The Basic Framework ◆ 23something. Price problems are about calculating the cost of items. These
are the base types of problems, and two-step and multistep problems are
mixtures of these problems with the addition and subtraction types.
Multiplicative Comparison Problems
There are three different types of multiplicative comparison problems (see
Figure 2.7 and Figure 2.8). The first type is where we are looking for the
bigger part. We are comparing two quantities and looking at a multiplica-
tive difference. For example, John has 40 marbles; his cousin David has 3
times as many as he does. In the second type of problem, we are looking
for the smaller part. For exam-
ple, David has 120 marbles, he
has 3 times as many as his cousin
John. How many does John have?
This is a tricky problem for
students because they want to
multiply instead of divide
because they see the word
times and want to solve it,
without thinking about the
entire problem. In the third
type of problem, we are look-
ing for the difference between
the two parts. This problem
tends to be the hardest to
understand. It is where both
quantities are given and stu-
dents have to think about the
relationship between the two
numbers. For example, Mike
has 15 toy cars. John has 3 toy
cars. How many times as many
toy cars does Mike have as John?
CGI and Fraction Word Problems
CGI also has a hierarchy of word problems for fractions and decimals.
This is amazing, although not many people are aware of it or use it in
their day-to-day curriculum approaches. Empson and Levi (2011) wrote
an incredible book describing the continuum of teaching fraction and
Make sure that you emphasize the dif-
ference between additive com parison
problems and multiplicative comparison
problems. Illustrative Math ematics has
a really good snake problem where they
discuss measuring different-sized snakes
that grow the same amount. They ask the
students to explain if the snakes grew the
same amount and expect students to be
able to discuss that there are a few ways
to look at it—both additively and
multiplicatively.
***See Illustrative Mathematics Snake
Problem 1 & 2
https://www.illustrativemathematics.
org/content-standards/tasks/356
https://www.illustrativemathematics.
org/content-standards/tasks/357Figure 2.7 Additive Comparison and Multiplicative
Comparison Problems