Most practice problems in your average algebra textbook follow a familiar pattern: they are generally one or two-step applications of the basic skill taught in the previous chapter, with the numbers shuffled around a bit. What they all have in common is that they are ‘closed’ problems. They have a single correct answer, and usually can only be solved in one or two ways. In contrast, open-ended problems are those which, while having a single solution admit of multiple solution methods, or those which have multiple correct solutions.

### The Benefit of Open-Ended Problems

Open-ended problems are highly valuable teaching tools for several reasons:

- they foster a deeper conceptual understanding, because students must come up with the solution techniques themselves instead of simply following the ‘recipe’ laid down in the solved examples earlier in the chapter
- they can help improve communication and teamwork skills, as students discuss among themselves the possible avenues to a solution, none of which may be THE right one
- they tend to be highly motivating, as students rise to the challenge of tackling an intriguing problem which seems solvable but does not suggest any obvious solution method
- they facilitate differentiating instruction, as students who find difficulty with one solution method (for example, tables of values) might find it easier to use another (e.g. the sketch of the graph of a function).

Despite their clear educational value, such problems are almost nowhere to be found in standard textbooks. Some of these feature one or perhaps two problems labeled open-ended, but on inspection they often turn out to be just a banal recycling of the standard two-step application problem (for example, one ‘open-ended’ textbook problem I recently came across simply said, “Construct and solve a quadratic equation”; this is not so much open-ended as empty-headed, a meaningless, pointless task that does not tend to increase understanding in the slightest).

### Creating Open-Ended Problems

During the past two years I have conducted an intensive search for rich open-ended problems I can use in my classrooms. By studying the examples I came across I feel I have begun to understand what goes into the construction of such problems. In this post I will share a few examples, and suggestions for constructing them in your own classroom.

I’ll start with a simple example from Algebra I. One of the units in this course covers linear equations and their graphs. Students learn about the concept of slope, the meaning of the y-intercept, how to derive the equation of a straight line from two points on the line, etc. Standard problem tasks include finding the equation of a line given its slope and a point on the line, using the equation of a line to find its slope and y-intercept, and others. These are all one or two-step, mechanical applications of the procedures taught in the textbook. But amid all the effort to master procedures, several crucial understandings are at risk of being overlooked. For example, how many students really understand how slope determines the direction of the line, or why specifying the slope of a line does not pick out a unique line we might be interested in?

One straightforward way of constructing an open-ended problem is to specify a set of constraints. In Algebra I, instead of having my students do the standard tasks of finding the equation of a line using two points on the line, I occasionally instruct my students to construct a set of lines each satisfying a certain constraint, such as having a certain slope or a certain y-intercept. If you ask students to come up with five lines that have the same slope, for example, it is up to them to realize that they must come up with different y-intercepts. Similarly, if you ask them to come up with five lines that pass through the same point, they must realize that such lines will have different slopes and will resemble the spokes on a wheel if plotted on the xy plane. While this task may seem trivial to teachers who have been working with lines and slopes for many years, I found that students have a surprisingly hard time with such a task, as they are used to memorizing procedures instead of thinking about the meaning of their work.

### Making a Problem’s Solution Practical

Another technique for creating good open-ended problems in order to reinforce thinking outside the box is to come up with a task in which standard, textbook solution methods can be applied, but they are so long and complicated that students are compelled to think of an alternative. For example, if you ask students to multiply 10,001×9,999 without using a calculator, the standard algorithm is fairly tedious. However, an enterprising student might notice that 10,001=10,000+1 and 9,999=10,000-1, so the expression becomes a difference of squares: (10,000+1)(10,000-1)=100,000,000-1=99,999,999. Adding a further constraint, for example a time limit, might make the search for alternative solutions even more compelling.

These examples just barely scratch the surface of open-ended problems, of course. And constructing such problems does take time and energy, precisely because they are non-routine. They can be challenging for the teacher as well as the student! Nevertheless, putting in the effort is well worth it, because the pedagogical benefits of such problems are too obvious to ignore.

## Open-Ended Problems:

Modeling Orbital Debris Problems

In this lesson students discuss the problems caused by human space pollution. This may not be a subject they have thought about previously and it provides a way for them to creatively describe the problem. Students can use graphs, spreadsheets, and computer programs to display their results.

Trout Pond Population

Students analyze a fish population using recursion, iteration, and algebra. In this lesson students use a variety of tools to figure out the long-term effects on the fish population. This lesson provides a way for students to create original means to use graphs, tables, equations, and other strategies to display information.

A Family Finance Experience

This lesson has students use family finance as a way to explore the use of mathematical functions. It’s a way for students to connect mathematical equations, graphs, etc. to real life experience.

Slopes of Lines

In this lesson students learn how to use mathematical equations to show the slopes of lines. They apply this knowledge to real world experiences, and graph the results.