20 ◆ The Basic Framework
example, Susie ate ¼ of her candy bar. Joe ate^1 ⁄ 5 of his candy bar. Who ate more?
Another example, Mary had $500 and Jane had $350. How much more money
does Mary have than Jane? Or, How much less money does Jane have than Mary?
In a difference problem, when you say less, it is considered a more difficult
version of the problem. There is another version of the compare the dif-
ference problem. For example, Jean has 20 marbles and Mike has 10. How
many more marbles does Mike need to have the same amount as Jean?
The second type of comparison story is where the bigger part is
unknown. In this type of story, we are looking for the bigger amount. For
example, Luke had $507 and Marcos had $109 more than Luke. How much does
Marcos have? How much do they have altogether? Another example, Luke had
$507. This is $109 less than Marcos. How much does Marcos have? There are
two types of this problem. When you say less and you are looking for
more, it is considered the harder part because it is counterintuitive. The
task is to find the bigger part.
The third type of comparison story is where the smaller part is
unknown. In this type of story, we are looking for the smaller amount.
For example, Luke had $507 and Marcos had $109 less than Luke. How much
does Marcos have? How much do they have altogether? Another example, Luke
had $507. This is $109 more than Marcos. How much does Marcos have? There
are two types of this problem. When you say more and you are looking
for the smaller part, it is considered the harder version because it is coun-
terintuitive. The task is to find the smaller part.
Multiplication Problems
There are several different types of multiplication problems (see Figure 2.5).
Equal group problems require students to look for the total amount of the
group. Array problems require students to look for the total amount in
the array. Area problems require students to solve for the total area. Rate
problems are about students calculating the amount of time it takes to do
something. 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.
Division Problems
There are several different types of division problems (see Figure 2.6).
Equal group problems require students to look for either the total amount
in each group or the total amount of groups. Array problems require
students to look for the amount in each row or the amount of rows. Area
problems require students to solve for the sides or the total area. Rate
problems are about students calculating the amount of time it takes to do