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Errors with Fractions and Decimals

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					Errors with Fractions
and Decimals
   Draw a representation for each task.
    1) In my sock drawer I have 4 blue socks,
      6 white socks and 2 black socks. What
      fraction represents the number of black
      socks that I have?
    2) I have a piece of string that is 3 feet
      long. I need to cut the piece into three
      equal pieces. How long is each piece?
Fractions

   Part of a group
    – A group of items in which the objects
      have different characteristics
   Part of a whole
    – A whole or “unit” that has broken into
      pieces

    Most errors are misconceptions related to
     part of a whole
      Fractions-
Representational Errors
   Working with shaded parts
    – Gretchen (p. 138)
         What’s the error?
Gretchen

   Interventions- Start with the Whole

4 2          1) Read “2 out of 3 parts”
             2) Draw a rectangle
9 3
                 3) Break it into 3 equal parts

                     4) Shade in two parts

                        5) This is “2 out of 3 parts”
                        which is also called two-thirds
  Gretchen (#2)

     Start with an unshaded rectangle

4   2
  
9   3
Gretchen (#3)

   Still, in traditional
    curricula students are
    given: “Write the
    fraction that shows the
    number of regions that
    are shaded.”

   What options do we
    have??
Fractions- Representational

    Working with shaded parts
     – Carlos (p. 139)
          What’s the error?
Carlos

   Equal area
    – Draw a number of squares on your paper
        Divide each square into fourths in a different
         way
        How many ways can you think of?
       Fractions- Symbolic

   Equivalent Fractions
    – Jill (p. 140)

           4 2         3 1      3 1       4 2
                                        
           9 3         9 3      8 4       8 4

Jill makes every…
                                         Why does
       4 into a ____       8 into a ____ she make
       3 into a ____       9 into a ____ this error?
       Fractions- Symbolic

   Equivalent Fractions
    – Jill (p. 140)

Help by
 1) Using fractional parts of regions
 2) Build an array with fractional parts of a set
 3) Look for a pattern in a list
 4) Make sets of equivalent fractions
      Equivalent Fractions

   Concrete Modification
    – Cover your hexagon with six
                                      4 2 2
      equal pieces
                                        
    – Can you cover only four-        6 2 3
      sixths of your hexagon?
    – How many triangles do you
      have on your hexagon?
    – Can you cover those triangles
      with another shape?
       Equivalent Fractions
   Representational Modification
   Using fractional parts of regions
    – Draw six parts (rectangle model)
    – For 4/6, shade four of the parts
    – “Can we use larger parts to cover exactly what
      we have?”
    – When students use thirds, the two-thirds will
      cover 4/6ths
    – Bring in mathematical rule
                                          4 2 2
    after students have experienced the     
      representation                      6 2 3
Equivalent Fractions
      2/3



2/3

4/6

6/9
Look for a pattern in a list
  10 5
    
  12 6           Pattern– dividing
   8 4            both the numerator
                 and the denominator
  10 5            by two
  6 3
    
  8 4
   4
     
   6
     Make sets of equivalent
     fractions
              *1   *2   *3   *4   *5   *6   *7   *8
Numerator     1    2    3     4    5   6     7    8
Denominator   3    6    9    12   15   18   21   24
Decimals- Symbolic

   Tonya (p. 144)
    – What’s the error?
Tonya

   Multiple representations
    – Hundreds chart
Tonya

   Multiple representations
    – Ordering decimals
          Put three in a row (least-to-greatest, greatest-to-
           least)
    2nd-        0.3560
    Small-      0.3500
    3rd-        0.3562
Tonya

   Multiple representations
    – Number line




       0                       1
Tonya

   Multiple representations
    – Models to decimals with base-10 blocks
Decimals- Concrete

   Use of base-10 blocks
    – Students learn     Common Decimal use
        Cube = 1            Cube = 0.01
        Rod = 10            Rod = 0.1
        Flat = 100          Flat = 1
        Block = 1,000

    – Students constantly mix them up
Decimals- Concrete

   Use of base-10 blocks
    – Place value when the task is given orally
          “Show 2.04”
             – 2 flats, 4 cubes
    – Place value when reading the task
          “2 and 4 hundredths”
    – Modify with…
        Place value chart with words and pictures as
         headings
        Practice using base-10s to show decimals
Operations with
Fractions and Decimals
Adding Fractions

   Covering the hexagon
    – Using your pattern blocks…
          Cover the hexagon in as many possible ways
          Fill out the chart for each solution that you found




             Pieces        Picture      Fraction
Pieces                  Fraction
6 triangles             6 * (1/6)
2 trapezoids            2 * (1/2)
3 rhombi                3 * (1/3)
1 trap, 3 tri           (1/2) + 3* (1/6)
2 rhombi, 2 tri         2*(1/3) + 2*(1/6)
1 trap, 1 tri, 1 rhom   (1/6)+(1/2)+(1/3)
Pieces          Fraction
4 tri, 1 rhom   4*(1/6) + 1/3
     Adding Fractions

        Clock Model
             1 1
              
             4 6
If students learn equivalent
fractions, then this method
is effective.
Adding Fractions- Errors
– Robbie (p. 151)
   Verticalorientation
   Use unit regions
     – Fraction strips
     – Fraction tiles
   Counting   pieces
     – Only if the denominator is the same
   Estimate   answers
Adding Fractions- Symbolic
   Modifications for fractions with unlike
    denominators (5th grade)
    – Estimate answers before adding
    – Emphasize adding requires having like denominators
    – Use a number line to focus on equivalent fractions
    Subtracting Fractions

       Johnny says that…   So, he claims
                            that…
1 1 1
   and (3)(4)  12
3 4 12                      1 1   1
                              
1 1 1                       a b a *b
     and ( 4)(5)  20
4 5 20
1 1 1                       Will his method
     and (5)(6)  30      always work?
5 6 30                      Why or why not?
               Adding Fractions-
                   Symbolic
   Errors adding                  Incorrect answers
    fractions                            4
                               1                    What
    – Steve runs 2 ½
                             2         4 5          did
      miles in the morning               4
      and 2 ¾ miles at
                               2                    these
                                       5    1       three
      night. How many          3         1
                             2         4    4     students
      miles did he run
                               4                     do?
      total?                             4    2
    – What is the correct
                                       4 4
                              ?          6    3
      answer?
Subtracting Fractions
   1    Draw two rectangles
 1
   4    Shade 1 ¼ of them lightly
    3
       Cross out ¾ of a rectangle
    4
        What’s remaining?
Adding Decimals- Concrete
   Use of base-10 blocks
    – At the track and field meet, triple jumpers’ scores are
      the sum of their two jumps. Monique jumped 3.4 and
      2.7 yards. What was her total score?
    – Students use base-10 blocks and record the following
      answers:
                              Use base-10 blocks. What
        6.1
                              were there errors?
        5.11

        51.1

        16.1

        16
Adding Decimals- Symbolic
   On paper
    – Harold (p. 167)
        What’s the error?
Adding Decimals- Symbolic
   On paper
    – Harold (p. 158)- Modifications
        Use base-10 blocks or c-rods
         – Exchange
         – “use as few blocks as possible!!”
       Use   a number line
Adding Decimals- Symbolic
   On paper
    – Harold (p. 167)- Modifications
        Use vertically lined paper (similar to
         division)
        Use meter sticks
Adding Decimals

   Math Applet
   National Library of Virtual
    Manipulatives

				
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