5.NF.A.2
*This standard is part of a major cluster
Standard
Solve word problems involving addition and subtraction of fractions referring to the same
whole, including cases of unlike denominators, e.g., by using visual fraction models or
equations to represent the problem. Use benchmark fractions and number sense of
fractions to estimate mentally and assess the reasonableness of answers. For example,
recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
Unpacked
5.NF.A.2 extends from 5.NF.A.1. In this standard, students are asked to solve problems
in the context of a word problem by using visual models or equations. In addition
students are asked to use benchmark fractions, number sense, and mentally estimate to
assess the reasonableness of answers.
*It is not necessary to find a least common denominator to calculate sums of
fractions, and in fact the effort of finding a least common denominator is a
distraction from algorithms for adding fractions (Progressions for CCSSM: Number
and Operations Fractions 3-5, The CCSS Writing Team, June 2012).
Estimation is one of the most effective ways to build understanding and procedural
fluency with fractions. There are different ways to estimate fraction sums and differences
(Siegler et al., 2010). Students might use benchmark fractions or the relative size of the
unit fraction. When using the relative size of the unit fraction, students decide how big
the fraction is, based on its unit (denominator), and apply this information to adding or
subtracting (Van de Walle, 2012). This standard refers to number sense, which means
students’ understanding of fractions as numbers that lie between whole numbers on a
number line. Number sense in fractions also includes moving between decimals and
fractions to find equivalents, also being able to use reasoning such as 7/8 is greater than
3/4 because 7/8 is missing only 1/8 and 3/4 is missing only 1/4, so 7/8 is closer to a whole
Also, students should use benchmark fractions to estimate and examine the
reasonableness of their answers. Benchmark fractions are ¼, ½, ¾, and 1. Example here
such as 5/8 is greater than 6/10 because 5/8 is 1/8 larger than 1/2 (4/8) and 6/10 is only 1/10
larger than 1/2 (5/10).
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 have 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:
Your teacher gave you 1/7 of the bag of candy. She also gave your friend 1/3 of the bag
of candy. If you and your friend combined your candy, what fraction of the bag would
you have? Estimate your answer, and then calculate. How reasonable was your estimate?
Estimate: 1/7 is really close to 0. 1/3 is larger than 1/7, but still less than 1/2. If we put them
together we might get close to ½.
*Note the task calls for the student to estimate, and then solve, so the visual model does
not have to be exact.
Solve and check for reasonableness:
1/7 + 1/3= 3/21 + 7/21 = 10/21. I know that 10 is half of 20, so 10/21 is a little less than ½.
Example:
Jerry was making two different types of cookies. One recipe needed 3/4 cup of sugar and the
other needed 2/3cup of sugar. How much sugar did he need to make both recipes?
Mental estimation:
A student may say that Jerry needs more than 1 cup of sugar but less than 2 cups. An explanation
may compare both fractions to 1/2band state that both are larger than 1/2 so the total must be
more than 1. In addition, both fractions are slightly less than 1 so the sum cannot be more than 2.
Example:
Using an area model to subtract
This model shows 1 ¾ subtracted from 3 1/6 leaving 1 + ¼ = 1/6 which a student can then
change to 1 + 3/12 + 2/12 = 1 5/12. 3 1/6 can be expressed with a denominator of 12. Once this is
done a student can complete the problem, 2 14/12 – 1 9/12 = 1 5/12.
• This diagram models a way to show how 3 1/6 and 1 ¾ can be expressed with a denominator of
12. Once this is accomplished, a student can complete the problem, 2 14/12 – 1 9/12 = 1 5/12.
Questions to check for understanding and increase rigor:
• Write a word problem that can be solved with ⅔ - ⅗?
• Is the sum of 4/5 and 7/8 under or over one? Explain how you know.
• There is ⅘ of the pepperoni pizza left over, and ¾ of the Hawaiian pizza left over.
Sydney says that there are 1 ½ total pizzas left. Do you agree? Explain why or
why not.
• Write a subtraction problem that has the same difference as ¾ - ⅛.
• Why doesn't 2/6 - 1/4 = 1/2?
• Why can you use mental computation to solve 5 - 3 1/2?
Is 3 ¾ + 2 ⅙ greater or less than 6? Explain.