All Fractions are Not
Created Equal
SARIC RSS Mini-Conference 2014
Laura Ruth Langham Hunter
AMSTI-USA Math Specialist
Flashback…
• Discuss your experiences with fractions in the
classroom when you were going through
school.
• What are some of the biggest challenges
students face with fractions today?
Common Misconceptions
Students…
• plot points based on understanding fractions as whole
numbers instead of fractional parts.
• see the numbers in fractions as two unrelated whole
numbers separated by a line.
• do not understand that when partitioning a whole shape,
number line, or a set into unit fractions, the intervals must
be equal.
• do not understand the importance of the whole of a
fraction and identifying it.
• do not count correctly on the number line.
• do not understand there are many fractions less than 1.
• do not understand fractions can be greater than 1.
• think all shapes can be divided the same way.
Why are Fractions so Hard?
• Fraction notation – numbers must be
considered in new ways
• Practices that simplify and/or mask the
meaning of fractions
• Overreliance on whole number knowledge
• Many meanings of fractions
Learning Outcomes
• Identify the importance of the whole
• Determine the unit fraction and it’s purpose in
working with fractions
• Identify different strategies for comparing and
ordering fractions
• Evaluate the four types of fractions
CCRS Math Practice Standards
1. Make sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
CCRS Content Standards
Grade 3: Number and Operations – Fractions
• Develop understanding of fractions as numbers.
Grade 4: Number and Operations – Fractions
• Extend understanding of fraction equivalence and
ordering.
• Build fractions from unit fractions by applying
and extending previous understandings of
operations on whole numbers.
• Understand decimal notation for fractions, and
compare decimal fractions.
CCRS Content Standards
Grade 5: Number and Operations – Fractions
• Use equivalent fractions as a strategy to add
and subtract fractions.
• Apply and extend previous understandings
• of multiplication and division to multiply and
divide fractions.
Mini Olympics
• Stations:
– Mini Javelin: Stand behind the line and throw a cotton
swab like a javelin.
– Long Thump: Press the edge of a penny with your
thumb onto the edge of the another coin to make the
lower coin jump forward.
– Discus Blast: Use a breath of air to blow a plastic
bottle cap across a long table while remaining entirely
behind the line.
• Use your teacher-provided measuring tool to
figure out the distance your item traveled.
Shift from Whole Number Reasoning
• What allowed for more precision in your
measurements?
• Why did you subdivide your “ruler”?
• What was the smallest subdivision that a unit
was divided into?
• What is the smallest possible subdivision the
unit could have been divided into?
• In conclusion, rational numbers are just an
extension of whole numbers.
Attending to Precision
Two important aspects of fractions provide
opportunities for the mathematical practice of
attending to precision:
• Specifying the whole.
• Explaining what is meant by “equal parts.”
Pizza Problem
José ate 1⁄2 of a pizza.
Ella ate 1⁄2 of another pizza.
José said that he ate more pizza than Ella, but
Ella said they both ate the same amount. Use
words and pictures to show that José could be
right.
Example from Illustrative Mathematics
What’s the Whole?
Emily
Raj
Alejandra
What’s the Whole?
What misconceptions do you think your
students might have about the size of the
whole?
What questions could you ask students to get
them to think deeper about the size of the
whole?
What is a Unit Fraction?
•
•
•
•
A fraction with the numerator of 1
¼ is a unit fraction
¼ + ¼ + ¼ + ¼ = 4/4 = 1 whole
4/5 is a fraction made up of 4 pieces of size
1/5 or 4 copies of the unit fraction 1/5
1/5 + 1/5 + 1/5 + 1/5 = 4/5
Pattern Blocks: If… Then…
• If the yellow hexagon is one whole, then…
• If the red trapezoid is one whole, then…
• If the blue rhombus is one whole, then…
• If the green triangle is one whole, then…
Part-Whole Relationship
“Looking at fractions as though they
represent two separate whole numbers
leads to misinterpretation of their meaning
and an inability to assess the
reasonableness of calculation results.”
(Investigations in Number, Data, and Space “Finding Fair Shares”)
Equal Parts?
• The parts of the whole must be equal to one
another, and all the parts combined must
equal the whole.
Part-Whole Relationship
• One key idea throughout the study of fractions is
that a fraction represents a quantity in relation to
a unit whole.
• Examples of unit wholes could include a single
object, an area, a linear measure, or a group of
objects.
• Although ½ can represent many different
quantities, depending on the size of the whole, ½
has the same relationship to any whole. It is one
of two equal parts that compose the whole.
Let’s Estimate!
Let’s reason about two numbers for a second…
12
7
13
8
Estimate the sum of these two numbers
A. 1
B. 2
C. 19
D. 21
Most students answered C or D!
They were not able

 to see the relation of each
number to 1 and thus missed the correct
estimation of 2.

Compare unit fractions in context
Kim, Les, Mario, and Nina each had a string 10
feet long.
Kim cut hers into fifths.
Les cut his into fourths.
Mario cut his into sixths.
Nina cut hers into thirds.
After the cuts were made, who had the longest
pieces of string?
Reasoning Strategies for
Comparing Fractions
• Same Denominators
– More same-sized parts
• Same Numerators
– Same number but different-sized parts
• Comparison to Benchmarks
– More or less than one-half of one whole
• Distance from Benchmarks
– Closeness to one-half or one whole
Snow Day
Alec and Felix are brothers who go to different
schools. The school day is just as long at Felix'
school as at Alec's school. At Felix' school, there are
6 class periods of the same length each day. Alec's
day is broken into 3 class periods of equal length.
One day, it snowed a lot so both of their schools
started late. Felix only had four classes and Alec
only had two. Alec claims his school day was
shorter than Felix' was because he had only two
classes on that day. Is he right?
Learning Outcomes
• Identify the importance of the whole
• Determine the unit fraction and it’s purpose in
working with fractions
• Identify different strategies for comparing and
ordering fractions
• Evaluate the four types of fractions
Thank You for Attending!
Contact Information
Laura Ruth Langham Hunter
LLangham@southalabama.edu