5.NF.B.3
*This standard is part of a major cluster
Standard
Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word
problems involving division of whole numbers leading to answers in the form of fractions or
mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For
example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3,
and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If
9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice
should each person get? Between what two whole numbers does your answer lie?
Unpacked
This standard calls for students to extend their work of partitioning a number line from third and
fourth grade. Students need ample experiences to explore the concept that a fraction is a way to
represent the division of two quantities. Students are expected to demonstrate their understanding
using concrete materials, drawing models, and explaining their thinking when working with
fractions in multiple contexts. For example, they read 3/5 as “three fifths” and after many
experiences with sharing problems, learn that 3/5 can also be interpreted as “3 divided by 5.”
When fractions represent division, the denominator acts as the divisor and the numerator is the
multiplier. This means that 3/4 is three times what you get when you divide a whole into four
parts. So if you were to divide 3 pancakes among 4 people (3 ÷ 4), each person would get 3
pieces or 3/4 of a pancake. Students need to understand that numerator is the same as the dividend
and denominator is the same as the divisor.
Fifth grade students should connect fractions with division, understanding that 5 ÷ 3 = 5/3.
Students should explain this by working with their understanding of division as equal sharing.
This standard refers to using visual models. There is substantial evidence to suggest the effective
use of models in fraction tasks is important (Cramer & Henry, 2002; Sibert & Gaskin, 2006). In
order to provide experiences that focus on modeling, using appropriate tools, and justifying the
model, it is likely that those experiences will extend far beyond textbook experiences. Students
often do not explore models beyond an area model (a model with portions shaded in), and it is
imperative that students are exposed and used to working with a variety of fraction models.
Sometimes students are able to make sense of one representation and not another. It can be
helpful to ask students to justify thinking using two different visual models. Because this standard
moves into the symbolic form of a process, using visual tools can help students make sense of
what is happening. What appears to be critical in learning is that the use of physical tools leads to
the use of mental modes, and this builds students understanding of fractions (Cramer & Whitney,
2010; Petit, Laird, & Marsden, 2010). Students also need to be exposed to real world contexts and
connections when working with fractions. One model may not serve the context. For example, if
students are being asked who walked the farthest, a linear model is more likely to support their
thinking than an area model (Van de Walle, 2012). Having students defend their solutions using
models, including simple student drawings, and having them connect the models to the symbolic
operations help students learn to think about fractions and the operations, as well as contribute to
mental methods (Van de Walle, 2012).
Example:
If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice
should each person get?
This can be solved in two ways. First, they might partition each pound among the 9 people, so
that each person gets 50 x 1/9 = 50/9 pounds.
Second, they might use the equation 9 x 5= 45 to see that each person can be given 5 pounds,
with 5 pounds remaining. Partitioning the remainder gives 5 5/9 pounds for each person.
(Progressions for the CCSSM, Number and Operation – Fractions, CCSS Writing Team, August
2011, page 11)
The above visual model could be interpreted as 9/9 + 9/9 + 9/9 + 9/9 + 9/9 + 5/9 = 50/9
OR
5
It could be seen as 5 wholes and 5/9 or 5 9
Example:
Ten team members are sharing 3 boxes of cookies. How much of a box will each student get?
When working this problem a student should recognize that the 3 boxes are being divided into 10
groups, so s/he is seeing the solution to the following equation, 10 x n = 3 (10 groups of some
amount is 3 boxes) which can also be written as n = 3 ÷ 10.
Using models or diagram, they divide each box into 10 groups, resulting in each team member
getting 3/10 of a box.
The above model illustrates dividing the three boxes into tenths, and then equally sharing the
parts with 10 people where each person gets one part of each box equaling 3/10 each.
Example:
The six fifth grade classrooms have a total of 27 boxes of pencils. How many boxes will each
classroom receive? Students may recognize this as a whole number division problem but should
27
27
also express this equal sharing problem as 6 . They explain that each classroom gets 6 boxes of
3
1
pencils and can further determine that each classroom get 4 6 or 4 2 boxes of pencils.
Your teacher gives 7 packs of paper to your group of 4 students. If you share the paper equally,
how much paper does each student get?
1
Each student receives 1 whole pack of paper and 4 of the each of the 3 packs of paper. So each
3
student gets 14 packs of paper.
Questions to check for understanding/ increase rigor:
• What is the relationship between 7 divided by 8 and ⅞?
• How do the numerator and denominator of a fraction compare with the dividend and
divisor of a division expression?
• Every kid in the car on its way to swim practice got ¾ of a sub. There were four kids.
How many subs did the mom get?
• Why do some people write ⅘ when they mean 4÷5 ?
• Draw pictures or build models to show why ⅛ = 1 ÷ 8, why 8/3 = 8 ÷ 3 =, and why ⅜ =
3÷8
•
15 friends want to share 3 watermelons equally. What fraction of a watermelon
will each friend get?