Creating Open-Ended Questions

Although there are more than 450 open-ended questions on this website, there are times when you will want or need to create your own. One way to create new items is to change closed-ended questions into open-ended ones. In the examples following, note how the revised questions are more conceptually oriented and require students to communicate their thinking processes.

Original Closed-Ended Item       Revised Open-Ended Item
Which of the following numbers are prime?
7, 57, 67, 117
      Fred thinks that 57 and 67 are prime because they both end in 7, which is a prime number. Dick says he is wrong. Who is correct and why?
What are the next three numbers in the following sequence?
1, 4, 7, 10, 13, ___, ____, ____
      Consider the following sequence: 1, 4, 7, 10, 13, ... . Is 100 a member of this sequence? Explain your reasoning.
Round 37.67 to the nearest 10th.       Generate three different numbers that when rounded to the nearest 10th give 37.7.
Find the LCM of 18 and 24.       Why can't 48 be the LCM of 18 and 24?

Our experience over the past twelve years indicates that, in general, teachers do not have time to create a large number of open-ended questions. But when they do, there are certain "heuristics" that can help.

Ask Students to Create a Situation or an Example That Satisfies Certain Conditions
Questions of this type require students to recognize the defining characteristics of the underlying concept. Students must take what they know about a concept and apply it to create an example. (In each of the examples below the student is asked to create a number or some kind of mathematical object that satisfies certain criteria.)

Sample Elementary-Level Questions

Make a 4-digit even number using the digits below. Explain why your number is even.

3     6     7     1     5
_____ _____ _____ _____

Give an example of an event that has a probability of 0. Explain how you know the probability is 0.

Draw a rectangle and label the sides so that the perimeter is between 19 and 20 units. Explain how you know the perimeter is greater than 19 and less than 20.

Draw a triangle that has the line of symmetry below.

Sample Middle School-Level Questions

Identify three numbers whose greatest common factor is 5 and whose least common multiple is 180. Describe how you found the numbers.

Create a set of data that would satisfy the following conditions:

The set includes 7 data points.
The range is 10 units.
The mean is greater than the median.

Show that your data set satisfies the conditions.

Fill in values for a and b to make the equation below true. Explain why your equation is true.

Draw a quadrilateral ABCD that has one and only one line of symmetry. Explain why your quadrilateral satisfies the given condition.

Sample High School-Level Questions

Write an irrational number whose square is smaller than itself. Explain why your number fits the criteria or argue that it is not possible to write such a number.

Write a data set consisting of 10 numbers so that the range is twice the median. Show that your data set satisfies the criteria.

Give the dimensions of a cone and a cylinder that have the same volume. Show that the two solids have the same volume.

Write an equation of a circle that contains the points (-4, -3) and (6,1). Graph your circle and explain why its equation satisfies the given condition.

Ask Students to Explain Who Is Correct and Why
These types of items present two or more views of some mathematical concept or principle and the student has to decide which of the positions is correct and why.

Sample Elementary-Level Questions

Of the coins made by the U.S. Mint in one year, 73% were pennies and 6% were quarters. Suppose you could have all of the coins of one type. Alex says you would get more money if you had all the pennies. Austin says you would get more money if you had all the quarters. Jenna says it depends on how many of the two coins were made. Who is right and why?

Jan says that when you find the sum , you have a lot of choices for a common denominator. Frank says there is only one choice for the common denominator. Who is correct and why?

Sasha and Brett are trying to decide how to write 5 as a decimal. Sasha thinks it is $0.5 and Brett thinks it is $0.05. Who is right and why?

Sample Middle School-Level Questions

The following responses were given when students were asked to evaluate 28 :

Michael: 28 = 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 16
Damon: 28 = 22 . 22 . 22 . 22 = 256
Dawn: 28 = 22 . 22 . 22 = 64
Which student is correct? Explain why that student is correct.

Casey claims he has divided the rectangle below into four equal areas. Terrell disagrees. Who is correct and why?

Seth and Jermaine use various methods to determine the circumference of the circle and the perimeter of the inscribed polygon. They each round their answers to the nearest whole tenth. Seth found the circumference of the circle to be 25.1, and Jermaine found the perimeter of the inscribed polygon to be 25.4. Can they both be correct? Why or why not?

Sample High School-Level Questions

Perry claims that 3 is not a zero of the polynomial below. Janice claims 3 could be a zero of the polynomial, depending on the value of a. Who is correct and why?

2x4 + ax3 + 3x2 - 5x + 10

Melanie claims that there is some value for a for which the system of linear equations below has no solution:

2x - 5y = 8
3x - 6y = a

Jeffrey disagrees and claims that there will be a solution for the system regardless of the value of a. Who is correct and why?

Kent calculated tan Ø and sin Ø (for a particular angle Ø) and claimed that tan Ø < sin Ø. Wally said this was impossible. Who is correct and why?

Ask Students to Solve or Explain the Problem/Solution in Two or More Ways
This heuristic is usually of little value to elementary teachers and only marginally useful for middle and high school teachers. The difficulty is that it is not always easy to determine what constitutes an alternative method. Also, it can be difficult to get students to think about solving a problem a different way once they have solved it one way. Their attitude is essentially, "Why find a second method when you already have one that works?" Still, some teachers like it.

Sample Middle School-Level Questions

Describe two different transformations that would each map square ABCD onto itself.

Give two different whole number values for x such that it would be possible to construct a triangle with the given lengths. Explain why your values for x will allow you to make a triangle.

Sample High School-Level Questions
Using different combinations from the list of number systems, write three true statements of the form All __________________________ are _____________________________.

Complex numbers
Irrational numbers
Natural numbers
Rational numbers
Real numbers
Whole numbers

Explain why your statements are true.

Using different combinations from the list of polygons, write three true statements of the form All __________________________ are _____________________________.


Explain why your statements are true.

When you create open-ended items, make sure they are really different from traditional items. For example, the following item is really no different than simply asking students to solve the equation:

Johnny solved 2x + 4 = 8 and got 2. Susie solved the equation and got 6. Who is correct and why?
The created item should require students to explain their reasoning, not simply to reproduce an algorithm.