Fraction (or Fractured?) Understanding
Fraction (or Fractured?) Understanding Debi DePaul
PROFESSIONAL LEARNING www.origoeducation.com
© ORIGO Education
1
Represen,ng Frac,ons Teachers and students need to consider the following: – The type of quan,ty (con,nuous or discrete?) that the model is intended to represent. – How the whole is defined. – What equal-‐sized means in the model. – What the frac,on indicates.
Represen,ng Frac,ons
Represen,ng Frac,ons
The number – a discrete (countable) quan,ty
The area – a con,nuous (measureable) quan,ty
The whole is determined by a defined count of a collec,on or set.
The whole is determined by the defined area or region.
The same number of items represents equal-‐sized parts.
The same area represents equal-‐sized parts.
The frac,on indicates the count of objects in the subset compared to the defined set of objects.
The frac,on indicates the area of the part compared to the area of the whole.
Set Model
Area Model
+
Represen,ng Frac,ons
1 2
+
1 2
+
1 3
+
1 3
+
1 3
Length Model
Represen,ng Frac,ons 0 1 Number Line Model
The length – a con,nuous (measureable) quan,ty
The distance – a con,nuous (measureable) quan,ty
The whole is determined by a defined length.
The whole is determined by a unit of distance from 0 to 1.
The same length represents equal-‐sized parts.
The same distance represents equal-‐sized parts.
The frac,on indicates the length of the part compared to the length of the whole.
The frac,on indicates the loca,on of a point in rela,on to the distance from 0 with regard to the defined unit.
1 2
0
1 3
1 2 3
1
A special type of length model.
© ORIGO Education
2
© ORIGO Education
3
con'nuous
Length model
con'nuous
con'nuous
Area model
Number line model
discrete
Type of Quan/ty
Set model
Type of Model
unit of distance from 0 to 1
determined by a defined length
same distance
same length
same area
same number of items
determined by a defined count of a collec'on or set
determined by a defined area or region
Meaning of Equal-‐Sized Parts
Whole
the loca'on of a point in rela'on to the distance from 0 with regard to the defined unit
the length of the indicated part compared to the length of the indicated whole
the area of the indicated part compared to the area of the indicated whole
the count of objects in the subset compared to the defined set of objects
What the Frac/on Indicates
Represen'ng Frac'ons
PROFESSIONAL LEARNING www.origoeducation.com
© ORIGO Education
1
Represen,ng Frac,ons Teachers and students need to consider the following: – The type of quan,ty (con,nuous or discrete?) that the model is intended to represent. – How the whole is defined. – What equal-‐sized means in the model. – What the frac,on indicates.
Represen,ng Frac,ons
Represen,ng Frac,ons
The number – a discrete (countable) quan,ty
The area – a con,nuous (measureable) quan,ty
The whole is determined by a defined count of a collec,on or set.
The whole is determined by the defined area or region.
The same number of items represents equal-‐sized parts.
The same area represents equal-‐sized parts.
The frac,on indicates the count of objects in the subset compared to the defined set of objects.
The frac,on indicates the area of the part compared to the area of the whole.
Set Model
Area Model
+
Represen,ng Frac,ons
1 2
+
1 2
+
1 3
+
1 3
+
1 3
Length Model
Represen,ng Frac,ons 0 1 Number Line Model
The length – a con,nuous (measureable) quan,ty
The distance – a con,nuous (measureable) quan,ty
The whole is determined by a defined length.
The whole is determined by a unit of distance from 0 to 1.
The same length represents equal-‐sized parts.
The same distance represents equal-‐sized parts.
The frac,on indicates the length of the part compared to the length of the whole.
The frac,on indicates the loca,on of a point in rela,on to the distance from 0 with regard to the defined unit.
1 2
0
1 3
1 2 3
1
A special type of length model.
© ORIGO Education
2
© ORIGO Education
3
con'nuous
Length model
con'nuous
con'nuous
Area model
Number line model
discrete
Type of Quan/ty
Set model
Type of Model
unit of distance from 0 to 1
determined by a defined length
same distance
same length
same area
same number of items
determined by a defined count of a collec'on or set
determined by a defined area or region
Meaning of Equal-‐Sized Parts
Whole
the loca'on of a point in rela'on to the distance from 0 with regard to the defined unit
the length of the indicated part compared to the length of the indicated whole
the area of the indicated part compared to the area of the indicated whole
the count of objects in the subset compared to the defined set of objects
What the Frac/on Indicates
Represen'ng Frac'ons
