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, notice how the revised questions are more conceptually oriented and require students to communicate their thinking processes:
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?
My experience over the past 13 years in teaching has shown me 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.
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.
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 1/4=1/6 , 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?