As a math teacher, I often became frustrated when I gave a math problem to my students only to have a small percentage of the whole class be able to answer the question correctly. Naturally, many of my students became frustrated too. Consider the following problem:

The word problem above is a very specific problem that only has one answer. The fact that there is only one answer is not a serious issue for me or for the students that need to solve it. The issue with a problem like this is the fact that there is only **one **way to answer it.

This is a very specific problem that requires a very specific solution. If I gave this problem at the beginning of a grade 6 transformational geometry unit or TLCP cycle (Teaching Learning Critical Pathway) there is a good chance that only a handful of students would be able to solve it correctly. This is the kind of problem that I would give at the end of the unit or TLCP cycle since it is a really good assessment **OF** learning type of problem. Then it should come as no surprise that this problem was taken from the 2008 EQAO math assessment.

But what if I wanted to use this problem to begin my transformational geometry unit? Well, maybe not in its current form but what if I could “open up” the problem so that it wasn’t so narrow and specific and that a lot more students could solve it. Consider the same problem but with some modifications:

This is an example of opening up a very specific math problem. This open problem has more entry points for students than the previous problem since students have a choice in how they can move and manipulate the mat. Open questions are questions that have more than one answer and are great for differentiating instruction in the math classroom. Open questions allow students to solve problems based on where they are at in their math development.

I actually used the “open” gym mat question last year when I helped a grade 6 teacher introduce her transformational geometry unit. At first, the teacher was hesitant. This approach was drastically different from how she usually introduced the unit. In years past, she would introduce each transformation in isolation. First, a note on translations. Second, examples and demonstrations of translations. Third, practice problems involving translations. The three-step process would be repeated for rotations and reflections (This is also how I used to teach math). Therefore, the notion of giving an open problem to her students that allowed them the opportunity to investigate and use any transformation without defining, modeling or practicing them was pretty daunting. However, the results were very eye-opening and informed the teacher’s next steps for the next few lessons. Here are a couple of the student solutions:

All of the students in the class participated and solved the problem in small groups. As you can see from the gallery, there was a range of solutions from the class that brought up some really good discussion during the reflect and connect portion of the lesson where groups were able to explain their solutions to the class and answer any questions about their transformations. Some topics/questions that were discussed were:

- efficiency in transformations.
- What is the most efficient/fastest way to get the mat to the desired position?
- What’s the purpose of the dotted line AB?
- points of rotation.
- Can an object/shape have more than one point of rotation?

This rich discussion was able to occur because of the openness of the question and the fact that students had the freedom to investigate and use their own math thinking to come up with a solution. It was also very powerful for the students to see that none of the groups came up with the same solution to the problem. The range of students’ solutions also allowed the me and teacher to determine appropriate action for the next couple of lessons.

For more information on differentiating math content using open and parallel questions please read the following article. (A very good read!)

Thomas, we have been working on opening up math questions in Intermediate classes. Thank you for including the photos and the detailed description of what happened in the class. I noticed the sticky notes. Did you use Bansho to consolidate, or do you think this activity would have been a good one to use with Bansho?

Hi Shelley,

I did use a Bansho to consolidate and annotated the student work with sticky notes as the explained their solutions and during class discussion. The list of key learnings was the summarization of the reflect and connect and co-constructed with the students.

how can you do the (Translate the square to (5,4)