Stephanie Billmyer
CCLM^2 Project
Summer 2012
DRAFT DOCUMENT. This material was developed as part of the
Leadership for the Common Core in Mathematics (CCLM^2) project at the University of Wisconsin-Milwaukee.
Grade: 4th
Domain: (NF) Number and Operations – Fractions
Cluster: Build fractions from unit fractions by applying and extending previous
understandings of operations on whole numbers.
Standard: (3) Understanding 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.
Explanation and Examples of the Standard
This standard builds upon students’ previous understandings of whole number values, and how they
are operated with, and how they are decomposed and manipulated.
In CCSSM (Common Core State Standards for Math) 4.NF.3a, students use strategies similar to what
they had used in the addition and subtraction of whole numbers. Fractions can be decomposed and
composed when they are referring to the same size whole. (This is building upon prior learning, and
reviewing that, ½ of one size whole is not equal to ½ of a different size whole.) One way I like to
explain this to my students is by taking an even number of pencils, many of different lengths. I might
have something like this:
Then I would choose two students to divide the pencils between, and as the teacher – purposely
make sure that as I’m dividing them between the students, make sure that one student gets most of
the shorter pencils. So the division could look like this:
Then I say, “Ok, now Joe has half the pencils and Amy has the other half.” Some students will agree
with me, but others will argue that it’s not fair because even though they both have the same number
of pencils, the amount of pencil (lead) that they each could use is not the same. One student has a
bunch of short pencils, so that student will not have as much pencil to write with in the end.
Therefore, it nicely opens up a conversation that when we’re talking about fractions (in this case
halves), we have to be using the same size wholes in order for the value or quantities to be the same.
Another eye-opening example is to bring in enough cookies for each of your students to receive a
fraction of a cookie (for example, a half or a third). Just make sure that you have a few cookies of
different sizes. As you divide the cookies into, say, thirds... some students will complain that they
have less cookie than another classmate. Have your class discuss why this is – understanding that
each of them has been given one-third of a cookie. Students will see that it’s important to have the
same size whole (the same size cookie) in order to give everyone an equal third.
In standard 4.NF.3b, just as when students had started by learning that to add 5 + 8, they could add
(1 +1 + 1 + 1 + 1) + (1 + 1 + 1 + 1 + 1 + 1 + 1 + 1) = 13... students begin to understand that fractions
can be added similarly. 3/9 + 2/9 can be decomposed to add (1/9 + 1/9 + 1/9) + (1/9 + 1/9) = 5/9.
The ‘unit fraction’ is an essential piece of vocabulary, which had been introduced in grade 3, and is
built upon in grade 4. A unit fraction is a fraction that represents one piece of the whole. If I eat one
piece of a pizza that is cut into 8 slices, then I have eaten 1/8 of the pizza. That is a unit fraction. If
Mason ate three times as much as me, then he ate 1/8 + 1/8 + 1/8 of the pizza. I can use the unit
fraction to understand that he ate 3/8 of the pizza. (In contrast, if I gave the same scenario to a
student who did not have the background understanding of unit fractions... he might say that Mason
ate 3 x 3/8 of the pizza, or 9/8 of the pizza, and that would be eating more than the whole pizza,
which is clearly not correct. Students will come to understand that b/b represents a value equal to 1
whole, and will begin to use that in their addition and subtraction as well. If I ask a student what
6/10 + 8/10 is... he might decompose each fraction to unit fractions, and count up each one – or, he
might decompose as follows, using the understanding that 10/10 will make one whole.
6/10 + 8/10 = 6/10 + (4/10 + 4/10)
= (6/10 + 4/10) + 4/10
= 10/10 + 4/10
= 1 + 4/10
= 1 4/10
Standard 4.NF.3d explains that students will be able to solve word problems involving fractions with
the same denominators when they have to add or subtract those fractions. Students should be able
to justify or describe their work using equations or visual models to represent their thought process,
and be able to contextualize the answer in terms of the situation of the problem. For example: “Mrs.
Billmyer had a pack of 50 pencils at the beginning of the year for her class to use. The students used
3/8 of them during the testing week, and have used another 2/8 of them since then. How much of the
pack does Mrs. Billmyer have left?” A student should solve this problem and understand that the
class has used 5/8 of the pack of pencils, and Mrs. Billmyer has 3/8 of her pack left. When a student
had perhaps added 3/8 + 2/8 and resulted in 5/8, the student should not think that this is how many of
the pencils are left... he should know that he then needs to subtract 8/8 (representing the whole pack)
– 5/8 to find the remaining amount (3/8). He also needs to understand that this is 3/8 of the original
pack of pencils, not just 3/8 of a pencil.
School Mathematics Textbook Program -- Investigations
Textbook Development:
The 4.NF.3 standard is only covered in one out of the nine units in the Investigations curriculum, but
within that unit, it is included in many of the lessons. Students begin by shading in halves and fourths
of a 4 x 6 array, and finding many different configurations of those. They then work to show eights,
thirds, and sixths of the same size array. The curriculum moves students to solving word problems
involving fractions of 24. After many lessons, a new array (a 5 x 12) is introduced, and students work
through similar practices with that. In the next few lessons, students practice finding combinations of
fractions that equal one whole. I found a lot of frustration in the transition to this last year, I thought
maybe because the students had not had enough background built up during their grade 3 curriculum
(because it was the first year they had had the Investigations curriculum), or because they needed
more practice finding fractions with different sizes of wholes. Students then move into adding and
subtracting fractions with like and unlike denominators. The curriculum then shifts to adding and
subtracting mixed numbers and fractions.
Conclusions and Suggestions:
While I do feel that the Investigations curriculum in 4th grade covers the Common Core Standards, I
do not feel that there is enough beginning fraction work done in 3rd grade. When examining the 3rd
grade curriculum, I find that the CCSSM are equally well covered at that grade level too. The
fractions standards are similarly only covered in one unit: Unit 7. I am then left to believe that we
need to examine how the third grade teachers are teaching those standards-based lessons, and
whether they feel that the curriculum is not thorough enough in certain areas, or if they are simply not
finding the time in the school year to get to that unit. I am a bit concerned that if that fractions work is
only done in Unit 7, and if that’s nearing the very end of the school year, that teachers may be rushing
through the fraction curriculum, or not able to get to it much at all. The foundation built in 3rd grade is
essential for the work we need to cover in 4th grade. And of course, the same progression of
understanding needs to be built to continue into 5th grade fraction work. I know that I did not finish the
Investigations curriculum that I needed to last year, and so there will be necessary knowledge and
understandings missing for my students as they progress into the grade 5 standards.
I think that some adjustments will continue to happen in a more natural way as more teachers teach
to the CCSSM beginning in Kindergarten. Students will begin to develop a deeper understanding of
mathematical knowledge, and the curriculum work that they do will progress deeper and farther. I
also believe, however, that we will have to examine the Investigations curriculum at each grade level
to see if there are units or lessons that we will be able to touch on lightly, or skip altogether, in order
to focus on the major concepts given at each grade level.