Grades 3-5 Math Problem Solving in Action_ Getting Students to Love Word Problems

The Basic Framework ◆ 25

decimal word problems called Extending Children’s Mathematics: Fractions &
Decimals: Innovations in Cognitively Guided Instruction.
Many state standards somewhat follow the trajectory; however, teach-
ers tend to start with whatever standards are in their grade. It is of the
utmost importance that teachers figure out where their students are and
then start from there. It is crazy-making to just start where the book does.
Often, when we start where the book does, instead of where the students
are, we get big problems. We say, “These students don’t understand frac-
tions.” The intervention should start at the level the student is at, and that
requires that we assess students and then provide intentional interventions.
The figure below (Figure 2.9) shows a continuum of a fraction hierarchy
that is followed in most state standards. There are a few differences in the
order in some states, but this is the general hierarchy. What is important
about this is that it provides a framework for testing prior knowledge
before jumping into a fraction unit.

Figure 2.9 Fraction CGI Problems

Level 1:

Start with

situations of 2

or 4 children,

as children’s

earliest

partitions

are based on

halving

Level 2:

Move to

situations of

three sharers

Level 3:

Move into

equivalent

fraction

problems

Level 4:

Move to

higher levels

of sharers

Level 5:

Comparisons

of

equal-sharing

situations

Level 6:

Adding

fractions

with like

denominators

Level 7:

Adding

mixed

numbers

with like

denominators

Level 8:

Subtracting

fractions

with like

denominators

Level 9:

Subtracting

mixed

numbers

with like

denominators

Level 10:

Multiplying

a fraction

by a whole

number

Level 11:

Adding

fractions

with unlike

denominators

Level 12:

Adding

mixed

numbers

with unlike

denominators

Level 13:

Subtracting

fractions

with unlike

denominators

Level 14:

Subtracting

mixed

numbers

with unlike

denominators

Level 15:

Multiplying 2

fractions

Level 16

Dividing

a whole

number by

a fraction

Level 17

Dividing

a fraction

by a whole

number

Level 18

Dividing 2

fractions

Level 19

More complex

multistep

problems