Library of Virtual Manipulatives

Let's Make A Deal: This is a strategy games where you pick one of three doors. Upon picking a door one of the other two doors are opened (a loser). You must then decide if you would prefer to stick to your original decision or change to the other door. I played 20 games where I changed my initial decisions and the results were that I won 13 games or 65% and lost 7 games 35% and then I played another 20 games where I stuck with my initial choice. The results from this strategy were I won 5 games 25% and lost 15 games 75%. This was another great game that would keep my students interest and provide a lesson on probabilities.

Lesson Plan for Cuisennaire Rods

The lesson I chose for this week is called Sculptures Big and Small. It is listed for grades 5-6. It was found on the manipulative CD. Students are to costruct and find the volume and surface area of different sculptures made from cuisennaire rods. The students then explore the volume of an orange rod to the equal amount of white rods. 1 orange rod is equal to 10 white rods. The Volume of the organde rod would be 10 cubic centimeters. They then move on to surface area and how the organge rod has six surfaces and they must figure out the survace area.

Library of Virtual Manipulatives

Factor Trees: The factor trees I found in Algebra grades 6-8. This allowed students to take a given number and find its factors. It began with one number but allowed for the user to find the factor of two numbers. After finding the factors it provided a Venn diagram to compare factors and find common factors. It also allowed for the user to make their own problems.

Library of Virtual Manipulatives

Coin Problem: This manipulative consisted of finding the counterfeit coin. You are provided with 8 coins and must determine which coin is the counterfeit. In the first two challenges when you are provided with 8 and 9 coins it informs you if the counterfeit coin is lighter or heavier than the others. When you move on to the 12 coin challenge you must find out if the counterfeit coin is lighter or heavier and then identify the counterfeit coin. The 12 coin challenge requires more strategy in order to solve the puzzle. This was a great manipulative.

Geoboards

Lesson: Peg Capture

This is a game that requires students to turn their geoboard into a coordinate grid. They use the coordinated in an attempt to get four markers lined up horizontally, vertically and diagonally. This is a strategic thinking based game.

Geoboards

Lesson: Line Segments

In this lesson students will try to make all of the different line segments they can find on their geoboards. Tis allows the students to identify line segments, compare lengths, ad oder line segments form larger to smaller. This lesson was taken from our CD.

Geoboards and Tessallations

Lesson Plan: Tessellating the Geoboard
In this lesson student will practice finding shapes that will tessallate their geoboards. This will allow students to identify tessallations, identify shapes that will be tessallations and how some shapes will tessallate their geoboard in more than one way.

Virtual Manipulative

The manipulative I experimented with this week was called Diffy. It was under Numbers and Operations for grades 6-8. The object of the manipulative is to complete a puzzle by finding the difference of given numbers. After a couple of rounds I Thought of the puzzle as somewhat boring and simple. Then I noticed you could choose what you wanted to subtract. The choices were whole numbers, decimals, money, fractions and integers. This allowed the puzzle to be more challenging and would allow students to work at their level. This manipulative also allows for students or teachers to make their own problems. If the number entered was wrong it disappeared and allowed for another number to be entered. If the answer provided was correct the number changed to the appropriate color of the puzzle. After completing the puzzle the screen began flashing and changing colors.

Lesson: Making Flags

The lesson Making Flags was taken from our manipulative cd. This lesson asks students to desing their own flags using color tiles. They can make a flag however they like and must use more than on color. After making their flag they then must determine the fractions and percents of each color in the flag. For example, a flag with 32 color tiles and 8 blues would be one fouth blue or 25%. This allows students to identify fractions, reduce fractions and identify the percent of a fraction.

Library of Virtual Manipulatives

Turtle Geometry
Turtle Geometry allows students to program the movements of a turtle to get to a certain point, through a particular obstacle or to match a certain shape. It also provides a blank work space so a teacher can set up a specific challenge or so students can challenge each other. The student must plan ahead and think about what they want to tell the turtle to do. For example, they can tell the turtle to turn left or right and how many degrees they want it to turn and they can also provide inputs for movement form 1 to 5 spaces. This was an interesting manipulative and I can see benefits in this however, I don't see this as one of the better mathematical manipulatives.

Private Universe Project Video 6: Problems and Possiblities

The final video demonstrated a 2 week summer institute for 18 students who all participated in the Rutgers study in one way or another. They were provided a problem called the Cat Walk. This consisted of a cat moving in a series of 24 pictures. The pictures had a background so the distance of the cat could be measured and the students were provided a time. They were then tasked with trying to identify the speed of the cat in the 10Th frame and the 20Th frame. Their methods for solving this problem as in previous videos slowly developed and improved as they went. They were able to present their ideas to their peers and the researchers allowing for student discovery and collaboration. The video demonstrated how students need opportunity to build personal representations in order to understand the standard scientific representations and that mathematical knowledge grows from personal and real life experiences. The students were all able to graph their time and velocity of the movement of the cat and they did this in several ways. They were also able to take these graphs and representations and put them into a real life experience. They expanded the distances first to 10 times and then to 50 times the original distance. This allowed them to visualize, feel and live the actual accelerations of the cat. They were then able to identify the motions of the cat and where it was at certain points and relate it to their graphs. They were then given a scenario that a CEO of a company had the opportunity to bet on the speed of the cat in the 10Th frame. The students were to tell the CEO if he should place the bet. The three groups presented and decided that the CEO should not place a bet on this because their were to many variables. This final video really demonstrated how their mathematical problem solving abilities had grown and as a student stated in the video "The Rutgers way the students present ideas and get to show the others what they discovered and a regular class would never do that."

Week 5 Virtual Manipulative

This week I chose the Function Machine found in Number and Operations grades 6-8. This was pretty cool because it showed the input and output of a machine. I was given four numbers and I had to feed them into the machine and the machine would then spit out a new number. On the right side there was a function table and it would automatically begin filling in the table with my input and outputs. I would then have to figure out the pattern and solve three blank outputs. This did tell me if I had made an error and allowed me to fix it. I believe this would be great as extra practice for students who have difficulty with input an outputs and it also provides a pretty good visualiztion of input and output however, it did become rather boring pretty quickly.

Private Universe Project Video 5

Building on Useful Ideas
The video started by demonstrating how several teachers were using manipulatives in the classroom and the different strategies they used to use manipulative to work with the students. A couple of teachers were using trains. They had a block of a specific length and had to use several smaller blocks to make the same length block. This was similar to the unifix cubes because the students were attempting to find as many combinations as possible. I thought it was very interesting how the video demonstrated the use of these manipulative across grade levels. The teachers wanted to allow the students to answer questions with "clarity" and would not accept "vague answers". They listened to their students and wanted to see how their students solved the problems. They wanted the students to have skills to figure things out with out giving up and allow them to be curious about problems. As Arthur Powell stated "we are getting students to take ownership of a problem, think deeply about it and exchange ideas with each other."
They also showed a teacher who began her day using different math related problems. She began the day with this to get her students thinking and all of the teachers were working on getting their students to think mathematically.
The video then went on to the previous core of student and how they were building upon strategies they learned in previous years. They were using Pascal's triangle to solve more complex mathematical problems. It is interesting to see the students go back to previous problems and use the same methods they used years ago to help them solve more complex problems. It's as if they use it as a reference then expand from there.

Private Universe Project in Mathematics: Thinking Like a Mathematician video 4

This video introduced a mathematician who demonstrated some of the real world uses for her skills. I also liked that she did not really enjoy mathematics until she entered the 9Th grade (Algebra) and the abstract concepts pulled her in.
The video then went on to classroom instruction and the students were introduced to the Towers of Hanoi. They began problem solving and were attempting to figure out how many moves it would take for 100 discs. They began small and worked up from a few discs to more and more discs. They began to identify patterns and this was the key for them to solve the problem because they were able to use the patterns to solve the problem. After figuring out how many moves it would take for 100 discs they were then tasked with determining how many seconds and how many years it would take to accomplish that task. Their ability to come up with and identify the patterns was pretty impressive and the students figured out it would take about 2 billion years to move 100 discs.
Next the video cut to a high school teacher and her methods of instructing students. She "encouraged her students to think." She did not force feed students answers but allowed them to investigate and problem solve. She also demonstrated to her students that it was acceptable to have different strategies to solve problems. She developed a community of students who accepted each others methods and strategies.
Finally, the video went back to the original core of students and demonstrated how they went back to previous problems and expanded upon them and how their strategies used to solve these problems evolved.

Color Tiles Lesson Plan

The lesson I chose for this week is Squares Within Squares. This was taken from the manipulative disc. The students are to build a 10 by 10 square and identify as many squares within the larger square as they can. I chose this because my students recently worked on a similar problem but on a smaller scale. They had to find how many square were within a series of three squares that were intersecting.
This lesson requires students to use patterns, problem solve and think beyond the obvious. Some square will be very easy to find while others will be much more difficult. It will also require the students to develop a strategy to record the squares they have found so they do not duplicate squares. I also believe this lesson could engage all the students in my class. It will entice the higher achieving students into solving the problem and also should not frustrate the lower achieving students.

Virtual Manipulatives Triominoes

When I first saw this manipulative it appeared to be an interesting puzzle or activity to try out. After beginning I noticed this did not present any problems or activities to work on. It was simply allowing me to connect triominoes and connect pieces for creating designs.
Triominoes are three sided peices with three different colors at each vertex. You can not connect pieces without the colors matching at the sides of the triominoes. The manipulative does allow you to flip, rotate and move the triomino. I was unsure of how this could be used with students but after working on it for a little while I thought that maybe students could be presented with a design and they would have to recreate it using the triominoes making sure they mathced sides so the pieces would connect (problem solving). Overall I really did not like this specific manipulative because I would have preferred it to present a problem or activity to be worked on.

Lesson Plan Choice

The lesson I will be using this week is the lesson named "Don't break the Wagon." This is a strategy game where the students will use a game board and pattern blocks. Their goal is to force the other player to place the last piece on the board and break the wagon. Students will learn to use strategic thinking and spatial reasoning.

Priveat Universe Project In Mathematics Video 3

Problems and Possibilities: Inventing Notations

This video demonstrated how giving students a chance to make their own mathematical notations without providing them with standard mathematical notations can be very surprising. They naturally develop their own system of documenting their possible outcomes. The teacher should be willing to discover the notions the students invent. Look to see how they represent their ideas and the teacher should also learning from the student's. We should take the time to listen to and view what they do. Each group within the video had a different way of reporting their findings. It was interesting to see a teacher express her concern when her students did not initially solve the problem using a list but actually used drawings and other representations She was initially intimidated because they did not solve the problem in the way she visioned they would. However, they still came up with valid answers using the drawings. Teachers must be in tune with what students are interested in and what they would like to do and how they would like to solve a problem. Our way is not the only way.
It was cool to see how some students used blocks to represent pizzas and others used lists. Some students used symbols inside circles and others just used abbreviations. I also like the fact that the researchers intentionally used two topping with the letter P to force students to think beyond just using the first letter of the topping. It really forced them to plan out their problem solving technique. The student who used the 1's and O's to make combinations of pizzas in a chart format was really amazing. He was also able to connect the pizza combinations with the 4 tower combinations.
These videos show that students do not need to be force fed mathematical concepts but given the freedom they can learn concepts on their own. It appears a much more natural way for students to learn and they can develop and discuss their own ideas. "We must give student the opportunity to show us" what they can do.

Virtual Manipulative Pythagorean Theorum

I chose the virtual manipulative geometry 6-8 proving the pythagorean theorum. In this I was provided 4 triangles and a square and had to fill in two different shapes. This word in turn prove the pyahtgorean theorum to be accurate. The first shape was a square (c) and the second shape was two squares of different sizes placed together (a and b). The four triangles and square that I was provided were the exact same size and would in turn fill in both of the shapes. The second shape (a and b) was much easier for me to figure out than the other shape (c). Shape c required manipulation and rotation of the square in order to fill it in. After some time I was able to figure it out and solve the problem. It was a really cool manipulative and I was recently working on this with my students and I really see this as a useful way to ensure the students understanding of the theorum.

Here are some of the patters my kids created.

This was nothing formal I had planned. I pulled out the pattern blocks from my bag while I was doing homework for another class. I looked up from my book and my kids had began making patterns. I grabbed the camera and took some pictures. The next day we sat down and made some more. My six year old insisted on fixing some of my designs because he felt that they were broken and my nine year old (Tyler) was much more patient and consumed with making designs.

Virtual Manipulatives Six Peg Puzzle

The six peg puzzle was another fun puzzle. Starting with four pegs was pretty basic and advancing up to six and eight required a little more problem solving. The main goal was to keep the pegs alternating until they reached the end of the board. Once that was done the puzzle became pretty basic. This is another great problem solving tool that is fun and intriguing. I think these would be even more beneficial if they were able to have actual manipulative's and not just the virtual one however, it is much easier to provide a classroom the website than numerous puzzles.

Virtual Manipulatives Towers of Hanoi

After explaining to my wife the directions and goal of Towers of Hanoi she began moving pieces. During the process she expressed frustration and felt at times it was impossible to solve. I believe she felt I was setting her up on a game that could not be solved. Once she felt comfortable that it was possible and a real game she immediately began working to solve the puzzle. From that point on she solved it very quickly with little frustration. She ended up solving the puzzle on her second turn in 15 moves. This game may initially be intimidating but once people get a grasp for it then becomes rather fun. It is a game I can definitely see my students enjoying.

Private Universe Project Video 2

In watching video 2 it was interesting to see how teachers approached the problem of the towers and the various strategies they used to solve the problem and justify their answers. It was thought provoking when the presenter stated that many teachers are very tense, nervous or even become angry with they way he presents his lessons. They are not prepared to justify their answers or verbalize their methods. He stated they are expecting to be told if they are right or wrong. He also noted the importance of having children express themselves and verbalize their thinking in class and how many kindergarten teachers are far more comfortable in his workshops than other teachers. Kindergarten classes tend to focus on collaboration and communication. As students advance in grades it becomes more of a listen to and respond to the teacher individually.
The students in the video were able to grasp the tower concept and appeared to be more open to justifying or proving their answers then the adults. However, the students had a harder time accepting the way in which their peers solved the problem. What was really amazing was how the students were able to advance their strategies beyond 4 and 5 towers and up to ten towers.

Private Universe Project in Mathematics

I viewed video 1 in Private Universe Project in Mathematics. It was interesting to see the students working on the same project as we did in class. I especially enjoyed watching the progression of the students and how they solved the problems presented to them. I was impressed with their commitment to their project and how they continued on their own time after the school shut down. The opportunity provided to these students was invaluable. It was proven in the study when they expressed how they would always remember what they were taught in 2nd grade mathmatics but they had forgotten a majority of the lessons taught in 9th grade Geometry class. The teaching styles definitely influenced their academic achievement. They seemed to grow an appreciation or love for mathematics from the Private Universe project. Their mathematic problems solving skills were allowed to grow and develop and were not forced upon them in a rote and rigid way.

Virtual Manipulatives

After visiting the virtual manipulatives I was really impressed with the options avialable for the various grade levels. I tested the How High manipulative for grade levels 6-8. I really enjoyed predicting the level of liquid from one tank to the other tank and I can definately see my students enjoying this as well. It also provided five different visual comparisons. rectangular to rectangular, cylinder to cylinder, cone to tank an cone to cone. I was also impressed with the ability to reset a problem or create a new problem. This feature will keep the interest of the student and allow them to practice a variety of problems.

Unifix Cubes

I completed the unifix cube project with my wife. After explaining the directions she immediately began making the combinations and their opposites. That was her immediate strategy however, she was not sure if she was finished. She eventually completed 14 combinations or 7 pairs.