# 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
– Gretchen (p. 138)
   What’s the error?
Gretchen

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

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

4   2

9   3
Gretchen (#3)

curricula students are
given: “Write the
fraction that shows the
number of regions that

   What options do we
have??
Fractions- Representational

– 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
– How many triangles do you
– 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
   “2 and 4 hundredths”
– Modify with…
 Place value chart with words and pictures as
 Practice using base-10s to show decimals
Operations with
Fractions and Decimals

   Covering the hexagon
   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

   Clock Model
1 1

4 6
If students learn equivalent
fractions, then this method
is effective.
– Robbie (p. 151)
 Verticalorientation
 Use unit regions
– Fraction strips
– Fraction tiles
 Counting   pieces
– Only if the denominator is the same
   Modifications for fractions with unlike
– 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?
Symbolic
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
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?
   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
Use base-10 blocks. What
 6.1
were there errors?
 5.11

 51.1

 16.1

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