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Computational Strategies
presented by
Kim Sutton
Creative Mathematics
4001 West End Road
Arcata, CA 95521
1-800-841-5193
www.creativemathematics.com
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© 2008 Kim Sutton
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Teach for Understanding
Two Models for Operations
Addition
Addition
•
Combining Quantities
•
Number Line Growth
Subtraction
Subtraction
•
Take Away
•
Difference Between
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© 2008 Kim Sutton
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Multiplication
Multiplication
•
Repeated Addition ( ___ groups of ___ = ___ )
4 groups of 3 = 12
•
Area or Array
4
3 12
Division
Division
•
Repeated Subtraction
In 12, how many groups of 3 are there?
•
Area to Length of Side Relationship
?
3 12
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Number Line Petite to 30
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The Real Estate Game
Name
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The Ten Most Important
Computational Strategies
• Commutative Property
The operations of addition and multiplication have the property
that allows for a 50% reduction of memorization. A + B = B + A
and A x B = B x A is the understanding of the commutative property. What is commonly missed is that students must repeatedly
practice this to recognize the commutative property. (Finger Flips)
• Even and Odd Rules of Computation
The even and odd rules allow for generalizations that can be
applied to more complex mathematical situations. There are only
four events when adding (no matter the number of addends):
even + even = even
odd + odd = even
odd + even = odd
even + odd = odd
There are only four events when multiplying (no matter the number of factors):
even x even = even
odd x odd = odd
odd x even = even
even x odd = even
• Properties of Zero
Zero is the identity element for the operations of addition and subtraction. It is painful to watch students compute with deep
thought addition and subtraction with zero. Dr. Lola May coined
the phrase, “Zero is a hero!” because of its unique properties.
Multiplying by zero negates any group of. Dividing by zero is
impossible.
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• Properties of One
Adding or subtracting one should conjure up a number line in a
student’s mind’s eye. One is the identity element of multiplication.
This is key later on for understanding equivalent fractions. One
will not change the other factor in a multiplication equation.
There are many names for one whole. This creates the understanding for equivalent fractions.
• Combinations that Make a Ten
Knowing addition combinations that make a ten is pivotal not only
for computation but also for place value understanding. To scaffold mathematical skills, addition combinations making a ten will
lead to multiples of tens to make a hundred and then to mental
math skills.
• Working with Tens, Hundreds, Thousands
Knowing the shortcuts to working with a ten will allow students to
compute with multiples of ten, hundreds and thousands. Working
with a ten using the properties of zero and one with digits along
with place value understanding.
• Addition Doubles and Doubles Plus/Minus One
Addition doubles can be quickly learned due to the “sing song”
pattern often before understanding addition. Knowing your doubles will facilitate times two facts in multiplication. Every even
number is the result of a double in addition or a times two fact.
The most difficult addition facts are doubles plus one.
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• Groups of Twos, Fives, and Tens
Students become very familar with skip counting from kindergarten on. Skip counting twos, fives, and tens are comfortable
patterns before formal multiplication is taught. Based on these
familar patterns, multiplication practice sequences should be 2, 5,
10, 0, 1, (11), 3, 6, 9, (12), 4, 8, 7 for the best results.
• Patterns of Three, Six, Nine
The patterns of digital roots create an exciting and useful pattern
for students. The digital roots of multiples of three are 3, 6, 9. If
this pattern is ignored students will lose the opportunities that
digital roots provide in checking all computation and factoring.
• Inverse Operations
To make powerful connections in computation, students must
make the triangular relationship with the operations.
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© 2008 Kim Sutton
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Zero, Zero
Ron Brown
Math Math Math
Zero, zero!
Zero, zero!
In the place value line
It can hold a space.
Zero, zero!
Zero has a power
In every place.
Zero, zero!
When you add a zero
The answer stays the same.
You don’t have to think
Or use your brain.
Zero, zero!
Subtract a zero
The answer stays the same.
You don’t have to think
Or use your brain.
Zero, zero!
Zero, zero!
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Examining the Addition Table
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Examining the Multiplication Table
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Circle The Number
Name
Date
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
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© 2008 Kim Sutton
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Take Two for Ten
MATH TOOLS
• playing cards
• a partner
This game is to practice combinations of playing cards that add up
to 10.
This game can be played as a solitaire
game or with a partner. Each player will shuffle his/her deck
of cards. The cards are placed in 10 piles face up in two rows
in front of each player.
Students are “scanning” for two addends that make a sum of
10. Play should not be stopped. If a player can not find two
cards that make a sum of 10, a pile should be shuffled and a
new top card exposed. An “ace” is a one and all face cards are
zero. If the deck is played correctly, the check will be eight
cards left that are all face cards. A “joker” can have a value of
10.
If played competitively, the partners will race through the deck
using a timer. First player getting to the check situation, is the
winner.
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Subtraction Strategies
Name
Date
1.
10
11.
10
2.
10
12.
10
3.
10
13.
10
4.
10
14.
10
5.
10
15.
10
6.
10
16.
10
7.
10
17.
10
8.
10
18.
10
9.
10
19.
10
10.
10
20.
10
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Subtraction Strategies
Name
Date
1.
13
2.
18
3.
15
4.
17
5.
14
6.
11
7.
16
8.
13
9.
17
10.
14
11.
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12.
17
13.
15
14.
12
15.
16
16.
17
17.
15
18.
12
19.
15
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15
18
20.
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Doubling
Numbers
14 18 4 10 0
6 14 16 8
16 2
6
14 10
12 16 18 4 14
10 12 2
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Doubles +1
Name
Date
1.
11.
2.
12.
3.
13.
4.
14.
5.
15.
6.
16.
7.
17.
8.
18.
9.
19.
10.
20.
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Math Tools:
• Kim’s Number Line
with colored dots
• Number Line Petite
• pattern stick books
• “magic finger of math”
• transparent chips
• double dice
• pointer(s) for number line
Mathematical Vocabulary
multiples
factor
product
skip counting
growth pattern
Math Literature
Cat Up a Tree
by John and Ann Hassett
(multiples of 5)
Arctic Fives Arrive
by Elinor Pinczes
(multiples of 5)
Building the Power of Groups
The first growth pattern that children meet is
counting numbers. A class number line is the
most important visual as a classroom tool.
Early learners will work with counting forward
and backward with counting songs and activities. Some of Ron Brown’s counting songs
include:
• Do You Like to Count?
(Math Concepts I & II)
• Let’s Count to 30 (Math, Math, Math)
• The Counting Creatures
(The Learning Ride)
To introduce the idea of multiples the teacher
will use objects that come in a constant of count
to create a picture in the mind’s eye. To introduce multiples of two, I recommend playing the
game called “The Stand Up Game.” One student
stands up. The teacher directs the activity by
asking, “How many students are standing?”
The relationship between the number of students standing to the number of eyes is made
through the meaning of multiplication in
groups.
A High Fiving Gift for Mom
by Judy Bradbury
(multiples of 5)
Reese’s pieces Count by
Fives
by Jerry Pallotta
(multiples of 5)
The Cheerio Counting Book
by Barbara McGrath
(multiples of 10)
As the game is played the teacher will add a
red dot above the multiples of two. Students
will add a red dot above the multiples of two on
the Number Line Petite (Number Line
Workbook). Students can also fill in the pattern stick for twos.
Double Bubble Trouble
by Judy Bradbury
(multiples of 2)
Introduce Cat Up a Tree by John and Ann
Hassett to introduce counting in groups of or
skip counting with the students. Students
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should build a five stick as the story is being read. Yellow dots will be added
for multiples of five on Kim’s Number Line. Other books can introduce other
skip counting patterns.
Students should construct the pattern stick books for the skip counting patterns. Kindergarten and first graders should have 1, 2, 5, and 10 sticks.
Second graders should have 1, 2, 3, 5, and 10 sticks. Students in third
through sixth grades should have 1-12 sticks.
The pattern sticks will help imprint the multiples through choral chanting.
This is an auditory form of skip counting. The teacher will model directing
the group through the skip counting of one multiple. The students will touch
the pattern stick with the “magic finger of mathematics” as they say and see
the patterns of skip counting.
A color-coded number line can be used as a larger visual tool for the classroom. This extends the skip counting. For primary students, the number line
would start at zero and go to one hundred. For older students, the number
line would start at zero and go to one hundred forty four. The number line is
color-coded so that all of the multiples of two would have the same colored dot
above them. Each different multiple would have a different colored dot above
it. The color patterns are:
2---red
3---green
4---orange
5---yellow
6---light blue
7---neon orange
8---neon green
9---black
10---navy blue
11---purple
12---gold star
The “Pattern Stick Game” is a simple game to play for the “over and over” practice with the meaning of multiplication and decisionmaking regarding addition
and the diffference between.
Each player needs a designated pattern stick (2-12), double dice, transparent
chips and a partner. Players will take turns. The first player will roll the double dice. The player can deside to add or compute the difference between the
numbers on the two dice. That answer is inserted into the meaning statement
for multiplication, “_____ groups of _____ = _____ .” The player must state that
complete equation. The product will be covered up. The objective is to be the
first to cover the complete stick.
Let’s say the teacher wants play to take place on the “two stick.” For example, a
player might roll a three and a four. Those two numbers can be added or the difference between can be computed. If the three and four are added, the player would
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say, “Seven groups of two equal fourteen.” The fourteen would be covered. If the
fourteen is covered, the player would say, “The difference between three and four is
one, one group of two is two.” This game should be played often.
The Random Number Game Boards will practice any operation using a drill
command for the operations of addition, subtraction, multiplication or division.
Each player should have chips and a game board. If the drill command is
groups of five, then each player writes the groups of five facts on the
response board. The products are written in the objects on the game board.
One game board is used between two students. Decide order of play. The first
player rolls the double dice. That player covers the resulting product of
inserting the total of what is rolled into the groups of five meaning statement.
Play continues until all objects are covered. The “bump it” rule can be used
when a players rolls a number where all the products for that combination
are covered. The random number CD can be used instead of dice or spinners.
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© 2008 Kim Sutton
Pattern Sticks
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Date
Name
Race Track Facts 0-12
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Date
Name
Race Track Facts 0-18
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First Difference Wins!
0
1
2
3
4
5
Difference Winners
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1
2
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25
5
Difference Winners
4
6
First Difference Wins!
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8
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Dogs Diggin’ for Bones (+)
Name
Date
13
18
13
12
15
4
17
9
17
5
16
11
7
12
14
15
18
14
6
8
10
10
3
18
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Dogs Diggin’ for Bones (x)
Name
Date
64
100
60
40
32
216
36
144
25
27
150
72
180
80
24
75
54
45
90
18
6
8
108
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Partners For Ten
Ron Brown
Addition
I need a ten.
I need a ten.
It’s so easy with the number
friends.
And, all I have to do is remember
them.
The partners for ten!
10+0
9+1
8+2
7+3
6+4
5+5
The partners for ten!
I need a ten.
I need a ten.
It’s so easy with the number
friends.
And, all I have to do is remember
them.
The partners for ten!
10+0
9+1
8+2
7+3
6+4
5+5
The partners for ten!
I need a ten.
I need a ten.
I need a ten.
I need a ten.
Ten!
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Hang Ten
Ron Brown
Math Math Math
When you add a single digit
With the number ten.
Here’s a little trick
That’s happening.
Write that single digit
In the one’s place now.
Then write a 1 down
In the ten’s place, wow!
10+0
10+1
10+2
10+3
=
=
=
=
10
11
12
13
Hang ten, hang ten, hang ten!
10+4 = 14
10+5 = 15
10+6 = 16
10+7 = 17
10+8 = 18
10+9 = 19
Hang ten!
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Let’s Tally Man!
Ron Brown
Math Beats
Let’s tally man!
Let’s tally man!
Tally 1, 2, 3, 4,
Diagonal
And make a group of five.
When you’re counting up things,
And you want to keep track
Use the tally system
It’s as easy as that.
When your tally is done
And you need an amount,
Count the groups by five
Just count, count, count.
Just make a little line,
For each thing that you count.
When you get to five,
You’ll be doing so fine,
Just make that mark a diagonal
Line.
If any are left,
Just add those ones to the score.
You’ll be doing so fine,
When it’s tally time,
Forever more.
Tally 1, 2, 3, 4,
Diagonal
And make a group of five.
Tally 1, 2,
Diagonal
And make
Tally 1, 2,
Diagonal
And make
3, 4,
a group of five.
3, 4,
a group of five.
Let’s tally man!
Let’s tally man!
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In The House With Eight
Ron Brown
Multiplication
Eight’s in the house!
Put your hands together
No time to wait.
We’re gonna learn the facts
That go with eight.
No time to lose.
No time to waste.
Get in the house with eight.
8x0 is 0
8x1 is 8
8x2 is 16
8x3
24
8x4
8x5
8x6
In the house
32
40
48
with eight!
8x7 is
8x8
8x9
8x10
8x11
8x12
56
64
72
80
88
96
In the house
You’re doin’ great.
In the house with eight!
(You’re in the house.
You’re doin’ great.
In the house with eight.
In the house with eight.)
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iPod Inspirations
Celebratory
Songs
Fluency Music
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
New Ideas Music
Motivational Music
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
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