Reasoning and explaining

Junior (College 3rd year) ・Mathematics ・MLA ・1 Sources

The following reflection includes a presentation of a well-reasoned artifact which is derived from the Newton's Method to solve the mathematical problem3√5.The artifact is also well explained to demonstrate important ideas and facts in solving various mathematical problems. The first part of the reflection will present a mathematical idea which will first be abstracted and the reasoning behind the argument quantitatively presented. The above stated arithmetic equation will be used to bring two complementary abilities to bear on problems involving both the abstract and quantitative part as well as construction of a viable argument and finally to critique others argument. The ability to bring about quantitative relationships will also be demonstrated.

Mathematical Practice Domains

The given situation will be represented symbolically and manipulated to represent symbols like they have their own life. The manipulation will be carried out in such a way that it will not necessarily attend to the referent of the symbols applied. When contextualizing, the manipulation of the numbers will be visible. On the other hand, the Quantitative reasoning will be about the habits of coming up with a coherent representation of the problem at hand which in this case is arithmetic equation x3 − 5 = 0 ; otherwise represented as an artifact. There will also be consideration of the units involved; while carefully attending to the meaning of quantities and not just how to compute them; The specific scenario will also entail knowing and flexibly using different properties of numbers operations as well as objects.

When approximating the value of 3√5, the Newton's method is the most appropriate method of solving the above stated mathematical equation. The basic method of solving such a problem would be x = ± 3√5.Howver when the Newton's method is applied, the equation x3 − 5 = 0 can be effectively assimilated as part of the solution to the arithmetic. The idea behind using the equation in solving the mathematical problem is derived from the fact that x3= 5 and therefore making x the subject of the formula x= 3√5 which is the problem at hand. The two mathematical problems are therefore one and the same and can be substituted.
To solve 3√5, the values of x are computed by first approximating the value of xo using a simple x and y graph. From the graph, the values of x will be estimated where the line crosses the x axis as at that point the value of y=0.Arithmeticaaly that approximate point is at (1.6,0).The point is arrived at from the equation x3 − 5 = y(substituting y with 0),x = 1.6.From here the value of X1,X2 and X3 can be determined with the point where X2 will be equal to X3 being the solution to the problem. After carrying out the numeric computations of the three x value the solution to the problem can be computed as x= 1.710.

The mathematical problem presented above connects very well with secondary school mathematics as it touches on various important topics studied in high school mathematics. The topics concerned are introduction to algebra where the equation of a line is taught. The equation of a line formed the basis of solving the mathematical problem solved above using the Newton's method. The other contextual topic applied while explaining the artifact is the surds, while trying to approximate the value of 3√5, the knowledge from surds was critical(Ďuriš et al,50). Finally, introduction to graphs knowledge was also used to approximate the initial value of x otherwise regarded as xo. All the topics highlighted indicate the application of the artifact to teach various high school mathematics topics.

Work cited

Ďuriš, Karol, et al. ""Comparison of the Analytical Approximation Formula and Newton's Method for Solving a Class of Nonlinear Black-Scholes Parabolic Equations."" Computational Methods in Applied Mathematics 16.1 (2016): 35-50.

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