The Last of the Complex Diagrams

If you tried the previous problems for the past two posts, I have two more for you. I use both of these problems at some point in time in my classes. They are both excellent for stressing the importance of organizing thoughts and work. I also ask my students for a written explanation of their problem solving strategy.



The triangle question really forces one to be very organized in order to find all 28 triangles. This is a great problem to have students work together on.



The square problem allows one to analyze a pattern. When discussing this problem have students make up a list of the number of individual squares, of squares made up of four smaller squares, of squares made up of nine smaller squares, etc. The students should see a pattern occurring. This will allow you to move into examining patterns and finding rules.

Killing many birds with one stone

If you started to create the "thinking" environment with the previous rectangle problem (by the way there are a total of 9 rectangles), then you want to continue along this line. I would suggest that the next day you have a second problem similar to the previous one for students to work on. I have created a triangle problem.



This problem allows me to accomplish several goals. First, I give it to the students as soon as class begins. This allows me a chance to take care of administrative details like attendance. Second, after they have worked on it for a few minutes, I ask them to check with their neighbor to see if they agree. If they don't agree I ask them to try to decide which answer is correct. This problem is more challenging than the previous, so usually students do not agree on the answer. Consequently, I have them discussing their methods and they are engaged. I will ask them what they think the answer is and I will write these answers on the board. Now, I have the opportunity to accomplish my third goal which is to stress the importance of organizing their thoughts and their work. I will number the individual triangles and then I will make a list of triangles I am combining to form another triangle. After we discuss various strategies and discuss methods of organizing their count, the students usually beg me for another one to try. I tell them we will do another one tomorrow and there will be prizes for those who get it correct. Consequently, I have them looking forward to coming to math class! So, this quick activity from the day before which had the purpose of creating a safe thinking environment has provided me many benefits including the engagement of my students in problem solving!

Create a "thinking" environment immediately

I have heard teachers comment that students can't (and do not) think - all they want is to know what the right answer is. Well, I disagree. I do believe many students are hesitant to share what they are thinking and there is the natural fear of being wrong in front of everyone. After all we do not like to be wrong in front of everyone, why should they? Consequently, it is important to create a safe and comfortable "thinking" environment in your classroom.

I start on day one by setting the expectations and guidelines. Now, I am not a big fan of handing out a page of rules and regulations. I do discuss the importance of respect and then I ask them the following question.



I give the students a few minutes to work on this (depending on the students math background I may need to discuss that a square is a rectangle) and then I write the numbers from 1 to 15 on the board. I ask for a show of hands how many students think there is one rectangle, two rectangles, etc. and I do a tally on the board. When I get to 15, I will ask if anyone has a different answer and will place that on the board, if there is one. This is the first day of school, so students tend to be fairly quiet and there is not much conversation going on. I will then ask if anyone would like to change their answer. I do this as I am sending them a message that it is okay to change your answer. Next I will count the tally marks and usually it does not equal the number of students - there are always a couple of students who hesitate participating in this activity. Consequently, I will announce that there appears to be a few people who hadn't decided yet and did they now want to vote. Finally, I do the problem (count the rectangles) and demonstrate my method. As I am doing this I will hear various comments like "Oh, I forgot that one" or "I didn't see that one". We will then discuss their strategies.

When I finish I point out three very important things:
1) If the students got the problem wrong, do they now understand? I discuss the importance of speaking up, asking questions, and never leaving the classroom confused.
2) It was NO big deal if they were wrong! We are all going to be wrong at some point in time. Everyone was polite and respectful and that is how this class will always run!
3) Non participation is unacceptable. I do stress that I will not embarrass them, but I can't read their minds so they will need to help me by participating and sharing what they are thinking.

Consequently, I have the built the foundation of a safe and comfortable "thinking" environment through an activity. I encourage the students to copy the problem down and have their parents try it.

In my next post I will provide the answer to the question and will discuss how to continue building this safe and comfortable "thinking" environment.