Michelle Painter and Jenny Kim and
CCLM^2 Project
Summer 2012
This material was developed for the Leadership for the Common Core in Mathematics (CCLM^2) project at the University of Wisconsin-Milwaukee.
CCSSM Analysis
4.NF.3abcd
Part 1. Standard
Grade: 4
Domain: Number and Operations - Fractions
Cluster: Build fractions from unit fractions by applying and extending previous understandings of operations on
whole numbers.
Standard: 4. NF.3.abcd
3. Understand a fraction a/b with a>1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same
whole.
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording
each decomposition by an equation. Justify decompositions, e.g. by using a visual fraction model. Examples:
3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
c. Add and subtract mixed numbers with like denominators, e.g. by replacing each mixed number with an
equivalent fraction, and/or by using properties of operations and the relationship between addition and
subtraction.
d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having
like denominators, e.g. by using visual fraction models and equations to represent the problem.
Part 2. Explanation and Examples of the Standard
Standard
Explanation: What a Student Will
Understand
Examples: Visual Models
4. NF.3
-students will understand that a fraction
with a numerator greater than 1 can be
decomposed (taken apart) into its smaller
unit fraction
¾=¼+¼+¼
Understand a fraction a/b with a>1 as a
sum of fractions 1/b.
¼
1
4
0
4. NF.3.a
Understand addition and subtraction of
fractions as joining and separating parts
referring to the same whole.
4. NF.3.b
Decompose a fraction into a sum of
fractions with the same denominator in
more than one way, recording each
decomposition by an equation. Justify
decompositions, e.g. by using a visual
fraction model. Examples: 3/8 = 1/8 +
1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1
+ 1 + 1/8 = 8/8 + 8/8 + 1/8.
¼
¼
2
4
3
4
1
-students will understand that fractions
represent a quantity of a whole that can be
joined and separated with addition and
subtraction through the use of multiple
visual models
-this model helps students see fractions as
separate quantities that when joined or
separated create a whole
-students will understand how to decompose
(take apart) a fraction with like
denominators in more than one way by
writing it in an equation and showing their
understanding with a visual model
¾=¼+¼+¼
2/3
¼
8/5
¼
1¾=1+
¼
1/4
+
2/4 = 4/4 + 1/4 +
2/4
1 = 4/4
1/4
2/4
Standard
What a Student Will Understand
Visual Models
4. NF.3.c
-students will understand how to change
fractions with like denominators from a
mixed number to an equivalent fraction with
the use of visual models and then be able to
add and subtract with those fractions
1¾=1+
Add and subtract mixed numbers with like
denominators, e.g. by replacing each
mixed number with an equivalent fraction,
and/or by using properties of operations
and the relationship between addition and
subtraction.
1/4
+
2/4 = 4/4 + 1/4 +
2/4= 7/8
1 = 4/4
1/4
2/4
2 ¾ - 2/4 = ?
2 ¾ = 4/4 + 4/4 + 3/4 = 11/ 4
11/4 – 2/4 = 9/4
(the above example uses decomposing of
numbers)
9/4 = 4/4 + 4/4 + 1/4= 1+1 + 1/4 =
¼
¼ ¼ ¼
¼ ¼ ¼ ¼ ¼
2¼
¼ ¼
1= 4/4
2= 8/4
(fraction strips can also be used as a visual
model)
2 ¼ + 2 ¼ = 2 + ¼ + 2 + ¼ = 2 +
2 + ¼ + ¼ = 4 + 2/4 = 4 ½
(the above example uses the commutative
property)
4. NF.3.d
Solve word problems involving addition and
subtraction of fractions referring to the
same whole and having like denominators,
e.g. by using visual fraction models and
equations to represent the problem.
-students will understand how to solve word
problems involving addition and subtraction
of fractions starting with like
denominators, leading to making the
denominators alike through consistent use
of visual models
-the problems will be represented by the
visual models and equations
Bill had 2/3 of a cup of juice. He
drank 1/2 of his juice. How much
juice did Bill have left.
1/3
1/6 1/6
2/3 = 4/6
1/3
1/6
1/6
1/2
1/6 1/6
1/2 = 3/6
1/6
4/6 – 3/6 = 1/6
1/6
Bill has 1/6 of a cup of juice left.
Part 3: School Mathematics Textbook Program
Text Development:
3rd Grade-Everyday Math
In the 3rd grade math curriculum
students are learning:
•mixed numbers (one type of visual
model-fraction circles-lesson 8.7)
•word problems with fractions (one
use of number line, counters,
drawings-lesson 8.8)
4th grade-Everyday Math
In the 4th grade math curriculum
students are learning:
•adding and subtracting fractions
with unlike denominators (one use of
visual model-pattern blocks-lesson
7.5)
•mixed numbers (appears in review
of basic fraction concepts-one
reference using pattern blocks, and
one reference using a number linelesson 7.1)
•word problems with fractions (very
few embedded into the math boxes
and homework, but doesn’t explicitly
appear in the teaching of the lessonunit 7)
5th Grade-Math Thematics
In the 5th grade math curriculum
students are learning:
•adding and subtracting fractions
with unlike denominators (very little
use of visual modeling-one reference
to a number line, some use of
pattern blocks-module 1)
•mixed numbers (very little use of
visual modeling, mostly pattern
blocks used-module 1)
•word problems with fractions
(limited problems embedded into
practice-module 1)
Conclusions and Suggestions:
Conclusion:
There is not nearly enough repeated practice in fractions or use of visual models such as tape diagrams, number
lines, or multiple examples of area models.
Suggestions:
•More time needs to be spent on explicitly teaching fraction concepts through the use of visual models so that
students develop a deeper understanding of fractions.
•More time needs to be spent on using visual fraction models rather than algorithm based models so that students
will deepen their understanding of fractions as a quantity and how a unit fraction relates to the whole.
•More exposure needs to be given to real world context problem solving situations so that students will develop a
solid understanding through application.
•District conversations with staff to needs to occur to reiterate the importance of the use of visual models in daily
instruction.
•Vertical conversations should occur with the grade levels before and after so that the progression of learning can
be seen.