june 2012 understanding is a measure of the quality and quantity of connections that an idea has...

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)

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)

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.

http://www.youtube.com/watch?v=vxJSCwcnac8&feature=related

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

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

Math Talk-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 – 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

http://www.giresd.net/14971045125715483/blank/browse.asp?a=383&BMDRN=2000&BCOB=0&c=54142

Begin with no regrouping 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


1010

1010

1010 1

1 1

1 1 1 1

1 1 1

1

10

35 +26

6 1


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


1010 10

1010

10 10

10

1 1 1 1 1

1

86-8


1010 10

1010

10 10

10

1 1 1 1 1

1

1111

1111

1

1

86-8

70 + 8 = 78 86-8=78


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


100

100

100

10

10 10 10

10 10 10

10 10 1

1

110

10

10

141x3


100

100

100

10

10 10 10

10 10 10

10 10 1

1

110

10

10

100

141x3

4 2 3141x 3=423


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


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


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


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


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


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

Drawing simple visual models to represent word problems.

Steps1. 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.

Model Drawing

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

Part-Part Whole Model: Addition and Subtraction

Scrooge had 17 pennies in his piggy bank. He also had 8 dimes. How many total coins did Scrooge have in his piggy bank?



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?



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?



Joe has $45. Tom has $28. How much more money does Joe have than Tom?

Work problem on your Comparison Model Handout

Comparison Model: Addition and Subtraction

Thomas makes $35 more dollars a week than Bobby. Bobby makes $82 dollars a week. How much money does Thomas make?


Comparison Model: Addition and Subtraction

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?


Part-Part Whole Model: Multiplication and Division

Ann saved $45 dollars in nine weeks. She saved the same amount each week. How much did she save each week?



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.


There are 8 white flowers. There are 3 times as many red flowers. How many red flowers are there?


Comparison Model: Multiplication and Division

There are 36 red flowers. There are 4 times as many red flowers as white flowers. How many white flowers are there?


Comparison Model: Multiplication and Division

Joe buys 27 toys. 2/3 of them are trucks. How many of them are not trucks.

Solve on a scrap sheet of paper.

Fraction Models

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?


Fraction Models

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?


Fraction Models

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.

Ratio Models

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?


Ratio Models

There were 250 people at a concert. Of these, 40% were children and the rest were adults. How many adults were at the concert?


Percent Models

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?


Percent Models

We know that ¾ of a sum of money is $72. What is the sum of money?


Algebra Models

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?


Algebra Models

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.

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.

Observation with Analysis

Purpose: Assure progress in terms of academic achievement

Activity:

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

Step-by-Step Model Drawing: Char Forsten

Resources

june 2012 understanding is a measure of the quality and quantity of connections that an idea has...

Documents

students practice

connections slide

singapore students

students visualization

independent practice

changes students attitudes

deeper thinking

math talks