How To Create Higher DOK Problems

Now that you’ve read about the three steps I often follow when trying to raise a math problem’s Depth of Knowledge level, I thought it might be useful to take you through the entire thought process of actually implementing those steps.  The experiences I’m describing took place while preparing with some other teachers to teach a lesson in an eighth grade classroom.

The first thing I needed to know was the standard I wanted to address.  In this case, it was the Pythagorean Theorem.  Next, I thought about what a typical Pythagorean Theorem problem looks like.  Knowing this is useful because often times you can take typical problem as the foundation and make tweaks to it.  Here are examples of what common problems look like:

From here, I started thinking about what changes I could make.  A typical problem gives two of the three side lengths for the triangle.  So, following the steps listed here, I thought about what would happen if two or three of the side lengths were missing and I asked students to give me multiple answers.

If three side lengths were missing, I thought that there might be too much flexibility.  Students could just list Pythagorean triples.  So, I thought it would be useful to define one side length and have students find the other two.

The next choice to make was deciding which side length to provide and which two to leave for them to discover.  I decided that if I defined one of the legs and left blanks for the other leg and the hypotenuse, it could leave the problem open to highlighting a misconception.  Specifically, I could imagine students finding two potential side lengths, but not realizing that the hypotenuse is always the largest side.  This would be useful to uncover.

I also realized that there was no way that one-digit side lengths were going to work for this problem so they had to be at least two-digits.  I figured that three-digits would be too long, so two-digits seemed ideal.

The next dilemma was what to do about the inevitable issue of irrational side lengths expressed with a square root symbol such as √7 or √34.  I seemed to have two choices:

• I could allow students to place the square root symbol on either of the missing side lengths
  • Pro: allows more flexibility
  • Con: this might make the problems and instructions a little more confusing
• I could place the square root symbol over one of the missing side lengths
  • Pro: much more straightforward
  • Con: potentially limits opportunities for creative problem solving.

Ultimately, I decided to allow students to place the square root symbol where they want… at least at first.

With the DOK 2 done, I was most of the way towards the DOK 3.  Generally, I now want to optimize the problem so that something has the greatest or least value possible or is close to a certain value.  More detail on this process is discussed here.  In this case, I thought it would be useful to make the length of one of the sides as long as possible.

At first I thought that the side I wanted to have the greatest length should be the hypotenuse.  Then I realized that if I required the leg to be as long as possible, it would be another chance to get a misconception to come out.  As a rule, a right triangle’s hypotenuse (the side opposite the right angle) must always be the longest side.  So, if I asked for the missing leg to be as long as possible, it really should be the triangle’s second longest side length.

If students didn’t fall for the potential misconception during the DOK 2 problem, surely some students will make the mistake now.  To be clear, it isn’t my intention to trick students.  My hope is that if anyone still has misconceptions and isn’t 100% clear that the hypotenuse always has to be the longest side, as a teacher, I want to know so I can do something about them.

With the problems now in their first draft, the next thing you need to do is actually attempt to solve them.  This works best with as many people as possible trying to solve them because sometimes you only think about your problem solving method and not how others might approach it.  So, we started solving the DOK 2 and then the DOK 3 problem, and I also tweeted them out to others to see how they approached it.

We immediately started reflecting on the choice to place the square root.  I’ve already discussed the pros and cons of the two options we saw.  As it turns out, people took it in even another direction: using the square root in the middle of the number.

Ultimately, we thought that it might be too confusing for the 8th graders so we decided to place the square root symbol over the leg on both the DOK 2 and DOK 3 problems.  If it didn’t give us the results we wanted, we could always take it off and change the directions.  Using this modified problem, when we found the answer to the DOK 3 we got the length of the hypotenuse to be 10 and the missing leg to be √84.

Here are links to the DOK 2 and DOK 3 problems we eventually came up with.

At this point, we thought we were basically done and were ready to be done.  However, someone in our group asked about how many answers were possible for the DOK 2 problem.  I said that I figured there were many, many answers.  I think I said 50.  Someone else thought closer to 10.  Others were less sure.  So we thought it would be a good idea to verify this.

We realized that the only possible values for the hypotenuse were 05 (has to be greater than the leg of 4), 06, 07, 08, 09, or 10 (the maximum value).  So, right there at the most there was only 6 possibilities.  So much for my supposed “many, many answers”!  To our great surprise, as we checked each one, only two of these were even possible without breaking other rules like using the same digit twice!  Clearly this impacts the DOK 3 problem as there are only two possibilities to choose from.

I share this narrowly avoided folly to give better perspective into potential issues that exist and the value of spending time actually doing the problems you make and/or use ahead of time.

When we used these problems with students, it was admittedly really rough.  For better and worse, so many gaps in their understandings came out.  Those gaps included:

• Students viewed a square root as something other than a number.  This was despite teaching this earlier in the year.
• Students did not understand the important balance between simplification and precision.  For example, if you have to choose between √9 and 3, you choose 3 because both expressions are equally precise but 3 is simpler.  However, if you have to choose between √8 and 2.82, you choose √8 because both expressions are not equally precise.  So, there was a lot of approximations and rounding without realizing the impact.
• Students had gaps in their understandings of what a right triangle is.  As an example, some students drew equilateral triangles (with or without a theoretical right angle) and assigned sides at random to be the legs or hypotenuse.  Non-traditional triangle orientation (such as the right angle being at the top of the triangle) continued to be a problem.
• As we predicted, students had issues realizing that there were impossible right triangles such as ones with leg lengths longer than the hypotenuse or leg lengths whose sum was longer than the hypotenuse.

I hope that this perspective on the thought process behind creating higher DOK problems has been useful.  What parts resonated with you?  What would you like to read more about?  Please let me know in the comments.