Embracing Technology in Mathematics

As we progress in our mathematical exploration, the problems we encounter will become more complex, requiring more than just mental arithmetic and paper-and-pencil calculations. In these instances, the use of technology becomes indispensable. Graphing calculators, spreadsheets, and mathematical software can greatly enhance our problem-solving abilities, allowing us to solve complex problems, visualize mathematical concepts, and analyze large data sets with relative ease.

Remember: technology in mathematics is like a new language – it may seem challenging at first, but with consistent use, you’ll soon wonder how you ever did without it.

Graphing Calculators

As we continue to explore the intersection of technology and mathematics, one powerful tool stands out: the graphing calculator. Despite its name, it’s not just for graphing! It’s an incredibly versatile tool, capable of much more.

Graphing calculators can solve both simple and complex equations. For instance, to solve the quadratic equation [latex]x^2 - 4x + 3 = 0[/latex], you’d typically need to factorize or use the quadratic formula. With a graphing calculator, you can simply input the equation and it will find the roots for you. Moreover, graphing calculators can numerically solve equations that aren’t easily solvable analytically. For example, finding the roots of the equation [latex]\text{sin}(x) = x^3[/latex] might be a daunting task without a graphing calculator.

Watch the following video to see how to solve an equation using a 10-94 graphing calculator.

You can view the transcript for “Solving Equations on a TI-84 Plus” here (opens in new window).

Graphing calculators like the name suggests can plot graphs for us. Plotting graphs is an area where graphing calculators truly shine. Understanding the visual representation of an equation can significantly enhance your understanding of its behavior.

For example, to plot [latex]y = 2x^2 - 3x - 2[/latex], you simply input the equation, and the graph is displayed. You can quickly determine key features like the y-intercept, x-intercepts (roots), and vertex (maximum or minimum point).

Watch the following video to see how to graph a function using a 10-94 graphing calculator.

You can view the transcript for “TI-84 Plus Graphing Calculator Guide: Graphing functions” here (opens in new window).

Graphing calculators are also capable of performing calculus operations. You can compute derivatives and integrals, and even evaluate limits. Suppose you want to find the integral of [latex]\cos(x)[/latex] between [latex]0[/latex] and [latex]\frac{π}{2}[/latex]. Using a graphing calculator, you can compute this definite integral with just a few button presses.

Watch the following video to see how to perform calculus operations using a 10-94 graphing calculator.

You can view the transcript for “TI-84 Plus Graphing Calculator Guide: Calculus” here (opens in new window).

Finally, graphing calculators can handle matrix operations and statistical analysis, both of which are invaluable in fields like engineering and data science. For example, you can calculate the determinant, inverse, or eigenvalues of a matrix. Or if you’re working with data, you can compute mean, median, standard deviation, and even create histograms or box plots.

Watch the following video to see how to perform matrix operations using a 10-94 graphing calculator.

You can view the transcript for “TI-84 Plus Graphing Calculator Guide: Matrices” here (opens in new window).

Here are some more basic tips and useful features for a 10-84 graphing calculator.

You can view the transcript for “Quick tips to learn for using your TI-84 Plus calculator” here (opens in new window).

You can view the transcript for “TI-84 Plus Graphing Calculator Guide: Fun and Useful Features” here (opens in new window).

Let’s use what we just learned about graphing calculators to solve some examples.

1. Use a graphing calculator to find the roots of the following equations:
   1. [latex]x^3 - 5x^2 + 6x - 1 = 0[/latex]
   2. [latex]e^x = 2x + 1[/latex]
2. Plot the following functions and describe their key features:
   1. [latex]y = 3x^3 - 2x^2 + x + 1[/latex]
   2. [latex]y = \frac{1}{x}[/latex]

Show Solution

1. The roots of these equations would be found using a graphing calculator’s zero-finding or intersection-finding functionality, which might differ depending on the calculator’s model or brand.
   1. For the cubic equation, the solutions are found by graphing the equation and finding the x-intercepts. The precise values might depend on the calculator’s numerical algorithms, but you’ll likely find that the roots are approximately [latex]x = 0.34[/latex], [latex]x = 1.0[/latex], and [latex]x = 2.67[/latex].
   2. For the equation [latex]e^x = 2x + 1[/latex], you’ll need to graph [latex]y = e^x[/latex] and [latex]y = 2x + 1[/latex] separately, and find the [latex]x[/latex]-values where the two graphs intersect. The calculator should provide you with an approximate solution: [latex]x ≈ 0.35[/latex].
2. Key features of a function’s graph can include intercepts, maxima, minima, asymptotes, and end behavior.
   1. The graph of [latex]y = 3x^3 - 2x^2 + x + 1[/latex] is a cubic function, so it has the general shape of an “S”. It crosses the [latex]y[/latex]-axis at [latex]y = 1[/latex]. The exact locations of the local maxima and minima can be found using a graphing calculator’s maximum and minimum finding functionalities.
   2. The graph of [latex]y = \frac{1}{x}[/latex] is a hyperbola with a horizontal and vertical asymptote at [latex]y = 0[/latex] and [latex]x = 0[/latex], respectively. It has two distinct sections (in quadrants [latex]1[/latex] and [latex]3[/latex]), reflecting at the origin. The graph gets closer to the axes but never touches or crosses them.

The graphing calculator is a formidable ally in your journey through mathematics. While it doesn’t replace the need to understand the underlying mathematical concepts, it certainly aids in visualizing, analyzing, and solving complex mathematical problems more efficiently.

Spreadsheets

Our journey through the intersection of technology and mathematics brings us to the versatile tool known as spreadsheets. Software such as Microsoft Excel or Google Sheets can dramatically streamline your problem-solving process, especially with large datasets.

Spreadsheets excel (pun intended!) at performing a series of calculations, from simple arithmetic to more complex mathematical functions. Input your data into cells and use formulas to perform calculations across these cells. For example, to calculate the sum of a column of numbers, just use the SUM function. For more complex tasks, like calculating a square root or a sine, there are functions like SQRT and SIN.

One of the major advantages of spreadsheets is the ease of organizing and analyzing data. For instance, you can keep track of grades for a class of students, calculate averages, find the highest or lowest grade, and even plot the grades on a graph. The SORT and FILTER functions allow you to manage your data effectively and make it more comprehensible.

Spreadsheets also offer a wide array of statistical functions. You can calculate measures of central tendency (mean, median, mode) and dispersion (standard deviation, variance). You can also use spreadsheets for regression analysis, t-tests, ANOVA, and more. These powerful features can greatly simplify statistical analysis.

Let’s watch a few videos on using Excel to solve problems.

You can view the transcript for “Excel: Getting Started” here (opens in new window).

You can view the  transcript for “Excel: Cell Basics” here (opens in new window).

You can view the transcript for “Excel: Functions” here (opens in new window).

Try using a spreadsheet to solve the following.

Remember, a spreadsheet is a powerful tool, but like any tool, its effectiveness depends on the skill of the person using it. So get hands-on, start experimenting, and before you know it, you’ll be a spreadsheet pro!

Mathematical Software

Finally, we have mathematical software, powerful tools designed to handle advanced mathematics. Software like Desmos, Wolfram Mathematica, or GeoGebra can help solve algebraic equations, perform calculus operations, create geometric constructions, and much more.

Desmos is a powerful, web-based mathematics tool that enables users to perform a wide range of operations, from basic arithmetic to advanced calculus and statistics. It excels in its graphing capabilities, providing real-time, interactive graphs that vividly demonstrate the behavior of mathematical functions. Whether plotting a simple linear equation or exploring intricate parametric plots, Desmos brings math to life in a user-friendly interface.

Wolfram Mathematica stands as a giant in the world of mathematical software. This comprehensive system enables everything from simple calculations to large-scale computations, programming, visualizations, and much more. Its symbolic computation abilities make it unique, offering exact solutions that other numerical-based systems can’t. It’s also renowned for its integrative capabilities, making it possible to perform operations across graphics, text, and data in a unified manner.

You can view the  transcript for “Mathematica Essentials: Intro & Overview (Wolfram Language)” here (opens in new window).

GeoGebra is a dynamic mathematics software that unifies geometry, algebra, and calculus. It allows users to construct figures and perform computations, offering a more tactile, interactive approach to understanding mathematical concepts. Whether constructing geometric proofs or exploring transformations, GeoGebra provides an invaluable platform for learning and instruction. Its suite of tools includes a graphing calculator, geometry calculator, spreadsheet, and 3D calculator, making it a versatile choice for mathematics education.

You can view the transcript for “GeoGebra Tutorial 1: The GeoGebra Interface” here (opens in new window).

Together, Desmos, Wolfram Mathematica, and GeoGebra provide a suite of tools that can enhance your mathematical understanding and proficiency. By leveraging these tools, you can develop a deeper, more intuitive understanding of mathematical concepts and their applications. But remember, while technology can assist, it’s your critical thinking and problem-solving skills that truly drive mathematical success.