a  fraction  if  both  the  numerator  and  the  denominator  

increase?      

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  23  

VIII.   Features  of  Each  Model    

A.   Fraction  Circle  

B.   Paper  Strip  

C.   Number  Line  

D.   Quotient   Three people share two cookies evenly. How much of a cookie will each person receive?

           

3223

10

Day  2  –  Mental  Images  and  Interpretations  of  Rational  Numbers   page  24  

Interpretations  of  Rational  Numbers    interpretation problem description

A.    Measure  Manny runs five-fourths of a mile before stopping. Did he run more or less than one mile?

Rational number is a quantity compared with a whole. Involves partitioning into equal parts. A number line representation captures this interpretation.

B.    Quotient  Quentin has two cookies. He wants to share them equally among three people. How much of a cookie does each person get?

Rational number as division. Involves partitioning into equal parts.

C.    Ratio  There are three boys for every two girls in Raina’s class. How many boys are in the class if there are 12 girls?

Rational number as a multiplicative comparison of two quantities. Comparisons can be part to part or part to whole.

D.    Operator  Olivia makes $24 an hour. How many dollars will she earn if she works two-thirds of an hour?

Rational number transforms another amount (numerical or geometric) by magnifying, shrinking, enlarging, reducing, expanding, or contracting it.

E.    Part-­‐whole  A pizza is cut into 3 equal sized pieces. Wael eats two of the pieces. How much of a whole pizza did Wael eat?

Rational number is a part of a whole. Partition continuous or discrete parts into equal-sized parts.