Transcript Three Meanings of Fractions

Chapter 12
To a c c o m p a n y H e l p i n g C h i l d re n L e a r n M a t h C d n E d , R e y s e t a l .
© 2 0 1 0 J o h n Wi l e y & S o n s C a n a d a L t d .
Guiding Questions
• What are three meanings of fractions, and what are models of
the part-whole meaning?
• How can you help children make sense of fractions, and how
can you use concrete and pictorial models to develop
children’s understanding of ordering fractions and equivalent
fractions?
• Describe how children can use estimation strategies for
adding and subtracting by rounding to whole numbers and
benchmark numbers to determine reasonableness of answers
to fraction and decimal problems.
• How can models assist the development of children’s
conceptual understanding of adding, subtracting, multiplying,
and dividing fractions or decimals?
Conceptual Development of
Fractions
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Three Meanings of Fractions
Models of the Part-Whole Meaning
Making Sense of Fractions
Ordering Fractions and Equivalent Fractions
Benchmarks
Mixed Numbers and Improper Fractions
Three Meanings of Fractions
Part-Whole
Quotient
Ratio
3 boys to 5 girls
Models of Part-Whole Meaning
Region
Set
Length
Area: special case of region model where parts are equal in
area but not necessarily congruent
Making Sense of Fractions
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Partitioning
Words
Counting
Symbols
Drawing a Model
Extending the Model
Benchmarks
Going From a Part to a Whole
Understanding Equivalence
Ordering Fractions and
Equivalent Fractions
• Concrete Models
• Pictorial Models
• Symbolic Representation
1/2 = 3/6
1/3 = 2/6
Benchmarks
• Students also need to be able to tell if a fraction is
near 0 or near 1. This will let them use the
benchmarks 0, 1/2 and 1 to put in order a set of
fractions such as 13/25, 2/31, 5/6, 4/11, and 21/20,
which would be a time-consuming task if they tried
to find a common denominator.
• Using benchmarks as shown on the following
number line makes the task rather easy:
Mixed Numbers and Improper
Fractions
A mixed number is a natural
symbolic representation of the
adjacent model.
You can add partitions in the model to show all the fourths, so
children can see that the initial counting is 9 fourths, or the
improper fraction 9/4.
Operations with Fractions
• The key to helping children understand
operations with fractions is to make sure they
understand fractions, especially the idea of
equivalent fractions.
• They should be able to extend what they
know about operations with whole numbers
to operations with fractions.
Adding Fractions with Like
Denominators
2/6 + 1/6 = 3/6
Addition of Fractions
Addition of Fractions
Addition of Fractions with
Unlike Denominators
Subtraction of Fractions
3/4 -1/4 = 2/4
or 1/2
Multiplication
• Whole Number Times a Fraction:
– You have 3 pans, each with 4/5 of a pizza.
How much pizza do you have?
Multiplication
Fraction times a whole number
• You have ¾ of a case of 24 bottles. How many
bottles do you have?
Multiplication of Fractions
Fraction Times a Fraction
• You own 3/4 of an acre of land, and 5/6 of this
is planted in trees. What part of the acre is
planted in trees?
Division of Fractions
• How many 2-metre lengths of rope can be made from a 10metre length of rope?
• Students can then experiment with 1/2 metre and 1/4 metre
sections of rope. Now ask students how they would draw a
picture to find out how many 3/4 metre pieces there are in a
6-metre length of rope.
Conceptual Development of
Decimals
• Relationship to Common Fractions
• Relationship to Place Value
• Ordering and Rounding Decimals
Relationship to Common
Fractions
Relationship to Place Value
• Often, you can use a place-value
grid to help students who are
having difficulty with decimals.
Consider, for example, this grid
for the number 32.43.
• Point out that the decimal can be
seen both as 32 and 43
hundredths and as 32 and 4
tenths and 3 hundredths. It can
also be read as 3243 hundredths.
What other ways could you
express this number in words?
Ordering and Rounding
Decimals
• Children should be able to
understand the ordering
and rounding of decimals
based directly on their
understanding of decimals
and their ability to order
and round whole numbers.
For decimals, this
understanding must include
being able to interpret the
decimals in terms of place
value and being able to
think of, for example, 0.2 as
0.20 or 0.200.
Operations with Decimals
• One advantage of decimals over fractions is
that computation is much easier, since it
basically follows the same rules as for whole
numbers.
Addition and Subtraction of
Decimals
Difficulty with adding or subtracting decimals arises mainly when the values
are given in horizontal format or in terms of a story problem and the
decimals are expressed to a different number of places (e.g., 51.23 + 434.7).
To deal with this difficulty, have the
children first estimate by looking at
the wholes (about 480). Some
children may need help in lining up
the like units and so may benefit from
using a grid.
Multiplication of Decimals
• To help students make
sense of multiplying a
decimal by a decimal,
rather than only
remembering a rule about
counting decimal places,
consider this decimal grid.
Division of Decimals
Have children talk
through a decimal
division problem so
that they can
evaluate the
reasonableness of
their answer.
For Class Discussion
The following slides will show you samples of
student thinking around fractions. As you
review each slide please take a few minutes to
share your observations with others.
Student Interviews
Interviewer: “Which fraction is more, 1/3 or
1/4? 2/5 or 2/7?”
Charles: Fifth
Month of Grade Four
Student Interviews
Interviewer: “Add these fractions:
3/8 + 2/8
2/3 + 1/4.”
Amanda: Fifth Month
of Grade Four
Student Interviews
Interviewer: “Add these fractions:
3/8 + 2/8 2/3 + 1/4.”
Amy: Fifth Month of Grade Four
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