BUILDING CONCEPTUAL
KNOWLEDGE IN MATHEMATICS:
GRADES 3 - 5
Fluency Strategies
December 7, 2012
Presenters: Andrea Tamarazio & Steven Graser
Erie 1 BOCES
MAD MINUTE!
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READY, SET, GO
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When asked to begin, take one minute to complete as
many questions as you can
ANTICIPATORY GUIDE
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Read each statement
Circle True / False
 Write an sentence / comment about each statement
 Complete Frayer Model for “fluency”
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SPRINT!!
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READY, SET, GO!

When asked to begin, take one minute to complete as
many questions as you can
Sprint Video
SPRINT VS. MAD MINUTE
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Take a minute to complete the graphic organizer

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Likes and Dislikes of each activity
Share Out
MATHEMATICAL SHIFTS
MATHEMATICAL PRACTICES
Make sense of problems and persevere in solving them
 Reason abstractly and quantitatively
 Construct viable arguments and critique the reasoning
of others
 Model with mathematics
 Use appropriate tools strategically
 Attend to precision
 Look for and make use of structure
 Look for and express regularity in repeated reasoning
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FLUENCY PRACTICE (COMMON CORE INC.)
Fluency in designed to promote automaticity by
engaging students in practice in ways that get their
adrenaline flowing. Automaticity is critical so
students avoid using up to many of their attention
resources with lower level skills when they are
addressing lower level problems. Computational
foundations enables deep understanding.
SAMPLE TASK
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What fluencies do students need to solve this
problem?
See Handout for problem
COMMON CORE STANDARDS
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Professional Development Needs
Curriculum
 Assessments
 Instruction (Teacher Capacity)
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What Common Core aligned PD has your district
provided in regards to the items listed above?
MATH FLUENCY
Experts agree that the ability to recall basic math
facts fluently is necessary for students to attain
higher-order math skills.
*Drop off in student performance in the middle and high school years can be
attributed to math instruction at the elementary level. Many students
memorize facts and can pass elementary assessments, however the lack of
computational fluency becomes apparent in the secondary grades.
WARM UP: MENTAL MATH ACTIVITY

7 Questions
Solve mentally
 Explain the process used to solve each problem
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JIGSAW ACTIVITY
In small groups, read assigned articles
 Complete Graphic Organizer for your article
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MENTAL MATH
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Mental Math is one component of Computational
Fluency
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Understanding Basic Number Combinations
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Understanding Base Ten Number System
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Fact Families
Inverse Functions; solve for missing addend, subtrahend, and
minuend.
Understand and Use Properties
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10 times more, 10 times less
In the number 220, the 2 in the hundreds place is 10 times
greater than the 2 in the tens place
Recognize and Use related problems
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Number Bonds
Dot Plates
Commutative and Associative
Making Connections between operations
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Inverse Functions
MATH MODELS
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Place Value & Standard Algorithm
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Bundles
Place Value Chart
Base Ten Blocks
Money
Number Disks with Place Value Chart
Fractions
Number Line
 Area Model
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Multiplication
Array
 Area Model
 Rekenrek (Number Rack)
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Word Problems
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Bar Diagram
FLUENCY VS. MEMORIZATION
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Memorization
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Single procedure and / or recall
Fluency
Understanding of operations and their relationship to
each other (inverse function; fact families)
 Number Relationships (compatible numbers – fives
and tens)
 Base Ten System
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For Example: recognizing the relationship between 5 + 9
and 50 + 90
COMPUTATIONAL FLUENCY
Computational Fluency includes three ideas: efficiency,
accuracy, and flexibility
 Efficiency implies that the student does not get bogged down
in too many steps or lose track of the logic of the strategy. An
efficient strategy is one that the student can carry out easily,
keeping track of subproblems and making use of
intermediate results to solve the problem.
 Accuracy depends on several aspects of the problem-solving
process, among them careful recording, knowledge of
number facts and other important number relationships,
and double-checking results.
 Flexibility requires the knowledge of more than one
approach to solving a particular kind of problem, such as
two-digit multiplication. Students need to be flexible in
order to choose an appropriate strategy for the problem at
hand, and also to use one method to solve a problem and
another method to double-check the results.
ASSESSMENT OF COMPUTATIONAL
FLUENCY
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The student draws on basic facts and number
relationships
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Student uses the structure of the base ten system
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2 + 5 = 7; 20 + 50 = 70; 200 + 500 = 700
Student recognizes related problems
Students relationships among related operations
Student knows what each number and numeral
means in a problem
Student can explain the steps he or she uses to solve
even recall problems
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Properties
Same student explains twice
Student can clearly record and track his or her
procedure
Student may have multiple approaches to solve a
problem
“DRILL & KILL” ≠ COMPUTATIONAL
FLUENCY
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Memorization of math facts is not the same as
understanding number concepts
Regardless of time and energy spent with “drill &
kill”, some students will not master facts
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Memory
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Cognitive
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Memory problem interferes with ability to retrieve basic facts
quickly
Short Term Retention
Understanding relationship between numbers
Solving word problems
Understanding number system & effective counting strategies
Visual / Spatial
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Misalignment of numerals in a column
Problems with Place Value
COMMON CORE FLUENCY
FACT FLUENCY INSTRUCTIONAL
FLOW
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Number Relationships & Concepts
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40 – 50% of Instructional Time
Contextualizing math fact problems
Creating arrays and drawing problems
Decomposing or using known facts
Using compensation strategies
Fact Strategy Development
30 - 40% of Instructional Time
 Doubles, Near Doubles, Combinations to 10
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Drill
15 – 20% of Instructional Time
 Fact Fluency related games and flash cards
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Fluency Assessment
Less than 3% of Instructional Time
 Timed Test
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CCLS FLUENCY RECOMMENDATION
Distribution of Instructional Minutes
NUMBER RELATIONSHIPS & CONCEPTS
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Contextualize Math Facts
Reading, Writing, and Symbolizing Math Problems
 High Number Toss Game
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Creating Arrays and Drawing Pictures (word to
picture; number to picture; picture to word or
number)
Place Value Literacy
Decomposing Numbers & Listing their Values
 Number Line Up
 Build a Number
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FLUENCY ACTIVITIES
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Adding Mentally
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Whole Numbers, Fractions & Decimals
1, 2, 3, sit on 20
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This can be modified for multiple grade levels – skip
counting, multiplication facts, etc
Take the 1 Out
 Fluency Counting
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This can be modified for multiple grade levels
Skip Counting by Fractions
 White Board Equivalent Fractions
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Written or Oral
FACT STRATEGY DEVELOPMENT (CHORAL
WHITE BOARD)
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One-More-Than & Two-More-Than Facts
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Facts with Zero
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Doubles / Near Doubles (Doubles Plus 1, Doubles
Plus 2)
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Compensation (Plus 1, Minus 1; Plus 2 Minus 2)
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Compatible Numbers (Combinations to 5 or 10)
DRILLS
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Around the World
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Math Fact Race
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Math Fact War
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BINGO
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I Have You Have
ACTIVITY SHARE OUT


Take time to create activities to use in your
classroom
What do you have in your classroom that you
would like to share?

Activities

Strategies
QUESTIONS / COMMENTS
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If you have any questions or comments, please
feel free to contact us.
Andrea Tamarazio
atamarazio@e1b.org
Steven Graser
sgraser@e1b.org