Transcript Visual Models in Math: Connecting Concepts with Procedures

Visual Models in Math
Connecting Concepts with Procedures for Fraction Addition and
Subtraction
Tuesday, March 3, 2015
Presented by Sara Delano Moore, Ph.D.
Director of Mathematics and Science at ETA hand2mind
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Visual Models in Math: Connecting
Concepts with Procedures
March 3, 2015: Fraction
Addition and Subtraction
Sara Delano Moore, Ph.D.
Director of Mathematics & Science
ETA hand2mind
Visual Models in Math:
Series Overview
• January 6: Connecting Concepts with Procedures
Overview
• February 3: Connecting Concepts with Procedures for
Whole Number & Decimal Addition & Subtraction
• March 3: Connecting Concepts with Procedures for
Fraction Addition & Subtraction
• April 7: Connecting Concepts with Procedures for
Whole Number & Decimal Multiplication & Division
• May 5: Connecting Concepts with Procedures for
Fraction Multiplication & Division
• Effective teaching of mathematics builds
fluency with procedures on a foundation of
conceptual understanding so that students,
over time, become skillful in using procedures
flexibly as they solve contextual and
mathematical problems.
PtA, page 42
• Being fluent means that students are able to
choose flexibly among methods and strategies
to solve contextual and mathematical
problems, they understand and are able to
explain their approaches, and they are able to
produce accurate answers efficiently.
PtA, page 42
Hands-On Learning Instructional Cycle
C
Concrete
R
A
Representational
Abstract
10
Key Ideas for Fraction
Addition & Subtraction
• Reminder: procedural focus in this series
• Equivalence
– Changing units
• Regrouping
– Composing & decomposing
• Strategies & Methods
– Moving from concrete to abstract
Equivalence
• A number can be
named/described in many
different ways.
• Depending on the
situation, it can be helpful
to name/describe a
number differently
• Equivalent fractions can
be created by
representing the same
value using a different
unit fraction
Regrouping Fractions
• A whole is created by a
complete set of unit
fractions
• A fractional number can
be composed or
decomposed by
regrouping the unit
fractions from which it
is built.
4
4
7
4
Strategies and Methods
•
•
•
•
Making a whole
Representing with the same unit
Partial sums for mixed numbers
Estimation is still helpful!
Moving from Concrete to Abstract:
Common Denominators
3 4
2 1 
5 5
3 4
(2  1)  (  ) 
5 5
7
3 
5
2
3  (1  ) 
5
2
4
5
Moving from Concrete to Abstract:
Common Denominators
3 4
2
2 1  4
5 5
5
0
1
2
3
4
5
Moving from Concrete to Abstract:
Common Denominators
1 3
2  
4 4
1 1 2
2 (  ) 
4 4 4
1 1 2
(2  )  
4 4 4
2
2 
4
1
1
2
Moving from Concrete to Abstract:
Uncommon Denominators
1 3
 
2 4
2 3 5
1
  1
4 4 4
4
4 6 10
2
  1
8 8 8
8
6 9 15
3
  1
12 12 12 12
What do I call it?
Moving from Concrete to Abstract:
Uncommon Denominators
1 3
1  
2 4
1 1 1
1 (  ) 
2 2 4
1 1 1
(1  )  
2 2 4
1
1 
4
3
4
Should one strategy come first?
• Same unit first means we can use our other
strategies to find a solution
• Hence, the standard algorithm typically finds
same unit first.
• It doesn’t matter what the same unit is,
except from an efficiency perspective.
The operation doesn’t change;
the way we record it does.
Visual Models in Math:
Series Overview
• January 6: Connecting Concepts with Procedures
Overview
• February 3: Connecting Concepts with Procedures for
Whole Number & Decimal Addition & Subtraction
• March 3: Connecting Concepts with Procedures for
Fraction Addition & Subtraction
• April 7: Connecting Concepts with Procedures for
Whole Number & Decimal Multiplication & Division
• May 5: Connecting Concepts with Procedures for
Fraction Multiplication & Division
Join our community on edWeb.net
Building Understanding in Mathematics
 Invitations to upcoming webinars
 Webinar recordings and resources
 CE quizzes
 Online discussions
Join the community
www.edweb.net/math
Recognition for your participation today!
Attending Live?
Your CE Certificate will be
emailed to you within 24 hours.
CE Certificate provided by
Viewing the Recording?
Join the community at www.edweb.net/math
Go to the Webinar Archives folder
Take the CE Quiz to get a personalized CE Certificate
Join us for the next webinar
Tuesday, April 7th – 4 PM Eastern Time
Visual Models in Math:
Connecting Concepts with Procedures for Whole Number &
Decimal Multiplication and Division
For an invitation to the next webinar
Join Building Understanding in Mathematics
www.edweb.net/math
Thank you!
www.hand2mind.com