a visual representation of details and actions which assists children with problem solving a tool to...

Bar Modeling- What is It?

A visual representation of details and actions which assists children with problem solving

A tool to help children think logically when making computations.

X

7 3

Aren’t These Like Part/Part/Total Diagrams?

• Part/Part/Total diagrams are not proportional

• PPT diagrams do not reinforce problem solving or number relationships

Part/Part/Total Ted has 6 toys. Mary has 2 toys. How many toys do they have altogether?

T

6 2

•When the parts are not represented in proportion, student do not reinforce the relationship between the numbers.

Bar Models Are Proportional

• Bar models foster number sense because bars are proportional and have meaning

10

8 2

10

4 6

• Fred has 6 toys. Mary has 2 toys. How many toys do they have altogether?

T

6 2

Let’s consider the same problem with bar modeling…

Like the Part/Part/Whole diagram, the bar model allows the students to put parts together to get the total number of toys, but the bar model• reinforces the notion that Ted has more than Mary. • requires students to be more thoughtful about where to put the numbers• It also allows for further exploration…

• How many more does Ted have than Mary?• How many could Ted give Mary so they could have the same, etc….

Bar Modeling- Scaffolding the Notion of Proportion

Number Model:

_____________+_____________=_____________

_____________=_____________+_____________

Bar Modeling

Number Model:

_____________+_____________=_____________

_____________=_____________+_____________

Bar Models Provide a Foundation For Algebraic Understanding

• Fred has 6 toys. Mary has 2 toys. How many toys do they have altogether?

T

6 2

T is the same as 6+2T=6+2

T=8

There are 8 toys altogether.

*Since the parts are the same size as the whole, it reinforces that equal means ‘same as’. As bar models get more complicated, the bar models will continue to provide a visual for the algebraic relationships

Bar Modeling Teaches The Importance of Language When Solving Math Problems

• Have you ever had students add when they should subtract?– John collects rocks. He started with 4. After his

vacation, he had 10. How many rocks did he collect on vacation?

Why do we add and subtract?


CHANGE

and Wholes

CompaRe

Parts Parts Parts Parts Parts Parts

Addition and Subtraction- Change

Beginning Change EndAddition Part Add (join) a Part Total

Subtraction Total Subtract (separate) a Part

Part

You start with an amount You add or subtract an amount

You end with a total amount

Addition Problems: ChangeBar Modeling

End

Beginning Change

Number Model:

_____________+_____________=_____________

Change-Addition

Start Change End

+ =

4 butterflies were sitting on a branch. 2 more landed on the branch. How many butterflies were on the branch now?

Addition Problems: ChangeBar Modeling

End(6)

Beginning(4)

Change(2)

Number Model:4+2=6


• Change problems can be presented in different ways so…• Bar Models will help students to:

– Think about what information they have been given and what information they will are being asked to figure out.

Beginning Change End

Addition(Change)

Part Add (join) a Part Total

B* 2 64 B* 62 4 B

* May require subtracting or ‘counting up’. The bar model will help students choose what to do.

Bar Modeling: Joining (Addition-Change)

Total Amount Unknown # Story•The team had 8 runs. They scored 2 more. How many do they have in all?

•8+2=R

Amount Joined Unknown •John has $2 in his piggy bank. He needs $10 to buy the new toy he wants. How much more does he need to save to have enough money?

•2+D=10

Initial Amount Unknown •Mary added 5 books to her library. Now she has 10 books in her library. How many books did she start with?

•B+5=10

R

8 2

10

2 D

10

B 5


Total Amount Unknown Picture

Amount Joined Unknown Picture

Initial Amount Unknown Picture


Total Amount Unknown Number Model

Amount Joined Unknown Number Model

Initial Amount Unknown Number Model

Bar Modeling- Subtraction-Change

Beginning

Change End

Number Model:

_____________-_____________=_____________

Change-Subtraction

Start Change End

- =

There were 4 butterflies sitting on a branch. 2 flew away. How many were left?

Bar Modeling- Subtraction-Change

Beginning(5)

Change(2)

End(3)

Number Model:

5-2=3


• Change problems can be presented in different ways so…• Bar Models will help students to:


Beginning Change EndSubtraction

(Change)Total Subtract (separate) a

PartPart

5 B 35 2 B

B* 2 3

* May requires adding. . The bar model will help students choose what to do.

Bar Modeling: Separating (Change- Subtraction)

Amount Remaining Unknown # Story•Suzie had $10. She spent $8 on lunch. How much money does she have left over?

•10-8=x

Amount Separated Unknown •John had to read 10 books over the summer. He only has 2 left to read. How many books did he read?

•10-x=8

Initial Amount Unknown •Jim ate 5 cookies out of the box. He only has 5 left. How many were in the box?

•X-5=5

10

8 x

10

x 8

x

5 5

Bar Modeling: Separating (Change-Subtraction)

Amount Remaining Unknown Picture

Amount Separated Unknown Picture

Initial Amount Unknown Picture

Bar Modeling: Separating (Change-Subtraction)

Amount Remaining Unknown Number Model

Amount Separated Unknown Number Model

Initial Amount Unknown Number Model

Bar Modeling- Part/Part Whole

Whole

PartPart

Number Model:

_____________+_____________=__________________________-_____________=______________

Part/Part/Whole

1st Part 2nd Part End

+ =

There were 4 butterflies and 2 ladybugs sitting on a branch. How many insects were sitting on the branch?

Bar Modeling-Part/Part/Whole

Whole(6)

1st Part(4)

2nd Part(2)

Number Model:4+2=66-4=26-2=4


• Part/Part/Whole problems can be presented in different ways so…• Bar Models will help students to:


1st Part 2nd Part WholeAddition

(Part/Part/Whole)I

Subtraction(Part/Part/Whole)

II

Bar Modeling: Part/Part/Whole

Whole Unknown # Story•There were 2 apples and 8 bananas in the fruit bowl? How many pieces of fruit were in the bowl?

•F=2+8

First Part Unknown •Jim had 10 brothers and sisters. How many sisters did he have if head 2 brothers?

•10=s+2

Second Part Unknown •Jim had 10 brothers and sisters. If he had 5 brothers, how many sisters did he have?

•10=5+B

F

2 8

10

S 2

10

5 x


Whole Unknown Picture

First Part Unknown Picture

Second Part Unknown Picture


Whole Unknown Number Model

First Part Unknown Number Model

Second Part Unknown Number Model

Warning, Warning!• There are 3 types of addition/subtraction problems: change, part/part/whole and comparison. • Change and part/part whole problems are based on the

concepts of parts building wholes. COMPARISON PROBLEMS DO NOT!

• COMPARISON PROBLEMS ARE BASED ON FINDING THE DIFFERENCE BETWEEN TWO PARTS

Bar Modeling- Multiplication

LARGER AMOUNT

Number Model:

_________-__________=_________ (more or less)

SMALLER AMOUNTDIFFERENCE

Bar Modeling- Comparison

Larger SmallerAmount Amount

- =

Jim has 5 butterflies. Mary has 2. How many more butterflies does Jim have than Mary?

Comparison

Larger Amount(5)

Number Model:

5-2=3

Smaller Amount(2) Difference

3 More


• Comparison problems can be presented in different ways so…• Bar Models will help students to:


Larger Amount Smaller Amount Amount more or Less Unknown

(Difference)

Subtraction(Comparison)

BB

Addition(Comparison)

B

* May require adding. Bar Models will help students figure out what to do.

Bar Modeling: Comparison- All(Finding The Difference)

Amount More (or Less) Unknown

c Difference

# Story

•Ted has 10 crayons. Neil has 4 crayons. • How many more crayons does Ted have than Neil?• How many fewer crayons does Neil have than Ted?

•10-4=c

Smaller Amount Unknown

6 Difference

•Ted had 10 crayons. Neil has 6 less than Ted. How many crayons does Neil have?

•10-6=c

Larger Part Unknown

5 Difference

•Ted Has 5 crayons. Neil has 5 more than Ted. How many crayons does Neil have?

•5+5=c

10

4

10

c

c

5

Bar Modeling: Comparison- Smaller Amount Unknown

Picture

Picture

Picture

Bar Modeling: Comparison- Smaller Amount Unknown

Number Model

Number Model

Number Model

Multiplication Bar Modeling

Number Model:

_____________x_____________=_____________

Bar Modeling- Multiplication

Total

Equal part Equal Part

Number Model:

_____________x_____________=_____________

Bar Modeling- MultiplicationStart Join

X=

Jim collects butterflies. There were 3 butterflies in each container. He had 2 containers. How many butterflies does Jim have in his collection?

Multiplication

Total(6)

Separate Equal Part(3)


Number Model:

6÷2=3


• Multiplication can be presented in different ways so…• Bar Models will help students to:


Total # of Parts Amount Per Part

6 B* 36 2 B*B 2 3

* May require dividing. . The bar model will help students choose what to do.

Bar Modeling: Joining Equal Groups (Multiplication)

Total Amount Unknown Example•Ted has 5 cartons of eggs. Each carton has 12 eggs in it. How many eggs does Ted have?

•E=12x5

Amount Per Group Unknown Example

•Ted bought 4 bags of apples. Altogether, there were 36 apples. How many apples were in each bag?

•4xA=36

# of Groups Unknown PictureTed bought $40 worth of books for his friends. Each book cost $8. How many books did he buy?

40=Bx8

e

12

12

12

12

12

36

A A A A

40

8 B

Bar Modeling: Joining Equal Groups (Multiplication)(Remember to split bottom bar in TA and APG examples)


Amount Per Group Unknown Picture

# of Groups Unknown Picture



Amount Per Group Unknown Number Model

# of Groups Unknown Number Model

Bar Modeling- Division

Total

Equal part Equal Part

Number Model:

_____________÷_____________=_____________

Bar Modeling- DivisionStart Separate

End

÷=

Jim caught 6 butterflies. She placed them in 2 containers. How many were in each container

Division

Total(6)



Number Model:

6÷2=3


• Division can be presented in different ways so…• Bar Models will help students to:


Total # of Parts Amount Per Part

5 B 36 2 B

B* 2 3

* May require adding. . The bar model will help students choose what to do.

Bar Modeling: Separating Equal Groups (Division)

Amount Per Group Unknown Example•Ted has 60 eggs.

•E=12x5

Amount Per Group Unknown Example

•Ted has 36 apples. He put them into 4 bags. How many apples did he put in each bag?

•36=4xA

# of Groups Unknown PictureTed bought $40 worth of books for his friends. Each book cost $8. How many books did he buy?

40=Bx8

36

e e e e e

36

A A A A

40

8 B



Amount Per Group Unknown Picture

# of Groups Unknown Picture



Amount Per Group Unknown Number Model

# of Groups Unknown Number Model

Problem Solving With Bar Modeling

• Visual representation of details and actions which assists children with problem solving

• Helps children logically think using visual models to determine their computations

• Fosters number sense because numbers are represented proportionally

• Teaches the importance of language within math problems

• Provides foundation for algebraic understanding

• Provides for differentiated instruction

• Empowers students to think systematically and master more difficult problems

• Makes multi-step and multi-concept problems easy to work

Bar Models Do NOT

• Work on every problem

• Specify ONE RIGHT model

• Specify ONE RIGHT operation

a visual representation of details and actions which assists children with problem solving a tool to...

Documents

unknownpicture slide

d b5 slide

total number of toys

bar modeling number

bar models

unknownnumber model

unknown mary

number relationships