4.NF.B.4.A
*This standard is part of a major cluster
Standard: Understand a fraction a/b as a multiple of 1/b. For example, use a visual
fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by
the equation 5/4 = 5 × (1/4).
Grade four expectations in this domain are limited to fractions with denominators
2,3,4,5,6,8,10,12, and 100.
Unpacked:
* Note this standard references fraction a/b as a MULTIPLE of 1/b. This is NOT
referencing the procedure or steps for standard algorithm for multiplication
of fractions.
1/b is the unit fraction, a fraction with one as the numerator. For example 1/4, 1/3,
etc. a/b is the understanding that a different numerator simply means it is a
multiple of the unit fraction. In whole-number learning, counting precedes and helps
students to add and later subtract. This is also true with fractions. Students should come
to think of counting fractional parts in much the same way as they might count apples or
any other objects (Van de Walle, 2013). To count or repeat a piece is called iterating.
Iterating is critical in developing this standard. Many tasks can be practiced and designed
such as simply counting a partitioned ribbon and asking students to count together “ onefourth, two-fourths, three-fourths, and four-fourths.”
In addition, this standard references the importance of using a fractions model. There is
substantial evidence to suggest that the effective use of models in fraction tasks is
important (Cramer & Henry, 2002; Siebert & Gaskin, 2006). Unfortunately, textbooks
rarely use manipulatives, and when they do, they tend to only be area models (Hodges,
Cadey, & Collins, 2008). This means that students often do not get to explore fractions
with a variety of models and/or do not have sufficient time to connect to the models to
the related concepts. In fact, what appears to be critical in the learning is that the use of
physical tools leads to the use of mental models, and this builds students’ understanding
of fractions (Cramer & Whitney 2010; Petit, Laird & Marsden, 2010).
Finally, this standard asks to record or model the conclusion with an equation.
In summary, students are asked to understand that fractions a/b are multiples of unit
fractions, model their understanding of this with visual models and with an equation.
This standard builds on students’ work of adding fractions where they worked with
decomposing fractions into unit fractions and extending that work into
multiplication. Example:
3/6 = 1/6 + 1/6 + 1/6 = 3 x (1/6) Number line:
Questions to Check for Understanding and Increase Rigor:
• Josh noticed that ⅓ + ⅓ + ⅓ +⅓ was the same as 4 x ⅓. Do you agree or
disagree with Josh’s observation? Show a visual representation to justify
your thinking.
• What are the multiples for 6/8? Write an equation and a visual model for
your conclusion.
• When multiplying a whole number by a fraction, what happens to your
product? Why is your product less than your original whole number? (also
works for 4.NF.4b)
• What two factors can be multiplied to equal a product of 6/8? Show a visual
representation to justify your thinking.