Upload
gertrude-foster
View
219
Download
0
Embed Size (px)
Citation preview
Singapore Math Computation
StrategiesJune 2012
“Understanding is a measure of the quality and quantity of connections that an idea has with existing ideas.” (VandeWalle,
Pg. 23)
Mathematical proficiency has 5 strands:
1. Conceptual understanding2. Procedural fluency3. Strategic competence4. Adaptive reasoning5. Productive disposition
Source: Common Core (Pg. 6)
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.
Why Singapore Math?
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
Why Singapore Math? Cont.
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.
C-P-A Concrete, Pictorial, Abstract
Manipulatives
How do you know when manipulatives are and are not needed?
Discuss.Catch thinking on chart paper.Write a summary paragraph.
1) Teacher modeling and thinking aloud about the strategy
2) Students practice with the teacher3) Students practice in small groups4) Students practice in partners5) Independent practice
I Do, We Do, We Do, We Do, You Do
Number Sense
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)
“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
Teacher Think Alouds
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!
Teacher Think Aloud Example
Math Talk – Think Aloud
Purpose: Modeling, communicating, promoting a more efficient strategy, promoting reasoning, moving to a more sophisticated level of thinking.
Math Talk – Ask a Math QuestionPurpose: “Unstick” someone, get help, clarify, promote deeper thinking, make connections
Math Talk – Partner PromptsPurpose: Promote productive math conversations
Example: Practicing questions in multiple-choice format Step 1: Solve individually. Write down your
answer. Step 2: Compare. Same or different? Step 3: Explain why you chose that answer.
Math Talk – Expand the Use of VocabularyPurpose: Use words that proficient mathematicians would use, make connections
Example: End of Year Jeopardy Review Game. 500 point question: What is addition?
Trying to Help Someone? Look at their work.
Do the model, the picture, and the equation match the question and each other?
Read or listen to their explanation. Ask a math question. Seek professional help – Ask a student
expert, the teacher, or other adult.
Place Value Strips Place Value Disks Place Value Chart Number-bond cards Part-whole cards Decimal Tiles Decimal Strips Gratiot Isabella ISD Maniplatives Link
Instructional Materials
Begin with no grouping Sequence1. Number bonds2. Decomposing numbers3. Left-to-right addition4. Place value disks and charts5. Vertical addition6. Traditional addition
Addition
Number Bonds
A Collection of Number Relationships- Relationships Among Numbers 1 Through 10
Spatial Relationshipspattered arrangements
One and Two More, One and Two Lesscounting on and counting back7 is 1 more than 6 and it is 2 less than 9
Anchoring Numbers to 5 and 10using 5 and 10 to build on and break from is
foundational for working with facts Part-Part-Whole Relationships
understand that a number is can be made of 2 or more parts
A Collection of Number Relationships - Relationships for Numbers 10 to 20
A Pre-Place Value Relationship with 1011 through 20, think 10 and some more
Extending More and Less Relationshipsi.e. 17 is one less than 18 like 7 is one less than
8
Double and Near-Double Relationshipsspecial cases of the part-part-whole constructuse pictures
2 Strategies to Emphasize
Build Up and Down through 10
Break Apart to Find an Unknown Fact
Number Sense and the Real World Estimation and Measurement
More or less than _______? Closer to _____ or _____?About _______.
More ConnectionsAdd a Unit to Your NumberIs It Reasonable?
GraphsMake bar graphs and pictographs
Basic Facts
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
Decomposing Numbers47
40 7
10 101010 1
1 1
1 1 1 1
33+56= Decompose each number by place value. Put the tens together and one ones
together. (30+50) + (3+6)
(30+50) + (3+6) 80 + 9 = 89
Left-to-Right Addition
30 50 63
35 +26
Place Value Disks and Charts
1010
1010
1010 1
1 1
1 1 1 1
1 1 1
1
35 +26
Place Value Disks and Charts
1010
1010
1010 1
1 1
1 1 1 1
1 1 1
1
10
35 +26
6 1
Place Value Disks and Charts
1010
1010
1010
1
10
36 +28 50 +14 64Add from left to right.
Vertical Addition
38 +85 123
Traditional Addition
Begin with no regrouping Sequence1. Number Bonds2. Place Value Disks and Charts3. Traditional Subtraction
Subtraction
Number Bonds
86-8
Place Value Disks and Charts
1010 10
1010
10 10
10
1 1 1 1 1
1
86-8
Place Value Disks and Charts
1010 10
1010
10 10
10
1 1 1 1 1
1
1111
1111
1
1
86-8
70 + 8 = 78 86-8=78
Place Value Disks and Charts
1010 10
10
10 10
10
1 1 1 1 1
1
1111
1111
1
1
54 - 28 26
Traditional Subtraction
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
Stages1. Number bonds2. Place value disks and charts3. The distributive property4. Area model5. Traditional multiplication
Multiplication
16 4 16 16 or 4 4 4
16 16
2 8 1 16 Vocab: Factor x Factor=Product
Number Bonds
141x3
Place Value Disks and Charts
100
100
100
10
10 10 10
10 10 10
10 10 1
1
110
10
10
141x3
Place Value Disks and Charts
100
100
100
10
10 10 10
10 10 10
10 10 1
1
110
10
10
100
141x3
4 2 3141x 3=423
Place Value Disks and Charts
100
100
100 10
1
1
110
100
45x3 (40x3) + (5x3) 120 + 15 = 135
The Distributive Property
6x33 30 3
6 180 18
6x33=180+18180+18=198
Area Model
15x26 20 6
10 200 60
5 100 30
15x26=(200+100)+(60+30) 300 + 90 =39015x26=390
Area Model
Traditional Multiplication
Begin by introducing division as repeated subtraction
Use number bonds to demonstrate the inverse relationship of multiplication and division
Sequence for teaching1. Number bonds2. Place value disks and charts3. The distributive property4. Partial quotient division5. Traditional long division6. Short division
Division
9 27 ?What should the second factor be?
Vocab: Dividend ÷ Divisor=Quotient
Number Bonds
64÷5 Step 1 Build with disks
Place Value Disks and Charts
10
10
10101010 1 1 1 1
64÷5 Step 2: The divisor tells us how many groups
we need: Draw 5 rows
1
2
3 4
5
Place Value Disks and Charts
10 1010101010 1 1 1 1
64÷5 Step 3: Begin with the tens. Do we have enough
tens to put one in every row? Yes
1
2
3 4
5
Place Value Disks and Charts
10
10
10
10
10
10
1 1 1 1
64÷5 Step 4: If there are any large disks left, trade
them for an equivalent value of smaller disks
1
2
3 4
5
Place Value Disks and Charts
10
10
10
10
10
10
1 1 1 1
1 1 1 1 1
1 1 1 1 1
64÷5 Step 5:Divide the ones equally among the groups
1
2
3 4
5
1 2 10+2=12 64÷5=12 r 4
Place Value Disks and Charts
10
10
10
10
10
1 1 1 1
1
1
1
1
1
1
1
1 1
1
345÷3= 345=300+40+5 unfriendly345=300+30+15(300÷3) + (30÷3) + (15÷3)= 100 + 10 + 5 = 115
The Distributive Property
4 504 -400 100 104 -100 25 4 - 4 1 0
100+25+1=126
Partial Quotient Division
157 3 472 -3 17 -15 22 -21 1
Traditional Long Division
169 2 3
4 676
Short Division
Wrap a Story around It
Purpose: Provide context, aid concept development, make real-world connections
Activity:
Article: Round-Robin Story Telling
Title: Round-Robin Story Problems 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.
Not done yet
Bar Modeling
Build, Draw, WritePurpose: Concept development, communicating ideasActivity: 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.
Observation with Analysis
Purpose: Assure progress in terms of academic achievement
Activity:
Why Before How: Singapore Math Computation Strategies: Jana Hazekamp
Resources