Fractional thinking concept map
*
these categories are not separate, they can overlap
significantly.
Introduction
Underlying the development of fractional thinking
is a number system that is different from the
numbers that students have already had experience with.
Fractions have different rules for naming,
quantifying, ordering, adding, subtracting, multiplying,
dividing, etc. Students will need to develop an
understanding of these
rules and be able to apply them when working with
fractions. Using a variety of visual and numerical
representations for fractions can support students to build
up experiences with the different areas of fractions (fractional
constructs).
In the development of fraction knowledge there
are important understandings and strategies that
children develop as they learn about fractions.
For instance fractions can be
about a relationship
between a part and a whole (part-whole relationship);
a result of a
division (quotient);
compared and ordered
(equivalence); and
like a function
(operation) performed on a quantity.
These are some of the different constructs that
fractions can take. The
order of the mind map clockwise from the intorduction is a suggested order
for instruction: Partitioning and divided quantities;
part whole fractions; Equivalence (comparing fractions);
Fractions and number lines; Fractions as operators; and Adding and subtracting
fractions. The subsequent concepts tend to
rely upon students having some understanding of the
previous concepts.
The following pages introduce some of these
important ideas, and provide links to related assessment
resources. The
resources are designed to provide diagnostic and
formative information about teaching and learning, to
guide the development of the next learning steps toward
developing a fuller understanding about the different
constructs of fractions.
Partitioning and Divided quantities
What is partitioning?
Partitioning involves dividing an object or set
of objects into parts. It can involve the
breaking up of quantities into equal or
non-equal parts. Partitioning into
equal-sized (non-overlapping) parts is an
important concept that lies at the heart of
understanding fractions, percentages and
decimals. The equal sharing of quantities
(called partitive division) can also
involve partitioning. Knowing that shapes, sets
and quantities can be partitioned into
equal-sized parts, and understanding the
importance of equal-sized partitions is
fundamental to recognising the part-whole
relationship between the numerator and
denominator in fractions.
When partitioning, it is important to use a
variety of representations of sets, shapes and
quantities to ensure that students are thinking
and responding to different issues, not simply
memorising images or procedures to solve
problems. This helps students develop a more
robust understanding of what a partition is.
Partitioning
sets of objects (discrete)
The methods students use to partition a set of
objects are as important as whether they can
actually partition. Some strategies are based
on counting and matching. Other strategies may
indicate multiplicative understanding, such as: arranging objects into an array; finding the
number of items per group by partitioning the
set into a known number of equal-sized groups (partitive
division); or finding the number of equal-sized
groups given the number of items per group (quotitive
division).
Using a counting strategy (partitive division)
Students who line the counters up and count out
the counters into groups then count the number
in each group are using a counting
strategy. For example, if students knew they were sharing into
three groups.
Example:
Share 12 counters into 3 equally-sized
groups.
How many in each group?
This indicates that they understand the concept
of equal-size and equal sharing with countable
items (see equal sharing resources NM1216, NM1219, NM1229, NM1239).
Grouping strategy (quotitive division)
Students may line up the counters and indicate
groups by circling pictures or grouping the
counters (this lends itself more to quotitive
division problems). for example, if students knew they
were sharing into groups of three.
Example:
How many groups can you share 12 counters
amongst, so that there is 3 in each
group?.
Arrays
Students may arrange counters or draw pictures
to illustrate an array. This may help
students visualise the groups that make up a
set. e.g., "If there are 12 counters and 3
equal-sized groups, what do we know about
breaking 12 into 3 parts?"
Encouraging students to think about how to set
out the counters to make breaking them into
groups easier may promote the use of
multiplicative strategies. Start with small
numbers divided into small groups, e.g., 6
counters into 2 groups, and then ask them to
explain how they worked it out.
Recognising division in partitioning
If student are encouraged to describe what they
have done with the numbers of the objects and
groups in the set they could start to recognise
the connection between partition and division.
For example ask how they could write what is
happening as a number sentence.
Students that recognise the multiplicative
relationship between the total and the number of
groups, and are using a reverse multiplication
(e.g., 6 × 3 = 18) or division (18 ÷ 6 =
3) strategy could try solving similar problems
with larger numbers without using counters. It
is also important that students have experiences
partitioning shapes.
Partitioning regions/shapes (continuous)
Students who have had limited experiences with
fractions and partitioning may rely on the
methods of cutting up they are familiar with.
If they have only ever divided up circles
(pizza) they may think this is the only way to
divide shapes up. Using different shapes (such
as squares or rectangles) which can be cut in
many different ways or square pizzas can be used
to challenge the students' concept of circular
representations of fractions. Hexagons, for
example, can be easier to partition accurately
into 3 or 6 equally-sized parts
than circles. The number of partitions can
relate to the shape to be partitioned. Some
shapes are more difficult to partition into
different numbers, e.g., a square into 3 parts,
a circle into 5 parts, etc.
It is important that students build up many
experiences of partitioning shapes starting
with:
| 1. |
Halving of basic shapes, then halving, multiple times to derive
other parts;
e.g., |
| 2. |
Partitioning a variety of shapes: squares,
rectangles, circles, hexagons;
e.g., |
| 3. |
Partitioning shapes into a different number of pieces,
e.g., 3, 5, 6, 7, 9, etc. |
By partitioning shapes into an odd number of
parts, and using a range of shapes, students
develop a better understanding of
partitioning. This variety ensures that they
are not only remembering partitions with certain
shapes, but that they are developing a strategy
to represent and understand fractions as a part
of a whole.
Partitioning two shapes (or more)
Two shapes can be partitioned
in a variety of ways. Asking students to
identify what each part is should promote rich
discussion. Students can be further probed by
comparing the two questions "What fraction of both squares would each person get?" and
then "How much of a (single) square would
each person get?"
For example:
Show how to share these two cakes amongst 3 people. |
 |
Students may partition them in the following
ways.
Student 1
Student 2
Here Student 1 has used a bigger initial chunk
than Student 2. Student 1 has realised that each
person will get at least half of a cake and then
examined what is left over. Research shows that
students with more sophisticated understandings
of fractions are more likely to use the biggest
chunks possible when sharing.
Asking students to name the parts they have
divided the shapes into:
Student 1: 1/2, 1/3 of a half (1/6);
Student 2: 1/3.
Asking how much of a shape each person gets:
Student 1: 1/2 and a 1/3 of 1/2 (1/2 +1/6);
Student 2: 1/3 and 1/3 (2 lots of 1/3 = 2/3).
Then, ask how much of all the cake does each
person get?
Student 1: 1/4 + 1/12;
Student 2: 2/6 = 1/3.
NOTE: Student 1 needs understanding of how to
add fractions and fractions as operators.
Students can discuss how the fractions are
different, and why they may be different. By
asking "what is the part?" and "what is the
whole? (shape, set or quantity)" students can begin to explore the idea
of relative thinking.
Students should recognise
that when the whole (also called the referent
whole) is two squares the share is
different than when it is one square.
For
example, each person will get 1/3 of the two
squares, or 2/3 of a square).
Students' may also
notice that the fraction is twice as big when
the referent whole is half as much.
Key ideas about partitioning and divided
quantities
| • |
| Partitioning can involve dividing a number of objects
into equal-sized groups
(a discrete context); or |
 |
| partitioning an object or shape into same-sized
pieces
(a continuous context). |
 |
|
| • |
Children with less developed partitioning skills will
often not pay attention to the equal-size of their
parts. |
| • |
Students' experience with partitioning usually begins
with partitioning sets of objects. Generally students
find this easier than partitioning shapes or
quantities. This is likely to be related to their
experience with countable objects, where they can break
sets into groups using a matching or counting strategy. |
| • |
Partitioning usually begins with a repeated halving
strategy that enables students to create fractions such
as quarters or eighths. The ability to create fractions
that involve an odd number of parts, for instance thirds
or fifths, develops later and requires practice. |
| • |
Different shapes can be more or less easy to partition.
For instance, for many students it is more difficult to
partition a circle into thirds than it is for a
rectangle. |
 |
|
| • |
Students who are more mature in terms of understanding
fractions are able to use larger size pieces or "chunks"
when partitioning. |
| • |
Activities involving partitioning should not be
restricted to the early years. |
| • |
Through partitioning, students can come to see that a
fraction can be used to indicate the result of a
division. In mathematics terms this is called a quotient. For instance 2/3 can be understood as
meaning 2 objects divided into 3 equal parts. This is an
important understanding of fractions. |
|
Resources
|
| Fractions as part-whole relationships
The concept of a fraction as a part-whole
relationship is where one or more equal parts of
a whole are compared with the total number of
these parts that it takes to make up the whole.
To understand fractions as part-whole
relationships, students need to recognise the
relationship between the bottom number (total
number of equal-sized parts that make up the
whole) and the top number (number of these parts
of interest). Understanding part-whole fractions can
also involve:
sets of countable objects (discrete),
shaded regions (continuous), and
quantities (either continuous or discrete).
Most students' first introduction to fractions
in the classroom as a part-whole comparison is
with unit fractions, e.g., half, quarter,
third (1/2, 1/4, and 1/3). A unit fraction is one part of a whole.
That whole may be partitioned into many parts,
but as long as it is only one of these parts of
interest it is called a unit fraction. For
example if a shape is partitioned equally into n parts then each unit part is called one
n-th which is written 1/n.
Most fractions are non-unit fractions. It is
important to introduce simple non-unit fractions
such as 2/3 and 3/4 at the same time as unit fractions to avoid
students developing misunderstanding about all
fractions from their limited experience with
unit fractions. Similarly, it is important to
encourage students to explore fractions with different numbers such as 11/23,
9/13, etc, improper
fractions (e.g., 7/6, 4/3, etc), and mixed
fractions 2 1/3, etc.
Fractional notation
Students beginning to understand fractions
should be encouraged to use words to describe
the parts, and delay the fractional notation
until they have developed some understanding of
what fractions represent.
For all fractions, the notation convention is
the bottom number (denominator) tells you how
many equal parts make up the whole. The top
number (numerator) tells you how many of
these parts are of interest.
Using part-whole understanding of fractions
we can say the rectangle is one-fifth (1/5) shaded because
there are five equal-sized parts and one of them
is shaded. |
or
|
In the fraction "one-fifth", the one shaded part is
described by the numerator and that four
of these equal-sized parts make up the whole
shape is described by the denominator.
The equal-sized parts could be called
"fifths".
When solving part-whole fraction problems
it is often helpful to remind students to ask
themselves "What is the whole (shape or set)?
and "What is the part (of the shape or set)?".
Fractions of shapes or regions (continuous)
A part-whole understanding of fractions involves
identifying what fraction of a shape (or region)
is shaded. This context is usually
continuous, when a part of the shape is shaded
as below. However if the shape has
already been partitioned into equal parts it is
essentially countable and therefore a discrete
problem.
Example,
What fraction of this rectangle is shaded?
Half - because there are two parts?
Quarter - because four of those parts would make up the whole
shape? |
 |
This problem is different from the 1/5 above
because the pieces cannot simply be counted to
construct a fraction. Students here must use
the shaded part and work out what fraction of
the whole it represents (how many times the
parts "goes into the whole"). They could be
encouraged to draw lines to create equal-sized
partitions to help them work out or explain
their strategy. Although some students may feel
they are not supposed to draw lines on to the
shapes, encouraging them to do so can help eliminate
misunderstandings.
Asking students to explain or show their working
can provide useful information about the
strategies they use and their understanding
behind their answer. It is important to be
aware of the strategies students employ to solve
part-whole fraction problems and to ensure
students are not developing a "narrow" solution
method for solving without understanding the
part-whole nature of fractions.
Unequal-sized pieces
Other similar questions, where the shapes are
not evenly partitioned (as in the second shape
below) can identify important misunderstandings.
For example, what fraction of each of these
shapes is shaded?
Another continuous context for fractions is
distance or length. Fractions can be shown on a
number line. This concept is explored further
in Fractions and number lines.
Fractions of sets
Fractions
can also represent relationships involving sets of countable objects (discrete). Here the whole
is considered to be the set and the part is the
selection (or number) of countable items.
For example: What fraction of the counters is
shaded?
Quantities
Part-whole fraction questions can involve
identifying the fraction of a quantity.
For example,
Bob has 5 marbles left from 15 at the start of
the day.
What fraction of marbles does he have
left?
For this question, the whole (referent whole) is 15 and
the part is 5.
So the part of the whole is 5/15.
Students may recognise that there are three
parts in the whole,
which means the part is a third (1/3).
Context
Context can be important for students to begin
to relate to their own experiences, but being
aware of the meanings that underlie the contexts
is important. Circular pizzas are often used as
a context for fractions, they tend to be cut
into quarters and eights, but if students are
asked a pizza question involving thirds or
sixths they may find it difficult. The
context
is meant to be realistic and one that students
can relate to, and
should therefore support the understanding
rather than hinder it.
Students need to be introduced to a variety of
shapes illustrating fractions as part-whole
relationships - squares, rectangles, and
hexagons are good starting shapes rather than
using just circles. This variety
encourages students to use different strategies
to solve the problems and develop their
understanding about fractional representations
rather than simply memorising common iconic
fractions shapes without understanding:
  
These may be well recognised as 1/3, 3/4 and
1/4 without understanding about the relationship
between the part and the whole.
Shapes like:   
require students to begin developing an
understanding about how much of the shape is
actually shaded.
Finding the whole from a part
Part-whole fraction problems can also involve
finding the whole when given a known part.
Asking fraction problems in this form ensures
that students are not simply developing a
formula for solving similar structured
part-whole fraction questions (always finding
the fraction of a whole). An example of a
resource that explores
this is Cuisenaires
and fractions (NM0134).
For example:
Bob has 5 marbles left.
If this was 1/3 of how
many he started with, how many marbles did he
start with?
The whole is unknown;
The part is 5
And 5 is 1/3 of the whole. So there are 3 of
these parts in the whole, if each is 5 the whole
is 15.
Example:
If  is
1/3 draw what the whole would look like?
If  is
1/4 what is  ?
________.
It is important that students are asked
questions in the form "what is the [whole]
given the [part]" - not only "what is
the [part]
of the [whole]". Otherwise they may develop
only a limited understanding of fraction
problems from having explored only simple
part-whole problems.
Using the part to find another part
Students can also be asked fractions problems of
the form:
If  is
2/3 what is 1/3?
This problem can be solved by students finding
(and/or drawing) 3/3 and
working out 1/3 or by students recognising that 1/3 is half of
2/3, so the shape would be half the number of
counters,
e.g., or 
or
If is
2/3 what is 1/4?
In this example, because the denominators are
different (and one is not a multiple of the
other), there is no immediate or obvious link
between the 3 and the 4 except for recognising
the common multiple of 12 and working with
complicated calculations to solve the problem.
This problem can be solved more easily by
students finding 3/3 then working out 1/4 of
that,
e.g., 3/3 is so 1/4 of that is 
NOTE: Students would need to know that any
fraction with the same number as denominator and
numerator is one (a whole), e.g., 4/4, 3/3, etc.
Important ideas about part-whole fractions
| • |
The size of each part in a fraction must be equal. |
| • |
The greater the number of parts needed to make a whole,
the smaller each individual part is. This means for instance, that 1/20 is smaller than 1/19. |
| • |
Different fractions can be used to represent the same
amount of a whole, depending on what size parts have
been used. |
|
Resources
|
| Equivalence:
comparing and ordering fractions
As students learn about fractions they come to
understand them as a system of numbers. In
particular, they realise that fractions have a
size and can be compared, ordered, and
represented as a point on the number line.
Strategies
for comparing fractions
Students employ a range of strategies when comparing or
attempting to compare the size of fractions.
Drawing pictures
Students will often draw pictures to compare the
size of fractions. These pictures can be useful
to support or explain their understanding.
However, they can sometimes be misleading,
especially when the fractions are close together
in size.
Identifying fractions with the same
denominator or numerator
Students will
often use their part whole understandings of
fractions to reason about fraction size. For
instance, when a student understands that the
size of the parts that make up a fraction depend
on the size of the denominator, they can quickly
see that fractions with the same denominator or
numerator can be easily compared.
|
Unit fractions
Students
can compare unit fractions by using their
understanding of the meaning of the denominator.
They can be aware that "the larger the
denominator the smaller the fraction",
because the whole is broken into more parts,
therefore each must be smaller. |
|
Fractions
with the same numerator
Student
comparing fractions with the same numerator
is a derivation of comparing unit
fractions. For example, if one unit
fraction is larger than another then any
common count of the parts will be larger,
e.g., 3/4 is larger than 3/7 because
1/4 is larger than 1/7, and there is the
same number of parts being compared. |
|
Fractions
with the same denominator
Students
can compare fractions with the same
denominator by comparing only the
numerator. This is almost no different
from comparing whole numbers as the
numerator is simply the count of
units. For instance,
5/7 is greater than 4/7, just because you have
more of the same size part. |
Benchmarking fractions to well known fractions
Students will sometimes use a familiar fraction
as a benchmark or reference point when comparing
fractions. For instance, they will note that
9/10 is greater than 7/8, as 9/10 is only 1/10
away from a whole, while 7/8 is 1/8 away.
A half can be a useful benchmark. Students
often develop the understanding that a fraction
is the same as 1/2 when the denominator is
exactly twice the numerator. They can use this
relationship to place a fraction in relation to
a half and then make comparisons with other
fractions. For instance, 2/3 is greater than
1/2, while 5/11 is less than a 1/2. This means
2/3 must be greater than 5/11.
Some students who are able to identify when a
fraction is greater or smaller than a half will
not be able to tell you how close the fraction
is to 1/2. Asking how far away a fraction is
from 1/2 can be a useful follow up question to
encourage further understanding.
Using equivalent fractions
When comparing fractions, students will
sometimes convert one or more of the fractions
to an equivalent fraction with the same (common)
denominator as the other fraction or
fractions.
|
For example,
Which fraction is larger 3/4 or 1/2?
Students may recognise that they can easily
compare fractions if they both have the same
denominator, and they may know that 2/4 is the
same as (equivalent to) 1/2. Using this
knowledge of equivalent fractions they can
compare 3/4 and 2/4. |
|
Example,
Which fraction is larger 2/3 or 3/5?
Students may look for the lowest common
multiple, in this case 15, and rename the
fractions equivalent fractions, so the question
becomes:
Which fraction is larger 10/15 or 9/15? |
One common way to find equivalent fractions is
work upwards from both fractions until a common
denominator is found (trial and error)
Another way is to recognise the relationship
between both denominators and have a sense of
what the denominator needs to be. Students then
either extrapolate or cross multiply to work out
what the numerator should be.
Some
students will also convert fractions to
equivalent decimals and percentages to make
comparisons. Students use many methods to
convert fractions to equivalent fractions or
decimals and percentages including
cross-multiplying. These methods can often rely
on learned processes that mask their true
understanding of fraction size.
Misconceptions
- Attempting to use whole number knowledge
Students will sometimes attempt to apply their
knowledge of whole numbers to compare
fractions. For instance, they will say 7/10 is
greater than 4/5 as 7 is greater than 4. These
students need to be challenged about the meaning
of fractions. Using concrete materials can help
them check their thinking.
|
Resources
|
Fractions
and number lines
The ability to place a fraction on a number line
follows from having a part-whole understanding
of fractions, and being able to compare and
order fractions. For some students using a
number line can help them visualise the size of
a fraction, and can provide them with a tool for
representing and comparing fractions.
There are two distinct types of resource about
fractions and number lines:
| 1. |
Writing or ordering fractions in pre positioned boxes
along number line. This can illustrate how a
number line can be used to represent
fractions of distance or length; or support the notion of
a fraction being larger or smaller than another.
For example: Write the fractions that would go in the
boxes below. |
| 2. |
Marking the fraction on an empty number line - this
involves measurement and judgement of a fraction as a
proportion of a length or distance.
For example: Show where 1/2, 2/3 and 3/4 would be on
the number line below. |
When using number lines it is also important to
explore:
| • |
odd looking fractions (e.g., 9/15, 11/23, etc); and |
| • |
number lines beyond 0-1 (e.g., 0-2, 1-3, 2-5, etc); |
|
•
top heavy fractions (e.g., 4/3, 7/6, 12/5, etc); and |
|
•
mixed fractions (e.g., 2 1/4, 4 2/3, etc). |
For example: Show where 11/23 and 12/7 would be
on the number line below.
Context
The number line relates to the context of
distance and length and questions like "how far
along.?" can access student's prior
understandings. This context is already a part
of student experience. It can also be introduced
using the context of a milk bottle, by asking
how full is (a given mark) and using common
fractional amounts used in conversation (i.e., a
quarter, half, third, etc). |
Resources
|
Adding and subtracting fractions is different from adding
whole numbers. Fractional parts can be
added, but it is working out the value (or
fractional name) of the sum that can be difficult.
For example, adding 1/2 +1/4 cannot be done by
using rules for whole numbers, i.e., 1/2 +1/4 =
(1+1)/(2+4) = 2/6 (or some derivation).
Adding fractional amounts involves working out
some degree of common unit (denominator) before
working out how many of these units there are
(numerator).
Equivalent fractions
One strategy is if one fraction could be a factor of the
other to use equivalent fractions,
e.g., adding 1/3 and 1/6,
1/6 is a factor of 1/3 because there are 2 of them in 1/3.
Therefore 2×1/6 is 1/3, so 2/6 = 1/3
So now there are common units (sixths) they can be added 2/6
+ 1/6 = 3/6 (which is also a half)
This could also be done by using drawings
and
in the array form:
and
for subtraction:
Equivalent
fractions (common multiple)
One very common strategy is to find a common multiple, use
that as the common denominator and then make
appropriate adjustments to the top
(numerator) by finding the fraction of the common
denominator, cross multiplying.
This involves multiplying both the top and
bottom number by the other multiple, and is based
on the concept that multiplying top and bottom by a/a is the same as multiplying it by one (multiplicative
identity), i.e., x × 1 = x
Example: Adding 3/4 + 2/3
Finding a common multiple of 4 and 3 - the most obvious and
lowest is 12
3/4 (×3/3) + 2/3 (×4/4) =
(3×3)/(4×3) + (2×4)/(3×4)
= 9/12 + 8/12 = 17/12 (or 1and 5/12)
Another way to adjust the numerator is by finding the
equivalent fractional value of the new common
denominator, i.e., asking "What is 3/4 of 12
(=9) and what is 2/3 of 12 (=8)?"
This will result in
9/12 + 8/12 which can be solved as above.
|
Resources
|
| • |
"multiplied by 2 and then divided by 3"; |
| • |
"divided by 3 and then multiplied by 2"; or |
| • |
other forms of a function. |
| 1. |
2 × 12 = 24 and then 24 ÷ 3 = 8. |
| 2. |
2 × 12 = 24 and then 12 ÷ 3 = 4 and then 4 × 2 = 8 |
| 3. |
 (divided
by 3) and then two of these partitions (multiplied by 2) |
| 4. |
Finding 1/3 of 12 (=4) and then finding 4 + 4 = 8 (1/3 +
1/3 = 2/3) |
Find the [fraction] of a [shape/ set /quantity].
Examples of some fraction operation problems:
| • |
Find 3/4 of this shape  |
| • |
What is 7/8 of 56? |
| • |
What is 5/6 of this set?  |
|
Resources
Level
2
|
Unitising with
fractions
Unitising is about selecting a unit of measurement with which to measure or interpret other quantities. Unitising is a strategy that can be used to work out another quantity that relates to the unit selected. It involves the understanding of partitioning and equivalence of fractions to develop flexible thinking about fraction situations. This flexibility can in turn support further understanding about equivalence and fractions as rational numbers (in particular quotients – the result of division). Unitising can be defined as the assignment of a unit of measure to a given quantity or "chunks" that make up a given quantity (Lamon, 2007).
For example 
The shaded part could represent 7 (circles), or 3 1/2 (columns), or 7/12 (of a dozen), 1 3/4 (bundles of 4), or 1 1/6 (bundles of 6 or rows) depending on what "chunk" or unit you use as your reference whole or unit. All the answers above can be argued as correct and for each the unit (or chunk) is different.
How many shaded |
Chunk |
7 |
7/1 |
circles |
3 1/2 |
7/2 |
columns |
7/12 |
7/12 |
dozens |
1 3/4 |
7/4 |
bundles of 4 |
1 1/6 |
7/6 |
bundles of 6 or rows |
Another example could be the unit of measurement for a case of 36 soft drinks. The unit could be the case, a dozen, six-pack, or the individual bottles. Depending on what unit of measure is selected, the overall quantity of that measure can be different.
Questions that involve unitising require students to identify (or decide) the unit of measurement, and use this to explore how many of those make up the total. They can be Number and Algebra fraction type problems (e.g., Cuisenaires and fractions (NM0134), Parts and wholes (NM0157)) or Measurement type problems involving repeated units (see Five dolls (MS2073) and Five swimmers (MS2074).
Click on the link to read more about Unitising and re-unitising.
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