Turning Regular Math Problems Into Open-Ended Problems

 

Many math problems have one expected answer and one obvious pathway to get there. While these problems have a place in mathematics instruction, teachers can often get more mathematical thinking from the same basic problem by turning it into an open-ended task.

An open-ended problem allows students to make choices, find multiple solutions, explain their reasoning, or investigate what happens when conditions change. The good news is that teachers don't have to create an entirely new lesson. Often, they can simply change the question being asked.

Step 1: Start With a Traditional Problem

Begin with a familiar problem from your textbook, worksheet, or lesson.

For example:

A rectangle has a length of 12 inches and a width of 5 inches. What is its area?

Students calculate:

12 × 5 = 60 square inches.

The problem is straightforward, but there is only one answer.

Step 2: Remove Some Information

One of the easiest ways to make a problem open-ended is to remove a given number.

Instead, ask:

Find as many rectangles as you can with an area of 60 square inches. What do you notice?

Now students might find rectangles measuring 1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, and 6 × 10.

The mathematics hasn't changed, but students now have to search for possibilities and recognize patterns.

Step 3: Change "What Is the Answer?" to "What Could Be?"

Consider a traditional algebra problem:

Solve: 3x + 5 = 20.

Instead, ask:

Create three different equations whose solution is x = 5. Explain how you know.

Students might create:

• 3x + 5 = 20
• 2x − 7 = 3
• 4x + 10 = 30

Now students are working backward and thinking about the structure of equations rather than simply following a procedure.

Step 4: Ask Students to Find Multiple Solutions

A problem can also become open-ended simply by changing the wording.

Traditional:

Two numbers have a sum of 20. What are the numbers?

Open-ended:

Find as many pairs of whole numbers as possible that have a sum of 20. What patterns do you notice?

Students can generate multiple solutions and then discuss how they know they have found them all.

Step 5: Add a "Convince Me" Component

Another powerful strategy is asking students to justify their answer.

Instead of:

Is 37 prime?

Try:

Is 37 prime? Convince someone who disagrees with you.

Or:

A student says that the sum of two odd numbers is always odd. Do you agree or disagree? Use examples, words, or mathematical representations to convince the student.

The emphasis shifts from getting an answer to constructing an argument. This is important.  The next blog will explore this topic in more detail.  Let me know what you think, I'd love to hear.  Have a great day.