DEVELOPING FRACTION CONCEPTS
Reason with shapes and their attributes.
CCSS.Math.Content.1.G.A.2
Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or
three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to
create a composite shape, and compose new shapes from the composite shape.1
CCSS.Math.Content.1.G.A.3
Partition circles and rectangles into two and four equal shares, describe the shares using the words halves,
fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or
four of the shares. Understand for these examples that decomposing into more equal shares creates smaller
shares.
Reason with shapes and their attributes.
CCSS.Math.Content.2.G.A.3
Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves,
thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that
equal shares of identical wholes need not have the same shape.
Reason with shapes and their attributes.
CCSS.Math.Content.3.G.A.2
Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For
example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area
of the shape.
Develop understanding of fractions as numbers.
CCSS.Math.Content.3.NF.A.1
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts;
understand a fraction a/b as the quantity formed by a parts of size 1/b.
CCSS.Math.Content.3.NF.A.2
Understand a fraction as a number on the number line; represent fractions on a number line diagram.
CCSS.Math.Content.3.NF.A.2.a
Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and
partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based
at 0 locates the number 1/b on the number line.
CCSS.Math.Content.3.NF.A.2.b
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the
resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
CCSS.Math.Content.3.NF.A.3
Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
CCSS.Math.Content.3.NF.A.3.a
Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
CCSS.Math.Content.3.NF.A.3.b
Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are
equivalent, e.g., by using a visual fraction model.
CCSS.Math.Content.3.NF.A.3.c
Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples:
Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line
diagram.
CCSS.Math.Content.3.NF.A.3.d
Compare two fractions with the same numerator or the same denominator by reasoning about their size.
Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results
of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Extend understanding of fraction equivalence and ordering.
CCSS.Math.Content.4.NF.A.1
Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with
attention to how the number and size of the parts differ even though the two fractions themselves are the
same size. Use this principle to recognize and generate equivalent fractions.
CCSS.Math.Content.4.NF.A.2
Compare two fractions with different numerators and different denominators, e.g., by creating common
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denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that
comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons
with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Basic Models of Fractions
Linear Model
This model will split a length into equal size parts. A number line is often used for a length model. A ruler or
string can be used to divide the length into equal parts. Cuisenaire rods can be used for teaching fractions as
a length model.
Example: Place
3
on this number line:
5
0
Example: Place
1
3
on this number line:
5
0
2
Area Model
This model will split a whole shape into equal parts. The fraction is often represented by a shaded part of the
whole. It is important to identify what the whole is (ex a circle, rectangle, pattern block hexagon, etc.) and
then divide it into equal parts. Most manipulatives related to fractions rely on the area model (pattern blocks,
fraction circles, fraction towers, fraction squares, etc.)
Example: Divide the rectangle and circle below into fifths and discuss with your partner the pros and cons of
doing it with a rectangle versus a circle.
Example: Shade
1
of the following rectangle:
8
Set Model
This model uses a set of objects as a whole and will split these objects into equal size subsets. Counters are
often used to illustrate the set model. Set models are found most frequently in real world examples (ex.
1
of
12
the students in our class are male).
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Example: Draw a set model illustrating
3
7
Example: Draw a set model illustrating:
1
of the 12 stars are black.
4
When are Children Ready to Study Fractions?
Piaget identified seven conditions or characteristics about fractions children need to understand before they
can obtain operational understanding of fractions.
1.
The whole can be divided. Young children do not see the whole anymore once it has been divided
into smaller parts.
2.
A fraction implies a determinate number of parts. Young children do not realize that if I want to divide
something among four people, then I want four parts not six.
3.
The subdivision of a whole must be exhaustive. Many times young children divide an object into three
equal parts to illustrate one third, but have some left over.
4.
There is a relationship between the number of parts and the number of cuts. Students eventually learn
that in order to cut something into fourths, we need three cuts.
5.
All of the parts have to be equal in some manner. Too often, children see an object divided into four
pieces and call each piece one fourth, even though the pieces are not congruent or equal in some
way.
6.
Children must realize that each of the fractional pieces are part of the whole, but are also wholes
themselves. After dividing an object into fourths, have the students count fourths--one-fourth, twofourths, etc. This should be extended to count things such as five-fourths, six-fourths, etc.
7.
Conservation of the whole is essential for operational understanding. That is, students must realize the
whole remains invariant. This operational under-standing occurs somewhere around the ages of 6 or 7.
Introducing Fractional Notation and Terminology
a
a is the numerator and b is the denominator.
b
a c
a
c
 then and are called equivalent fractions.
If
b d
b
d
1
Examples of equivalent fractions:
   
2
1
2 is called a mixed number.
3
1
1 3 3 1 7
1 ( 2 3) 1 7

It can be changed to an improper fraction: 2  1  1     
or 2 
3
3 3 3 3 3
3
3
3
In the fraction
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Activities
Developing Fractional Concepts
1.
Circle which ones represent fourths:
A
2.
B
C
D
E
Circle which of the following models represents one-third? If a model does not represent onethird, explain why it does not.
3.
Find three models in this room that could be used to represent three-fourths? What models did you use?
Length? Area? Set?
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Whole-to-Part Activities
1.
Solve the following problems using Cuisenaire Rods.
a. If the brown Cuisenaire Rod represents one whole, what rod would equal three-fourths.
b. If the blue rod is one, what rod would equal two-thirds?
c. If the black rod is one, what rod would be one-half?
d. What basic model of fractions did you use?
2.
a.
Solve the following problems using Pattern Blocks.
If the yellow hexagon is one unit, what would represent one-third?
b.
If the yellow hexagon is one unit, what would the trapezoid represent?
c.
If the blue rhombus were one unit, what would the trapezoid represent?
d.
What basic model of fractions did you use?
3.
If the counters below represent one unit, what would represent two-thirds?
d.
What basic model of fractions did you use?
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Part-to-Whole Activities
4.
a. If the rectangle below represents 3/5, what would represent one whole?
b. If the rectangle below represents 5/4, what would represent one whole?
5.
a.
Solve the following problems using Cuisenaire Rods.
If the purple rod is two-thirds, what rod represents the whole?
b.
If the dark green rod is three-fourths, what rod represents the whole?
c.
If the brown rod is 4/3, what rod is one?
6.
a.
Solve the following problems using Pattern Blocks.
If a red trapezoid equals one-sixth, what could represent one whole?
b.
If a red trapezoid represents three-fifths, what could represent one whole?
c.
If a blue rhombus represents one-third, what could represent one whole?
7.
a. If the diagram below represents three-fifths, what would represent one whole?
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b. If 12 counters represent four-thirds, what represents one whole?
Part-to-Part Activities
8.
a.
Solve the following problems using Cuisenaire Rods.
If the purple rod is 1/2, what is 3/4?
b.
If the blue rod is 1 1/2, what is 2/3?
9.
a. If the rectangle below is 2/3, what is 1/2?
b. If the rectangle below is 4/3, what is 1/2?
10.
a. If a set of four counters represents one-third, how many counters are needed to represent one half?
b. If a set of three counters represents one sixth, how many counters are needed to represent one-half?
One fourth?
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FRACTION APPLICATIONS
1. All but
1
of the students in a high school attended the basketball game. If 1,050 students were in the
8
stadium, how many students attended the school?
2. According to the Container Recycling Institute, 57 billion aluminum cans were recycled in the United
States in 1999. That amount was about 5/11 of the total number of aluminum cans sold in the United
States in 1999. How many cans were sold in the United States in 1999?
3. Kids belonging to a Boys and Girls Club collected cans and bottles to raise money by returning them for
the deposit. If 54 more cans than bottles were collected and the number of bottles was 5/11 of the total
number of beverage containers collected, how many bottles were collected?
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