4.NF.A.2
*This standard is part of a major cluster
Standard: Compare two fractions with different numerators and different denominators, e.g., by creating
common 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.
Unpacked Standard:
This standard calls students to compare fractions by creating visual fraction models or finding common
denominators or numerators. Students’ experiences should focus on visual fraction models rather than
algorithms, and continue the work from 4.NF.1. Students must JUSTIFY conclusions by using and creating
fraction models, and therefore limiting understanding to a process does not allow students to develop the
understanding intended for this standard. A deep understanding of 4.NF.1 should be in place prior to work in
this standard. After that is established, integration of the two standards is appropriate.
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Understanding WHY fractions are equivalent, and developing the conceptual understanding of
equivalency is of CRITICAL importance in fraction understanding. If students are only able to replicate
the process, major gaps in learning will occur as more difficult fraction concepts are introduced such as
comparing, ordering, adding, and subtracting fractions.
Teachers may be compelled when teaching 4.NF.2, to state “In order to compare these fractions, we have to find
a common denominator. After all we can’t compare apples to oranges.” This well-intentioned statement is
essentially false. A correct statement might be “Remember to compare we might use a benchmark fraction, or
compare equal-sized parts. The algorithm is simply designed to find common denominators so that the parts are
now equal in size, and easier to compare.”
When tested, models may or may not be included. Students should learn to draw fraction models to help them
compare. Students must also recognize that they must consider the size of the whole when comparing fractions
(ie,. and 1/8 of two medium pizzas is very different from 1/2 of one medium and 1/8 of one large).
Example:
Using an area model to compare to a benchmark fraction, and then make broader observations.
Example:
Using set models to compare. (it is important that students are able to compare fractions as a set, and are not
limited to thinking of fractions in terms of area models).
Example: Understanding that fraction comparisons are only valid when comparing the same size whole:
In Example A, students might have a misconception in regard to the total amount of buttons, and the conceptual
understanding that the fraction represents the part to whole relationship, and not the total number of buttons,
which has been the basis for whole number comparison. This misconception may cause confusion for students
when trying to conceptualize ½ of the earth to ½ of an apple. This concept becomes important when attempting
to create visual models to prove the comparisons.
Example A:
Example B:
Asking students to model how much coffee each person would get if they were only to receive 1/3 of each size,
and then compare the results. Guiding a discussion in regard to what they notice about 1/3 of each size could
lead into the understanding that while all three cups can be divided into three equal parts, the actual size of each
third is different depending on the size of the whole.
Examples understanding the relationship when the numerators are the same or the denominators are the same:
Example A:
In this type of example, students might reason that while the numerators are the same, the sizes of the pieces are
different and they can therefore compare them.
Example B:
In this type of example, a student might reason that since each fraction is divided into equal size pieces (the
denominator), they can then reason about the amount of shaded parts (the numerator).
Example Using a Number Line:
Students may use a number line to identify fractions in order to compare them. Students might place each
number on their own number line, or generate common denominators or numerators to place them all on the
same number line.
Questions to check for understanding and increase rigor:
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Write three fractions between ½ and 1 whole. Model and explain your thinking.
With a partner, list five fractions between 0 and ⅘? Together, create a way to model and share your thinking with your
classmates.
If two fractions have the same numerator, how will this help you compare the fractions? Explain
How do benchmark fractions help you compare fractions? Model and explain
If you were hungry and you offered your friend a choice between ⅗ of a cookie and 7/10 of a cookie, which fraction of a
cookie would you hope she took? Explain.
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What strategies could you use to compare ¾ and 2/12? Explain your reasoning
Think of a real-world example when ¾ would be smaller than ½. (Students should think of examples where the whole is
different. For example, ¾ of a cupcake is less than ½ of an entire cake) Model and explain your thinking.