Integer Rod OperationsInteger Rod Operations

Multiplying and DividingMultiplying and Dividing

Representing Fractions Using Bars How do we represent fractions using integer

bars? Part to whole Whole changes as necessary to make equivalents

A train is two rods put together – ALL trains must have at least one E in them

We will ALWAYS use the least number of bars possible to make a representation

Do NOT draw more lines on representations than necessary

Six Steps Required

1. Represent the fraction with the smallest and least number of rods possible

2. Race the denominators to a tie. This will ALWAYS take 3 rows – the new common denominator is at the bottom

Six Steps Required - Continued

3. Represent the fraction using the “race” as a guide using the common denominator rod and the least number of rods possible for the numerator

4. Do the operation

Six Steps Required - Continued

5. Simplify the representation –least number of rods possible

6. Interpret the representation in #5 as a fraction number answer

Race Representation: Multiplication

Use one common denominator bar

The numerator will represent the SECOND factor only

Do NOT represent the first factor

Do the Operation: Multiplication Use one common denominator bar Place the numerator of the second

factor directly above the common denominator

Look at the first factor in the problem Treat the numerator of the second factor

as the denominator of the first factor Place a bar above it that represents the

numerator for the first factor Total of 3 rows

Simplify the Representation: Multiplication Use one original common

denominator bar Place the top bar from the step

above directly above the common denominator bar

Represent all with the least number of rods possible

Total of 2 rows

Multiplication – Concrete2 1

?3 2

13

A.

B.

C.

D.

E.

F.

A.

B.

C.

D.

E.

F.

3 1?

5 2

310

Multiplication – Semi-Concrete2 1

?3 2

13

A.

B.

C.

D.

E.

F.

A.

B.

C.

D.

E.

F.

3 1?

5 2

310

WRRL

R R RL L

G

G

R

G

G

L

L

R

W

E

E

L

LY

Y Y

R

E

E

R R R R R

Y

YL

Multiplication – Semi-Abstract2 1

Problem: ?3 2R W

A. L R

B. 2L 3R G

LC.

GR

LD.

GR

E. G

1F.

3

3 1Problem: ?

5 2L W

A. Y R

B. 2Y 5R E

YC.

EL

YD.

EL

E. E

3F.

10

Do the Operation: Division Use one common denominator bar Place the divisor (the factor) directly

above the common denominator Place the dividend (the product)

directly above the divisor (the factor)

Total of 3 rows

Simplify the Representation: Division

Use the divisor (the factor) as the new common denominator

Place the dividend (the product) directly above the divisor (the factor)

Represent all with the least number of rods possible

Total of 2 rows

Division – Concrete1 2

?2 3

34

A.

B.

C.

D.

E.

F.

A.

B.

C.

D.

E.

F.

3 1?

4 5

154

Division – Semi-Concrete1 2

?2 3

34

A.

B.

C.

D.

E.

F.

A.

B.

C.

D.

E.

F.

3 1?

4 5

154

W R

P

P

P

G

G G

R L

R R RL L

L

G

L

L

W

P

EY P

PY

LP Y

P P P

E

E

Y Y

E

EE

E

P PY Y

E

E E

E

Y

Division – Semi-Abstract1 2

Problem: ?2 3

W RA.

R LB. 3R 2L G

L PC.

G GL P

D. G GL

E. P3

F. 4

3 1Problem: ?

4 5L W

A. P Y

B. 5P 4Y 2E

EY PC.

2E 2EEY P

D. 2E 2EEY

E. P

15F.

4