Download Figure It Out: How to Achieve Computational Fluency

August 7, 2013
Elementary Math Summit
Beth Edmonds
Math Intervention Teacher
Wake County Public School System
[email protected]
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Participate actively.
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Have an open-mind.
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To become more familiar with Common Core
State Standards.
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To explore computational strategies.
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To investigate resources and tools.
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To help students achieve computational
fluency.
Solve these math problems:
28 + 32 =
200 – 86 =
15 x 7 =
90 ÷ 6 =
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Kindergarten: Add and subtract within 10,
demonstrating fluency for addition and
subtraction within 5.
1st Grade: Add and subtract within 20,
demonstrating fluency for addition and
subtraction within 10.
2nd Grade: Fluently add and subtract within
100, using place value and the properties of
operations.
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3rd Grade: Fluently add and subtract within
1,000; Fluently multiply and divide within
100.
4th Grade: Fluently add and subtract multidigit whole numbers using the standard
algorithm.
5th Grade: Fluently multiply multi-digit whole
numbers using the standard algorithm.
Fluency includes three ideas:
 Efficiency means that the student can carry
out the strategy easily.
 Accuracy depends on precise recording,
knowledge of number relationships, and
checking results.
 Flexibility requires the knowledge of more
than one approach to solve the problem and
to check the results.
“Developing Computational Fluency with Whole Numbers in the
Elementary Grades” – Susan Jo Russell
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An understanding of the meaning of the
operations and their relationships to each other.
The knowledge of a large “repertoire” of number
relationships.
An understanding of the base ten system and the
place value system.
“Developing Computational Fluency with Whole Numbers in the
Elementary Grades” – Susan Jo Russell
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How many dots did you see?
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How did you see that number?
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How many dots did you see?
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How did you see that number?
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How many dots did you see?
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How did you see that number?
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“The most common and perhaps most
important model” for number relationships is
the ten-frame.
“Students must reflect on the two rows of
five, the spaces remaining, and how a
particular number is more or less than 5 and
how far away from 10.”
(Van de Walle, 2006)
 http://illuminations.nctm.org/acti
vitydetail.aspx?id=75
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“Every child’s journey in math begins with
counting. But it is the next step - moving
from counting to adding - that is critical.
Unfortunately, many kids never make the
transition from counting numbers one at a
time to thinking abstractly and efficiently in
groups. No wonder they find math difficult!”
(Greg Tang)
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Ask: Where are groups of counters that could
help us count more efficiently?
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Counting On is a difficult skill for many
children.
A child will begin with the first number and
count on or begin with the larger number and
count on.
Once mastered some children do not
progress beyond the Counting On strategy.
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http://www.brainpopjr.com/math/add
itionandsubtraction/makingten/
“Knowing how to make ten is a fundamental
building block for more complex mental math
in our base-ten system.”
Games are an important way to practice
making 10.
Make ten by subtracting the required amount from
one addend and adding it on to the other addend.
Example: 9 + 4 can become . . . .
Make ten by subtracting the required amount from
one addend and adding it on to the other addend.
Example: 9 + 4 can become 10 + 3.
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Commutative Property of Addition:
If 8 + 3 = 11, then 3 + 8 = 11.
Associative Property of Addition:
To add 2 + 6 + 4, the second two numbers
can be added to make a ten.
So, 2 + 6 + 4 = 2 + (6 + 4) = 2 + 10 = 12.
Common Core State Standards, Grade 1
57+34=
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Adding By Place:
50+30=80, 7+4=11, 80+11=91
Keeping One Number Whole and Adding On the
Other Number in Parts:
57+30=87, 87+4=91
Compensation Involves Adjusting Numbers to
Make Them Easier to Work With:
(57+3)+34 = 60+34 = 94
94-3 = 91
(57+3)+(34-3) = 60+31 = 91
Solve: 48 + 25
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48 + 25
40 + 20 = 60
8 + 5 = 13
60 + 13 = 73
Did you use important math knowledge
about:
◦ Place value (how to break two-digit numbers into
tens and ones)?
◦ Addition (how to take the addends apart and put
the parts back together in any order)?
Relearning to Teach Arithmetic: Addition and Subtraction (TERC)
+20
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48
+5
68
48 + 25 =
73
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Larger Numbers: 652 + 131?
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Fractions: ¾ + ½ ?
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Decimals: 1.2 + 0.9 ?
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“Subtraction facts prove to be more difficult
than addition. (Van de Walle, 2006)
How can we apply what we know about
addition to subtraction?
◦ Place Value
◦ Properties of Operations
◦ The Relationship Between Addition and Subtraction
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Join or Separate on a Mat
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Join or Separate Bars of Cubes
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Hops on a Number Line
0
1
2
3
4
5
6
7
8
9
10
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Connecting Subtraction to Addition
Knowledge
Start with 9 counters. Place them under a
cover. Remove 6 from under the cover so
they are visible. Ask: How many are covered?
(Van de Walle, 2006)
What goes with the part I
see to make the whole?
Six and what makes 9?
What goes with 6 to make
9?
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Children use known addition facts to
determine the unknown quantity or part.
It is essential for addition facts to be
mastered first if the think-addition strategy is
to be used effectively.
(Van de Walle, 2006)
72 – 49 =
Think: 49 + ____ = 72
49 + 1 = 50
(1)
50 + 20 = 70 (20)
70 + 2 = 72
+1
49 50
(2)
1+20+2 = 23
+20
+2
70
72
72 - (40+9)
72 – 40 = 32
72 – 9 = 23
72 – 49 =
72 – 40 = 32
32 – 10 = 22 (Made an easier problem –
Subtracted 1 too many)
22 + 1 = 23 (Added 1 back to the answer)
(Partners for Mathematics Learning, Elementary K-2 Number Module 2008-2009)
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Subtraction determines the space between
two numbers. If both numbers are adjusted
equally, then the answer to the problem stays
the same.
Example:
81
- 57
+3
+3
84
- 60
24
92
- 35
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Consider: “You can’t take a big number from
a little number.” This is not true because it
will produce a negative number, but it can be
done.
What should we say instead? “If we have only
2 ones, then we need to get access to more
ones. We need to ungroup one of the tens to
get access to more ones.”
(“Supporting the Struggling Student in Math: Emphasize the
Mathematics, not the Tricks” Valerie Faulkner, 2008)
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Designed by the teacher to help students develop an
understanding of number relationships while working
on computation strategies
Each number sentence is shown one at the time.
The students discuss how to solve each one.
The teacher encourages students to find a way to use
the previous number sentence to solve the next one.
The students will be composing and decomposing
numbers flexibly as they work.
Provide opportunities for students to use number
lines and hundred boards in a meaningful way.
(Partners for Mathematics Learning, Grade 2, Module 6)
16 + 10 = ______
16 + 10 = ______
16 + 9 = ______
16 + 10 = ______
16 + 9 = ______
16 + 19 = ______
16 + 10 = ______
16 + 9 = ______
16 + 19 = ______
16 + 29 = ______
To explore the use of
adding 10 and the
understanding that
adding 9 is like
adding 10 and then
subtracting 1.
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“Because figuring in our heads is such an
important life skill, it should have a regular role
in your classroom math teaching.”
Choose problems that can be solved mentally or
have students find an estimate rather than the
exact answer.
Have students explain their ideas and listen to
their classmates.
(“Marilyn Burns: Mental Math”, Instuctor Magazine, 2007)
26 + 57 =
 If you have 65, how much more is
needed to make 100?
 How could you rename 578 as the
sum of two smaller numbers?
 Which answer is closest to the
product?
148 x 21 (1,000; 2,000; 3,000; 4,000)
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(“Marilyn Burns: Mental Math”, Instuctor Magazine, 2007)
 www.gregtangmath.com
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Greg Tang’s goal is to revolutionize the way
children learn math.
He uses rhymes, books, and games to show
that there is much more to computation than
memorization.
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Children must move from additive
understanding to multiplicative
understanding (the ability to think about and
operate with collections as a unit).
It is important that children see the
relationship between addition and
multiplication and the relationship between
multiplication and division.
“Experiences with making and counting
groups, especially in contextual situations,
are extremely useful.” (Van de Walle, 2006)
0
1
2
3
4
5
6
7
8
9
10 11 12
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The Commutative Property:
The array is a powerful illustration of the order
property. Turnaround facts should always be
learned together. (Ex. 2 x 8 = 8 x 2)
The Zero Property:
Make up story problems to go with these
problems.
(Ex. 5 x 0 = 5 hops of zero on the number line)
(Van de Walle, 2006)
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The Identity Property:
“One is simple as can be, it’s known as the
identity. The answer to identify? It’s the one to
multiply!”
(Greg Tang, The Best of Times)
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The Distributive Property:
It is very useful to relate one basic fact to
another. The array model can be used to
illustrate that a product can be broken up into
two parts.
(Van de Walle, 2006)
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7 x 5 = (5 x 5) + (2 x 5)
= 25 + 10 = 35
5x5
2x5
Seven doesn’t take much
time,
even though it is prime.
Here is all you have to
do,
first times 5 then add
times 2!
(Greg Tang, The Best of Times)
(Van de Walle, 2006)
4 x 63
4 x 63
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63 + 63 + 63 +63 (Repeated Addition)
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2 x (2 x 63)
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(Double Twice)
(4 x 60) + (4 x 3) (Partition 63 into 60 + 3
and use the Distributive Property)
(Partners for Mathematics Learning, 3-5 Number & Operations
Module 2008-2009)
How is 4 x 5 related to 4 x 50?
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“Avoid saying ‘add a zero’ because when you
actually add a zero, you are not changing a
number’s value - Here you are not ‘adding’ a
zero, you are multiplying by ten!”
(“Supporting the Struggling Student in Math: Emphasize the
Mathematics, not the Tricks” Valerie Faulkner, 2008)
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The biggest problems created by teaching the
steps of “standard” algorithms are:
The focus is on individual digits, not on
whole quantities and relationships among the
quantities.
The central goal is mastering a series of steps
rather than understanding the meaning and
use of the mathematical operation.
Relearning to Teach Arithmetic: Addition and Subtraction (TERC)
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Teachers should provide drill in the use and
selection of computational strategies after
they have been developed.
Practice: Problem-based activities in which
students develop flexible and useful
strategies.
Drill: Repetitive non-problem-based
activities. Students have a strategy they
understand and know how to use. Drill helps
to make the strategy more automatic.
(Partners for Mathematics Learning, Elementary K-2 Number Module
2008-2009)
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Different Strategies
Mental Math
Number Talks
Games
Computation Probes (Intervention Central)