PROFESSIONAL LEARNINGwww.origoeducation.com

Fractions: Choosing the Best Model to Develop

a Concept

© ORIGO Education 1

Show 1 4 in each of these models.

For each model, consider the following questions:

• What is the whole?

• What does equal-sized mean?

• What does the fraction indicate?

• What attribute is the focus?

Set Model: Area Model:

Length Model:

Number Line Model:

0 21

© ORIGO Education 2

Teachers  and  students  need  to  consider  the  following:  

•  The  type  of  quan7ty  that  the  model  is  intended  to  represent  (con7nuous  or  discrete?).  

•  How  the  whole  is  defined.  •  What  equal-­‐sized  means  in  the  model.  •  What  the  frac7on  indicates.  

Represen7ng  Frac7ons  

The  number  –  a  discrete  (countable)  quan7ty.

The  whole  is  determined  by  a  defined  count    of  a  collec7on  or  set.  

Represen7ng  Frac7ons    –  Set  Model  

The  same  number  of  items  represents  equal-­‐sized  parts.

The  frac7on  indicates  the  count  of  objects  in  the  subset  compared  to  the  defined  set  of  objects.  

The  area  –  a  con7nuous  (measureable)  quan7ty.

Represen7ng  Frac7ons    –  Area  Model  

The  whole  is  determined  by  the  defined  area  or  region.  

The  same  area  represents  equal-­‐sized  parts.

The  frac7on  indicates  the  area  of  the  part  compared    to  the  area  of  the  whole.  

The  whole  is  determined  by  a  defined  length.  

Represen7ng  Frac7ons    –  Length  Model  

The  length  –  a  con7nuous  (measureable)  quan7ty.

The  same  length  represents  equal-­‐sized  parts.

The  frac7on  indicates  the  length  of  the  part  compared  to  the  length  of  the  whole.  

Represen7ng  Frac7ons    –  Number  Line  Model  

1 20 1

1 2

+

0 11 3

1 3

1 3

+ 1 3

+ 1 3

+1 2

+

A  special  type  of  length  model.  

2

The  length  –  a  con7nuous  (measureable)  quan7ty.

The  whole  is  determined  by  a  unit  of  distance    from  0  to  1.  

The  same  distance  represents  equal-­‐sized  parts.

The  frac7on  indicates  the  loca7on  of  a  point  in  rela7on  to  the  distance  from  0  with  regard  to  the  defined  unit.  

© ORIGO Education 3

Repr

esen

'ng  

Frac

'ons

 Ty

pe  o

f  M

odel

 Ty

pe  o

f  Qua

n/ty

 W

hole

 M

eani

ng  o

f    Eq

ual-­‐S

ized

 Par

ts  

Wha

t  the

 Fra

c/on

 Indi

cate

s  

Set  m

odel

 di

scre

te  

dete

rmin

ed  b

y  a  

defin

ed  c

ount

 of  a

 co

llec'

on  o

r  se

t  

sam

e  nu

mbe

r  of

 ite

ms  

the  

coun

t  of  o

bjec

ts  in

 the  

subs

et  c

ompa

red  

to  th

e  de

fined

 se

t  of  o

bjec

ts  

Are

a  m

odel

 co

n'nu

ous  

dete

rmin

ed  b

y  a  

defin

ed  a

rea  

or  re

gion

 sa

me  

area

 th

e  ar

ea  o

f  the

 indi

cate

d  pa

rt  

com

pare

d  to

 the  

area

 of  t

he  

indi

cate

d  w

hole

 

Leng

th  

mod

el  

con'

nuou

s  de

term

ined

 by  

a  de

fined

 leng

th  

sam

e  le

ngth

 th

e  le

ngth

 of  t

he  in

dica

ted  

part

 co

mpa

red  

to  th

e  le

ngth

 of  t

he  

indi

cate

d  w

hole

 

Num

ber  lin

e  m

odel

 co

n'nu

ous  

unit  

of  d

ista

nce  

from

 0  

to  1

 sa

me  

dist

ance

 th

e  lo

ca'o

n  of

 a  p

oint

 in  

rela

'on  

to  th

e  di

stan

ce  fr

om  0

 w

ith  re

gard

 to  th

e  de

fined

 uni

t  

© ORIGO Education 4

Suppose that four people are going to share 3 square pizzas. If the pizzas are shared equally, how much pizza does

each person get?

Each share is of one pizza.

© ORIGO Education 5

Step Up 1. Complete these to show how you would solve each problem. Draw lines and color parts to show each share.

2. Each large rectangle below is one whole. Complete the sentence, draw lines, and color parts to show each share.

a. Two slices of banana bread are shared equally by 3 people. How much of one whole slice will be in each share?

2 ÷ = slices shared by

people

b. Three pies are shared equally by 8 people. How much of one whole pie will be in each share?

÷ = pies shared by

people

a. 3 5 is the same as

divided by

b. 4 7 is the same as

divided by

Step AheadComplete the sentences so that the quotient in the division sentence is the same as the product in the multiplication sentence. Then use the diagram to show why they are the same.

3 ÷ 4 =

3 ×

=

Stepping Stones Student Journal Grade 5/ Module 11/ Lesson 1

© ORIGO Education 6

Frac%on

s  as…  

Num

erator  

Den

ominator  

Meaning  of  3

/4  

Part  of  a

 Who

le  

numbe

r  of  equ

al-­‐sized

 parts  indicated  

numbe

r  of  equ

al-­‐sized

 parts  in  th

e  who

le  

3  parts  ou

t  of  4

 equ

al-­‐

sized  parts  

Num

bers  or  Measures  

numbe

r  or  cou

nt  of  

equal-­‐sized

 parts  (u

nit  

frac:o

ns)  

numbe

r  of  equ

al-­‐sized

 parts  (unit  frac:on

s)  

need

ed  to

 create  the  

who

le  

• 3  coun

ts  (rep

e::o

ns)  

of  th

e  un

it  frac:o

n  1/4  

• 1/4  +  1/4  +  1/4  

Quo

%en

ts  

numbe

r  of  item

s  in  th

e  who

le  

numbe

r  of  shares  or  

equal-­‐sized

 parts  

• 3  ÷  4  

• the  result  whe

n  divide

d  or  shared  

Inte

rpre

ting

Frac

tions

© ORIGO Education 7

© O

RIGO

Edu

catio

n.

ORIGO Stepping Stones 5

Comparing Fractions4

Reasoning withUnit Fractions

Using CommonDenominators

Using CommonNumerators

Using Benchmarks

1 3 or

1 5

10 12 or

4 6

12 18 or

4 7

2 3 or

4 5

4 3 or

7 6

16 12 or

5 3

2 6 or

6 15

6 5 or

5 6

4 5 or

7 8

12 10 or

26 20

5 12 or

10 16

3 7 or

5 8

5 6 or

3 4

7 4 or

8 6

3 5 or

6 7

7 8 or

4 3

© ORIGO Education 9

1.5 2 15

21

38

167100

44 1.16 0.7

67100

010 2.6 .3

1 23 2.0 9

4 1.05

3 12

© ORIGO Education 11

unit fractions

common fractions

proper fractions

improper fractions

mixed numbers

decimal fractions

0 1 2 3