Virtual Manipulatives : Right Triangle Solver

I was looking through the Virtual Manipulatives and came upon the Right Angle Solver. I started to play around with it and found it very helpful. While it starts out like a normal problem, it lets me choose or identify the term in the picture. After that it let me choose, it gave me a few methods to solve the problem. As I was highlighting the methods it showed me examples of each method. If I chose the wrong method to use, it would give me a reason why I could not choose it. I was able to see the method used and correctly fill in the blanks. This is a very helpful tool to further the understanding of properties of right triangles.

Private Universe : Inventing Notations

This video deals with the students presented with another problem that would challenge them just as much as the other problems. Since this was not the first problem, they were able to jump right in start to organize their thoughts on paper. While one group seemed to get a bit flustered in a short amount of time because of the degree of difficulty, the other group argued about which would be the best way to solve the problem. The level of interest seemed to be a lot higher for every single student in the groups. From the beginning of the problem they had the idea of getting the correct answer but knew that they would have to have conclusions about how they came up with the answer in order to convince others. They also knew that just because they had an answer and everybody agreed, it did not mean that it was the correct one. One student said that everyone needed to be convinced because it was possible that maybe the one student who wasn't convinced could have the correct answer. I was very impressed with the intensity the students work on the problem. The fact that they were willing to eat lunch and then come back to work on the problem meant that they were into it.
The focus was brought to a student because of his unique way of the solving the problem. The fact that he was using binary code to solve the problem was very impressive. He was asked were he came up with the idea, but it just popped in his head. As they worked more with the problem he started to realize a few more things. It shows that children at a young age are able to come up with ideas and think like mathematician without the use of variables or any other algebraic symbol.

Video : Private Universe Are You Convinced?

Watching video 2 reminded me of the process we went through in order to come up with strategies and answers to the problem. What I noticed is that the teachers had pretty much the same thinking process that the students had when they were working on the problem. Both used different strategies such as drawing the towers, patterns, but usually trial and error. Both were asked to give reasons in order to convince others why there knew that there was not anymore combinations. The 4 story problem forces anyone use inductive and deductive thinking. Towards the end I like that the students where able to see a pattern within their own patterns. All four students shared how they solve the problem and were willing to share their strategies. They also had to realize that coming up with a pattern would be the best strategie to solving a problem with 4 cubes or 10 cubes.

Peg Puzzle : Virtual Manipulatives

I've seen this type of problem before, for example, using frogs jumping over each other. I have always enjoyed this problem especially because I can never seem to remember how to solve it. So every time I see the peg puzzle, it is like starting all over. The idea seems to be simple enough that is until someone gets to about 6 pegs. The pattern seems to be obvious but after solving the puzzle for the first time, repeating the process is not easy. After repeating the process you can start figuring out other patterns that are not so obvious. Using this would help a student look beyond just a simple pattern. Also using the virtual manipulative is more convenient than having actual peg puzzles in class.

Towers of Hanoi : Manipulative

I explained the directions and goal of the Towers of Hanoi to my girlfriend. We started from the beginning using two disks. She completed it with ease, so we moved on to 3 disks. She got it in one try, and told her to count the number of steps. We moved on to 4 disks and she started getting frustrated. After a few tries, I told her to focus on her moves and think about when there was only 2 and 3 disks. She said that she felt that there was a pattern but couldn't exactly see it. She ended up getting the 4 disks within 43 steps. She felt it was too long and I decided to tell her that it's only supposed to take 31 steps. We wrote down how many steps it took from 1 to 4 disks. She started to see the pattern and we figured out how many steps it would take all the way to 7 disks. This game is very helpful when analyzing patterns. While it can be frustating, it forces the person think in an enjoyable manner.