June 2012
“Understanding is a measure of the quality and quantity of
connections that an idea has with existing ideas.”
(VandeWalle, Pg. 23)
1.
2.
3.
4.
5.
Conceptual understanding
Procedural fluency
Strategic competence
Adaptive reasoning
Productive disposition
Source: Common Core (Pg. 6)
Definition:
Number sense is a “good intuition about
numbers and their relationships. It develops
gradually as a result of exploring numbers,
visualizing them in a variety of contexts, and
relating them in ways that are not limited by
traditional algorithms.” (Van de Walle, Pg. 119)
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According to a 2007 Trends in International
Mathematics and Science Study, Singapore
students are among the best in the world in math
achievement.
The Singapore model drawing approach bridges
the gap between the concrete and abstract
models we tend to jump to in the US.
Model drawing reinforces students’ visualization
and understanding of math processes.
Model drawing can be used effectively to solve
80 percent of problems in all texts.
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Computation is about students comprehending
what they are doing, not following a set of rules.
Students need to understand both what to do
and why.
Students will have a variety of strategies to solve
problems.
Changes students attitudes toward math and
problem solving.
Language based learning- think alouds and math
talks
Can be used as a supplement to an adopted
curriculum
Singapore Math Video
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Instruction begins at the concrete level with
manipulatives to build understanding of basic
concepts and skills. (begin with proportional
manipulatives)
Then students are introduced to the pictorial
stage: model drawing.
Students are not introduced to formulaic or
algorithmic procedures, the abstract, until
they have mastered model-drawing.
Discuss.
Catch thinking on chart paper.
Write a summary paragraph.
1)
2)
3)
4)
5)
Teacher modeling and thinking aloud about
the strategy
Students practice with the teacher
Students practice in small groups
Students practice in partners
Independent practice
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Purpose: Modeling, communicating, promoting a
more efficient strategy, promoting reasoning,
moving to a more sophisticated level of thinking.
“I’m thinking…” “I’m wondering…” “What are
you thinking?” “How did you figure that out?” “Is
there another way?” “Why did you choose this
way?” “How do you know this answer is correct?”
“What would happen if?”
Model clear, explicit language about concepts
Mathematical thinking and language promote
more understanding than memorization or rules
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346+475=
First, I’m going to add the hundreds. That
means 300+400=700. Now I will add the
tens, four tens (40) plus seven tens (70)
equals eleven tens. I can make 110. So now I
have 700+100+10. Now I will add the ones
and 6+5 makes 11. This is one ten and one
1 so I have 700+100+10+10+1. My answer
is 821.
Place Value Talk is critical!
Purpose: “Unstick” someone, get help, clarify,
promote deeper thinking, make connections
Purpose: Promote productive math
conversations
Example: Practicing questions in multiplechoice format
 Step 1: Solve individually. Write down your
answer.
 Step 2: Compare. Same or different?
 Step 3: Explain why you chose that answer.
Purpose: Use words that proficient
mathematicians would use, make connections
Example: End of Year Jeopardy Review Game.
 500 point question: What is addition?

Look at their work.
Do the model, the picture, and the equation match
the question and each other?
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Read or listen to their explanation.
Ask a math question.
Seek professional help – Ask a student
expert, the teacher, or other adult.
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Place Value Strips
Place Value Disks
Place Value Chart
Number-bond cards
Part-whole cards
Decimal Tiles
Decimal Strips
Gratiot Isabella ISD Maniplatives Link
Begin with no regrouping
 Sequence
1. Number bonds
2. Decomposing numbers
3. Left-to-right addition
4. Place value disks and charts
5. Vertical addition
6. Traditional addition
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Spatial Relationships
pattered arrangements
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One and Two More, One and Two Less
counting on and counting back
7 is 1 more than 6 and it is 2 less than 9
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Anchoring Numbers to 5 and 10
using 5 and 10 to build on and break from is
foundational for working with facts
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Part-Part-Whole Relationships
understand that a number is can be made of 2 or
more
parts
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A Pre-Place Value Relationship with 10
11 through 20, think 10 and some more
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Extending More and Less Relationships
i.e. 17 is one less than 18 like 7 is one less than
8
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Double and Near-Double Relationships
special cases of the part-part-whole construct
use pictures
Build Up and Down through 10
Break Apart to Find an Unknown Fact
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Estimation and Measurement
More or less than _______?
Closer to _____ or _____?
About _______.
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More Connections
Add a Unit to Your Number
Is It Reasonable?
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Graphs
Make bar graphs and pictographs
Big Ideas:
1. Number relationships provide the foundation
for strategies that help children remember
the basic facts. (i.e. relate to 5, 10, and
doubles…)
2. “Think addition” is the most powerful way to
think of subtraction facts.
3. All of the facts are conceptually related. You
can figure out new or unknown facts from
those you already know.
4. What is mastery? 3 seconds or less
10
10
10
10
1
1
40
1
1
1
7
1
1
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33+56=
Decompose each number by place value.
Put the tens together and one ones together.
(30+50) + (3+6)
30
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50
3
6
(30+50) + (3+6)
80
+ 9 = 89
35
+26
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10
10
10
10
10
10
1
1
1
1
1
1
1
1
1
1
1
35
+26
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10
10
10
10
10
10
10
1
1
1
1
1
1
1
1
1
1
1
35
+26
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10
10
10
10
10
10
10
1
6
1
36
+28
50
+14
64
Add from left to right.
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38
+85
123
Begin with no regrouping
 Sequence
1. Number Bonds
2. Place Value Disks and Charts
3. Traditional Subtraction
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10
10
86-8
10
10
10
10
10
10
1
1
1
1
1
1
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10
10
86-8
10
10
10
10
10
10
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
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10
10
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86-8
10
10
10
10
1
10
1
1
1
1
1
70
+
1
1
1
1
1
1
1
1
8
= 78
86-8=78
1
1
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54
- 28
26
Before “memorizing” multiplication facts,
students must first understand the concept of
multiplication----they must know it is
repeated addition with special attention to
place value
 Stages
1. Number bonds
2. Place value disks and charts
3. The distributive property
4. Area model
5. Traditional multiplication
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16
4
16
16
or
4
4
16
2
4
16
8
Vocab: Factor x Factor=Product
1
16
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141x3
100
10
100
10
100
10
10
10
10
10
1
10
10
1
10
10
1
10
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141x3
100
100
10
100
10
100
10
10
10
10
10
1
10
10
1
10
10
1
10
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141x3
100
100
1
100
1
10
100
4
141x 3=423
2
10
1
3
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45x3
(40x3) + (5x3)
120 + 15 = 135
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6x33
30
6 180
3
18
6x33=180+18
180+18=198
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15x26
20
6
10 200
60
5
30
100
15x26=(200+100)+(60+30)
300
+
90 =390
15x26=390
Begin by introducing division as repeated
subtraction
 Use number bonds to demonstrate the inverse
relationship of multiplication and division
 Sequence for teaching
1. Number bonds
2. Place value disks and charts
3. The distributive property
4. Partial quotient division
5. Traditional long division
6. Short division
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9
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27
?
What should the second factor be?
Vocab: Dividend ÷ Divisor=Quotient
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64÷5
Step 1 Build with disks
10
10
10
10
10
10
1
1
1
1
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64÷5
Step 2: The divisor tells us how many groups we
need: Draw 5 rows
10
10
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1
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2
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3
4
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5
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10
10
10
10
1
1
1
1
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64÷5
Step 3: Begin with the tens. Do we have enough
tens to put one in every row? Yes
10
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1
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2
10
10
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3
4
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5
10
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10
10
1
1
1
1
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64÷5
Step 4: If there are any large disks left, trade them
for an equivalent value of smaller disks
10
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1
10
10
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3
4
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5
10
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1
1
1
10
2
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1
10
1
1
1
1
1
1
1
1
1
1
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64÷5
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Step 5:Divide the ones equally among the groups
1
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1
2
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3
4
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5
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1
1
10
1
10
1
1
10
1
1
10
1
1
10
1
1
1
1
2
10+2=12
64÷5=12 r 4
1
345÷3=
 345=300+40+5
unfriendly
345=300+30+15
(300÷3) + (30÷3) + (15÷3)=
100 + 10 +
5 = 115
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4 504
-400 100
104
-100
25
4
- 4
1
0
100+25+1=126
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157
3 472
-3
17
-15
22
-21
1
169
2 3
4 676
Drawing simple visual models to represent word
problems.
 Steps
1.
Read the entire problem.
2.
Rewrite the question in sentence form, leaving a
space for the answer.
3.
Determine who and/or what is involved in the
problem.
4.
Draw unit bar(s).
5.
Chunk the problem, adjust the unit bars, and fill in
the question marks.
6.
Correctly compute and solve the problem.
7.
Write the answer in the sentence, and make sure
the answer makes sense.
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Janet picked 3 daisies and 2 sunflowers from
her garden. How many total flowers did Janet
pick from her garden?
Work problem on your Part-Part Whole Model
Handout
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Scrooge had 17 pennies in his piggy bank.
He also had 8 dimes. How many total coins
did Scrooge have in his piggy bank?
Work problem on your Part-Part Whole Model
Handout
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Liz earned $500 as her weekly pay. She paid
$413 to cover her bills for the week. How
much money did she have left to spend?
Work problem on your Part-Part Whole Model
Handout
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A movie theater has 1,250 seats. If 756
people attended today’s matinee, how many
seats will there be left in the movie theater?
Work problem on your Part-Part Whole Model
Handout
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Joe has $45. Tom has $28. How much more
money does Joe have than Tom?
Work problem on your Comparison Model
Handout
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Thomas makes $35 more dollars a week than
Bobby. Bobby makes $82 dollars a week.
How much money does Thomas make?
Work problem on your Comparison Model
Handout
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When planting her flower garden, Tanya
placed her flowers in 3 rows. She put 8
flowers in each row. How many flowers did
she plant in her garden?
Work problem on your Part-Part Whole Model
Handout
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Ann saved $45 dollars in nine weeks. She
saved the same amount each week. How
much did she save each week?
Work problem on your Part-Part Whole Model
Handout
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The movie theater has 12 rows of chairs with
36 chairs in each row. How many chairs are
in the movie theater? If there are 250 kids in
our school, can we all go to the movie at the
same time?
Work problem on blank piece of paper.
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There are 8 white flowers. There are 3 times
as many red flowers. How many red flowers
are there?
Work problem on your Comparison Model
Handout
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There are 36 red flowers. There are 4 times
as many red flowers as white flowers. How
many white flowers are there?
Work problem on your Comparison Model
Handout
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Joe buys 27 toys. 2/3 of them are trucks.
How many of them are not trucks.
Solve on a scrap sheet of paper.
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Susie spent 2/5 of her money of a purse. The
purse cost $15. How much money did she
have before she bought the purse?
Solve on a scrap sheet of paper.
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From a 5th grade Singapore text.
Mrs. Chen made some tarts. She sold 3/5 of
them in the morning and ¼ of the remainder
in the afternoon. Is she sold 200 more tarts
in the morning than in the afternoon, how
many tarts did she make?
Solve on a scrap sheet of paper.
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In a class of 35 students, the ratio of girls to
boys is 3:4. How many more boys than girls
are there?
Solve on scrap paper.
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There are 30 dogs, cats, and hamsters
altogether at a pet store. The ratio of dogs to
cats to hamsters is 5:3:2. How many dogs
and cats are there at the pet store?
Solve on scrap paper.
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There are 30 dogs, cats, and hamsters
altogether at a pet store. The ratio of dogs to
cats to hamsters is 5:3:2. How many dogs
and cats are there at the pet store?
Solve on scrap paper.
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There were 250 people at a concert. Of
these, 40% were children and the rest were
adults. How many adults were at the concert?
Solve on scrap paper.
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Of the 60 students in the third grade, we
know that 60% are girls. We also know that
50% of the girls have blue eyes and 25% of
the boys have blue eyes. How many of the
students in third grade have blue eyes?
Solve on scrap paper.
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We know that ¾ of a sum of money is $72.
What is the sum of money?
Solve on scrap paper.
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Susie bought 26 treats for her 2 cats, Joe and
Moe. She gave Joe 4 more treats than Moe.
How many treats did each cat receive?
Solve on scrap paper.
Purpose: Concept development, communicating
ideas
Activity: Work with a partner or in a small
group. Choose one of the problems that we
wrapped a story around.
1- Build a Model
2- Draw a Picture
3- Write an Equation
4- Write your answer in a complete
sentence.
5- Explain.
Purpose: Provide context, aid concept
development, make real-world connections
Activity:
Article: Round-Robin Story Telling
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Author(s): Joseph Martinez and Nancy Martinez
Source: Instructor (1990). 109.6 (Mar. 2000): p70.
Round-robin storytelling works well as a math activity
with elementary-school students. Provide each small
group with a story beginning such as: Jon and
Michelle make paper hats. Their standard hat is paper
with a tissue tassel. It costs $.30 to make and sells
for $1.25. Within each group, students add to the
story For example: They also make a fancy hat that
sells for $2.50. The last student in the group wraps
up the story and poses the problem: Jon and Michelle
sell 60 standard hats, and 40 fancy hats. How much
profit will they make? Each group works to solve its
problem. The groups then swap stories.
Purpose: Assure progress in terms of academic
achievement
Activity:
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Why Before How: Singapore Math Computation
Strategies: Jana Hazekamp
Building Number Sense: Catherine Jones Kuhns
The Singapore Model Method for Learning
Mathematics: Mastery of Education Singapore
The Parent Connection for Singapore Math:
Sandra Chen
8-Step Model Drawing: Singapore’s Best
Problem-Solving MATH Strategies: Bob Hogan &
Char Forsten
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Step-by-Step Model Drawing: Char Forsten