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Part 1: For this section of the midterm, I am discussing the 5 roles that graphing calculators played as a tool in the classrooms in the study by Doerr & Zangor (2000) in terms of two virtual manipulatives, Diffy and Circular Geoboard, both of which came from the National Library of Virtual Manipulatives (NLVM) hosted by USU.
Diffy
Diffy is a virtual manipulative on the NLVM website which models a situation of taking positive differences of four different numbers, then taking the differences of those differences, as seen below:
In this virtual manipulative students can look at problems given or create their own problems to try using whole numbers, integers, fractions, decimals or money as the original entries in the four corners. Now, I would like to look at the different possible roles for this tool (coming from the Doer & Zangor article)
Computational Tool
For the most part, I do not see diffy as a computational tool. The students are required to do all of the subtraction in a separate format (be that paper and pencil or a calculator), and although diffy does organize the differences so students can see what it is they need to subtract, it does not do any of the computational work that students would otherwise do when exploring this situation. In fact, when you get into some of the more difficult number sets (such as money and fractions) the computations become quite lengthy and it would be helpful to have a calculator to do them.
Transformational Tool
The organization I described above is key in Diffy's ability to be used as a transformational tool. If you can imagine trying to give students this sort of concept (differences of 4 numbers, then differences of their differences, and so on and so forth..) without the framework and structure provided by diffy, you see just how helpful this virtual manipulative really is. Without diffy, students would have to spend much of their time trying to organize all of the differences they are required to find, instead the students will be able to focus more on making conjectures and thinking about proving them than worrying about organization.
Data Collection and Analysis Tool
Diffy definitely has some capabilities for data collection, even though it may not seem so at first. In the 'create problem' mode, students can collect data about what happens as different numbers are put into the corners to begin with. In this way, students can use Diffy as a way to attempt to manipulate inputs of data to see if they can get specific patterns out. This is very similar to the sort of thing students were doing with a graphing calculator in the Doerr & Zangor study.
Visualizing Tool
Diffy can't be used quite so neatly as a visualizing tool as thegraphing calculator can, however, as I talked about in thetransformational tool section, diffy does give students the opportunityto see what is happening as opposed to thinking about it solely intheir head. The only way for students to see patterns between examples would be to write them out or complete different difference squares in different windows or tabs.
Checking Tool
Diffy can definitely be used as a checking tool. In the most basic sense, diffy will check your subtraction, since it makes an ugly beeping noise when you enter an incorrect difference. However, diffy can also be used to check specific cases . For example, a student may conjecture that it will not end up being all zeros in the center if there are zeros in three corners and a large number in the other. They could then create a problem with those properties and check to see if it is right.
Circular Geoboard
The circular geoboard, also found on the NLVM website, is a virtual manipulative in which students can use virtual rubber bands to create lines, angles, triangles, and other geometric figures on a board with pegs in a circular array as seen below:
Math 101 Midterm Exams
Math 101 Midterm Exam
As you can see, bands can be used as asingle line, or be stretched to cover an area such as a triagngle,these areas can be colored so students can more easily differentiatewhen you have multiple figures on the board.Computational Tool
This virtual manipulative can be used to compute the angle measure of many different angles in the absence of a protractor. For example, since a circle is 360 degrees, and there are 48 pegs on the outer circle, the angle between each two sets of pegs is 360/48 = 7.5 degrees. So, students can find the measure of any central angle using this method (either on the outer circle, or one of the inner circles). Students can also compute other angles on the board by using other circular angle theorems.
Transformational Tool
Using the circular geoboard, tasks of proving many theorems about circular angle theorems can become much more interactive, and instead of simply applying theorems or drawing pictures, students can use the circular geoboard to derive theorems like the fact that the measure of an inscribed angle is half the measure of the arc it subtends. With this, students are able to create and manipulate triangles and angles to help them make and prove conjectures, whereas with paper and pencil, students are often limited to using a protractor, or simply applying theorems.
Data Collection and Analysis Tool
Data collection and analysis is one area of use in which the circular geoboard is somewhat lacking. Although students can create multiple angles and/or triangles on the board at one time. There is really no satisfactory way to collect data, or to compare different sets of angles easily that I have found in my use of the circular geoboard.
Visualizing Tool
Students can see how the arc an angle subtends changes the angle, andhow different triangles/angles relate to each other, because they canhave multiple representations on the same geoboard, even overlapping ifyou can translate the triangles or angles correctly from one spot on the board to another.
Checking Tool
Although not as obvious as a first, the circular geoboard can be used as a testing ground for conjectures formed about angle relationships. For example, if students conjecture that a regular pentagon has angles of 108 degrees, the students can create a pentagon, and then use other angle relationships on the geoboard to confirm or disprove their conjecture.
Part 2:
For Part 2, I was asked to create a Geometer's Sketchpad document in response to the following prompt:
Math 101 Midterm Review
I recently flew over Kansas and noticed someinteresting geometric designs in the crop fields below. There were manysquare fields that appeared to be one mile by one mile. Sometimes theseone-mile squares were inscribed with one circle; at other times theone-mile squares were divided into four equal squares and then acircle was inscribed into each of these four smaller squares. In allcases, the farmers were only planting inside the circles. I couldn’thelp but wonder, “Which approach allows the farmers to utilize the mostland?” Create a GSP document in which you explore this situation.Within the document writeup your exploration and illustrate and proveyour answer to the question. Create a link from your midterm web pageto this GSP document.