Exploring Geometric Tasks: Area And Perimeter Manipulation, Integer Variance, And Invariance And Change

Task X.Y.Z

Question 1

a). Considering invariance in measurement, practically doing measures we will use conventional units, these are meters and its derivatives, kilometers, millimeters. We do measurements by comparison with a specified standard measure; this in most cases gives errors that are always involved. Exactly stating, any measurement usually has 3 parts: the said units that are involved, and a specified error interval for which the quantity shall be confidently placed. The purest measure is compared in at most two lengths which are exactly in an optimal structure. Involvement of any error should not be there. For this, I have a keen interest in looking at this task that focuses on Integer variance.

– Drawing a triangle with vertices of integer coordinates. Imagining a square drawn in an out form direction on each side. Will all the vertices of the squares really have integer coordinates?

b). In the collection of evidence, we might have used trial and error, from which we can only get evidence, not proof. For proof provision we cover all possibilities as outlined below:

The vertices on a triangle based on a coordinate planes are unlikely to be integers, and in looking at anything in all the triangles it is advisable to use co-ordinates such as (a, b) or (a_1, b₁) or (a₂, b₂), where the letters are integers or any other real number.

To check for the points that the vertices of the triangle lie we check some simplifications. Starting from the origin as one of the vertexes and setting the next on any of the axis lines that there are. So having the two vertices of the triangle as (0, 0) and (0, 1) or (0, 0) and (1, 0). So the third one can be denoted by    (x, y). Here the vertices (x, y) can be arbitrarily filled by any of the two variables. Dynamic geometric software we can get to drag about the third point, but this gives no proof that most relationships always hold. It is basically a tool that tries large numbers of cases faster. With dynamic geometric software, only a limited amount of cases are cases that can be reflected on. This suggests an incorrect conjecture. As we all know in geometry we are to be thinking of all possible cases but not a few cases.

c). in revisiting the problem above by all means we look at one case of same vertices as we had above. Say (0, 0), (0, 1) together with (a, b). A square can be drawn at each side of the triangle in an outward direction. Vertex too of the drawn squares can be joined to that of its neighbor square. By finding the coordinates of the other vertices of each square we can then proceed and find the areas of the triangles so involved. Newly drawn triangles must have asides of length =1; the sides forming the base for each. Seeing that the triangles have similar heights, and then all three have similar areas. If we dealing with the 4^th triangle, we can imagine a different pair of axes and their origin at the point (along with the side of the original triangle too (1,0). This 4^th triangle with the other two definitely have equal length base (these are the sides of different other squares), and similar heights and these will definitely have similar areas. Therefore, we will have therefore areas of all the triangles similar in value.

1. d) Considering a correlated task in the integer vertices, I will turn my main focus on gradient calculations. Here again, I consider similar vertex as in the case above; (0, 0), (0, 1) and (x, y). From here any gradient calculation would be ; meaning that actually, we will have different values provided the values for the vertex (x, y) are defined.

Task X.Y.Z

a). For areas and volumes to be calculated, the things that are taken into consideration are the dimensions of the sides of the figures in question. Therefore for us to develop a concrete discussion as to how we are to arrive in the volumes and areas we are looking for then we will have to start from then dimension reasoning. This is the reason as to why I am choosing the shortest paths task. This will enable us to understand the importance of volumes and area calculation from dimensions, distances, and paths. For this we pick the task with this set of the question;  

Let’s imagine an ant’s walk on the cuboids depicted below, we want to find the shortest path that the ant will walk on AB.

b). How do we imagine an ant’s shortest possible distance? Should we consider drawing anything close to a net? If we can trace the path of the ant on the so said net, then a straight line AB must be realized. However, I see 3 possibilities on how to draw the net. From the figure below I see 3 ‘overlapping’ nets from the cuboid in the first figure above, for the 3 possible routes to be displayed. It’s well important spending some time to imagine how the figure below does relate to the cuboid in the first figure above, then we can use it in imagining the 3 dashed paths found on the surface of the cuboid. You can wish to experiment with the interactive file ‘Shortest Path’.

c). considering the dimensions of the variation in this task, it does not matter if that you have restricted your attention to cuboids. For instance, points A and B might fail to beat any of the vertexes. The so variations encountered in just a dimension of the cuboids of Figure 6.2c above would suggest an additional task in finding a rule of thumb for deciding on the route that would turn out to be the shortest for any cuboid when the ant decides to go between diagonally opposite corners. Would it be possible for the shortest path on the surface of a cuboid to involve travel on more than three faces?

Seemingly a short step to move from cuboids to cylinders since both can be seen as examples of prisms.

A cylinder can be thought of as prism which has a circular cross-section and not a polygonal one.

Task X.Y.Z

N.B: Comment; There will be no workings appendix and therefore all the workings are shown in the document itself.

Question 2

Invariance and Change:

The topic question above students is supposed to look into suggesting that they closely focus on invariance and transformation since a transformed figure or object is as well as a changed one. The fundamentally important idea of invariance characterizes different geometrical transformations. There exist different transformative actions; Isometries, and these keep the same shape and size of the transformed object. Other transformations preserve shapes but not necessarily sizes. There too exist transformations where the list of transformations is combined to give different shapes.

a). The main focus that I expect students to have in as far as invariance and change is concerned is on Shears. If we are to consider a Cartesian plane then having x and y-axes then we can draw a figure say a square figure with two sides parallel to the x-axis and the other two sides parallel to the y-axis. Shear kind of transformation can, therefore, be demonstrated by say moving every point parallel to the x-axis and a distance that is equal to that that is half its distance from the said x-axis. The realization is that the points will move to the right by amount spelled out by the height that lies above the x-axis.

The task that is to be chosen here that focuses on shears enables students to understand the importance of having hinges as the holding parts of the joining parts of objects that would be rather expected to be movable. Consider the seats that are used mostly in the African settings that are movable and that are recommended by a single person to sit on. Such seats have hinges that allow for the complete flattening of the seat for easier carriage into and out of the house. The seats too can be modified to be in a state of a seat that can be used for resting. Understanding these enables students to see the importance of shear in the society. This will, therefore, develop our major discussion idea.

b). considering an actual task shear task for your students we take this;

Imagining what happens to each of the elements that follow in Figure 9.2b when sheared before doing any drawing! It all takes some little thinking about, but it is possibly doable.                                             

What lengths will change and what lengths will stay the same? What will the ratio of the changed length be to the original length in each specific case?

Shortest paths task

Which angles would change? What incidences would change? What pairs of parallel lines would no longer be parallel? What is going to happen to the square’s area?

Major comment: The transformation to be considered is one example of a shear. Every point on the –axis will remain fixed whereas all the other points will move right. What usually happens under a shear, by the idea of an inclined line indicating how much each point would move according to its distance from the x-axis.

Answering the questions intuitively we get to have to have that the lengths parallel to the x-axis remain the same whereas those that are parallel to the y-axis get longer. The reason as to why they get to be longer is because of the fact that the lines are stretched out in the process of the shear to match the newly found shape and angle inclinations.

All the angles will get to change since the entire shape of the figure will change. Therefore for the new shape to be matched we get to have new angles inclinations. As a student, therefore, looking at the ratios given for you to do your shears, one should be able to deduce the actual angle change so described.

The indices on the lower side will remain in the same location that they were unlike the indices on the upper side of the sheared diagram the diagram to be sheared and this should be evident to you as a student when you will be doing your shear.

The parallel lines that are disrupted from being parallel are the lines that will be parallel to the y-axis because they will have been pushed to match the new shape.

A shear is always an inquisitive transformation. One of the lines stays unchangeable, and if any point is far off from this line, the further it will have to move. Points that are above the line will move towards a similar direction, points below the line will have to move in a different way.

Parallel lines will have to stay parallel; angle sizes must always change; the height of the figure which is perpendicular to this unchangeable line will always stay in the same position. Whereas the area to stays the same but the perimeter automatically changes.

There will always be confusions between a shear and a stretch. This is 4 equal rods that are hinged at the ends forms a square but can also be fouled over to form a rhombus.

What transformations are involved? What transformations can we involve? Shearing can be used to achieve the so required angle, but the height will stay the same, and tilted rods will definitely become longer thereby will need a scaling down in the direction of the tilted rod to help re-establish the similar lengths.  Therefore, the sideways push is a shear that is combined automatically with a stretch.

All of the things that I have talked about in writing can be demonstrated practically by for instance by the use of papers or plastic objects that are flat and are hinged. This will enable the perfect shear. In the shear that is to be demonstrated, critical thinking will show that a stretch is also involved. Therefore there is no shear without the existence of a stretch. And therefore as students, you are expected to experiment with different material and read largely too in order to agree that there are several applications in the field of physics, especially in motion where shear is being used. Shear aids in most daily activities. Like in the case of the African seat I mentioned above. Another application that students can explore is in the tent set up during a camp.