Strategies to support student learning of fractions

Report
STRATEGIES TO SUPPORT
STUDENT LEARNING OF
FRACTIONS
Cynthia Santosuosso
The Research…


Only 24 % of thirteen and seventeen-year-old students
identified 2 as the estimated sum for 12/13 + 7/8,
while a greater percentage identified 19 or 21 as the
estimated sum.
Only 50% of eight grade students successfully
arranged 2/7, 1/12, and 5/9 from least to greatest.
Based on the NAEP (National Assessment of Educational
Progress).
Why are fractions so challenging?




The way that fractions are written
Classroom practices designed to help students make
sense of fraction values mask the meaning of
fractions.
Students’ overreliance on whole number knowledge
The many meanings of fractions, such as measure
and ratio
Fraction sense



A deep and flexible understanding of fractions that
is not dependent on any one context or type of
problem
Tied to common sense
Allows students to reason about fractions
Fraction Sense
Workshop Format
This workshop is based around teaching scenarios.
From each of these scenarios will be drawn:
key ideas about fractions
how to communicate these ideas to students
Scenario: Illustrating fractions
Grades Taught: Third, fourth, and fifth
Ms. Chu reviews some basic fraction concepts
with her fifth graders. While examining their
work, she notes that the drawings of more
successful students varied, depending on the
details of the problem. Students who were less
successful relied solely on drawing circles (or
pizzas), regardless of the problem type or
context.
Consider….

What can Ms. Chu do to expand her students’
drawings?
The Math…Area Model


Fractions are based on parts of an area or region
Circles
 Most
common
 Emphasize amount that is remaining to make up whole
 Not as flexible as other area or region because don’t
allow for different-sized units



Rectangles
Pattern Blocks
Paper Folding
The Math…Linear Model

Lengths are compared instead of areas
 Fraction
Strips
 Cuisenaire Rods
 Number Line
 Important to note that each number on number line
denotes the distance of the labeled point from zero, not
the point itself
 Often difficult for students to view as comparing lengths
The Math…Set Model


Whole is understood to be a set of objects &
subsets make up fractional parts
Two color counters often used to model fractional
parts
Key Idea
Understanding the relationship between
representations of the same idea is core
to fraction sense.
Classroom Activities to Support this Idea
Provide students with a fraction (1/10) and have them
draw an example of each type of fraction model. For
example:
 One cake shared among ten people (area model using
a rectangle)
 A pizza divided into ten equal pieces (area model
using a circle)
 One piece of gum from a package of gum that has ten
pieces (set model)
 A segment marked 0 to 1 on a number line where the
segment is divided into ten equal sections (linear
number line)
Classroom Activities to Support this Idea
Have students write word problems based on Fraction
Representation Cards.
For example, if a student chose the “set model” card,
he would write a fraction problem that could be
solved using the representation on the card. Once
corrected, another student could answer the problem
Scenario: Modeling with fractions
Grades Taught: Third, fourth, and fifth
Students are using paper strips to make fraction
kits. Ms. Alvarez notices Tyler labeling pieces as
1/7, which was not a directed step. He
explained “When I counted them, there were
seven, so I knew they were sevenths.”
Consider…

Is Tyler necessarily correct?

Why / Why not?

What action, if any, do you take?
The Math…


Tyler lost one strip (1/8).
Determining the size of a strip in relation to the
whole entails considering the number of copies of
the strip needed to cover the whole.
Key Idea
The label on a fraction strip is not a fixed
name, but a description of the relationship
between the strip and the whole.
Classroom Activities to Support this Idea
Pattern Block Fractions
Fraction Strips
Scenario: Comparison of fractions
Taught: Third and fourth grade
Consider:
Is he correct?
 After checking that Bruce understands what the
“>” symbol means, what action, if any, do you
take?

The Math…

No he is not correct.
The correct equation is 1/6 < 1/4 because
one sixth is less than one quarter.

Key Idea:
The more pieces a whole is
divided into, the smaller each
piece will be.
Classroom Activities to Support this Idea
Fraction Strips
Classroom Activities to Support this Idea
Stacked Number Lines
Classroom Activities to Support this Idea
Benchmark Fractions (0, ½, 1)
2/7______3/5
8/9______1/3
6/7______7/6
Scenario: Comparison of Fractions
Taught: Grades 3 and 4
A group of students are investigating the books they have in
their homes.
Steve notices that 1/2 of the books in his house are fiction
books, while Andrew finds that 1/5 of the books his family owns
are fiction.
Steve states that his family has more fiction books than Andrew’s.
Consider…

Is Steve necessarily correct?

Why / Why not?

What action, if any, do you take?
The Math…


Steve is not necessarily correct because the amount
of books that each fraction represents is dependent
on the number of books each family owns.
For example…
Key Idea
The size of the fractional amount
depends on the size of the whole.
Classroom Activities to Support this Idea



Provide opportunities for students to develop their understanding of the
importance of context in fraction comparison tasks.
Use materials and diagrams to demonstrate with clear examples, as in the
previous tables.
Question the student about the size of one whole:
 Is one half always more than one fifth?
 What is the number of books we are finding one fifth of? How many
books is that?
 What is the number of books we are finding one half of? How many
books is that?
Scenario: Labeling fractions
Grades Taught: Third and fourth
Luke labels the shaded portion of this fraction as
¼.
Luke labels the shaded portion of this fraction as
1/3.
Cont.
He explains, “It’s just like the other
one, only this time it’s one out of
three, instead of one out of four.”
Consider…

Is Luke necessarily correct?

Why / Why not?

What action, if any, do you take?
The Math…
Luke is not correct. Representing a fractional part of
an area using fraction notation involves determining
the whole area and then considering the size of the
shaded part of the area in relation to the size of the
whole.
Key Idea
Students understand that the size
of a fractional part is relative to
the size of the whole.
Classroom Activities to Support this Idea
Provide opportunities for students to work with
irregularly partitioned, and unpartitioned areas,
lengths, and number lines.
Solution: Measurement Strategy


Consider the number of copies of the shaded part are
needed to cover the whole. This is the denominator.
One of the copies is shaded (the numerator).
Solution: Part-Whole Strategy
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
Divide the whole into parts of equal size.
The whole is divided into equal parts by inserting
the dotted line.
Scenario: Equivalent fractions
Taught: Third and fourth grades
Which shape has 1/3 of its area shaded?
Sarah insists that none of the shapes have 1/3 of
their area shaded.
Consider:
Do any of the shapes have 1/3 of their area
shaded?
 What action, if any, do you take?

The Math…
Key Idea:
Equivalent fractions have the same
value.
Classroom Activities to Support this Idea
Fraction Strips
Classroom Activities to Support this Idea
Stacked Number Lines
Scenario: Addition of Fractions
Taught: Grades 4 and 5
You observe the following equation in Emma’s
work:
1/2 + 2/3
Is Emma correct?
= 3/5
Consider…
You question Emma about her understanding and
she explains:
“I ate 1 of the 2 sandwiches in my lunchbox, Kate
ate 2 of the 3 sandwiches in her lunchbox, so
together we ate 3 of the 5 sandwiches we had.”


What, if any, is the key understanding Emma
needs to develop in order to solve this problem?
The Math…
Emma needs to know that the 1/2 relates to a
different whole than the 2/3 .

If it is clarified that both lunchboxes together
represent one whole, then the correct recording is:
1/5 + 2/5 = 3/5

Emma also needs to know that she has written an
incorrect equation to show the addition of fractions.

Key Idea
When working with fractions, the whole
needs to be clearly identified.
Classroom Activities to Support this Idea
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
Question the student about their understanding.
 The one out of two sandwiches refers to whose lunchbox?
 Whose lunchbox does the two out of three sandwiches
represent?
 Whose lunchbox does the three out of five sandwiches
represent?
Use materials or diagrams to represent the situation. For
example:
Key Idea
When adding fractions, the units need to be the
same because the answer can only have one
denominator.
Classroom Activities to Support this Idea
Use visuals to illustrate how to find a
common denominator.
Classroom Activities to Support this Idea
Estimate the sum with benchmark fractions; check for
reasonableness. 0, ½, and 1 can serve as reference
points for estimation.
½ + 2/3= 1 1/6
½+½=1
Scenario: Mixed Numbers
Taught: Fourth and Fifth Grade
Anna says 7/3 is not possible as a fraction.
Consider…..

Is 7/3 possible as a fraction?

What action, if any, do you take?
The Math…
Key Idea
A fraction can represent more than one
whole. The denominator tells the number of
equal parts into which a whole is divided. The
numerator specifies the number of these parts
being counted.
Classroom Activities to Support this Idea
Use fraction strips or a number line to model
1/3
1/3
1/3
=1
1/3
1/
1/3
1/3
=1
1/3
=1/3
1+1+ 1/3 =2 1/3
Scenario: Multiplication of Fractions
Taught: Fourth and fifth grades
Katie solves the following problem:
2/3 x 5/7= 14/15
Consider…
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
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Is Katie correct?
What is the possible reasoning behind his answer?
What, if any, is the key understanding he needs to
develop in order to solve this problem?
The Math…
Katie is not correct. She has written the cross product
as the numerator of the product and the other cross
product as the denominator.
Key Idea
Students must grasp the concept behind
multiplying fractions, beyond the
algorithm.
Classroom Activities to Support this Idea
Use grid pencils and colored pencils to demonstrate
visually.
Remember that “x” means “of.”
2/3 x 5/7 = 10/21
Scenario: Division of Fractions
Taught: Fifth grade
You observe the following equation in Bill’s work:
2½÷½= 1¼
Consider…..
Is Bill correct?
 What is the possible reasoning behind his
answer?
 What, if any, is the key understanding he
needs to develop in order to solve this
problem?

The Math…


No he is not correct. The correct equation is
2½÷½= 5
Possible reasoning behind his answer:
1/2 of 2 1/2 is 1 1/4.


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He is dividing by 2.
He is multiplying by 1/2.
He reasons that “division makes smaller” therefore the answer
must be smaller than 2 1/2.
Key Idea
To divide the number A by the number B is to
find out how many lots of B are in A.

For example:
 8÷2=4:

There are 4 lots of 2 in 8
2 ½÷5=1/2: There are 5 lots of 1/2 in 2 1/2
Classroom Activities to Support this Idea
Successful Strategies
•
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Varied visual representation (area, linear, set)
Fraction strips
Number lines
Benchmark fractions
Let’s Work on Our Own Fraction Sense!
Fractured Numbers-Problem of the Month
3-2-1 Reflection

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3 ideas or pieces of knowledge you have found
useful.
2 things you want to try with students.
1 question you’re still asking.
Sources
Institute of Education Sciences. Developing Effective
Fractions Instruction for Kindergarten Through
Eighth Grade. What Works Clearinghouse.
2010.
McNamara, Julie. Beyond Pizzas & Pies. Math
Solutions. Sausalito, 2010.
Spangler, David. Strategies for Teaching Fractions:
Using Error Analysis for Intervention and
Assessment. Corwin. Thousand Oaks, 2011.

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