Multiplicative Comparison Problems
Multiplicative Comparison problems involve a comparison of two quantities in which one is described
as a multiple of the other. The relation between quantities is described in terms of how many times
larger one is than the other. The number quantifying this relationship is not an identifiable quantity. In
the following problem, the animals’ weights are measurable quantities, but the other quantity (three
times as much) simply describes the relation between the measurable quantities:
The first-grade class has a hamster and a gerbil. The hamster weighs 3 times as much as the
gerbil. The gerbil weighs 9 ounces. How much does the hamster weigh?
Multiplicative Comparison problems can also be constructed for any of the three basic problem types,
as seen below. Multiplicative compare problems appear first in the CCSS-M in fourth grade, with whole
number values. In Grade 5, unit fractions language such as “one third as much” may be used.
Multiplicative
Comparison:
4th Grade
(4.OA.2)
Smaller Unknown:
The giraffe is 18 feet tall.
She is 3 times as tall as the
kangaroo. How tall is the
kangaroo?
Multiplicative
Comparison:
5th Grade
Smaller Unknown:
The giraffe is 18 feet tall.
The kangaroo is 1/3 as tall
as the giraffe. How tall is
the kangaroo?
Larger Unknown:
The giraffe in the zoo is 3
times as tall as the
kangaroo. The kangaroo is
6 feet tall. How tall is the
giraffe?
Larger Unknown:
The kangaroo is 6 feet tall.
She is 1/3 as tall as the
giraffe. How tall is the
giraffe?
Mutliplier Unknown: The
giraffe is 18 feet tall. The
kangaroo is 6 feet tall. The
giraffe is how many times
taller than the kangaroo?
Multiplier Unknown:
The giraffe is 18 feet tall.
The kangaroo is 6 feet tall.
What fraction of the height
of the giraffe is the height
of the kangaroo?
The comparison is based on one set being a particular multiplier of the other. In an additive
comparison problem, the underlying question is what amount would be added to one quantity in order
to result in the other. The underlying question for multiplicative comparison problems is what factor
would multiply one quantity in order to result in the other.
It is important to introduce multiplicative comparison problems to our primary students as they can
be extended in later years to include multiplication and division of fractions. Thus, including these
problems with whole numbers in the primary grades lays the groundwork for developing an
understanding of multiplication and division of fractions in later grades.
Resources
Carpenter, T. P., Fennema, E., Franke, M., Levi, L., & Empson, S. B. (1999). Children’s mathematics:
Cognitively guided instruction. Portsmouth, NH: Heinemann.
Common Core State Standards Initiative (CCSSI). (2010). Common core state standards for
mathematics. (Grade 4 Operations and Algebraic Thinking, 4.OA., p. 29). Washington, DC: National
Governors Association. Retrieved from http://www.corestandards.org/the-standards.
Common Core State Standards Writing Team. (2011). Progressions for the Common Core State
Standards in Mathematics: K, Counting and Cardinality; K-5 Operations and Algebraic Thinking (draft
05.29.11). Retrieved from: http://ime.math.arizona.edu/progressions/#
Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012-2013 School Year