X
7
3
A visual representation of details and actions
which assists children with problem solving
A tool to help children think logically when
making computations.
• Part/Part/Total
diagrams are not
proportional
• PPT diagrams do not
reinforce problem
solving or number
relationships
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 foster number sense because bars are
proportional and have meaning
10
8
2
10
4
6
Let’s consider the same problem with bar modeling…
• Fred has 6 toys. Mary has 2 toys. How many toys do they
have altogether?
T
6
2
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….
Number Model:
_____________+_____________=_____________
_____________=_____________+_____________
Number Model:
_____________+_____________=_____________
_____________=_____________+_____________
Bar Modeling
Number Model:
_____________+_____________=_____________
_____________=_____________+_____________
• Fred has 6 toys. Mary has 2 toys. How many toys do they have altogether?
T
6
2
T is the same as 6+2
T=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
• 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?
Beginning
Addition
Part
Subtraction
Total
Change
Add (join) a Part
Subtract (separate) a
Part
End
Total
Part
Number Model:
_____________+_____________=_____________
Start
Change
+
End
=
4 butterflies were sitting on a branch. 2 more landed on
the branch. How many butterflies were on the branch
now?
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.
Addition
(Change)
Beginning
Change
End
Part
Add (join) a Part
Total
B*
2
6
4
B*
6
2
4
B
* May require subtracting or ‘counting up’. The bar model will help students choose
what to do.
Total Amount Unknown
R
8
2
Amount Joined Unknown
10
2
•8+2=R
•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
D
Initial Amount Unknown
10
B
# Story
•The team had 8 runs. They scored 2 more. How many do they
have in all?
5
•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
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
Number Model:
_____________-_____________=_____________
Start
Change
-
End
=
There were 4 butterflies sitting on a branch. 2 flew away. How many were left?
Bar Modeling- Subtraction-Change
Number Model:
5-2=3
• 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.
Subtraction
(Change)
Beginning
Total
5
5
B*
Change
Subtract (separate) a
Part
B
2
2
* May requires adding. . The bar model will help students choose what to do.
End
Part
3
B
3
Amount Remaining Unknown
10
8
x
Amount Separated Unknown
10
x
# Story
•Suzie had $10. She spent $8 on lunch. How much money does
she have left over?
•10-8=x
•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
8
Initial Amount Unknown
x
5
•Jim ate 5 cookies out of the box. He only has 5 left. How many
were in the box?
•X-5=5
5
Amount Remaining Unknown
Picture
Amount Separated Unknown
Picture
Initial Amount Unknown
Picture
Amount Remaining Unknown
Number Model
Amount Separated Unknown
Number Model
Initial Amount Unknown
Number Model
Bar Modeling- Part/Part Whole
Number Model:
_____________+_____________=_____________
_____________-_____________=______________
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
Number Model:
4+2=6
6-4=2
6-2=4
• Part/Part/Whole 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.
1st Part
2nd Part
Addition
(Part/Part/Whole)
Subtraction
(Part/Part/Whole)
I
I
Whole
I
Whole Unknown
F
2
8
•F=2+8
First Part Unknown
10
S
2
Second Part Unknown
10
5
# Story
•There were 2 apples and 8 bananas in the fruit bowl? How
many pieces of fruit were in the bowl?
x
•Jim had 10 brothers and sisters. How many sisters did he have if
head 2 brothers?
•10=s+2
•Jim had 10 brothers and sisters. If he had 5 brothers, how many
sisters did he have?
•10=5+B
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
• 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
Number Model:
_________-__________=_________ (more or less)
Larger
Amount
Smaller
Amount
-
=
Jim has 5 butterflies. Mary has 2. How many more butterflies
does Jim have than Mary?
Number Model:
5-2=3
• Comparison 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.
Larger Amount
Subtraction
(Comparison)
Addition
(Comparison)
Smaller Amount
Amount more or Less
Unknown
(Difference)
B
B
B
* May require adding. Bar Models will help students figure out what to do.
Amount More (or Less) Unknown
10
c
4
Difference
Smaller Amount Unknown
10
# 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
•Ted had 10 crayons. Neil has 6 less than Ted. How many
crayons does Neil have?
•10-6=c
6
Difference
c
Larger Part Unknown
c
5
5
Difference
•Ted Has 5 crayons. Neil has 5 more than Ted. How many
crayons does Neil have?
•5+5=c
Picture
Picture
Picture
Number Model
Number Model
Number Model
Multiplication Bar Modeling
Number Model:
_____________x_____________=_____________
Bar Modeling- Multiplication
Number Model:
_____________x_____________=_____________
Start
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?
Number Model:
6÷2=3
• Multiplication 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.
Total
# of Parts
Amount Per Part
6
6
B
B*
2
2
3
B*
3
* May require dividing. . The bar model will help students choose what to do.
Total Amount Unknown
e
1
2
1
2
1
2
1
2
1
2
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
36
A
A
A
A
Example
•Ted bought 4 bags of apples. Altogether, there were 36
apples. How many apples were in each bag?
•4xA=36
# of Groups Unknown
40
8
B
Picture
Ted bought $40 worth of books for his friends. Each book
cost $8. How many books did he buy?
40=Bx8
Total Amount Unknown
Picture
Amount Per Group Unknown
Picture
# of Groups Unknown
Picture
Total Amount Unknown
Number Model
Amount Per Group Unknown
Number Model
# of Groups Unknown
Number Model
Bar Modeling- Division
Number Model:
_____________÷_____________=_____________
Start
Separate
End
÷
=
Jim caught 6 butterflies. She placed them in 2 containers. How
many were in each container
Division
Number Model:
6÷2=3
• Division 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.
Total
# of Parts
Amount Per Part
5
6
B*
B
2
2
3
B
3
* May require adding. . The bar model will help students choose what to do.
Amount Per Group Unknown
Example
•Ted has 60 eggs.
36
•E=12x5
e
e
e
e
e
Amount Per Group Unknown
36
A
A
A
A
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
40
8
B
Picture
Ted bought $40 worth of books for his friends. Each book
cost $8. How many books did he buy?
40=Bx8
Amount Per Group Unknown
Example
•Ted has 60 eggs.
36
•E=12x5
e
e
e
e
e
Amount Per Group Unknown
36
A
A
A
A
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
40
8
B
Picture
Ted bought $40 worth of books for his friends. Each book
cost $8. How many books did he buy?
40=Bx8
Total Amount Unknown
Picture
Amount Per Group Unknown
Picture
# of Groups Unknown
Picture
Total Amount Unknown
Number Model
Amount Per Group Unknown
Number Model
# of Groups Unknown
Number Model
•
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
• Work on every problem
• Specify ONE RIGHT model
• Specify ONE RIGHT operation