n Before you calculated, could you tell
whether the number of students left in
the classrooms would be more or less
than one-half of the total number of students? Explain.
Open Questions for Every Grade
n The answer is 10. What might the question be?
n How are 5 and 10 alike? How are they different?
n Choose two numbers to add. What is the sum?
n Create a sentence using the words and numbers and, more,
n What operation did you or could you
use to solve your problem? Why that one?
n Would it be easier to solve the
problem if one more student had left the
n The answer is . What might the question be?
n How are 80 and 800 alike? How are they different?
n Create a sentence using the words and numbers product,
8, almost, and
n The product of two numbers is almost 30. What might the two numbers be?
n How could you use mental math to
solve your problem?
n The answer is 30π. What might the question be?
n How are the formulas for the circumference and the area of a circle alike?
How are they different?
n Create a sentence using the words surface area, volume, greater, and 300.
n The sum of two integers is a negative integer very far from zero. What
might the integers be?
n How did you solve your problem?
How many students are still in their
Notice that the questions focus on the
common elements—until the last one,
in which the teacher asks the students
to describe their specific strategies.
n The answer is
2. What might the question be?
n How are calculating powers and calculating logarithms alike? How are they
n Create a sentence using the words irrational, repeating,
4, and greater.
n An irrational number is approximately 8. What might it be?
on strategies and the meaning of multiplication would apply to both tasks.
(The problem involves subtraction and
is suitable for mental math calculations
because 99 is so close to 100.)
Creating Parallel Tasks
Just as there are techniques to creating
open questions, there are techniques to
creating parallel tasks.
Strategy 1: Let students choose between
two problems. The teacher might give
students a choice between two problems
at different levels of difficulty:
n Choice 1: There are 427 students in
Tara’s school in the morning. Ninety-nine
of them left for a field trip. How many
students are still in their classrooms?
n Choice 2: There are 61 students in
3rd grade. Nineteen of them are in the
library. How many students are still in the
classrooms? (This problem also uses subtraction and is suitable for mental math,
but it involves smaller values for students
who are not ready for work with 3-digit
Many teachers shy away from differentiation in math because they do not see
how to do it. These two strategies—
creating open questions and creating parallel tasks—show how to differentiate
math instruction in a manageable way.
By doing so, teachers can make all students feel like part of the larger community of learners as all contribute to a
rich discussion of mathematics. EL
1Small, M. (2005). Prime: Number and
operations. Toronto: Nelson Thomson
2My books Good Questions: Great Ways
to Differentiate Mathematics Instruction
(Teachers College Press, 2009) and More
Good Questions: Great Ways to Differentiate
Secondary Mathematics Instruction (with Amy
Lin, Teachers College Press, 2010) provide
many models of open questions built
around the big ideas in each strand of mathematics for each grade band.
Strategy 2: Pose common questions for
all students to answer. The teacher could
ask all students the following questions,
no matter which task they completed:
Marian Small is Dean Emerita of the
University of New Brunswick in Canada
and president of One Two… Infinity;