CCGPS Mathematics
Unit-by-Unit Grade Level Webinar
Third Grade
Unit 6: Representing and Comparing Fractions
January 22, 2013
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What’s Unit 6 all about?
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Organizing thinking
Quantity representation- part of whole emphasis
Understanding unit fractions
Composition/Decomposition of fractions
Representing data and solve using data
Modeling problem solving
Multiple solution paths
What should students bring from before?
• Foundational fractions understanding- equal
parts/shares
• Familiarity with number line, ruler
• Foundational ability to decompose quantities
• Foundational organizational ability
• Foundational ability to problem solve
• Use of the 8 Standards for Mathematical Practice
What’s Unit 6 all about?
• Building from Kindergarten
http://commoncoretools.me/wpcontent/uploads/2011/05/ccss_progression_cc_oa_k5_2011
_05_302.pdf
http://commoncoretools.me/wpcontent/uploads/2012/02/ccss_progression_nf_35_2011_08
_12.pdf
What’s Unit 6 all about?
• Organizing thinking
What’s Unit 6 all about?
Quantity representation
Extending Children's Mathematics Fractions and Decimals:
Innovations in Cognitively Guided Instruction
By Susan B. Empson and Linda Levi
Quantity representation/Unit fractions
How much is shaded?
Extending Children's Mathematics Fractions and Decimals:
Innovations in Cognitively Guided Instruction
By Susan B. Empson and Linda Levi
They think…
• Fractions are pieces.
“A fourth is a little pie shape.”
• Fractions are always smaller than a whole.
“4/3? That’s impossible!”
• Fractions values are determined by counting parts
“It’s 1/3 because 1 part out
of 3 parts is shaded.”
Organizing Thinking:
Three approaches to teaching fractions
• Introduce procedures and explain concepts
• Emphasis on student discovery, with no conceptual analysis
of discoveries
• Discuss and extend concepts and procedures that come up
in children’s problem solving
(from Saxe et al., 1999)
Quantity Representation
• Meaning of fractions -- what does 1/3 mean?
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1 thing shared equally by 3 people, each person gets 1/3
1 candy bar for every 3 people
1 ÷ 3 = 1/3
1 part, with 3 equal parts to make a whole
These meanings generalize to improper fractions, too…..4/3
Quantity Representation
3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole
is partitioned into b equal parts; understand a fraction a/b as the quantity
formed by a parts of size 1/b.
For example: ¼ is the quantity created by partitioning the
whole into 4 equal parts,
and ¾ is the quantity formed by combining 3 parts of the
size ¼.
Understanding unit fractions
18
What’s Unit 6 all about?
• Unit 1, Third grade
What’s Unit 6 all about?
• Unit 2, Third grade
Quantity Representation
• Fractional units can be combined
– 1 third from one candy bar plus 1 third from another candy bar is 2 thirds
– 1/3 + 1/3 = 2/3
• Fractional units can be combined no matter how many there are
– 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 10/4
• Fractional numbers “fill in” the whole-number line
– 2 1/4 cookies is more than 2 cookies but less than 3 cookies
• A fractional amount can be expressed in many ways
Composition/Decomposition
What’s Unit 6 all about?
• Quantity representation/Understanding unit fractions
Understanding unit fractions
Seven “units” plus 4 “units” gives 11 “units”
Grade 2  Grade 7  Grade 9
70 + 40 
24
7
3
+
4
3
 7x+3x
Extending Children's Mathematics Fractions and Decimals:
Innovations in Cognitively Guided Instruction
By Susan B. Empson and Linda Levi
Reasoning about relationships…
Extending Children's Mathematics Fractions
and Decimals:
Innovations in Cognitively Guided Instruction
25 By Susan B. Empson and Linda Levi
Conceptual Understanding Matters
“Fractions offer students the
opportunity to grapple with
fundamental mathematical
relationships that constitute the
core of Algebra”
26
Extending Children's Mathematics Fractions and Decimals:
Innovations in Cognitively Guided Instruction
By Susan B. Empson and Linda Levi
What’s Unit 6 all about?
• Decomposition of number/modeling problem solving
http://www.k-5mathteachingresources.com
What’s Unit 6 all about?
Decomposition of number/modeling problem solving
What’s Unit 6 all about?
Decomposition of number/modeling problem solving
What’s Unit 6 all about?
Decomposition of number/modeling problem solving
What’s Unit 6 all about?
Modeling problem solving
Modeling problem solving
What’s Unit 6 all about?
• Multiple solution paths
What about teaching?
A teacher asks:
How do you envision how math class would be taught with the Common Core
Standards? I think many teachers teach math in a fairly “traditional” way –
instructing students on how to do (whatever) and then assign problems to be
completed. How is our “mode of business,” if you will, going to change?
Bill McCallum says:
Dear Teacher, I don’t see the standards as dictating any particular teaching
method, but rather setting goals for student understanding. Different people
have different ideas about what is the best method for achieving that
understanding. That said, I think it’s pretty clear that classrooms
implementing the standards should have some way of fostering understanding
and reasoning, and classrooms where students are just sitting and listening are
unlikely to achieve that.
One of the strongest results in recent research
is that the most important feature in effective
teaching is giving students "opportunity to
learn". Teachers can set expectations, time,
kinds of tasks, questions, acceptable answers,
and type of discussions that will influence
students' opportunity to learn. This must involve
both skill efficiency and conceptual
understanding.
Hiebert, James; Grouws, Douglas
The Effects of Classroom Mathematics Teaching on Students' Learning 2007
Horizontal and vertical connections
• Strategies apply everywhere, in multiple contexts
• Integration of content areas ensures connections
and relational thinking
• Multiple steps builds understanding and deeper
thinking
• Work the culminating task collaboratively with
colleagues so you know where your kids need to
go, and what they might have difficulty with
Evidence that Attending to Student Thinking
Can Make a Difference
• Students constantly surprise us…
– Kindergarten data
– Fractions
– Algebraic thinking
Extending Children's Mathematics Fractions and Decimals:
Innovations in Cognitively Guided Instruction
By Susan B. Empson and Linda Levi
Development of Children’s Mathematical Thinking
19 Children are taking a mini-bus to the zoo. They
will have to sit either 2 or 3 to a seat. The bus has 7
seats. How many children will have to sit 3 to a seat
and how many can sit 2 to a seat?
How will children solve it?
How about a kindergartener?
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59% had a correct strategy
51% had correct answer
33% 1st graders
26% 2nd graders
Extending Children's Mathematics Fractions and Decimals:
Innovations in Cognitively Guided Instruction
By Susan B. Empson and Linda Levi
Focus on Making Student Thinking Explicit
• need to be able to use student strategies as the center of the
workgroup conversation, as one of the tools teachers
interact with
• expertise shared by students
• change in power structures, from teacher as expert
• Journaling, anchor charts provide an explicit trace of the
group’s thinking
• extends to other communities of practice in other content
areas
Extending Children's Mathematics Fractions and Decimals:
Innovations in Cognitively Guided Instruction
By Susan B. Empson and Linda Levi
Focus on Making Student Thinking Explicit
• Ask children how they solved problems. Probe their
understanding.
• Introduce symbols, number sentences, and mathematical
language to go with strategies
• Don’t rush using and manipulating symbols- use the
multiple models available first
Extending Children's Mathematics Fractions and Decimals:
Innovations in Cognitively Guided Instruction
By Susan B. Empson and Linda Levi
Resources
• http://maccss.ncdpi.wikispaces.net/Third+Grade
• http://www.k-5mathteachingresources.com/
• https://www.teachingchannel.org/videos/common-core-statestandards-elementary-school?fd=1
• http://www.ocde.us/CommonCoreCA/Pages/Best-and-NextPractices-in-Mathematics.aspx
Call for Student Work Samples
As part of the continuing implementation of CCGPS in the year
2013 - 2014, the current GADOE mathematics units are being
augmented. We would like your assistance with this critical
process.
Student work samples are a vital component of the frameworks
which only you can provide. If you have been using the GADOE
frameworks and have student work which you would be willing to
share, please send it our way. We will remove any identifiers, and
include selected student work samples in the revised frameworks
which are slated to be released July 1, 2013. If your student work
sample is selected for inclusion we will notify you of its placement
in the units via email.
Call for Student Work Samples
Submission guidelines:
Attach the work sample(s) to an email in any format. Whatever works for you,
works for us.
Indicate the grade level, unit, and task in the body of your email, and on the
work sample in the upper left corner.
You may cover any student/school identifiers if you wish, and we will do the
same if any remain.
Send the email with student work attached to the appropriate team member.
To be considered for inclusion, work samples must be submitted by May 17,
2013.
We look forward to seeing your students’ work.
Career-Based Mathematics Task Challenge
As part of the continuing implementation of CCGPS in the year 2013 - 2014, the current
GADOE mathematics frameworks and units are being reviewed, revised, and augmented. We
are offering an opportunity for educators to assist in this critical process.
The challenge:
Create a career-based mathematics task using guidelines provided to supplement and/or
address gaps in the existing CCGPS frameworks units.
If your task is selected for addition to a unit, you will receive a $200 honorarium per task. All
work is to be original using support structures provided by the Georgia Department of
Education Mathematics Team.
If you are interested in participating in this challenge, please view the task creation guidelines
at http://ccgps-task-submission-guidelines.wikispaces.com/, and get started! Task submission
period begins now and closes May 1, 2013. We look forward to seeing your tasks.
2013 Resource Revision Team
As part of the continuing implementation of CCGPS, the current CCGPS mathematics
frameworks and units are being reviewed, revised, and augmented. The Georgia Department
of Education is seeking qualified math educators to become part of the 2013 CCGPS
Mathematics Resource Revision Team which will assist in this critical process.
The scope of the CCGPS Mathematics Resource Revision Team work will include, but is not
limited to:
• evaluating newly submitted tasks
• assessing the need for additional tasks
• assessing the order of current units and tasks
• editing of current units and tasks
• creating additional tasks to address gaps, if necessary
2013 Resource Revision Team
All work will be completed collaboratively with support structures provided by the Georgia
Department of Education Mathematics Team.
All work is to be completed at the Georgia Department of Education, June 3rd-June 6, and
June 10-13, 2013.
Team members will be compensated for contracted work in the amount of $2000 and travel
expenses will be reimbursed.
If you are interested in becoming a part of the CCGPS Mathematics Resource Revision Team,
please respond to the appropriate Georgia Department of Education contact below by March
1, 2013. In your response, please indicate grade level interest, why you would like to be
part of this team, related experience, and the contact information for two references.
Grades K-5
Turtle Toms tgunn@doe.k12.ga.us
Mathematics Program Elementary Specialist
Can you help us capture this?
http://www.surveymonkey.com/s/WZKG5G2
Thank you!
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Thank You!
Please visit http://ccgpsmathematicsK-5.wikispaces.com/ to provide us with
your feedback!
Turtle Gunn Toms
Program Specialist (K-5)
tgunn@doe.k12.ga.us
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