BUILDING CONCEPTUAL
KNOWLEDGE IN MATHEMATICS:
GRADES 3 - 5
Fluency Strategies
December 7, 2012
Presenters: Andrea Tamarazio & Steven Graser
Erie 1 BOCES
MAD MINUTE!
READY, SET, GO
When asked to begin, take one minute to complete as
many questions as you can
ANTICIPATORY GUIDE
Read each statement
Circle True / False
Write an sentence / comment about each statement
Complete Frayer Model for “fluency”
SPRINT!!
READY, SET, GO!
When asked to begin, take one minute to complete as
many questions as you can
Sprint Video
SPRINT VS. MAD MINUTE
Take a minute to complete the graphic organizer
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
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
What fluencies do students need to solve this
problem?
See Handout for problem
COMMON CORE STANDARDS
Professional Development Needs
Curriculum
Assessments
Instruction (Teacher Capacity)
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
JIGSAW ACTIVITY
In small groups, read assigned articles
Complete Graphic Organizer for your article
MENTAL MATH
Mental Math is one component of Computational
Fluency
Understanding Basic Number Combinations
Understanding Base Ten Number System
Fact Families
Inverse Functions; solve for missing addend, subtrahend, and
minuend.
Understand and Use Properties
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
Number Bonds
Dot Plates
Commutative and Associative
Making Connections between operations
Inverse Functions
MATH MODELS
Place Value & Standard Algorithm
Bundles
Place Value Chart
Base Ten Blocks
Money
Number Disks with Place Value Chart
Fractions
Number Line
Area Model
Multiplication
Array
Area Model
Rekenrek (Number Rack)
Word Problems
Bar Diagram
FLUENCY VS. MEMORIZATION
Memorization
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
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
The student draws on basic facts and number
relationships
Student uses the structure of the base ten system
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
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
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
Memory
Cognitive
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
Misalignment of numerals in a column
Problems with Place Value
COMMON CORE FLUENCY
FACT FLUENCY INSTRUCTIONAL
FLOW
Number Relationships & Concepts
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
Drill
15 – 20% of Instructional Time
Fact Fluency related games and flash cards
Fluency Assessment
Less than 3% of Instructional Time
Timed Test
CCLS FLUENCY RECOMMENDATION
Distribution of Instructional Minutes
NUMBER RELATIONSHIPS & CONCEPTS
Contextualize Math Facts
Reading, Writing, and Symbolizing Math Problems
High Number Toss Game
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
FLUENCY ACTIVITIES
Adding Mentally
Whole Numbers, Fractions & Decimals
1, 2, 3, sit on 20
This can be modified for multiple grade levels – skip
counting, multiplication facts, etc
Take the 1 Out
Fluency Counting
This can be modified for multiple grade levels
Skip Counting by Fractions
White Board Equivalent Fractions
Written or Oral
FACT STRATEGY DEVELOPMENT (CHORAL
WHITE BOARD)
One-More-Than & Two-More-Than Facts
Facts with Zero
Doubles / Near Doubles (Doubles Plus 1, Doubles
Plus 2)
Compensation (Plus 1, Minus 1; Plus 2 Minus 2)
Compatible Numbers (Combinations to 5 or 10)
DRILLS
Around the World
Math Fact Race
Math Fact War
BINGO
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
If you have any questions or comments, please
feel free to contact us.
Andrea Tamarazio
atamarazio@e1b.org
Steven Graser
sgraser@e1b.org
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